Table 2 Design table with a summary of questions, hypotheses, power, analyses, and interpretation

To what extent is age at menarche associated with adolescent depression?

H1a) We hypothesised that an earlier age at menarche would be associated with elevated depressive symptoms at age 14

Power was calculated in R based on simulated data (1000 replicates). For this analysis, we simulated a depressive symptoms variable (M = 5.71, SD = 4.93; values taken from Sequeira et al.). Results indicated 95% power to detect an effect of Cohen’s D ≥ 0.08 at N = 12,000 and ≥ 0.06 at N = 13,000 for the association between age at menarche and depressive symptoms

Linear regression models with age of menarche as the independent variable and depressive symptoms at age 14 as the dependent variable

For this analysis, we also added depressive symptoms at age 8 (“pre-pubertal symptoms”) as a covariate

Perform a test of inferiority for the estimate against D = 0.23, the smallest effect size of interest (SESOI)

A negative and statistically significant association between age at menarche and depressive symptoms at age 14 would suggest that adolescent girls with earlier age at menarche show elevated depressive symptoms. A non-significant association would suggest an absence of evidence for this

If the inferiority test reveals that our estimate is consistent with effects more extreme than the SESOI, this would suggest that the association is of a meaningful magnitude

Hypothesis 1a would be supported if (1) the coefficient for the effect of age at menarche on 14-year depressive symptoms was significantly less than 0 (one-tailed test; alpha 5%) in the pre-pubertal symptoms-adjusted model and (2) we failed to reject the null hypothesis that this effect in the population was at least as large as the SESOI (one-tailed test; alpha 5%)

If we can reject an effect size at least as large as the SESOI, then the effect would be declared practically equivalent to 0, irrespective of the result of the NHST

We would explicitly remain undecided on the hypothesis if we could not reject an effect of exactly 0 (in the NHST) or an effect at least as large as the SESOI. Both these conditions apply to all remaining hypotheses

To what extent is age at menarche associated with adolescent depression?

H1b) We hypothesised that an earlier age at menarche would be associated with higher rates of depression diagnoses during adolescence

For this power analysis, we simulated a binary depression diagnosis variable, based on a pre-specified prevalence and association with age at menarche (varied across scenarios at 2%, 6%, 10%, 14%; D = 0 to − 0.26 in increments of 0.02). Results indicated 95% power to detect an effect of Cohen’s D ≥ 0.14 at both N = 12,000 and N = 13,000 for depression prevalence of 6% or higher

Logistic regression models with covariates and age of menarche as independent variables and depression diagnoses during adolescence (ages 10–17) as the dependent variable

For this analysis, we would also add depression diagnoses during childhood (ages 0–8; “pre-pubertal depression status”) as a covariate

Perform a test of inferiority for the estimate against D = 0.23, the smallest effect size of interest (SESOI)

A negative and statistically significant association between age at menarche and depression diagnoses during adolescence would suggest that adolescent girls with earlier age at menarche show higher rates of depression. A non-significant association would suggest an absence of evidence for this

If the inferiority test reveals that our estimate is consistent with effects more extreme than the SESOI, this would suggest that the association is of a meaningful magnitude

Hypothesis 1b would be supported if (1) the coefficient for the effect of age at menarche on odds of depression diagnoses was significantly less than 0 (one-tailed test; alpha 5%) and (2) we failed to reject the null hypothesis that this effect in the population was at least as large as the SESOI (one-tailed test; alpha 5%)

Does age at menarche associate with symptoms or diagnoses in other domains of mental health, independent of depression?

H2.1–4a) We hypothesised that age at menarche would be associated with symptoms of anxiety (H2.1a), CD (H2.2a), ODD (H2.3a), and ADHD (H2.4a) at age 14, independent of depressive symptoms

For this power analysis, we simulated anxiety, CD, ODD, and ADHD symptoms, with distributions based on the same variables in MoBa data at 8 years. Results indicated 95% power to detect an effect of Cohen’s D ≥ 0.12 at both simulated sample sizes (N = 12,000 and N = 13,000)

Separate multiple linear regression models with covariates and self-reported age at menarche included as independent variables, and either symptoms of anxiety, CD, ODD, or ADHD as the dependent variable

Add depressive symptoms as a covariate in each model to examine whether any associations are independent of co-occurring depressive symptoms

Then, add a measure of each domain at age 8 as a covariate in the age 14 model for that domain, to examine whether associations are independent of pre-pubertal symptoms (“fully adjusted” models)

Perform a test of equivalence for the estimate against D =  − 0.22–0.22, the equivalence bounds based on our smallest effect size of interest (SESOI)

Any association between age at menarche and anxiety, CD, ODD, or ADHD that significantly differed from 0 at age 14 would suggest that there was an association with age at menarche in that domain. No significant associations would suggest an absence of evidence for this

If the equivalence test reveals that our estimate is consistent with effects outside of the region of practical equivalence, this would suggest that the association is of a meaningful magnitude

Each of the hypotheses 2.1–4a would be supported if (1) the coefficient for the association between age at menarche and that domain of 14-year symptoms in the fully adjusted model was different from 0 (two-tailed tests, 5% alpha) and (2) we failed to reject the null hypothesis that the association in the population was at least as extreme as the SESOI in either direction (two one-tailed tests, 5% alpha)

Does age at menarche associate with symptoms or diagnoses in other domains of mental health, independent of depression?

H2.1–3b) We hypothesised that age at menarche would be associated with diagnoses in other domains: anxiety disorders (H2.1b), conduct disorders (H2.2b) including CD and ODD, and ADHD (H2.3b) during adolescence, independent of depressive disorders

This power analysis was similar to H2.1–4a above, apart from different equivalence bounds and the use of two-tailed tests. Results indicated 95% power to detect an association of Cohen’s D ≥ 0.20 at N = 12,000 and ≥ 0.18 at N = 13,000 between age at menarche and diagnoses of anxiety, ADHD, and conduct disorders (CD and ODD), for diagnosis prevalence of 2% or higher

Separate multivariate logistic regression models with covariates and self-reported age at menarche included as independent variables and either diagnoses of anxiety, ADHD, or conduct disorders as the dependent variable

Add depression diagnostic status (ages 10–17) as a covariate in each model to examine whether any associations are independent of comorbid depression

Then, add pre-pubertal diagnostic status (ages 0–8) as a covariate in the age 14 model for that domain, to examine whether associations are independent of prior diagnoses (“fully adjusted” models)

Perform a test of equivalence for the estimate against D =  − 0.22–0.22, the equivalence bounds based on our smallest effect size of interest (SESOI)

Any association between age at menarche and anxiety disorders, ADHD, or conduct disorders would suggest that there was an association with age at menarche in that domain. No significant associations would suggest an absence of evidence for this

If the equivalence test reveals that our estimate is consistent with effects outside of the region of practical equivalence, this would suggest that the association is of a meaningful magnitude

Each of the hypotheses 2.1–3b would be supported if (1) the coefficient for the association between age at menarche and diagnoses in that domain in the fully adjusted model was different from 0 (two-tailed tests, 5% alpha) and (2) we failed to reject the null hypothesis that the association in the population was at least as extreme as the SESOI in either direction (two one-tailed tests, 5% alpha)

What is the evidence for a causal link between age at menarche and depression?

H3a) We hypothesised that an earlier age at menarche would result in elevated depressive symptoms at age 14

Power was calculated for simulated genetic instruments with an R² for the instrument-exposure association of 0.05, 0.075, and 0.1. Results indicated 95% power to detect an average causal effect (using two-stage least squares regression; 2SLS) of Cohen’s D ≥ 0.2 when the R² of the instrument is 0.075 or above at either simulated sample size (N = 9500 and N = 10,500)

2SLS regression with the genetic risk score for age at menarche as the instrument, reported age at menarche as the exposure, and depressive symptoms at age 14 as the outcome

2SLS regression with depressive symptoms at age 8 as the outcome, as a negative control outcome

Perform a test of inferiority for the estimate against D = 0.25, the smallest effect size of interest (SESOI)

Two-sample multivariable MR with genetic instruments for age at menarche and childhood body size or adult body mass index as instruments, to estimate the direct effect of age at menarche on depressive symptoms

Two-sample MR sensitivity analyses: MR-Egger, MR-PRESSO, weighted median, and contamination mixture

In the 2SLS regression, a negative and statistically significant estimate of the link between genetically predicted age at menarche and depressive symptoms at age 14 would suggest that earlier age at menarche was causally linked with adolescent depressive symptoms. A non-significant association would suggest an absence of evidence for this

If the inferiority test reveals that our estimate is consistent with effects more extreme than the SESOI, this would suggest that the association is of a meaningful magnitude

Hypothesis 3a would be supported if (1) the coefficient for the causal effect of age at menarche on 14-year depressive symptoms was significantly less than 0 (one-tailed test; alpha 5%), (2) we failed to reject the null hypothesis that this causal effect in the population was at least as large as the SESOI (one-tailed test; alpha 5%), and (3) we failed to reject the null hypothesis that this causal effect in the population was at least as large as the upper bound of the negative control estimate (one-tailed test; alpha 5%)

If the hypothesis was supported, consistent results across the two-sample MR sensitivity analyses would provide further evidence of causality

What is the evidence for a causal link between age at menarche and depression?

H3b) We hypothesised that an earlier age at menarche would result in higher rates of depression diagnoses during adolescence

Power was calculated for simulated genetic instruments with an R² for the instrument-exposure association of 0.05, 0.075, and 0.1. Results indicated 95% power to detect a causal effect (using logistic 2SLS) of Cohen’s D ≥ 0.24 when the R² of the instrument is ≥ 0.075 and depression prevalence ≥ 10% at either simulated sample size (N = 9500 and N = 10,500)

Multivariate logistic (two-stage) regression with the genetic risk score for age at menarche as the instrument, self-reported age at menarche as the exposure, and depression diagnoses during adolescence (ages 10–17) as the outcome

Perform a test of inferiority for the estimate against D = 0.25, the smallest effect size of interest (SESOI)

In the 2SLS regression, a negative and statistically significant association between genetically predicted age at menarche and depression diagnoses during adolescence would suggest that earlier age at menarche caused higher rates of adolescent depression. A non-significant association would suggest an absence of evidence for this

If the inferiority test reveals that our estimate is consistent with effects more extreme than the SESOI, this would suggest that the association is of a meaningful magnitude

Hypothesis 3b would be supported if (1) the coefficient for the causal effect of age at menarche on depressive disorders was significantly less than 0 (one-tailed test; alpha 5%) and (2) we failed to reject the null hypothesis that this causal effect in the population was at least as large as the SESOI (one-tailed test; alpha 5%)

Is there evidence of causal links between age at menarche and other domains of mental health?

H4.1–4a) We hypothesised that age at menarche would show a causal relationship with symptoms in other domains: anxiety (H4.1a), CD (H4.2a), ODD (H4.3a), and ADHD (H4.4a) at age 14

The simulations for this power analysis were essentially identical to H3a (but with different equivalence bounds and the use of two-tailed tests). Results indicated 95% power to detect an average causal effect (using 2SLS) of Cohen’s D ≥ 0.2 when the R² of the instrument is 0.10 at either simulated sample size, and 80% power to detect Cohen’s D ≥ 0.2 with the R² of the instrument at ≥ 0.075 for N = 9500, as well as with the R² of the instrument at ≥ 0.05 for N = 10,500

2SLS regression model with covariates and genetically predicted age at menarche included as independent variables and either symptoms of anxiety, CD, ODD, or ADHD at age 14 as the dependent variable

Run the same models with other symptom domains at age 8 as dependent variables, as negative control outcomes

Perform a test of equivalence for the estimate against D =  − 0.20–0.20, the equivalence bounds based on our smallest effect size of interest (SESOI)

Two-sample MR sensitivity analyses: MR-Egger, MR-PRESSO, weighted median, and contamination mixture

Any significant links between age at menarche and symptoms of anxiety, CD, ODD, or ADHD at age 14 would suggest that causal effects of age at menarche extended to those other symptom domains. Non-significant estimates would suggest an absence of evidence for this

If the equivalence test reveals that our estimate is consistent with effects outside of the region of practical equivalence, this would suggest that the association is of a meaningful magnitude

Each of the hypotheses 4.1–4a would be supported if (1) the coefficient for the effect of age at menarche on a domain of 14-year symptoms was different from 0 (two-tailed tests, 5% alpha), (2) we failed to reject the null hypothesis that the causal effect in the population was at least as extreme as the SESOI in either direction (two one-tailed tests, 5% alpha), and (3) we failed to reject the null hypothesis that this causal effect in the population was at least as large as the upper bound of the negative control estimate (one-tailed test; alpha 5%)

If a hypothesis was supported, consistent results across the two-sample MR sensitivity analyses would provide further evidence of causality

Is there evidence of causal links between age at menarche and other domains of mental health?

H4.1–3b) We hypothesised that age at menarche would show a causal relationship with rates of diagnoses in other domains: anxiety disorders (H.4.1b), conduct disorders (H4.2b) including CD and ODD, and ADHD (H4.3b) during adolescence

The simulations for this power analysis were essentially identical to H3b (but with different equivalence bounds and the use of two-tailed tests). Results indicated 95% power to detect a causal effect (using logistic 2SLS) of Cohen’s D ≥ 0.26 when the R² of the instrument is ≥ 0.075 and diagnosis prevalence is 14% in either simulated sample size (N = 9500 and N = 10,500), and 80% power for Cohen’s D ≥ 0.20 in the same scenarios

Multivariate logistic (two-stage) regression model with covariates included as independent variables and either diagnoses of anxiety, conduct disorders, or ADHD during adolescence (ages 10–17) as the dependent variable

Perform a test of equivalence for the estimate against D =  − 0.20–0.20, the equivalence bounds based on our smallest effect size of interest (SESOI)

Any significant links between age at menarche and anxiety disorders, disruptive behaviour disorders, or ADHD during adolescence would suggest that causal relationships with age at menarche extended to those other mental health domains. Non-significant estimates would suggest an absence of evidence for this

If the equivalence test reveals that our estimate is consistent with effects outside of the region of practical equivalence, this would suggest that the association is of a meaningful magnitude

Each of hypotheses 4.1–3b would be supported if (1) the coefficient for the effect of age at menarche on that diagnosis was different from 0 (two-tailed tests, 5% alpha) and (2) we failed to reject the null hypothesis that the causal effect in the population was at least as extreme as the SESOI in either direction (two one-tailed tests, 5% alpha)