Statistical analysis of high-dimensional biomedical data: a gentle introduction to analytical goals, common approaches and challenges
BMC Medicine volume 21, Article number: 182 (2023)
In high-dimensional data (HDD) settings, the number of variables associated with each observation is very large. Prominent examples of HDD in biomedical research include omics data with a large number of variables such as many measurements across the genome, proteome, or metabolome, as well as electronic health records data that have large numbers of variables recorded for each patient. The statistical analysis of such data requires knowledge and experience, sometimes of complex methods adapted to the respective research questions.
Advances in statistical methodology and machine learning methods offer new opportunities for innovative analyses of HDD, but at the same time require a deeper understanding of some fundamental statistical concepts. Topic group TG9 “High-dimensional data” of the STRATOS (STRengthening Analytical Thinking for Observational Studies) initiative provides guidance for the analysis of observational studies, addressing particular statistical challenges and opportunities for the analysis of studies involving HDD. In this overview, we discuss key aspects of HDD analysis to provide a gentle introduction for non-statisticians and for classically trained statisticians with little experience specific to HDD.
The paper is organized with respect to subtopics that are most relevant for the analysis of HDD, in particular initial data analysis, exploratory data analysis, multiple testing, and prediction. For each subtopic, main analytical goals in HDD settings are outlined. For each of these goals, basic explanations for some commonly used analysis methods are provided. Situations are identified where traditional statistical methods cannot, or should not, be used in the HDD setting, or where adequate analytic tools are still lacking. Many key references are provided.
This review aims to provide a solid statistical foundation for researchers, including statisticians and non-statisticians, who are new to research with HDD or simply want to better evaluate and understand the results of HDD analyses.
The goal of the topic group TG9 “High-dimensional data” (HDD) of the STRATOS (STRengthening Analytical Thinking for Observational Studies)  initiative is to provide guidance for planning, conducting, analyzing, and reporting studies involving high-dimensional biomedical data. The increasing availability and use of “big” data in biomedical research, characterized by “large n” (independent observations) and/or “large p” (number of dimensions of a measurement or number of variables associated with each independent observation), has created a need for the development and novel application of statistical methods and computational algorithms. Either large n or p may present difficulties for data storage or computations, but large p presents several major statistical challenges and opportunities . The dimension p can range from several dozen to millions. The situation of very large p is the focus of TG9 and this paper. Throughout the paper, “p” will refer to the number of variables and the term “subject” will be used broadly to refer to independent observations, including human or animal subjects, or biospecimens derived from them; or other independent experimental or observational units. Researchers who design and analyze such studies need a basic understanding of the commonly used analysis methods and should be aware of pitfalls when statistical methods that are established in the low-dimensional setting cannot, or should not, be used in the HDD setting.
This overview, a product of STRATOS topic group TG9, provides a gentle introduction to fundamental concepts in the analysis of HDD, in the setting of observational studies in biomedical research. The focus is on analytical methods; however, issues related to study design, interpretation, transportability of findings, and clinical usefulness of results should also be considered as briefly discussed throughout this paper.
The STRATOS initiative and the STRATOS topic group TG9 “High-dimensional data”
The STRATOS initiative (www.stratos-initiative.org) is a large collaboration involving experts in many different areas of biostatistical research. The objective of STRATOS is to provide accessible and sound guidance for the design and analysis of observational studies . This guidance is intended for applied statisticians and other data analysts with varying levels of statistical training, experience and interests. TG9 is one of nine topic groups of STRATOS and deals with aspects of HDD analysis.
Main issues addressed by TG9 often overlap with those of other TGs, but in the work of TG9 there is always a focus on the HDD aspect. Sometimes TG9 guidance will build upon that of other TGs to adapt it for relevance to HDD (see the “Discussion” section), but also completely new issues arise and require novel statistical approaches.
High-dimensional data are now ubiquitous in biomedical research, very frequently in the context of observational studies. Particularly omics data, i.e., high-throughput molecular data (e.g., genomics, transcriptomics, proteomics, and metabolomics) have provided new insights into biological processes and disease pathogenesis and have furthered the development of precision medicine approaches . Rapidly expanding stores of electronic health records contain not only standard demographic, clinical, and laboratory data collected through a patient history, but also information from potentially many different providers involved in a patient’s care . Data may be derived from multiple sources and can be represented in many different forms. Collectively, these data can be leveraged to support programs in comparative effectiveness and health outcomes research, and to monitor public health. Many statistical methods that are discussed here may be applied to health records data as well as to omics data, but our primary focus here is on the analysis of omics data.
Simultaneously, advances in statistical methodology and machine learning methods have contributed to improved approaches for data mining, statistical inference, and prediction in the HDD setting. Strong collaborations between data and computational scientists (e.g., statisticians, computational biologists, bioinformaticians, and computer scientists) and other biomedical scientists (e.g., clinicians and biologists) are essential for optimal generation, management, processing, analysis, and interpretation of these high-dimensional biomedical data .
Credibility and importance of research findings from biomedical studies involving HDD can be better judged when there is understanding of various approaches for statistical design and analysis along with their strengths and weaknesses. While this overview directly aims to improve understanding, simultaneously this guidance implies what information is necessary to report to fully appreciate how a study was designed, conducted, and analyzed. Whether study results prompt further pre-clinical or early clinical work, or translation to clinical use, ability to judge quality, credibility, and relevance of those results is critical. It is important to avoid sending research programs down unproductive paths or allowing flawed research results such as poorly performing prognostic models or therapy selection algorithms generated from HDD to be implemented clinically . Historically, research involving biomarkers and prognostic modelling has been criticized for lack of rigor, reproducibility, and clinical relevance [7,8,9,10], and for poor reporting [11, 12]. At least as many deficiencies are also common in biomedical research involving HDD. The goal of STRATOS TG9 is to reduce these deficiencies, and improve rigor and reproducibility, by providing widely accessible didactic materials pertinent to studies involving HDD.
In any observational study, including in the HDD setting, study design plays a crucial role in relation to the research question. A first important point is the precise definition of the target population and the sampling procedure. The subjects included in a study (or biospecimens derived from them) may be selected from the population by a random or other statistically designed sampling procedure (e.g., case–control, case-cohort), or may simply represent a “convenience” sample. It is therefore important to understand whether the subjects are representative of the target population, how the variables associated with subjects were measured or ascertained, and whether there are potential confounding factors. Failure to account for confounding factors or minimize bias in subject or variable ascertainment can lead to useless or misleading results.
Outcome-dependent sampling is rather common in observational studies, particularly for those investigating risk factors for relatively uncommon diseases or outcomes. Examples include classical matched or unmatched case–control designs along with two-phase sampling from a cohort (case-cohort or nested case–control). Another often-used strategy oversamples long survivors, or, for continuous outcomes, subjects with high and low values of the outcome variable. When any such sampling strategies are employed, it is important to use inferential procedures [13, 14] that properly account for the sampling design.
Laboratory experiments generating high-dimensional assay data should adhere to the same best practices as traditional controlled experiments measuring only one or a few analytes, including randomization, replication, blocking, and quality monitoring. Arguably, careful design might be even more important in the setting of HDD generation because HDD assays may be especially sensitive to technical artifacts. Even when a study is primarily observational yet involves analysis of stored biospecimens using omics assays, good design principles should be followed when performing the assays. Best practices include randomizing biospecimens to assays batches to avoid confounding assay batch effects with other factors of interest. For unmatched case–control studies, balancing (randomizing) cases and controls into batches may provide important advantages for reducing the influence of batch effects . For matched case–control studies or studies involving analysis of serial specimens from each subject, grouping matched or longitudinal sets within the same assay batch can be a convenient way to control for batch effects.
Another fundamental aspect of design is sample size, which refers to the measurement of different subjects, which are referred to as biological replicates. Whenever there is interest in making inference beyond an individual subject, e.g., assessing differential gene expression between groups of subjects with different phenotypes or exposed to different conditions such as treatments, biological replicates are required. In the HDD setting, standard sample size calculations generally do not apply. If statistical tests are performed one variable at a time (e.g., differential expression of each gene comparing two groups), then the number of tests performed for HDD is typically so large that a sample size calculation applying stringent multiplicity adjustment would lead to an enormous sample size. Alternative approaches to controlling false positive findings in HDD studies are discussed in section “TEST: Identification of informative variables and multiple testing.” If the goal is to develop a risk or prognostic model using HDD, typical recommendations about the number of events required per variable break down . Other sample size methods that require assumptions about the model are challenging to implement considering the complexity of models that might be used in HDD settings [17, 18], as discussed in section “PRED2.4: Sample size considerations.” In reality, HDD studies are often conducted with inadequate sample size, which is an important reason why many results are not reproducible and never advance to use in practice .
It is important to distinguish technical from biological replicates. Technical replication refers to repeating the measurement process on the same subject. It should not be confused with sample size. Technical replicates are useful for evaluating the variability in the measurement process, which may be comprised of multiple steps each potentially contributing to the total error in the measurement  (Fig. 1) described the many steps in gene expression microarray analysis of mouse brains. Technical replication could theoretically be carried out at any of those steps. Sometimes measurements are repeated using an alternative non-high-throughput measurement technique (e.g., RT-PCR assay to measure expression or Sanger sequencing of a specific gene) as a form of measurement validation, but this must not be confused with other forms of validation such as clinical validation of a prediction model (see section “PRED2: Assess performance and validate prediction models”). In presence of budget constraints, if the goal is to compare different biological conditions, it is advisable to invest in biological replicates. When biological samples are inexpensive compared to the cost of the measurement process, pooling is sometimes recommended as a way to reduce costs by making fewer total measurements . However, caution is advised, as assumptions may be required about assay limits of detection or the correspondence between physical pooling and additivity of measurements . The context of any technical replication must be carefully described along with any methods of summarizing over replicates in order to interpret results appropriately.
Design of a study should ideally be placed in the context of an overarching analysis plan. Each individual study should be designed to produce results of sufficient reliability that its results will inform next steps in the research project.
Structure of the paper
This paper is organized with respect to subtopics that are most relevant for the analysis of HDD, particularly motivated by typical aims of biomedical studies but also applicable more generally. These subtopics are initial data analysis (IDA and Preprocessing, section “IDA: Initial data analysis and preprocessing”), exploratory data analysis (EDA, section “EDA: Exploratory data analysis”), multiple testing (section “TEST: Identification of informative variables and multiple testing”), and prediction (section “PRED: Prediction”). For each subtopic, we discuss a list of main analytical goals. For each goal, basic explanations, at a minimally technical level, are provided for some commonly used analysis methods. Situations are identified where performance of some traditional, possibly more familiar, statistical methods might break down in the HDD setting or might not be possible to apply at all when p is larger than n. Strengths and limitations of competing approaches are discussed, and some of the gaps in the availability of adequate analytic tools are noted when relevant. Many key references are provided. It should be noted that throughout this paper we are concerned almost exclusively with cross-sectional or independent observations rather than longitudinal observations.
Topics in the paper are organized into sections according to the structure summarized in Table 1, followed by a discussion of the importance of good reporting to improve transparency and reproducible research in the “Discussion” section and a summarizing discussion in the “Conclusions” section.
IDA: Initial data analysis and preprocessing
Initial data analysis (IDA) is an important first step in every data analysis and can be particularly challenging in HDD settings. IDA is a term for all steps of data inspection and screening after the analysis plan and data collection have been finished but before the statistical analyses are performed [23, 24]. It focuses on understanding the context in which the data were collected, on data cleaning (see section “IDA1: Identify inconsistent, suspicious or unexpected values”), and on data screening (see section “IDA2: Describe distributions of variables, and identify missing values and systematic effects due to data acquisition”). Data cleaning refers to identifying and possibly correcting errors. Data screening includes reviewing the characteristics of the data that could affect the subsequent analysis plan, for example, describing distributions of variables, by checking assumptions required for model fitting and hypothesis testing, describing missing values, and identifying the need for adjustments of systematic effects due to data collection. Systematic effects may include batch effects that are caused, e.g., by different technologies used for collecting the data or even by different technicians performing laboratory experiments, see section “IDA3.2: Batch correction” for details. Further, initial steps may include simplification of data, e.g., by excluding or collapsing variables, if deemed appropriate. Insights about the data gained from these screening steps might lead to refinement or updating of an analysis plan to ensure that the data are consistent with any assumptions or requirements of the proposed analysis strategies (see section “IDA4: Simplify data and refine/update analysis plan if required”). However, IDA should always be conducted independently of the analysis needed to address the research questions, in order to avoid biasing conclusions.
The term “data preprocessing” is often used in biomedical research involving analysis of HDD, especially in the omics field, to denote certain initial data cleaning and screening steps falling within the more general category of “initial data analysis.” Data preprocessing refers to the process of transforming “raw” data, obtained directly from measurement instrument, into quantifications that are suitable for the subsequent statistical analysis. This includes detection and handling of incomplete, incorrect or inaccurate values, application of normalization methods that aim to remove systematic biases (e.g., assay batch effects), and transformations of variables .
A first step of the data cleaning and screening process is often to standardize the names or terms of variables and observations, especially for omics data compiled using different technologies. This type of standardization helps facilitate other, more complex downstream analyses and interpretation of results, as well as better online dissemination and archiving of data.
The IDA material is organized for ease of discussion, but the IDA process is typically iterative. Preprocessing is discussed in section “IDA3: Preprocessing the data,” but after preprocessing one may need to go back to the data cleaning and screening steps described in sections “IDA1: Identify inconsistent, suspicious or unexpected values” and “IDA2: Describe distributions of variables, and identify missing values and systematic effects due to data acquisition.” Note also that some model-based methods used for the identification of informative variables incorporate normalization into the data analysis model (see section “TEST: Identification of informative variables and multiple testing”).
IDA1: Identify inconsistent, suspicious or unexpected values
Identification and handling of incomplete, incorrect, or inaccurate values is logically a first step in IDA. Attention is directed toward distinguishing aberrant values that clearly originate from the data collection or generation process from those that might reflect true biological variability. Both visual and analytical inspections of the data are used for the detection of such values.
IDA1.1: Visual inspection of univariate and multivariate distributions
Graphical displays are helpful to both understand the structure of the data and detect potential anomalies. For HDD, it is rarely feasible to conduct a detailed examination of the distribution of every variable individually. Visual displays might be constructed only after variables of interest have been identified, for example because a gene is differentially expressed between two experimental conditions or because a particular variable is identified to have an unusual distribution by calculation of summary statistics or has an outlier. A practical alternative is to first calculate scores (summary statistics) for each variable or pair of variables, and then select both typical and interesting atypical variables, with respect to distributions of the scores, for more detailed inspection of their univariate or bivariate distributions. Types of scores to be used in these analyses should include those that capture specific features of the distributions, including measures of location, dispersion, skewness, kurtosis for univariate distributions, linear relationships for bivariate distributions, and metrics to detect outliers or influential values (Table 2).
IDA2: Describe distributions of variables, and identify missing values and systematic effects due to data acquisition
IDA2.1: Descriptive statistics
For understanding the structure of data, often univariate measures for location and scale of the variables are informative. In the HDD setting, graphical display is often helpful to scan these measures across the large number of variables, both for detecting anomalies in the data and for a general exploration of variable distributions and their consistency with assumptions required for certain analysis methods. An example of the use of boxplots and of smooth histograms for exploratory purposes can be found in .
Standardization of data values is often performed prior to data analyses. Typically, this refers to normalization with respect to scale and location (e.g., subtract mean or median and divide by standard deviation). This can be helpful to give variables similar weight, especially if they are measured on different scales. However, standardization removes information about absolute magnitude of effects, so it should not be used when the actual magnitude of differences is of interest (e.g., differences in mean expression values between two groups). Another caution is that HDD will typically contain a certain number of variables that are uninformative because they do not vary much across observations, with variability essentially reflecting noise in the data. Standardization of such variables can exaggerate the noise to give these variables undue influence in analyses that is on par with that of truly informative variables. It is often preferred to drop such uninformative variables at the start of analyses (Table 3).
IDA2.2: Tabulation of missing data
Missing values are ubiquitous in real-world data and may have major implications for choice of analysis methods and interpretation of results . In fact, most multivariable and multivariate analyses methods have as their default requirement that values of all variables are available for all subjects, i.e., all observations are “complete.” An important early step in any analysis is tabulation of the missing values, i.e., the identification of the number of missing values per subject and per variable, respectively, to provide an overview of the missing data structure. In multi-omics integrative studies, high-dimensional data from different data types are collected for the same subjects. In such studies, small sample size caused by experimental and financial constraints, which can also vary between data types, can be the reason for missing data, the absence of which has to be taken into account in the subsequent statistical analysis.
IDA2.3: Analysis of control values
Laboratory assay measurements can be affected by technical artifacts related to factors such as reagent lots, equipment drift, or environmental conditions. Sometimes these artifacts can be detected, and potentially adjusted for, through use of control and reference standard samples, which have expected distributions of measurements. For single-analyte assays, a calibration process is typically performed to adjust raw measurements and produce final reported values (Table 4).
IDA2.4: Graphical displays
Systematic artifact effects arising from data acquisition processes can often be detected with graphical displays that visualize the data in a comprehensive manner. A widely used graphical representation for multivariate data is a principal components plot, which is also useful in exploratory data analysis, as described in section “EDA: Exploratory data analysis” (Table 5).
IDA3: Preprocessing the data
Data generated by omics assay technologies typically require preprocessing by specially tailored methods that are based on understanding of the sophisticated instrumentation and scientific underpinnings of the technologies. Omics data are some of the most frequently encountered HDD in biomedical settings and are the focus in this paper. However, similar challenges exist with other types of HDD in biomedical research. Notably, high-dimensional imaging data are becoming commonplace, with examples including those generated by digital radiography, PET scans, and magnetic resonance imaging. In the following, we explain the main principles of data preprocessing using omics data examples.
Omics technologies are highly sensitive to experimental conditions and can exhibit systematic technical effects due to time, place, equipment, environmental conditions, reagent lots, operators, etc. In general, the first step of preprocessing aims to obtain an “analyzable” signal from the “raw” measurements. Subsequently, the signal is separated from possible systematic technical effects. The corrected signal may then be transformed to fulfill certain distributional properties, e.g., approximating a normal distribution. Note that sometimes the transformation may be applied before correcting the signal.
Preprocessing aimed at removal of systematic effects is often conducted as a separate step, as part of the IDA process, before the statistical analysis for answering the research question is undertaken. If the data have already been corrected for systematic effects and retain only the signals of interest (e.g., treatment effects), then the preprocessed (“normalized”) measurements for the biological samples can be analyzed using statistical methods that are easily accessible to researchers. However, conducting normalization as a separate step has important disadvantages. For instance, the normalized values are estimates and often carry with themselves some uncertainty, which should be taken into account in the analysis of the normalized data. However, this complicates the statistical analysis.
If inferential analysis is of interest, e.g., when comparing groups of samples to assess for biological differences, then a preferred approach is to consider normalization as part of a comprehensive statistical analysis model. The model is then used both to remove systematic technical differences and to quantify biological effects of interest (e.g., treatment effects). In that case, the uncertainty related to the normalization part of the analysis is naturally included in the estimates of uncertainty (standard errors) of the quantities of biological interest.
IDA3.1: Background subtraction and normalization
Omics data are prone to perturbations due to systematic effects induced by the measurement technology, also referred to as the assay platform. Many of these effects are unique to the assay platform, but there are some commonalities. A biological sample may have its gene expression profile measured using a single microarray or gene chip or its protein profile measured using a mass spectrometry system. The set of experimental conditions that gives rise to profiles such as these will be referred to here as an experimental run. However, even for the same sample, measurements obtained in different runs may differ due to factors such as different amounts of biological material input to the measurement system, settings on the instrumentation, environmental conditions in the laboratory, and so forth. These “between-run” differences may confound the “between sample” comparisons of scientific interest. Thus, these nuisance run effects should be removed to allow valid comparisons among data obtained in different runs. A generic preprocessing step aimed at removing between-run differences is often termed normalization. Even before normalization methods are applied, data generated by omics technologies generally require correction to subtract background noise from measurements to reveal their signal components. In Table 6 we introduce some basic methods for background subtraction and normalization.
IDA3.2: Batch correction
Another example of a systematic effect that is common to many technologies is a “batch effect.” The effect may arise when groups of biological samples (“batches”) have something in common in the way they are processed, e.g., same day or time of day, on same instrument, same operators, but these aspects are different for other groups of samples. Besides these measurement conditions, factors at play prior to measurement can cause batch effects. For example, clinical centers might differ in their standard operating procedures for processing, handling, and storing biospecimens, giving rise to pre-analytic factors that could influence downstream measurements. Patient characteristics, co-morbidities, or concomitant medications could additionally vary by batch, and may give rise to different distributions of measured values that have biological basis. Batch effects are widespread . The challenge for batch correction is removal of nuisance effects such as those due to pre-analytic or technical factors while not inadvertently removing true biological differences. To facilitate appropriate correction, batch information such as dates, instrument, operator, and specimen collection sites should be recorded and patient factors might need to be taken into account in analyses. Above all, it is critical to avoid poor study designs in which important patient characteristics (including outcomes) are confounded with nuisance batch effects, as this could make it impossible to remove nuisance batch effects adequately.
Preprocessing of omics data aimed at removal of the aforementioned artifact effects poses several challenges. For instance, normalization is often data-driven and uses methods based on assumptions about the nature of the biological mechanisms. If those assumptions do not hold, then the methods might not work as intended. An example of a commonly made assumption in experiments involving genome-wide expression data is that most genes are not differentially expressed under the compared conditions. It may be challenging to verify whether such assumptions are correct.
The dependence of systematic effects on the platform raises an important issue for novel technologies, for which sources of measurement variation may not be fully established or understood. Out of convenience, preprocessing approaches developed for one platform have often been applied to other platforms. For example, normalization methods developed for microarrays are also used for proteomic  and metabolomic  mass spectrometry experiments. This might be reasonable in some settings, but the assumptions required for adequate performance of a normalization method should always be reviewed carefully for appropriateness prior to its application to another technology.
In addition, it is worth noting that preprocessing may need to be tailored to the analysis goals. For instance, it is problematic to remove batch effects when constructing a classification rule. This is because the measurements for a new sample presented for classification will most likely include effects from batches not represented in the data used to construct the classification rule. Consequently, a classification rule should be constructed using data that have not been batch corrected so that robustness to batch effects can be built in (Table 7).
IDA4: Simplify data and refine/update analysis plan if required
The findings from the IDA steps can have substantial impact on the choice of appropriate analytical methods for subsequent statistical analyses. Therefore, the analysis plan should be refined or updated as necessary and according to the relevant findings from the IDA analysis .
Recoding primarily refers to transformations of the (original, raw) data, which allow for easier handling for a specific purpose. This is particularly useful in HDD settings, in which simple representation of the information can be challenging and sometimes even impossible due to the large number of variables (Table 8).
IDA4.2: Variable filtering and exclusion of uninformative variables
Variable filtering refers to the exclusion of variables that are considered uninteresting, before the statistical analysis to address the main research question is even started. This practice is widespread in HDD analysis where any steps to reduce the dimensionality and complexity of models at the outset are appreciated. If many irrelevant variables are filtered out, the multiple testing problem (see section “TEST: Identification of informative variables and multiple testing”) is diminished, and the statistical power of subsequent analysis steps can substantially increase. However, as discussed below, caution is required when applying certain filtering strategies that may introduce bias (Table 9).
IDA4.3: Construction of new variables
Sometimes it is useful to construct new variables as an initial step of data analysis by combining the variables that are available in the dataset in a meaningful way, using expert knowledge. For example, in medical studies investigating factors affecting health, often, overweight status is an important variable to consider in the analysis. Because weight and height must be considered together in assessing whether an individual is overweight, constructed variables like body mass index (BMI) have been used. The importance of fat distribution has also been recognized, and it has motivated the combined measured of waist-hip ratio (WHR). Instead of relying on the ability of statistical methods and algorithms to construct such variables implicitly, e.g., during a statistical modelling process, it is useful to be informed by expert knowledge and to include these constructed variables directly into analyses.
Not all constructed variables are derived using expert knowledge. Some, like principal component scores (see section “IDA2.4: Graphical displays”), are constructed in an unsupervised manner meaning that they are constructed to capture features of the data based only on the explanatory variables without using dependent variables such as outcomes. These constructed variables are sometimes used as explanatory variables when building prediction models (see section “PRED: Prediction”), and they can also be used for exploratory data analysis (see section “EDA: Exploratory data analysis”). As discussed in section “IDA2.4: Graphical displays,” plots of (typically the first two) principal components are often helpful for detecting peculiarities in the data or problems such as batch effects. Some constructed variables are derived using outcomes or other dependent variables. Examples of outcome-informed constructed variables include supervised principal component , or partial least squares (PLS) scores (see section “PRED1.3: Dimension reduction” for further discussion). Sometimes new variables are constructed by discretization of continuous variables, but this practice is problematic and should generally be discouraged (Table 10).
IDA4.4: Removal of variables or observations due to missing values
The simplest approach to deal with missing data is a “complete case analysis.” That is, if a single variable is missing for an observation, the observation is fully excluded from the dataset. Basing analyses on only complete cases at best only leads to loss of statistical power, but at worst can lead to substantially biased analyses. Impact of missing data will depend on how many cases have missing data, how many variables have missing values, how many values are missing, and whether the likelihood of missing values in a variable is related to the value of that variable or other variables. When few observations have missing values for few variables, then the impact on results of subsequent analyses may be limited, but when the number is large, the impact can be substantial.
A typical strategy for dealing with missing data is to exclude variables from the analysis that have a large number of missing values. Obviously, the possible relevance of such variables is neglected. Only when the missingness (the events that lead to a value being missing) is independent of both unobserved and observed values, i.e., the data are missing “completely at random” (MCAR), are the results of the complete case analysis (using likelihood-based methods) unbiased. When missing values depend on the unobserved values themselves (e.g., it is more likely that the measurement of a variable is missing when the value of the biomarker is very high or very low), then the missing values are said to be “missing not at random” (MNAR), and the resulting complete case analysis is biased.
Between the two extreme situations of MCAR and MNAR, there is a third possibility: missing values are called “missing at random” (MAR), when the missingness is independent of the unobserved values after controlling for the other available variables. One way to diagnose whether data are MCAR or MAR is to tabulate a missing value indicator against the values of other variables. As an example, if the value of a biomarker (e.g., gene expression level) is missing with higher frequency in males than in females, but within these strata, the missing values are missing completely at random, then it is likely a situation of MAR and not MCAR.
In HDD settings, when a large number of variables must be considered, complete case analysis may require exclusion of too many observations. To avoid this, common approaches involve first removing variables for which more than a specified percentage (e.g., 5 or 10%) of observations are missing and then removing observations for which more than a specified percentage (e.g., 2%) of variables have missing values. For studies with more complex designs, additional considerations may apply. For example, it is common in case–control studies to remove variables for which there is larger imbalance (e.g., more than 5 or 10% difference) in the percentage of missing values between cases and controls.
For MAR situations, methods more sophisticated than complete case analyses or dropping variables are recommended to use the information from all observations in the study and obtain less biased results. An example method is multiple imputation, which is described below. Although imputation is a useful strategy, it should be understood that no single approach for dealing with missing data is fully satisfactory. Thus, the best approach is to carefully select variables that are both informative and feasible to collect when designing studies and then work diligently to collect those data as completely as possible in order to minimize the amount of missing information. In the context of LDD, a framework for the treatment and reporting of missing data was proposed .
For HDD data, performing a simple multivariable regression in high dimensions is typically not feasible. Therefore, most procedures for handling missing data in the HDD setting either involve a phase for selecting for imputation only those variables that are deemed important or trying to use some regularized regression  instead of standard multivariable regression. The handling of missing data in HDD settings is an active topic of research. Many tailor-made imputation algorithms have already been developed; for an early overview in the context of for gene expression measurements, see  (Table 11).
EDA: Exploratory data analysis
When performing statistical analyses, it is important to distinguish between exploratory data analysis (EDA) and confirmatory data analysis, as this has important consequences both for the selection of appropriate analytical methods and for the correct interpretation of the results. The starting point for confirmatory analysis is a hypothesis to be evaluated, whereas, in EDA the goal is to provide an unbiased view of the data. Insights from EDA may then lead to development of new hypotheses that can be evaluated in subsequent confirmatory analyses on independent data.
Caution is necessary when performing statistical inference (e.g., feature selection as described in section “TEST: Identification of informative variables and multiple testing”) or model performance assessment following EDA when decisions to remove or modify observations from the analysis might depend on the observed relationships one is trying to confirm. For example, if outlier observations are removed from a dataset, the performance of a prediction model built only on the remaining observations is most probably an overly optimistic estimate of what the model performance would be on an independent dataset, which might contain different outliers.
Two major analytical goals for EDA are (1) to identify interesting data characteristics such as variables with extreme values, associations between variables, or representative subjects with usual values of variables, and (2) to gain insight into the structure of the data. Note that many of the methods used in EDA are also applied in IDA (like PCA; see section “IDA2.4: Graphical displays”). In this section, we focus on methods that are more specific to EDA. Note that many of the methods described in this section are generally designed and suitable for continuous data; only some can also be applied for discrete data.
EDA1: Identify interesting data characteristics
EDA can assist a researcher to identify interesting data characteristics that may lead to generation of specific scientific hypotheses that can be more fully evaluated in subsequent studies. Through EDA, a researcher might identify variables exhibiting extreme values or study subjects (observations) having extreme values of one or more variables or unusual combinations of values of two or more variables. EDA might also reveal intriguing associations between variables (e.g., levels of a certain protein tend to differ between two phenotypic classes). The two main classes of exploratory methods for identifying such interesting data characteristics are graphical displays and inspection of descriptive univariate and multivariate summary statistics. Graphical displays are discussed in sections “IDA2.1: Descriptive statistics,” “IDA2.4: Graphical displays,” and “EDA1.1: Graphical displays,” whereas descriptive statistics were already described in section “IDA2.1: Descriptive statistics” as tools for the initial data analysis (IDA). It should be noted that due to the potential for identification of many false positive signals in the HDD setting, findings from large-scale comparisons of descriptive summary statistics are often tempered by application of multiple testing methods as described later in section “TEST: Identification of informative variables and multiple testing,” even though the original intent was exploratory analysis.
To identify interesting data characteristics in low-dimensional data via visual or graphical methods, it is usually possible to inspect simple summary statistics and graphical displays of distributions of variables one, two, or three at a time, but for HDD this approach quickly becomes infeasible. For instance, the number of scatterplots for all pairs of p variables is p(p − 1)/2, which already exceeds 1000 when p exceeds 45. Visual identification of interesting characteristics of HDD typically requires specialized graphical displays or reduction of data dimensionality.
EDA1.1: Graphical displays
As mentioned in section “IDA2.4: Graphical displays,” one can use principal components (PCs) for exploratory analysis by first summarizing the information included in all variables through calculation of PC scores (which are linear combinations of the original variables) and then plotting in two or three dimensions the first several PC scores that capture the majority of variability in the data. This may allow identification of clusters of observations or individual observations with unusual configurations of variables warranting further inspection.
Another goal for HDD visualization is to produce a display in lower dimensions that preserves the distances (more generally degrees of “dissimilarity”) between observations such that the closest points remain the closest and the furthest remain the furthest. Alternative data reduction techniques have been developed to achieve this goal. These methods aim to translate the data in such a way that dissimilarities among points in the lower-dimensional space are as proportional as possible to those quantified in the original (high-dimensional) space. One such technique, multidimensional scaling, is described below. A variation of multidimensional scaling not discussed here is correspondence analysis, which is suitable for categorical variables and shows the relationships between variables based on data specified in a contingency table. Cox and Cox  provide descriptions of both multidimensional scaling and correspondence analysis (Table 12).
EDA2: Gain insight into the data structure
A global depiction of data to identify structure, including patterns or motifs, is another major goal of exploratory data analysis for HDD. Here, data structure is understood in a general sense, it refers to many aspects of the data that concern the arrangement or interrelation of the observations or variables of a dataset. Although a natural first step is to look at marginal distributions (e.g., univariate and bivariate) of all variables across observations, this approach is generally not feasible for HDD for reasons discussed above. Further, some structure may involve many different variables and not be discernible by examination of univariate, bivariate, or even trivariate distributions.
The data visualization techniques described in section “EDA1.1: Graphical displays” are often supplemented with additional approaches geared toward detection of certain kinds of structure, for example clusters. The goal of cluster analysis is to identify subgroups of observations or variables that are similar to each other, but different from others. Identification of prototypical observations to characterize each cluster might be of interest. The structure might also be multi-level. In this section, we focus on techniques that are useful to uncover structure that might be missed by examining only marginal distributions or low-dimensional representations of HDD.
EDA2.1: Cluster analysis
The goal of a cluster analysis is to assemble objects (observations or variables) into subgroups, termed clusters, such that similarities between members within the clusters are high (or, equivalently, distances are small), compared to similarities between members from different clusters. Sometimes, the goal is only to find dense, i.e., heavily populated, regions in the data space that correspond to modes of the data distribution. Alternatively, there may be interest in fully characterizing the structure. Cluster analyses typically require choice of a similarity metric (or, alternatively, distance metric) for pairs of objects (sometimes also for pairs of clusters), a clustering algorithm, and a criterion to determine the number of clusters. Some clustering approaches that have been successfully used for low-dimensional data, e.g., mixtures of low-dimensional parametric probability distributions such as multivariate normal mixtures, either cannot be applied at all or perform very poorly in the HDD setting. Approaches not suitable for HDD are not further discussed here.
For comparing similarity of objects (either variables or observations), the Pearson correlation coefficient or Euclidean distance are the most popular metrics. The Pearson correlation does not depend on the scale of the variables, but the Euclidean distance does. If each of the variables characterizing an object is first standardized across the set of objects (subtract mean and divide by standard deviation), then use of Pearson correlation and Euclidean distance metrics will produce equivalent results. The measure should be chosen deliberately. If only relative levels of the values are important, then Pearson correlation is suitable, but if absolute values matter, then Euclidean distance is appropriate. It is important to note that both metrics tend to be more heavily influenced by a few large differences or deviations than by a series of small ones because the values are squared. An important modification of the Pearson correlation is the Spearman (rank) correlation, where values of observations are first replaced by their corresponding ranks before calculating the Pearson correlation. With this adjustment, the results are less heavily influenced by extreme data values.
In high-dimensional spaces, data are typically quite sparse. This means that distances between objects become large, a phenomenon often referred to as the curse of dimensionality. Therefore, the distance metrics may be prone to exaggeration by a few distant objects. Strategies to help avoid this problem include use of data reduction or variable selection before clustering (see section “IDA2.4: Graphical displays” for graphical displays for dimension reduction and section “PRED1.2: Variable selection.” for variable selection and dimension reduction in the context of improving prediction models).
Clustering algorithms can be divided into hierarchical and partitioning methods. In hierarchical clustering, observations are iteratively grouped together into larger clusters (agglomerative hierarchical clustering) or clusters are subdivided into smaller clusters (divisive hierarchical clustering). Centroid-based so-called partitioning algorithms aggregate the observations around specific points (the centroids) such that observations related to the same centroid are as similar as possible, and observations related to different centroids as different as possible. Hierarchical clustering algorithms provide a clustering for any number of clusters, whereas partitioning methods require an initial choice about the number of clusters present in the data. The most popular clustering algorithms are described in Table 13.
Other methods for cluster analysis applied to biomedical data include fuzzy clustering and SOMs (self-organizing maps). In fuzzy clustering, objects can belong to multiple clusters. In SOMs (a type of neural networks first introduced by Kohonen ), a meaningful topology (special relationships) between the cluster prototypes is assumed. This means that the clusters can be visualized as a two-dimensional “map,” so that observations in proximate clusters have more similar values than observations in clusters that are more distant. Since the assumptions for SOMs are not guaranteed to hold, the interpretation can easily be misleading, such that SOMs should only be used by experts in this field. In addition, SOMs can be very sensitive to starting node configurations.
For HDD, the computer runtime of such partitioning algorithms can present a challenge. For example, PAM cannot be applied if the number of objects to be clustered is very large, i.e., for clustering variables in omics data or for clustering observations in large health records data. This challenge motivated development of the algorithm CLARA (Clustering Large Applications) , which works on subsamples of the data. Distribution-based clustering methods provide another alternative where probabilistic distributions for the observations within the clusters are assumed (e.g., multivariate Gaussian in each cluster, but with different means and potentially different variances). Parameters of the mixture distribution are typically estimated with EM-type (expectation–maximization) iterative algorithms . However, not only, but particularly for HDD, the distributional assumptions are often difficult to verify and the algorithms may not converge to a suitable solution. Therefore, clusters might not be identified at all, or the results could be misleading due to incorrect assumptions about the data distributions.
Results produced by clustering algorithms are difficult to evaluate and often require subjective judgement. The validity of the results depends on the notion of a cluster, which varies between clustering algorithms, and this ambiguity carries through to estimation of the number of clusters (Table 14).
Some clustering methods have been specifically developed to handle the typical large storage requirements and long run times for HDD settings. For example, CAST (Cluster Affinity Search Technique)  is especially useful for large numbers of observations or variables. Iteratively, clusters are constructed as follows. Choose a randomly selected observation not already assigned to a cluster and assign it to a newly defined cluster. Then repeat the following two steps until the set of observations assigned to this new cluster no longer changes. Add unassigned observations with average similarity to the current cluster members above a predefined threshold, and remove observations with average similarity below this threshold.
Another method is subspace clustering , where first subsets of variables are identified (called subspaces) and clusters are determined by defining regions of values based only on these variables. Then, iteratively, lower-dimensional subspaces are combined to higher-dimensional ones. In biclustering (or two-way clustering), first introduced by Hartigan , simultaneously variables and observations are selected to generate clusters that do not depend on all variables at the same time. Again, heuristic and stable algorithms are required to find approximate solutions in acceptable time (see, e.g., ).
Many traditional clustering methods are best suited for continuous variables, but there are several examples of HDD that are not continuous. One example is count data such as generated by RNA-Seq. Some examples of clustering methods that have been specifically developed for count data include those of Witten  and Si et al. , which are based on Poisson or negative binomial distributions. Cluster analysis based on deep learning has also been proposed . That approach trains a deep neural network, extracts the resulting hidden variables, and uses them as the basis for clustering using standard methods like k-means.
EDA2.2: Prototypical samples
Often it is useful to construct prototypical observations that represent subgroups of observations. Prototypical observations are, for example, identified by some clustering algorithms. The motivation is to allow visualization or provide a summary of relevant characteristics of subgroups of observations. These summaries can be interpreted in the biomedical context, for example as a description of the characteristics of a typical patient who responds well to a particular therapy. Prototypical samples can be selected as central observations in their respective subgroups, or they can be newly constructed. When applying a k-means algorithm to separate observations into K clusters, centroids of each cluster are natural choices for prototypes. Similar to the principles of many cluster analysis approaches (see section “EDA2.1: Cluster analysis”), the construction of prototypical observations is done such that they are simultaneously as similar as possible to the observations of the same subgroup (cluster) and as different as possible from the observations of the other subgroups. Bien and Tibshirani  provide a nice overview of available methods, although their review is limited to classification problems. Prototypical observations can also be used to represent classes and then to predict the class of a new observation based on the similarities with these prototypical samples (see also section “PRED: Prediction”).
TEST: Identification of informative variables and multiple testing
In HDD analysis, one is often interested in identifying, among a large number of candidate variables, “informative variables.” These are associated with an outcome or with a set of other phenotype variables that characterize the study subjects. For example, one might wish to characterize which single-nucleotide polymorphisms are more often present in patients who experience severe side effects from a particular drug compared to patients without severe side effects. In drug sensitivity screens performed on bacterial cultures, one might aim to identify bacterial genes with expression significantly associated with degree of sensitivity to a new antibiotic. When comparing individuals with a particular disease to healthy volunteers, one might wish to identify circulating proteins that are present in different abundance. In all these cases, evaluation of the associations might be accomplished by conducting many statistical hypothesis tests, one per candidate variable. This represents a multiple testing situation.
Multiple testing scenarios commonly encountered in biomedical studies with HDD are divided here into three categories. Scenarios that consider each candidate variable individually and perform a similar evaluation or statistical test for each include the following three cases: (i) Identification of variables among a set of candidates that are associated with a single outcome or phenotype variable, i.e., related to outcome or phenotype classes (categorical) or correlated with a continuous phenotype variable or time-to-event outcome. (ii) Identification of candidate variables with a trajectory over time affected by experimental factors or exhibiting a prescribed pattern. (iii) Identification of candidate variables that are associated with a prespecified set of other variables, i.e., where the candidate variables are considered as dependent variables and the set of prespecified variables as independent “predictor” variables. To illustrate the concepts, much of the discussion here will focus on a simple example of scenario (i) in which two classes are being compared with respect to a very large number of variables. Methods discussed for scenario (i) that can be extended straightforwardly to scenarios (ii) and (iii) are noted.
Scientific goals may go beyond simply providing a list of individual variables exhibiting associations with an outcome, a phenotype, a collection of prespecified variables, or patterns over time. Frequently, there is interest in more globally characterizing the variables that were included in the identified list. For example, genes are organized into interconnected biological pathways. Expression of two different genes might exhibit similar associations because they are both regulated by certain other genes, because one lies downstream of the other in the same biological pathway, or because their products serve similar biological functions. Established organizational structures might be described by gene taxonomies such as Gene Ontology , KEGG , or BioCarta . Gene set enrichment analysis (see section “TEST3: Identify informative groups of variables”) refers to approaches that exploit these expected associations. They were first proposed in the omics field for use with HDD gene expression data. Although these enrichment analysis strategies could be applied in a variety of HDD settings, subsequent discussion of these methods will be based on examples with high-dimensional gene expression data for which the concept of enrichment is intuitively clear.
TEST1: Identify variables informative for an outcome
TEST1.1: Test statistics: Hypothesis testing for a single variable
Before discussing multiple testing procedures, it is helpful to briefly review basic concepts in statistical hypothesis testing involving a single variable. A hypothesis test aims to decide whether the data support or refute a stated “null hypothesis.” Typical examples of simple null hypotheses are that the distribution of a variable is not different between two or more groups or that a variable is not associated with another variable. A hypothesis test is based on some statistic that will reflect strength of evidence for or against the null hypothesis. Knowing the distribution of the test statistic (e.g., normal distribution or binomial distribution) allows one to construct a hypothesis test based on that statistic for which the probability of drawing an incorrect conclusion is controlled. Type I error refers to erroneously rejecting the null hypothesis when it is actually true. Type II error refers to failing to reject the null hypothesis when it is actually false. Statistical power is defined as one minus the type II error. In general, one wants to control the probability of a type I error, denoted α, at a small value, while maintaining acceptably high power (or low type II error). A conventional choice of \(\alpha\) for the single variable setting is 0.05, which means that the probability of a false positive decision, i.e., falsely rejecting the null hypothesis when it is true, is 0.05.
Hypothesis testing is often operationalized by calculation of a p-value from the observed data, which estimates the probability of observing a value of the test statistic that is at least as extreme as that observed, assuming that the null hypothesis was true. (Note the correct definition of a p-value stated here, in contrast to the common misinterpretation of a p-value as the probability that H0 is true). A significance test is performed by comparing the computed p-value to the prespecified α level. When the p-value is less than or equal to α (e.g., 0.05 in the conventional setting), the null hypothesis is rejected; otherwise, it cannot be rejected.
It should be mentioned that sometimes the goal of a scientific study is to estimate certain parameters of interest, for example means or correlations, rather than to test hypotheses. In estimation settings, it is generally desired to provide intervals of uncertainty, such as confidence intervals, to accompany parameter estimates. Although errors in hypothesis testing have some relation to confidence interval coverage probabilities, most of the multiple testing procedures discussed in this section are not readily applicable to multiple estimation. Multiple estimation procedures are beyond the scope of the present discussion.
The t-test is an example of a widely used statistical test for a single variable. It is the basis for the modelling approaches described below that are extensions of hypothesis testing to multiple variables. Extensions particularly developed for HDD include limma, edgeR, and Deseq2, as discussed in section “TEST1.2: Modelling approaches: Hypothesis testing for multiple variables.”
Calculation of a p-value usually requires assumptions about the distribution of the test statistic. Sometimes that distribution can be derived from assumptions about the distributions of the variables. For example, the statistic of the t-test can be shown to have a t-distribution when the variables are normally distributed, and the within-group variances are the same for the classes being compared. Similar requirements hold for F-tests in analysis of variance and statistics associated with standard linear regression analysis. Although one can never be certain if these assumptions hold for real data, many test statistics can be shown by theoretical arguments to have an approximate normal distribution when sample size is sufficiently large (referred to as “asymptotic” approximation). An example asymptotic property is that a t-statistic has an approximate normal distribution for large samples size, even if the data are not normally distributed. Nonetheless, extra caution is necessary in the setting of HDD where the requirements for sample size to qualify as “large” are far greater. Extremes of a test statistic’s distribution are particularly prone to departures from data distributional assumptions, and this is exactly where accuracy is needed most when calculating the very small p-values upon which many multiple testing procedures for HDD rely.
When validity of assumptions required for familiar statistical tests is uncertain, for example that the data follow a normal distribution for the t-test or F-test, alternative tests broadly referred to as nonparametric tests may be preferable. Wilcoxon rank sum (equivalent to Mann–Whitney U) and signed rank tests are nonparametric alternatives to the two-sample t-test and paired t-test, respectively; the Kruskal–Wallis test is an alternative to the F-test in one-way ANOVA. These nonparametric tests are robust to outliers and do not require data to be normally distributed; nor do they require that their distribution is fully characterized by two parameters in the way that a mean and variance characterize a normal distribution. Many nonparametric tests are based on ranks of observed data rather than their actual values. Permutation tests, as described in Table 15 and below, comprise another class of nonparametric tests and are more generally applicable than rank-based tests.
A word of caution is in order to emphasize that correct permutation of the data is critical to validity of a permutation test. The permutations must preserve any structure in the data that is unrelated to the null hypothesis. For instance, if the goal is to test whether the mean of a variable is different between groups, but it is thought that the variances are different, then the simple permutation test described for the two-group comparison will not be appropriate because the permutations will change the variances as well as the means. If the groups are paired, e.g., variables are measured both before and after each subject receives an experimental drug, then the permutations would have to preserve that pairing by randomly “flipping” the before and after measurements within patients. Correct permutation might not be easy, or even feasible, for regression models with multiple predictors. For example, naively permuting the outcomes in a logistic or Cox regression model with many predictors to provide test statistics for individual predictor variables (adjusted for the other variables) would not provide valid permutation p-values because the correlation structure of the data, e.g., correlations of the outcome with other variables that are not the focus of the test, would not be preserved. Anderson and Legendre  discuss appropriateness and performance of various permutation testing strategies in the context of testing partial regression coefficients in multivariable regression models.
Nonparametric methods have advantages and disadvantages. In the context of statistical tests, their main advantages include their applicability in situations where little is understood about the likely distribution of the data, and their robustness to oddities in the data such as outliers. The main disadvantage of nonparametric methods is their reduced statistical power, particularly for small samples sizes, compared to a parametric test when distributional assumptions of that test are actually met. For HDD settings, parametric tests have additional appeal, when reasonably justified, due to the possibility to “borrow information” across variables by modelling relationships of parameters (e.g., means or variances) across variable-specific distributions; modelling approaches such as those discussed in section “TEST1.2: Modelling approaches: Hypothesis testing for multiple variables” can greatly increase statistical power for testing multiple hypotheses.
TEST1.2: Modelling approaches: Hypothesis testing for multiple variables
In the scenarios (i)-(iii) described in the introduction of section “TEST: Identification of informative variables and multiple testing”, the number of statistical analyses performed is equal to the number of variables. For omics data, the number of variables is often in the range of tens of thousands or even millions. Direct application of standard hypothesis testing approaches to each variable in the setting of HDD is problematic. As an illustration, consider conducting several thousand statistical tests (one per candidate variable), each using the classical α level of 0.05 to test for significance of an association between a single variable and an outcome or phenotype of interest. If the truth were that none of the candidate variables had an association with the outcome or phenotype of interest, then, on average, testing 20,000 variables would lead to 1000 false positive test results (0.05 times the 20,000 variables tested), clearly an unacceptably large number that would limit interpretability of the results. Control of the number of false positives, often termed “false discoveries” in the setting of HDD, is critical.
Several challenges are encountered in multiple testing for HDD omics data. One is that in order to control false positives when a very large number of statistical tests are performed, small α levels must be used, which limits statistical power. Another challenge is the mathematical difficulty of dealing with joint distributions of certain variable types such as counts, which are commonly generated by newer omics technologies such as RNA-Seq. Furthermore, sample sizes are often insufficient to rely on classical statistical asymptotic (large sample size) theory to provide tractable approximate distributions of test statistics required to appropriately control type I and II errors. Finally, the classical approach of limiting false positives by controlling the overall probability of any false positive findings is overly stringent when extremely large numbers of tests are performed. These challenges have spawned a wealth of innovative statistical approaches for multiple testing with HDD, which are described in the sections that follow.
The earliest technologies for high-dimensional gene expression analysis based on microarray platforms quantified gene expression by fluorescence intensities. After logarithmic transformation, these continuous intensity values are typically well approximated by a normal distribution. Many of the early methods developed for statistical analysis of microarray data relied on normally distributed data, the simplest example being use of t-tests to identify lists of differentially expressed genes with varying degrees of type I error control. Sample size in these early studies was usually relatively small, making it difficult to adequately control false discoveries and still maintain sufficient statistical power. Some of these methods were ad hoc or limited to simple experimental settings such as two-group comparisons, but advances in statistical methodology led to improved approaches for the analysis of HDD gene expression data (Table 16).
Sometimes a researcher is interested in identifying genes for which expression is not different between conditions, opposite the more typical goal to identify differentially expressed genes. This requires reversing the usual role of the null and alternative hypotheses. However, since it is impossible to statistically rule out very tiny effects, the null hypothesis that is tested for each gene is that its effect is larger than some user-specified minimum size. When implementing this procedure to identify genes with negligible effect, mean parameter shrinkage functions must be turned off.
TEST2: Multiple testing
Methods described in the previous section provide useful approaches to improve statistical power for testing individual variables (genes) and to appropriately model commonly encountered omics data. However, a final step is required to control false positives in HDD settings. Several multiple testing correction methods and their utility for HDD are discussed in this section.
TEST2.1: Control for false discoveries: Classical multiple testing corrections
A simple table illustrates the types of errors that can be encountered in multiple testing . When testing m hypotheses, these are either true or false, and either rejected or not rejected, yielding four possibilities, which are displayed in Table 17 along with the numbers of hypotheses falling in each category.
In Table 17, m represents the number of tests conducted; R represents the number rejected hypotheses; V represents the number of tests for which type I errors were committed, or the number of false positives; and U represents the number of tests that correctly rejected the null hypothesis, or the number of true positives. Further, m0 represents the total number of true null hypotheses; m1 the total number of false null hypotheses; and m1 − U represents the number of tests for which type II errors were committed. The goal of a multiple testing procedure is to control V while not too severely limiting U. If R = 0, then no type I error can be committed. If m0 = m, then rejection of any test constitutes a type I error and represents a false positive result.
Classical multiple testing corrections that aim to control false discoveries by using more stringent (smaller) “critical” levels for significance testing may work well in situations with a few dozen tests or less. However, they can be problematic for HDD because they may be too stringent and severely limit statistical power for detecting associations that truly exist, particularly when sample sizes are not large.
The simplest approach to controlling false discoveries is the classical Bonferroni correction, where the critical level is adjusted by dividing it by the number of tests performed (see Table 18). Bonferroni correction is very stringent for several reasons. First, it is designed to control what is known as familywise error rate (FWER), which refers to globally controlling the probability that any of the tests results in a false discovery. In terms of the notation in Table 17, controlling the FWER at level α means requiring P(V > 0) ≤ α. Despite its conservativeness, Bonferroni adjustment has become the standard approach for genome-wide association studies to control the genome-wide significance level. This enforces stringent control on the probability that any of the hundreds of thousands of genomic variants typically studied is falsely identified as associated with the phenotype of interest. Second, a simple Bonferroni correction is conservative in that it does not leverage information about potential correlations between the test statistics; nor does it account for the ordering of the p-values when applying the significance-testing threshold. When evaluating p-values in order from smallest to largest, it is natural to require smaller critical levels for declaring significance earlier in the list. These limitations of the Bonferroni correction have motivated development of modified approaches that are less stringent, as discussed next.
Some adjusted versions of Bonferroni correction that take p-value ordering into account have been proposed. Some, such as those proposed by Hochberg  and Hommel , require assumptions about the joint distribution of the p-values such as the nature of correlations, and those are not discussed here. However, the approach proposed by Holm  provides a simple improvement on the Bonferroni method that allows critical values for significance testing to depend on the ordering of the p-values while, like Bonferroni, requiring no assumptions about the joint distribution of the p-values. Holm’s approach is described in Table 18.
Several other methods of controlling the FWER have been proposed that require additional assumptions about the nature of correlations between test statistics or might only control false positives under a global null in which all hypotheses are null. Such tests are not guaranteed to always control the FWER when these assumptions do not hold and will not be discussed further here.
An appealing aspect of multiple testing procedures that control FWER is that one can make statements about the probability that an individual test falsely rejects the null hypothesis. Because the probability that any test among a collection of tests falsely rejects must be at least as large as the probability that a single randomly chosen test falsely rejects, control of FWER at level α automatically guarantees control of the type I error at level α for each individual test.
An important caveat about any multiple testing correction method that is based on p-values is that it relies on the validity of the p-values or the validity of the corresponding test procedures. As noted in the discussion of test statistics above in section “TEST1: Identify variables informative for an outcome,” ensuring sufficient accuracy of p-values based on specific (parametric) distributions can be challenging in HDD settings. Permutation tests can provide distribution-free options for multiple testing in some situations. They also offer the flexibility to handle HDD with variables of different types, e.g., variables could be a mix of categorical, count, or continuous data. However, permutation tests can be problematic for multiple testing in HDD settings as well, as it can be very computationally intensive to accurately compute p-values that might be very small.
Multivariate permutation tests are permutation tests that are applied for testing multiple hypotheses simultaneously. For each hypothesis, a test statistic is calculated, for example for simultaneously comparing the distribution of many omics variables between two phenotype classes. As in the univariate case, class labels are randomly reassigned to the observations (keeping the full profile of measurements intact for each observation), and then a p-value for each variable is computed as the number of permutations on which the corresponding calculated test statistic is as extreme or more than the test statistic calculated of the original data. The popular Westfall-Young permutation procedure, as an example, is described in Table 18. Multiple testing procedures can be applied to the collection of permutation p-values to control false discoveries just as if the p-values had been computed assuming parametric distributions for the variable.
TEST2.2: Control for false discoveries: Methods motivated by HDD
Various multiple testing correction methods have been developed that are more appropriate for HDD than the classical Bonferroni-type methods. Usually, these approaches aim for a false discovery control that is less stringent than familywise error control, such as limiting the percentage of false discoveries (rather than aiming to avoid any false discoveries) in exchange for greater power to detect true discoveries. Many multiple testing methods for HDD are combined with methods such as those just discussed in section “TEST1.2: Modelling approaches: Hypothesis testing for multiple variables” that borrow information across variables (or tests) or that exploit correlations between candidate variables to increase statistical power. The growing amount of HDD stimulated development of a variety of innovative multiple testing procedures more appropriate for these data than traditional approaches.
To describe the various multiple testing approaches for HDD and the false discovery criteria that they control, it is helpful to focus again on one of the most frequent goals in omics data analysis, which is the identification of differentially expressed genes between two or more classes or conditions. The notation used in this section follows that defined in Table 17.
Aiming to control type I error in terms of the FWER through application of classical Bonferroni-type methods becomes extremely challenging with increasing dimension of HDD due to low statistical power, as already discussed. These challenges motivated consideration of alternatives to classical control of type I error, most commonly control of the false discovery rate (FDR). The popular FDR is in principle the expected proportion of false discoveries among the rejected tests and described in more detail below. The methods differ by the type of error they aim to control but share some operational aspects. Once the acceptable magnitude of error (e.g., FDR) has been specified, the (raw, uncorrected) p-values are calculated and next the variables are usually ranked based on their associated p-values. Those with p-values below a certain threshold are included in the list of the positive findings (rejecting their associated null hypotheses). This threshold can be fixed for all p-values, or it may depend on the ranking of p-value. Equivalently, the p-values can be adjusted and then compared to the desired level of error control. There are several methods for FDR control, which define in a different way the adjustment applied to the p-values and the threshold to which those p-values are compared.
As is common in statistics, some methods require additional assumptions and the claimed properties are only valid when those assumptions are met. In multiple testing, an important distinction is between methods that achieve weak control and those that achieve strong control. Weak control means that the method achieves the stated error control only when there are no true positives (i.e., all null hypotheses are true). In contrast, strong control means that the method achieves the stated control no matter how many of the null hypotheses are true or false. Only methods that provide a strong (general) control are discussed here. In multiple testing, it is also common to encounter assumptions about the dependence among variables or p-values; the assumption of independence among variables is unrealistic for omics data, where variables are often positively correlated.
In the following, we first define metrics to quantify false positives and then briefly present some of the methods that have been proposed to control them, focusing only on the essential concepts. We point the more technical reader to comprehensive reviews of multiple testing methods by Dudoit et al.  and more recently by Goeman and Solari . A practical introduction providing illustrative examples with implementation in the R language is available in the book of Bretz et al. .
The FDR is a popular extension of the concept of type I error for HDD. Using the notation described in Table 17, FDR is the expected (average) value of Q, i.e., FDR = E(Q), where Q = V/R if R > 0 and Q = 0 if R = 0 . Q is sometimes also called FDP (false discovery proportion). Since the case R = 0 is very uncommon in practical HDD applications, the FDR can be roughly thought of as the proportion of false positives among declared positives (i.e., among rejected tests). Controlling FDR is less stringent than controlling FWER, as FDR control inherently allows for some false positives. The goal of FDR control is to identify as many positive test results as possible, while accepting a relatively low proportion of false discoveries. In practice, common choices for FDR control are 5 or 10%.
The Benjamini–Hochberg procedure  is the most widely used method for controlling the FDR. It is described in Table 19. Notably, the adjusted threshold value used by the Benjamini–Hochberg method is identical to that used by the Bonferroni and Holm methods for the variable with the smallest p-value, but it is much larger for the others. It is generally true that lists of discoveries generated by procedures that control the FDR are much longer than those generated by methods that control the FWER at the same level. Yet, like the Bonferroni method, the original FDR method is conservative, effectively controlling the FDR at level α·m0/m ≤ α if the variables are independent. Many methods were proposed to improve the power of FDR by estimating this unknown proportion of true null hypotheses (m0/m) from data and using it to adapt the threshold value (see ). The original FDR , which was proposed for independent variables but proven to be valid under the assumption of a positive correlation of the p-values, was extended by Benjamini and Yekutieli  to handle more general dependencies. This more general procedure has lower thresholds and is more conservative. Several other methods were proposed to control FDR and some error rates closely related to FDR were defined . Figure 15 illustrates how the Bonferroni and the Benjamini–Hochberg correction work.
Many extensions and modifications of the FDR have been proposed. The most common criticism of FDR is that it controls only the average proportion of false positives, which might be very variable: in practice, the actual proportion Q of false positives derived from an analysis might differ substantially from the targeted FDR threshold, but the FDR methods do not provide an estimate of this variability. Readers are referred to Goeman and Solari  for discussion of methods that aim to control the false discovery proportion with a specified confidence. Other methods have been proposed for control of local FDR, a concept that allows a more powerful interpretation of q-values at the level of single hypothesis and not as a property of a list of variables . In practice, the FDR-controlling approaches are some of the most widely used methods for multiple testing for omics data, despite some recognized limitations.
TEST2.3: Sample size considerations
Determination of an appropriate sample size for a study that will involve conducting an extremely large number of statistical tests is very challenging. Sample size methods must be tailored to the desired approach and criteria for error control. Both false positive (type I error) and false negatives (type II) errors need to be considered. Early in the emergence of omics data, sample size methods focused on FDR control , particularly for microarray technology . A recent review mainly focusing on sequencing experiments also provides useful guidance .
TEST3: Identify informative groups of variables
The multiple testing problem is less severe when the interest is shifted to groups of variables instead of single variables, as described in the introduction of section “TEST: Identification of informative variables and multiple testing” as example (iii) of the main scenarios. In most cases, the groups are prespecified (e.g., genes belonging to the same biological pathway, genes with the same molecular function, or mutations on the same arm of a chromosome). A variable can belong to more than one group, and often the variables belonging to the same group are positively correlated. This type of analysis has the potential of having greater statistical power and greater between study reproducibility than a variable-by-variable analysis.
The methods tailored for the analysis of groups of variables can be divided into two broad classes [115, 116]: The first class are competitive methods, which attempt to identify which variable groups have a stronger association with the outcome (or phenotype) than the other groups. The second class are self-contained methods, which try to identify which of the variable groups contain at least one variable that is associated to the outcome. Example approaches are described below. The popular gene set enrichment analysis (GSEA) and over-representation analysis (ORA) are mixed approaches, while topGO is a competitive method and the global test a self-contained method. In all cases, FWER or FDR can be controlled using any of the methods already described in section “TEST2: Multiple testing” on multiple testing. When applying multiple tests for groups of variables, multiplicity refers to the multiplicity of these groups, not of individual variables. In order to also examine data from a single patient or a small number of samples in experiments, methods have been developed that score individual samples based on gene sets. Singscore  is one such approach. It is a rank-based single sample method that generates scores that are stable across a range of sample sizes (Table 20).
It is often of interest to build a prediction model that takes so-called “predictor variables” (sometimes also referred to as “independent variables”) as input and returns a prediction for a target variable of interest (sometimes also referred to as “dependent variable”) as output. This target variable, which refers either to the present state of the patient or to the future, may be a (binary or multi-categorical) class membership (e.g., treatment responder versus non-responder), a continuous variable (e.g., blood pressure or tumor size after therapy), an ordinal variable (e.g., WHO tumor grade), or a time-to-event (e.g., the overall survival time). Statistically more challenging cases of target variables that are not discussed in this paper are zero-inflated variables (typically continuous with additional frequent 0 values), continuous bounded variables (e.g., with values in [0,1]), or time-to-event variables in the presence of competing risks.
In the HDD setting, the number of candidate variables available to build the prediction model may be very large. This property has implications for construction of prediction models (section “PRED1: Construct prediction models”) and assessment and validation of their performance (section “PRED2: Assess performance and validate prediction models”). Detailed guidance for training, testing, and validation of HDD prediction models is provided by the IOM (Institute of Medicine of the National Academy of Sciences, USA) within a report that identifies best practices for the development, evaluation, and translation of omics-based tests into clinical practice . However, that report does not contain detailed guidance on statistical approaches for construction of prediction models and assessment of their performance. In addition, statistical methodology has seen substantial developments during the last decade.
Many methods to assess model performance and validate prediction models have been developed for low-dimensional data and then adapted to HDD, so a good starting reference is the explanation and elaboration paper of the TRIPOD (Transparent Reporting of a multivariable prediction model for Individual Prognosis Or Diagnosis) reporting guideline . This section explains, expands, and elaborates on existing guidance to more comprehensively cover issues in prediction modelling with HDD.
PRED1: Construct prediction models
Researchers developing a prediction model primarily focus on how well the model predicts the outcome of interest, especially for new observations, e.g., for patients whose data were not used to build the prediction model. While this is the main concern, often the researchers are also interested in the interpretation of the model, for example identifying which variables contribute most to the prediction and in what way. From this perspective, models involving only a limited number of predictor variables (denoted as “sparse models”), which clearly distinguish informative variables from non-informative variables, may be preferred to models making use of all variables measured for all observations. This is a particularly big challenge in the HDD setting, where many candidate variables are available. Beyond the issue of interpretability, sparse models may be easier to apply in clinical practice, because fewer variables have to be measured or determined to use them than for non-sparse models. In the case of gene expression, for example, the measurement of, say, 10 genes can be easily performed in any lab using PCR techniques, while the measurement of genome-wide expression requires the use of high-throughput methods (see, e.g., ).
A model is said to be “complex” if it reflects many patterns present in the available data, for example, by considering many predictor variables or capturing non-linear effects. Overly complex models risk overfitting the data, i.e., adhere too specifically to the data at hand and identify spurious patterns randomly present in the data used for model development that will not be present in independent data (see, e.g., ). An overfitted model usually exhibits suboptimal prediction performance when subjected to appropriate unbiased evaluation methods, and interpreting such models can be misleading. In contrast, a model that is not complex enough underfits the data. It misses important patterns that might have been useful for the purpose of prediction. When fitting prediction models, in particular (but not only) in the HDD setting, the challenge thus is to identify the optimal level of model complexity that will yield interpretable models with good prediction performance on independent data (see, e.g., [128, 129]).
The most straightforward statistical approach to construct a prediction model using several predictor variables simultaneously while taking into account their correlation is fitting a multivariable (generalized) regression model, for example a simple linear regression model in the case of an approximately normally distributed target variable. In linear regression, the regression coefficients are fitted such that the sum (for the n observations) of squared errors (i.e., of squared differences between the true value of the target variable and the predicted value) is minimal. Mathematically, this basic linear regression amounts to solving a system of n equations with p + 1 unknowns, where p stands for the number of predictor variables. Such a regression model, however, cannot be fitted if the number p + 1 of coefficients to fit (the intercept and one coefficient for each variable) exceeds the dataset size n. This dimension problem is complicated by the frequently occurring situation in which some of the p variables are highly correlated, i.e., they provide similar information. These correlations can cause instability with regard to which variables are deemed important contributors to the model and, thus, can influence model interpretability and performance.
Because the number of predictor variables p is usually larger than the number of patients n in HDD settings, basic regression models cannot be fitted directly. In this section, we briefly review some key strategies to deal with the dimension problem: variable selection, dimension reduction, statistical modelling (mainly through regularization methods), and algorithmic approaches (at the interface between statistics and machine learning). First, however, we discuss a preliminary step, variable transformation, that can be particularly helpful in the context of HDD analyses.
PRED1.1: Variable transformations
As mentioned in section “IDA3: Preprocessing the data,” data may be transformed to obtain certain distributional properties required for the methods that might be used in preprocessing or in downstream analyses of the preprocessed data. For example, (approximate) normal distributions for errors are a prerequisite for the application of tests such as the t-test or methods based on linear models such as ANOVA and linear regression . Transformations may also be helpful to dampen the influence of peculiar or extreme observations and may put variables on scales that are more amenable to analysis. For example, one could transform a bounded variable to an unbounded range or convert multiplicative effects to additive effects. It is often preferable to apply suitable transformations first and then work with transformed variables (Table 21).
Note that centering and scaling were discussed in section “IDA3.1: Background subtraction and normalization” (referring to normalization), but there the transformation was applied to all values of an observation (a subject, e.g., a patient) to adjust for potential systematic effects and make different observations more comparable, whereas here the transformation is related to all values of a variable.
PRED1.2: Variable selection
Variable selection refers to identification of a subset of predictor variables from all available variables, for purposes of building a prediction model. Note that terms as variable selection, selection strategy, or stepwise procedures are often used in the statistical literature, whereas use of the terms feature selection, wrapper, and filter is more common in the machine learning community. Multiple strategies have been proposed in the statistical and machine learning areas; for recent reviews, see, e.g., Heinze et al.  and Singh et al. . If available, subject matter knowledge should be included in the variable selection process. In many cases, however, variable selection is performed in a data-driven way, either with filter methods or with wrapper methods. In filter methods, the candidate predictor variables are considered successively independently of each other. Those satisfying a given criterion (for example, those associated with the target variable or those showing sufficient variability across all patients) are selected, while the others are ignored in the remaining analyses (for a comparison of filter methods in classification tasks with HDD data, see ). In contrast, wrapper methods select a subset of variables that, taken in combination, yield good performance accuracy (when used for prediction modelling with a considered method). The performance is assessed, e.g., through cross-validation (see section “PRED2.2: Internal and external validation”). Note that an embedded variable selection is also performed intrinsically with model building methods such as lasso and boosting (see section “PRED1.4: Statistical modelling”).
When the variable selection process uses outcome data, care must be taken to avoid optimistic bias in apparent model performance estimates due to multiple testing issues such as those described in section “TEST: Identification of informative variables and multiple testing.” It is critical that any data-driven variable selection steps are included as part of the model building process when model performance is assessed using any internal validation method, see Sachs and McShane  for a discussion of the use of “incomplete” cross-validation approaches and the bias inherent in such flawed approaches. Section “PRED2.2: Internal and external validation” provides a further discussion. With an emphasis on LDD, the topic group TG2 “Selection of variables and functional forms in multivariable analysis” of the STRATOS initiative raised several issues needing more research about the properties of variable selection procedures. Authors stressed that it is not straightforward which variable selection approach to use under which circumstances . Obviously, problems mentioned are strengthened in HDD.
PRED1.3: Dimension reduction
Data reduction has many purposes, including easier data handling (see also sections “IDA2.4: Graphical displays” and “EDA2.1: Cluster analysis” for aspects regarding data reduction). Concerning prediction, data reduction can help to reduce redundant information that may lead to instability of prediction models, as noted at the beginning of section “PRED1: Construct prediction models.” Data reduction may also facilitate explanation and interpretation by reducing the number of variables to consider. Note, however, that it may yield variables without a meaningful interpretation from a medical point of view .
In contrast to variable selection, the idea of dimension reduction is not to select variables but to build (a small number of) new variables, often called components, that summarize the information contained in the original variables. They can then be used as predictor variables for model building—possibly with a low-dimensional method. However, portability and feasibility of models generated using dimension reduction versus variable selection can be substantially different. To predict outcome using a model containing only a few selected variables, it is sufficient to measure these selected variables, while a model including derived components may require the measurement of all original variables. Consider, for example, deriving a prediction model from gene expression data generated using a microarray that measures 20,000 genes. There is a huge practical difference between using a model requiring input of expression levels of only 10 selected individual genes compared to using a model requiring input of 10 combination scores (components), each of which potentially requires knowledge of expression levels for all 20,000 genes.
The most well-known and widely used dimension reduction approaches are principal component analysis (PCA, see also section “IDA3: Preprocessing the data” for a description) and partial least squares (PLS), where the components are defined as linear combinations of the original variables . While PCA constructs components that have maximal variance and thus capture the signals of all types contained in the data, PLS constructs new variables that have maximal covariance with the target variable of interest. PLS is said to be a supervised method, where the term “supervised” refers to the fact that the target variable determines the construction of the components. Note that dimension reduction can be combined with variable selection.
For HDD analysis, a knowledge-based data reduction may also be useful. There is often external knowledge available about the entities to be investigated, such as knowledge of signaling pathways when analyzing gene expression data, or knowledge on conserved regions when analyzing DNA sequencing data (see also section “TEST3: Identify informative groups of variables” for incorporating information about functional relationships between genes in multiple testing). Attempts to re-discover such knowledge from the data at hand when performing data reduction then will typically be less reliable compared to using a data reduction strategy that explicitly incorporates external information, even if the latter itself also is to some extent unreliable (Table 22).
PRED1.4: Statistical modelling
Several modifications of traditional regression methods are available to address common challenges encountered in HDD settings with p > n. There is no unique mathematical solution for the standard regression parameter estimates. Traditional regression aims to find the parameters that minimize a sum of squared errors, which can be viewed as minimizing a type of “loss function.” Various modifications to this loss function can be made to permit a unique solution for the regression parameters in the HDD setting. The modifications described in this section impose mathematical constraints on regression coefficients. These constraints effectively limit the number of predictor variables included in the model or the magnitudes of their effects or both. Estimates obtained with such constraints are often referred to as “shrunken.” Some of these constraints can be shown equivalent to adjusting the covariance matrix (e.g., ridge regression; see ), but a variety of other constraints can be applied through specification of different loss functions; lasso  and elastic net  are two examples. Other methods, such as boosting , iteratively fit regression models that minimize a specified loss function at each stage. These various approaches usually lead to different models, each of which is optimal according to its corresponding criteria.
Numerous modifications of these basic approaches have been developed in the literature (especially for lasso, due to its variable selection property). Goals can be to recover desirable mathematical properties (e.g., the adaptive lasso  uses adaptive weights for penalizing different coefficients and estimates the correct model under some constraints) or to adapt the lasso to specific problems (e.g., the group lasso  allows predefined groups of variables to jointly be selected or not) (Table 23).
Boosting can be seen both as a statistical method, when a statistical model is fitted, and as an algorithmic approach, when it is implemented as a black box. In the latter case, the prediction updates are unrelated to an underlying statistical model, and only aim at minimizing a loss function . Several machine learning algorithms have been developed to provide prediction rules . The prediction model is constructed without variable selection or dimension reduction as a preliminary step, in a fully data-driven way, i.e., (in contrast to statistical methods) without assuming a particular model for the dependence between target and predictor variables. These algorithmic approaches may allow more flexibility to handle aspects such as non-linear or interaction effects, but often they are also less interpretable.
Machine learning algorithms comprise a diverse collection of methods. They include, among others, methods based on consideration of nearest neighbors in the predictor space (such as kNN), decision trees for classification and for regression (tree-based methods based on recursive partitioning of the predictor space), random forests (ensembles of decision trees, i.e., sets of decision trees whose predictions are averaged), and more complex approaches such as deep learning (neural networks with different structures and typically a huge number of parameters). In the HDD setting, many of these machine learning methods have been successfully used, but one must be particularly careful if the methods require the estimation of a large number of parameters, which applies especially to deep learning. Here, the overfitting problem discussed above becomes even more severe. Unbeknownst to users, some software developed to implement complex algorithms could have faulty designs that result in incorrect or overfitting results; hence, algorithms must be carefully tested  (Table 24).
PRED1.6: Integrating multiple sources of information
A major challenge for HDD, both for omics data and for electronic health records, is the integrative analysis of different data types. For instance, multiple types of omics data including proteomic, transcriptomic, and genomic, may be measured on the same subject. For health records data, various variable types are combined, such as blood values, urine values, cardiography measurements (ECG or EKG), categorical diagnostic measurements, or a variety of demographic variables. This has implications for visualization and use of clustering methods, which are often designed for a single data type. Conducting and interpreting joint analyses of disparate variable types can be challenging. Richardson and coauthors  distinguish between “horizontal integration” applied to the same type of data across multiple studies and “vertical integration” applied to different types of data on the same sample of subjects. The distinction between horizontal and vertical refers to the fact that, usually, data from high-throughput experiments are organized with samples represented by columns and variables by rows.
Regarding horizontal integration, the meta-analytic approach of pooling summary measures of association is the most used approach. For other applications, such as clustering, in order to deal with different normalizations and platforms for the different datasets, centering, and standardization  or specific methods should be considered; for clustering, see for example Huo et al. . Vertical data integration is typically model-based and the model used considers the specific characteristics of the data to be integrated and of the research question (whether exploratory or predictive).
In biomedicine, integration of multiple omics data types can provide deeper biological insights compared to individual omics in terms of disease subtyping, biomarker identification, and understanding of molecular mechanisms in diseases. For example, two different tissues from the same or different organism may carry an identical DNA sequence for a particular gene, but the gene may be inactivated by methylation in one of the tissues and not in the other; or the aberrant expression of one gene regulating the function of another downstream in the same biological pathway might be evident by observing the altered expression of the downstream gene at the RNA or protein level.
Richardson and coauthors  reviewed some vertical integrative analysis approaches, including integrative clustering and regression. The integrative clustering approach of Shen and coauthors , called iCluster, involves projection, via regression modelling, of the data onto scores representing a set of latent biological subtypes assumed common across data types. Resulting predicted biological subtype scores are clustered to identify latent subtype membership, and estimated coefficients from the fitted regression models can provide insights into data features that associate with certain subtypes. Mo and coauthors subsequently developed iCluster + to allow for other non-continuous, non-Gaussian data types . More complex Bayesian mixture modelling approaches have also been developed to offer greater flexibility to accommodate mixed data types (e.g., discrete mutation indicators in combination with continuous RNA or protein expression measures), provide metrics reflecting uncertainty about estimated underlying structure, and allow for elucidation of potentially different structure from different data types [165,166,167,168]. Integrative regression techniques are useful for supervised analyses of integrated data types, such as building a regression model for prediction of an outcome or phenotype. These methods allow to utilize structure inherent in different data types (e.g., DNA sequence location, functional categories of proteins, metabolic or signaling pathways) to effectively reduce the high dimensionality of the predictor variable space to facilitate development of more parsimonious and interpretable models relating the multi-omics data to outcomes or phenotypes of interest. Multi-omics integration methods using autoencodeurs in a deep learning setting are reviewed by Benkirane and coauthors . For more details, readers are referred to Richardson and coauthors  and references therein.
Although many prediction models for clinical outcomes have been developed based either on clinical data or (more recently) on high-throughput molecular data (e.g., omics), far fewer models have been developed to incorporate both data types through vertical integration. The paucity of such models in the literature and in clinical use persists despite suggestions that a suitable combination of clinical and molecular information might lead to models with better predictive abilities (e.g., [170, 171]).
In many medical specialties, there are some widely available and accepted clinical predictors with predictive value already validated in several independent populations. Strategies to combine such established clinical predictors with different data types, including high-dimensional omics data, have been proposed ; some examples have been published [173, 174], but applications are still rare. Volkmann and coauthors  investigated whether better use of the predictive value of clinical data has an influence on the added predictive value of molecular data. This concept can also be extended to multi-omics data .
Conceptually, it is obvious that incorporation of important clinical variables can potentially lead to better prediction models; thus, those variables should be considered in combination with molecular data. De Bin et al.  present strategies to combine low- and high-dimensional data in a regression prediction model, analyzing the influence of the complex correlation structure within and between the two data sources. In some situations, predictive value of molecular data might be fully captured through the clinical variables, thereby eliminating the need for the molecular data in the prediction model .
PRED2: Assess performance and validate prediction models
Perhaps even more than constructing predictive models and algorithms, evaluating their performance and validating them are key challenges. For HDD, not only the choice of suitable measures to assess and compare model performance (see section below), but also the way of computing these measures is generally not straightforward.
PRED2.1: Choice of performance measures
Prediction performance is typically assessed by comparing the true and the predicted values of the target variable. The comparison is based on specific metrics, mainly depending on the nature of the target variable. Typical metrics include mean squared error or mean absolute error for continuous target variables, area under the curve (AUC) or Brier score for binary target variables, and calibration plot and time-dependent Brier score for time-to-event variables. Such measures can be used to quantify the performance of a model (or algorithm) or to compare different models constructed using the same dataset. In most biomedical applications, the goal of a comparative assessment is to select a final model [177, 178]. Models of absolute risk that depend on covariates have been used to design intervention studies, to counsel patients regarding their risks of disease or future disease-related events, and to inform clinical decisions. Several criteria related to “calibration” and “discriminatory power” have been proposed [179, 180]. Often the main interest will be in the added value of biomarkers or gene signatures relative to an existing clinical prediction model. Several performance measures are available to quantify the added value .
For a clinical task, several very different models with equivalent prediction performance may be available. Not only, but especially in this situation, other aspects of the models can play an important role. Particularly noteworthy aspects of a model are sparsity, stability, interpretability, and practical usefulness [7, 182]. Regarding sparsity, when selecting a final model from among several with comparable prediction performance, selection of the most parsimonious (e.g., the model with smallest number of predictor variables) is preferred. Stability refers to the degree to which small changes in the data may produce large changes in the predictor output. A majority of predictors derived from HDD suffer from poor stability, irrespective of the method used to fit them, although some methods are more affected than others (see  for an overview of stability measures, and Sauerbrei et al.  for stability investigations of regression models for LDD and HDD). For HDD, the stability problem is due to the myriad ways to combine a set of predictor variables to derive similar performing predictors. If the stability is found to be low, then interpretation of specific model components (the list of selected predictor variables, relationships between predictor variables, etc.) should be avoided. In terms of interpretability of the model, strong prior biological knowledge may also be taken into account, similar as for the aim of data reduction described above (Table 25).
PRED2.2: Internal and external validation
Whatever measure of model performance has been chosen, computing it on the same dataset that was used for constructing the model may lead to a dramatic over-estimation of the performance. Instead, one should assess prediction performance using independent data, i.e., data not used to construct the model [189,190,191]. One classical procedure is to split the given dataset into a training set and a test set, and then to construct the model using only the training set and evaluate the model using only the test set. This is one type of “internal validation,” in contrast to “external validation,” where data from independent patient cohorts are used .
Due to the typical instability of predictors developed using HDD, this sample splitting procedure is very risky, as in most cases the specific split heavily influences the result. Resampling techniques, such as cross-validation, subsampling and bootstrapping, can be less risky, although even those methods cannot avoid impact of biases in the data introduced by faulty designs such as those that would confound batch effects with outcome variables. The common idea behind these procedures is to repeatedly use a part of the dataset as training dataset, i.e., to construct a prediction model, and the other (non-overlapping) part as test dataset to evaluate the constructed model. This process is repeated several times for different splits into training and test data to produce a more stable and representative estimate of model performance.
For such approaches, a bias in the performance estimates must be considered (see also ). This bias occurs because the training data sample size is smaller than for the full dataset, and therefore prediction models built on the training dataset tend to have somewhat worse performance than a final model built on the full data. The latter is typically used for further evaluation. This bias becomes larger the smaller the training dataset is compared to the full dataset. This aspect is less relevant if the sample size of the full dataset and thus of the training dataset is very large.
One misleading practice is use of resampling procedures for multiple different prediction modelling methods or for different parameter values, and then reporting results for only the model with best performance. This practice leads to over-optimism in model performance because it neglects to acknowledge and account for the fact that the reported model was the result of another optimization process . Such studies aiming to find “best” models occur quite frequently in the context of HDD. While it would be naive to expect that investigators will not try multiple approaches to develop a prediction model, the key is transparency in reporting how many models were actually produced and evaluated, and appropriately accounting for the additional selection step. One should either validate the final selected model using an independent dataset (see Table 26), or when such a dataset is not available, embed the selection process in the cross-validation procedure, i.e., perform a so-called nested cross-validation procedure [190, 202]. Figure 19  shows a schematic representation of a suitable process for developing a predictor, here specified for omics data, in which the discussed aspects are adequately taken into account.
PRED2.3: Identification of influential points
Identification of possible influential observations, defined as those for which inclusion or exclusion in model development might substantially alter characteristics of the final model , is an important aspect of prediction modelling that is often neglected in HDD settings, and even frequently in low-dimensional settings. Model alterations can be related to variable selection (see, e.g., ), functional forms (e.g., ) or parameter estimation (e.g., ). Influential points can be outliers in some of the variables (observations suspiciously different from the rest, such that they are probably generated by a different mechanism ), but they do not need to be.
For HDD, identification of influential points is particularly difficult, due to data sparsity (the so-called “curse of dimensionality”) and, more generally, the increased difficulty in identifying data patterns, especially by graphical methods. Many available methods for influential point detection are extensions of traditional low-dimensional tools such as Cook’s distance  and the CFBETA / DFFITS measures . Examples of adaptations to the high-dimensional framework of the former are Zhao et al.  and Wang and Li , of the latter Walker and Birch  and Rajaratnam et al. . Focusing more on statistical models (see section “PRED1.4: Statistical modelling”), methods like those of Shi and Wang  and Hellton et al.  investigate the effect of influential points on the choice of the tuning parameters, again adapting existing low-dimensional approaches (the aforementioned DFFITS measure and the resampling approaches in De Bin et al. , respectively). The latter is an example of how cross-validation and subsampling can be used to detect influential points by tracking large changes in the estimates when one (or a few) observation is omitted. Although influential points and outliers can strongly affect the results of analyses in HDD settings, systematic checks for them seem to be often ignored in the literature, despite of the availability of various techniques [216, 217].
In a study on the classification of breast cancer subtypes, Segaert et al.  stress that classical statistical methods may fail to identify outliers and argue for robust classification methods that flag outliers. They propose the DetectDeviatingCells outlier detection technique. Specifically for HDD, Boulesteix et al.  propose a rank discrepancy measure that considers the difference between gene rankings for the original data and for a pretransformation that tries to eliminate the effect of extreme values. For survival data, Carrasquiha et al.  propose a rank product test to identify influential observations, and more techniques have been proposed recently. Fan  released the R package HighDimOut, which contains three high-dimensional outlier detection algorithms. However, none of the approaches seems to have gained popularity in practice. More guidance and comparisons of available approaches are needed.
PRED2.4: Sample size considerations
Recent guidelines for calculating sample size when developing a risk prediction model  are not specifically tailored for applications involving variable selection or shrinkage methods (such as LASSO or Ridge Regression). This is the situation in high-dimensional settings, where variable selection or dimension reduction is needed to identify prognostic variables or components. The available methods for sample size planning [17, 18] and references therein are based either on simulations that require assumptions of feature independence or on the availability of a pilot dataset or preliminary data, but these methods are hardly used in practice. Moreover, penalized estimation has often been proposed for situations with potentially large overfitting problems, while recent evidence suggests that it yields unstable results, especially for small sample sizes , when overfitting is a major concern.
A practical sample size method for planning a preliminary study of a prognostic biomarker is suggested for microarray technology , which can be used in more general settings. When a gene signature is already available from previous exploratory studies, a formal calculation for a predictive model, including the gene signature and standard prognostic covariates, can be performed according to available guidelines, taking also into account the need for external validation [16, 200].
Good reporting to improve transparency and reproducible research
Reporting of studies involving HDD can be particularly challenging and at the same time especially important due to the many potential pitfalls in the collection and analysis of complex HDD as described herein. Complete and transparent reporting of these studies is critical to allow independent researchers to evaluate how a study was designed, conducted, and analyzed so that quality and relevance of the findings can be judged and interpreted in appropriate context. Provision of data and computer code may be required to achieve full transparency.
Guidelines for reporting of many types of health research have already been developed and are largely applicable in HDD settings. Simera et al.  introduced the EQUATOR (Enhancing the QUAlity and Transparency Of health Research) network as an umbrella organization for the reporting of studies in the health sciences. Most relevant for HDD data are the REporting recommendations for tumor MARKer prognostic studies (REMARK)  and TRIPOD for the reporting of multivariable prediction models for individual prognosis or diagnosis . For both reporting guidelines, more detailed “explanation and elaboration” papers have been published [125, 223], which also include several sections on statistical analyses. Furthermore, the two-part REMARK profile, a structured display that summarizes key aspects of a study, has been proposed to improve completeness and transparency of reporting, specifically of statistical analyses. The TRIPOD checklist distinguishes between model development and validation. Both guidelines were developed for markers and for models based on clinical data, with no more than a few dozen potential predictors in mind.
In an article stressing the importance of registering diagnostic and prognostic research, Altman  clearly expresses that non-reporting and misleading reporting do not just mislead researchers in the field, they also diminish the evidence base underpinning clinical practice and harm patients. To improve on such an unacceptable situation of non-transparency in studies, several initiatives including data pooling, registers, and journal requirements for protocols were started, see Peat et al.  for a detailed discussion with an emphasis on prognosis research.
Obviously, reporting of artificial intelligence and machine learning methods come with a large number of additional challenges. Concerns have been raised that they are overhyped in clinical medicine (see, e.g., ) and, if not used with proper expertise, have methodological shortcomings, poor transparency, and poor reproducibility . There is a strong need for applications of machine learning techniques to adhere to established methodological standards already defined in prediction model research .
In this section, we first summarize the content and the key messages of this overview paper. We also briefly present the relationships of the other topic groups of the STRATOS initiative to the HDD-focused TG9 group and discuss the importance of further collaboration.
Biomedical research has always relied on a combination of observational studies, carefully controlled laboratory experiments, and clinical trials, but the types of data generated and analyzed in these studies continue to evolve and now more often include HDD. The high dimensionality may result from new technologies such as omics assays, which are capable of comprehensive interrogation of biological specimens, or from increased ability to merge data from multiple information systems such as electronic health records or registries. HDD present many new challenges for statistical design and analysis of biomedical research studies. This overview provides a gentle introduction to basic concepts and useful strategies for design and analysis of studies involving HDD. Key points are summarized in the discussion that follows.
Study design for prospectively planned investigations and vigilance to detect (and avoid when possible) confounding in observational studies remain as important for studies involving HDD as for other studies. Consequences of inattention to these aspects can be particularly damaging when HDD are involved. While HDD may provide greater opportunity for discovery of new biological and clinical concepts and associations, they might also be more susceptible to influence of ancillary confounding variables and technical artifacts. Therefore, initial examination of data for technical artifacts such as batch effects, inconsistent, extreme, or suspicious values is critically important but simultaneously more challenging as the data dimension increases. New data visualization, detection, and correction or normalization methods have been adopted for HDD, as were described in section “IDA: Initial data analysis and preprocessing” of this overview. Techniques for data visualization and exploration such as those described in section “EDA: Exploratory data analysis” of this overview are also important to provide biological insights and support development of new scientific hypotheses from HDD. The initial steps and exploratory data analyses described in sections “IDA: Initial data analysis and preprocessing” and “EDA: Exploratory data analysis” are optimally performed when equipped with a good understanding of the data sources and data generation methods, for example assay technologies that produce omics data, and interpreted in collaboration with other scientists knowledgeable in the technology, biology, and clinical aspects.
Statistical analysis methods that were developed for traditional settings where the number of independent observations or subjects is substantially larger than the number of variables acquired are widely used by classically trained statisticians and others in a variety of applications, and their widespread use is supported by ready availability of software. Emergence of many new types of HDD has exposed the limitations of many traditional methods. Often, methods rely heavily on distributional assumptions such as normality, which may be unrealistic for data types generated by novel technologies such as omics assays. Many methods owe their robustness to such assumptions to large sample size, yet the notion of what qualifies as “large n” is dramatically different for HDD where even the most basic requirement n > p is not satisfied. Much traditional statistical methodology for addressing multivariate data has focused heavily on mathematically tractable joint distributions such as multivariate Gaussian or assumed that sample sizes were large enough that this served as a good approximation. As these are not reasonable assumptions for many types of HDD, many researchers opt for an alternative strategy of examining each of many variables one-at-a-time. Yet, naively taking such an approach is fraught with danger of generating many false discoveries due to the extremely large number of variables examined. Traditional strategies for controlling false positive findings, such as controlling the FWER, are often impractical or overly stringent in view of the goals of many studies involving HDD, and this recognition has stimulated development of novel approaches for false discovery control. Section “TEST: Identification of informative variables and multiple testing” of the overview highlighted some of these many challenges and summarized some useful strategies to address them.
The last few decades have seen substantial progress in development of prediction modelling methodology, especially as applicable to HDD, and increased availability of free software to implement these methods has fueled their use. Available methods include a variety of statistically based approaches as well as a number of purely algorithmic approaches such as many machine learning methods. Prediction models developed from HDD have intrigued many researchers under the impression that with sufficiently large volumes of data one should be capable of predicting virtually anything. Numerous dramatic claims of performance have been made; unfortunately, these claims do not always withstand careful scrutiny. Section “PRED: Prediction” provides a review of several popular prediction modelling methods for HDD, and it stressed the importance of following proper procedures to assess and avoid model overfitting that leads to prediction models that do not perform well outside of the data from which they were developed. Poor study design and faulty prediction modelling approaches that lead to spurious and overfitted models along with wildly inaccurate claims of their performance persist in the biomedical literature. Guidance provided in section “PRED: Prediction” aims to reduce this problem and promote successful development of useful prediction models.
Within the STRATOS initiative, there are currently nine topic groups (TGs), mainly concerned with LDD. Table 27 presents the relationship of the other STRATOS topic groups to TG9 group and how TG9 guidance will build upon that of other TGs to adapt it for relevance.
This overview aimed to provide a solid statistical foundation for researchers, including statisticians and non-statisticians, who are newly embarking on research involving HDD or who are merely wanting to better evaluate and understand results of HDD analyses. Common approaches for the statistical analysis of high-dimensional biomedical data are described in 24 method tables; see Table 28 for a list of these tables. New methods to generate HDD or combine existing data resources to yield HDD will continue to evolve, and there will be continued need to develop new and improved computational and statistical analysis strategies to address new types of data and novel questions to be answered from those data. Basic concepts and strategies presented in this overview will remain relevant, and their wider grasp by the biomedical research community will hopefully lead to continued improvement in the quality, reliability, and value of studies involving HDD. Most importantly, strong collaborations between statisticians, computational scientists, and other biomedical researchers such as clinicians, public health experts, laboratorians, technology experts, bioinformaticians, and others that are relevant to each project, are essential to produce the most reliable and meaningful data and results.
Availability of data and materials
Akaike information criteria
Analysis of variance
Average silhouette width
Area under the curve
Bayesian information criteria
Body mass index
Cluster Affinity Search Technique
Clustering Large Applications
Methods for adjustment of batch effects
Differential gene expression analysis of RNA-seq data
Exploratory data analysis
Enhancing the QUAlity and Transparency Of health Research
False discovery proportion
False discovery rate
Familywise error rate
Gene set enrichment analysis
Initial data analysis
Institute of Medicine of the National Academy of Sciences, USA
Linear Models for Microarray Data
- MA plot:
Median absolute deviation to the median
Mean absolute (prediction) error
Missing at random
Missing “completely at random”
Missing not at random
Mean squared (prediction) error
Non-Metric Multidimensional Scaling
Partitioning around medoids
Principal component analysis
Positron emission tomography
Partial least squares
REporting recommendations for tumor MARKer prognostic studies
Relative log expression
Root mean squared error
Receiver operating characteristic
Stochastic neighbor embedding
STRengthening Analytical Thinking for Observational Studies
Supervised principal components
Surrogate variable analysis
Support vector machine
Topology-based Gene Ontology enrichment analysis
Transparent Reporting of a multivariable prediction model for Individual Prognosis Or Diagnosis
T-Distributed Stochastic Neighbor Embedding
Uniform Manifold Approximation and Projection
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We thank Milena Schwotzer for administrative assistance.
The views expressed in the paper do not necessarily represent views or policies of the National Cancer Institute, National Institutes of Health, or the U.S. Department of Health & Human Services.
Open Access funding provided by the National Institutes of Health (NIH). WS was partially supported by grant SA580/10–1 from the German Research Foundation (DFG). FA was partially supported by the Italian Ministry of Education, University and Research project PRIN 2017, prot. 20178S4EK9_004. RDB was partially supported by the Norwegian Research Council research-based innovation center BigInsight, project no 237718. ALB’s group was partially supported by individual grants from the German Research Foundation (DFG, BO3139) and by the German Federal Ministry of Education and Research (BMBF) under Grant No. 01IS18036A.
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Rahnenführer, J., De Bin, R., Benner, A. et al. Statistical analysis of high-dimensional biomedical data: a gentle introduction to analytical goals, common approaches and challenges. BMC Med 21, 182 (2023). https://doi.org/10.1186/s12916-023-02858-y
- High-dimensional data
- Omics data
- STRATOS initiative
- Analytical goals
- Initial data analysis
- Exploratory data analysis
- Multiple testing