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Sample size and power calculations in Mendelian randomization with a single instrumental variable and a binary outcome

Sample size and power calculations in Mendelian randomization with a single instrumental variable... Background: Sample size calculations are an important tool for planning epidemiological studies. Large sample sizes are often required in Mendelian randomization investigations. Methods and results: Resources are provided for investigators to perform sample size and power calculations for Mendelian randomization with a binary outcome. We initially provide formulae for the continuous outcome case, and then analogous formulae for the binary outcome case. The formulae are valid for a single instrumental variable, which may be a single genetic variant or an allele score comprising multiple variants. Graphs are provided to give the required sample size for 80% power for given values of the causal effect of the risk factor on the outcome and of the squared correlation between the risk factor and instrumental variable. R code and an online calculator tool are made available for calculating the sample size needed for a chosen power level given these parameters, as well as the power given the chosen sample size and these parameters. Conclusions: The sample size required for a given power of Mendelian randomization in- vestigation depends greatly on the proportion of variance in the risk factor explained by the instrumental variable. The inclusion of multiple variants into an allele score to explain more of the variance in the risk factor will improve power, however care must be taken not to introduce bias by the inclusion of invalid variants. Key words: Mendelian randomization, sample size, power, binary outcome, allele score Key Messages Resources are provided for investigators to perform sample size and power calculations for Mendelian randomization with a binary outcome. The sample size required for a given power level is greater with a binary outcome than a continuous outcome, and is highly dependent on the proportion of the variance in the risk factor explained by the instrumental variable. V The Author 2014. Published by Oxford University Press on behalf of the International Epidemiological Association 922 This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited. Downloaded from https://academic.oup.com/ije/article/43/3/922/761826 by DeepDyve user on 20 July 2022 International Journal of Epidemiology, 2014, Vol. 43, No. 3 923 Introduction needed in a Mendelian randomization study to obtain a given power level. We initially present formulae with a con- Sample size calculations are an important part of experimen- tinuous outcome (this reviews material previously covered tal design. They inform an investigator of the expected power by Freeman et al. ) and then analogous formulae with a bin- of a given analysis to reject the null hypothesis. If the power ary outcome. We concentrate on estimates from the ratio of an analysis is low, then not only is the probability of re- (or Wald) method, as this method makes few parametric as- jecting the null hypothesis low, but when the null hypothesis sumptions, relying only on a linear relationship between the is rejected, the posterior probability that the rejection of the 1 conditional expectation of the outcome (or in the binary null hypothesis is not simply a chance finding is low. case, the logistic function of the probability of the outcome) Mendelian randomization is the use of genetic variants and the risk factor. If the imprecision in the estimate of as instrumental variables for assessing the causal effect of a 2 the genetic association with the risk factor is negligible, then risk factor on an outcome from observational data. estimates of power and sample size from the ratio method Genetic variants are chosen which are specifically associ- also correspond to those from assessment of the causal rela- ated with a risk factor of interest, and not associated with tionship of the risk factor on the outcome by testing the as- variables which may be confounders of the association be- 3 sociation between the genetic variant and outcome. tween the risk factor and outcome. Such a variant divides Other estimation approaches are possible with a binary the population into groups which are similar to treatment outcome but these either give equivalent estimates to the arms in a randomized controlled trial. Under the instru- 5,6 ratio method with a single IV (the two-stage predictor sub- mental variable assumptions, a statistical association be- stitution method ) or are not recommended for general tween the genetic variant and the outcome implies that the 7 use in applied practice. These include the two-stage re- risk factor has a causal effect on the outcome. However, sidual inclusion method, due to inconsistency for a param- as genetic variants typically explain a small proportion of eter with a natural interpretation, and the generalized the variance in risk factors, the power to detect a signifi- method of moments (GMM) and structural mean models cant association between the variant and outcome in an 8 (SMM) methods, due to potential lack of identifiability of applied Mendelian randomization context can be low. the causal parameter (S Burgess et al., unpublished data). Sample size analysis is particularly important to inform whether a null finding is representative of a true null causal relationship, or simply a lack of power to detect an effect Power with a continuous outcome size of clinical interest. With a single IV and a continuous outcome, the IV esti- Sample size calculations have been previously presented mates from the ratio, two-stage least squares (2SLS) and for Mendelian randomization experiments with continu- limited information maximum likelihood (LIML) methods ous outcomes. Calculations based on asymptotic statistical coincide. The estimator can be expressed as the ratio be- theory have been presented with a single instrumental vari- tween the coefficient from the regression of the outcome able (IV), whether that IV is a single genetic variant or an (Y) on the genetic variant (G), divided by the coefficient allele score. An allele score (also called a genetic risk from the regression of the risk factor (X) on the variant: score) is a single variable summarizing multiple genetic variants as a weighted or unweighted sum of risk factor- GY b ¼ : ð1Þ IV increasing alleles. A simulation study for estimating GX power has also been presented with both single and mul- The asymptotic variance of this IV estimator is given by tiple IVs. These approaches have shown good agreement. the formula: However, in many cases, the outcome in a Mendelian ran- varðR Þ domization experiment is binary (dichotomous), such as varðb Þ¼ ð2Þ IV N varðXÞq GX disease. In this paper, we present power calculations for where R ¼ Y – b . X is the residual of the outcome on sub- Mendelian randomization studies with a binary outcome. Y 1 We assume the context of a case-control study where the traction of the causal effect of the risk factor, and q is GX causal parameter of interest is an odds ratio, although the the square of the correlation between the risk factor X and 17 2 the IV G. The coefficient of determination (R ) in the re- calculations are also valid for other study designs. gression of the risk factor on the IV is an estimate of q . GX The IV in these calculations could either be a single genetic Methods and Results 10 variant or an allele score. We give results for the asymptotic variance of IV estima- The asymptotic variance of the conventional regression tors with a single IV, and for the resulting sample size (ordinary least squares, OLS) estimator of the association Downloaded from https://academic.oup.com/ije/article/43/3/922/761826 by DeepDyve user on 20 July 2022 924 International Journal of Epidemiology, 2014, Vol. 43, No. 3 between the risk factor X and the outcome Y is given by We use these formulae to construct power curves for the formula: Mendelian randomization using a significance level of 0.05. In Figure 1 (left), we fix the squared correlation q varðÞ R GX varðb Þ¼ : ð3Þ OLS at 0.02, meaning the variant explains on average 2% of the N varðÞ X variance of the risk factor, and vary the size of the effect The sample size necessary for an IV analysis to demon- b ¼ 0.05, 0.1, 0.15, 0.2, 0.25, 0.3 and the sample size strate a non-zero association for a given magnitude of causal N ¼ 1000 to 10 000. In Figure 1 (right), we fix the size of effect is therefore approximately equal to that for a conven- the effect at b ¼ 0.2 and vary the squared correlation tional epidemiological analysis to demonstrate the same q ¼ 0.005, 0.01, 0.015, 0.02, 0.025, 0.03 and the sam- GX magnitude of association divided by the q value for the GX ple size as before. In each of the figures, the power to de- IV. If the significance level is a and the power desired to tect a positive causal relationship is displayed; this tends to test the null hypothesis is 1b, then the sample size required 0.025 as the sample size tends to zero. We see that the to test a causal effect of size b using IV analysis is: power increases as the causal effect increases, and as the IV a 2 ðz þz Þ varðÞ R ð1 Þ b Y explains more of the variance in the risk factor (the q GX Sample size ¼ ð4Þ 2 2 varðÞ X b q parameter or the expected value of the R statistic GX increases). where z is a quantile function, so that z is the 100a per- Similar formulae to these have been made available in centile point on the standard normal distribution. If the an online tool for calculating either power for a given sam- significance level is 0.05 and the power is 0.8, then the ple size or sample size needed for a given power, taking the sample size to test for a change of b standard deviations in causal effect (b ) and squared correlation (q ) param- Y per standard deviation increase in X is: GX eters, as well as the variance of the risk factor and out- 7:848 Sample size¼ : ð5Þ come, and the observational (OLS) coefficient of the risk b q 1 GX factor from regression on the outcome. For a given sample size N, the power to detect a causal ef- fect (in the same direction the true effect) can be calculated as: Power with a binary outcome Uðb q N  zÞð6Þ GX 1 ð1 Þ With a single IV and a binary outcome, the same IV esti- mator as in the continuous outcome case can be eval- where U is the cumulative distribution function of the uated, except that a logistic model is typically used in the standard normal distribution. This is the inverse function regression of the outcome on the genetic variant. The of the quantile function (U(z ) ¼ a). asymptotic variance of this estimator can be approximated β = 0.05 ρ = 0.005 GX β = 0.10 ρ = 0.010 β = 0.15 GX β = 0.20 1 ρ = 0.015 GX β = 0.25 1 2 ρ = 0.020 GX β = 0.30 ρ = 0.025 GX ρ = 0.030 GX 0 2000 4000 6000 8000 10000 0 2000 4000 6000 8000 10000 Sample size Sample size Figure 1. Power curves varying the sample size with continuous outcome and a single instrumental variable. Left panel: for a fixed value of the IV strength (q ¼ 0.02) and different values of the size of the causal effect (b ¼ 0.05, 0.1,…, 0.3). Right panel: for a fixed value of the causal effect GX (b ¼ 0.2) and varying the size of the IV strength (q ¼ 0.005, 0.01,…, 0.03) GX Power (%) 0 20 40 60 80 100 Power (%) 0 20 40 60 80 100 Downloaded from https://academic.oup.com/ije/article/43/3/922/761826 by DeepDyve user on 20 July 2022 International Journal of Epidemiology, 2014, Vol. 43, No. 3 925 using the delta method for the ratio of two estimates. We use these approximations to calculate the number The leading term in the expansion is: of cases needed to obtain 80% power in a Mendelian ran- domization analysis with a binary outcome for different varðb Þ GY varðb Þ¼ : ð7Þ IV values of b and q , assuming a 1:1 ratio of cases to con- ^ GX GX trols. The results are displayed in Figure 2. We note that Although further terms from the delta method could be when the genetic variants explain a small proportion of the included, these are usually much smaller in magnitude. In variance in the risk factor, large sample sizes are required the simulation example later in the paper, if the association to detect even moderately large causal effects with reason- between the risk factor and IV is estimated using data on able power. the entire sample of control participants, the second and An R script for performing sample size and power cal- third terms in the expansion are two orders of magnitude culations is provided in the Appendix (available as smaller than the leading term (Figure 1). The asymptotic Supplementary data at IJE online). This code enables the variance of the coefficient b from logistic regression is: GY calculation of the sample size required for a chosen power level given the values of b and q , as well as the power 1 1 GX varðb Þ¼ P ð8Þ GY given the values of b , q and the chosen sample size. E g PðY¼1jG¼g ÞPðY¼0jG¼g Þ 1 GX i i i i A calculator using this code is available online. where i indexes individuals. This expression is obtained by differentiation of the log-likelihood. If the probability of Validation simulation an event does not depend greatly on the value of the gen- etic IV, then P(Y ¼ 1 j G ¼ g )  P(Y ¼ 1) which is the ratio In order to validate the estimates of sample size and power, of cases to participants in the sample. This approximation we simulate data on a genetic variant, a continuous risk will be reasonable if the genetic variant does not explain a factor and an outcome. The data-generating model for in- large proportion of the variance in the risk factor, and/or dividuals indexed by i is: the effect of the risk factor on the outcome is not extreme. We assume (without loss of generality) that the mean of G g  Nð0,1Þð13Þ 2 2 is 0 and the variance is 1, so that E(R g ) ¼ N, where N is x  Nðg q ,1  q Þ i i i GX GX the sample size. The square of the coefficient b is ap- y  Binomialð1, expitðb þ b x ÞÞ GX i i 0 1 proximately equal to var(X) q . This gives: GX where expit(x) ¼ (exp(x)/1 þ exp(x)) is the inverse of the logit function and b is the log odds ratio per unit (which varðb Þ¼ : ð9Þ IV N varðXÞq PðY ¼ 1Þ PðY ¼ 0Þ here equals 1 standard deviation) increase in the risk fac- GX tor. The genetic variant is modelled by a standard normal The sample size required to detect an effect of size b per distribution; it can be regarded as a standardized weighted standard deviation increase in X for 80% power with a sig- allele score. The parametric relationship between X, G and nificance level of 0.05 is therefore q ensures that the proportion of variance in the risk fac- GX tor explained by the instrumental variable in a large sample 7:848 Sample size ¼ ð10Þ is q . We also simulate data with a dichotomous risk fac- 2 GX b q PðY ¼ 1Þ PðY ¼ 0Þ 1 GX tor; details are given in the Web Appendix (available as Supplementary data at IJE online). where the effect b is a log odds ratio. If there are to be We set b ¼3 so that the outcome has a prevalence of an equal number of cases and controls, P(Y ¼ 1) ¼ about 5% in the population from which the case-control P(Y ¼ 0) ¼ 0.5, and: sample is taken. We take three values of b ¼ 0.1, 0.2, 0.3, 31:392 three values of q ¼ 0.01, 0.02, 0.03, three sample sizes Sample size ¼ : ð11Þ GX b q 1 GX (10 000, 20 000, and 30 000 cases), and two values of the The corresponding power to detect a causal effect of size ratio of cases to controls (1:1 and 1:2). For each set of par- b with a significance level of 0.05 is: ameter values, we calculate the estimate of the power from equation (12) using a significance level of 0.05, and com- pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Uðb q ðNPðY¼1ÞPðY¼0ÞÞ1:96Þ: ð12Þ pare this with the number of times the 95% confidence 1 GX interval for the ratio estimate excludes the null based on Similar power curves to Figure 1 in the binary outcome 10 000 simulated datasets. setting are given in the Web Appendix (available as The 95% confidence interval for the ratio method used Supplementary data at IJE online). in calculating the power of the simulation method is Downloaded from https://academic.oup.com/ije/article/43/3/922/761826 by DeepDyve user on 20 July 2022 926 International Journal of Epidemiology, 2014, Vol. 43, No. 3 2 2 ρ =1% ρ =0.5% GX GX 2 2 ρ =2% ρ =1.0% GX GX ρ =3% ρ =1.5% GX GX 2 2 ρ =5% ρ =2.0% GX GX 2 2 ρ =8% ρ =2.5% GX GX ρ =3.0% GX 1.1 1.2 1.3 1.4 1.5 1.1 1.2 1.3 1.4 1.5 Odds ratio per SD increase in risk factor Odds ratio per SD increase in risk factor Figure 2. Number of cases required in a Mendelian randomization analysis with a binary outcome and a single instrumental variable for 80% power with a 5% significance level and 1:1 ratio of cases:controls varying the size of causal effect [odds ratio per standard deviation (SD) increase in risk fac- 2 2 tor, exp(b )] for different values of IV strength. Left panel: q ¼ 1%–8%. Right panel: q ¼ 0.5%–3.0% GX GX constructed using Fieller’s method, and so does not rely factor; details are given in Web Table A1 (available as Supplementary data at IJE online). In comparing estimates on the same asymptotic assumption as the analytical method for estimating the power. Previous simulations of power with equal numbers of cases, greater power is have shown that confidence intervals from Fieller’s method achieved when there is a case:control ratio of 1:2 than with maintain nominal coverage levels even with weak instru- a ratio of 1:1. However, when the total sample size is fixed, ments. To obtain a case-control sample of the necessary the estimate of power is greatest when the numbers of size, we initially simulate data for a large number of indi- cases and controls are equal. This can be seen by compar- viduals, and then take the required number of cases and ing estimates with 30 000 cases and a ratio of 1:1, and controls from this population. with 20 000 cases and a ratio of 1:2. In response to concerns from a reviewer that the power estimates may not be valid with a discrete instrumental Simulation results variable (such as a single nucleotide polymorphism) or Results from the validation simulation are given in when there is confounding, additional validation simula- Table 1. The Monte Carlo standard error (the expected tions were performed in these scenarios. Results are given variation from the true value due to the limited number of in the Web Appendix (Web Tables A2–A4, available as simulations) in the simulation estimates of power is at Supplementary data at IJE online). No substantial differ- most 0.5%. The coverage levels of the 95% confidence ences were observed from the validation simulation in the interval from Fieller’s method are close to 95% throughout main paper when the instrumental variable was discrete. (between 94.8 and 95.9 for the 54 scenarios). When there was confounding, estimates from the analyt- We note that estimates of power from the formula of ical formula slightly overestimated power, particularly equation (12) are similar to those from the simulation ap- when the confounding was in the same direction as the proach. There is no apparent systematic bias in the esti- causal effect. However, this overestimation was slight (on mates from the analytical formula, with simulation average less than 1% when the confounding was in the op- estimates being greater and less than those from the for- posite direction, and less than 2% when the confounding mula a similar number of times (when rounded to nearest was in the same direction). As the magnitude of confound- 0.1%, the estimate from the simulation was less 24 times ing is not possible to estimate in applied practice, conserva- and greater 19 times). Estimates from both approaches are tive estimates of the correlation and causal effect parameters used in power calculations are recommended, no more different than would be expected due to chance alone. Similar results are obtained with a dichotomous risk particularly if confounding is thought to be substantial. Number of cases required for 80% power 2K 10K 20K 30K 40K 50K Number of cases required for 80% power 5K 20K 40K 60K 80K 100K Downloaded from https://academic.oup.com/ije/article/43/3/922/761826 by DeepDyve user on 20 July 2022 International Journal of Epidemiology, 2014, Vol. 43, No. 3 927 Table 1. Validation simulation to compare estimates of power in a Mendelian randomization analysis with a continuous risk fac- tor and a binary outcome from analytical formula and simulation study with a 5% significance level varying the size of causal ef- fect (b ), the IV strength (q ), the sample size and the ratio of cases to controls GX Case:control ratio ¼ 1:1 10000 cases 20000 cases 30000 cases Formula Simulation Formula Simulation Formula Simulation b ¼ 0.1 10.5% 10.2% 16.9% 16.6% 23.1% 22.4% b ¼ 0.2 29.3% 28.4% 51.6% 51.2% 68.8% 69.5% q ¼ 0.01 1 GX b ¼ 0.3 56.4% 56.4% 85.1% 85.0% 95.7% 95.7% b ¼ 0.1 16.9% 17.2% 29.3% 28.9% 41.0% 41.1% b ¼ 0.2 51.6% 51.0% 80.7% 80.2% 93.4% 93.6% q ¼ 0.02 1 GX b ¼ 0.3 85.1% 84.9% 98.9% 98.9% 99.9% 100.0% b ¼ 0.1 23.1% 22.9% 41.0% 40.8% 56.4% 57.0% b ¼ 0.2 68.8% 68.5% 93.4% 93.3% 98.9% 99.0% q ¼ 0.03 1 GX b ¼ 0.3 95.7% 95.5% 99.9% 99.9% 100.0% 100.0% Case:control ratio ¼ 1:2 10000 cases 20000 cases 30000 cases Formula Simulation Formula Simulation Formula Simulation b ¼ 0.1 12.6% 12.9% 21.0% 21.4% 29.3% 28.9% b ¼ 0.2 37.2% 37.5% 63.7% 64.4% 80.7% 81.1% q ¼ 0.01 1 GX b ¼ 0.3 68.8% 68.2% 93.4% 93.3% 98.9% 98.8% b ¼ 0.1 21.0% 21.2% 37.2% 37.8% 51.6% 51.6% b ¼ 0.2 63.7% 63.9% 90.4% 90.7% 97.9% 97.9% q ¼ 0.02 1 GX b ¼ 0.3 93.4% 93.2% 99.8% 99.8% 100.0% 100.0% b ¼ 0.1 29.3% 29.0% 51.6% 51.4% 68.8% 68.8% b ¼ 0.2 80.7% 80.8% 97.9% 97.7% 99.8% 99.9% q ¼ 0.03 1 GX b ¼ 0.3 98.9% 98.9% 100.0% 100.0% 100.0% 100.0% Discussion score approach. With an allele score, power can be further increased by the use of relevant weights for the variants. In this paper, we have provided information on sample Provided that weights are not derived naively from the sizes and power calculations in a Mendelian randomiza- data under analysis, the allele score approach avoids some tion analysis with a single IV and a binary outcome. We of the problems of bias from weak instruments resulting have shown in the continuous setting how the power de- from using many IVs. A disadvantage of the inclusion of pends on the magnitude of causal effect and the proportion many variants in an IV analysis, whether in a multiple IV of variance in the risk factor explained by the IV. With a or an allele score model, is that one or more of the variants binary outcome, the precision of the coefficient in the re- may not be a valid IV. If a variant is associated with a con- gression of the outcome on the IV is reduced compared founder of the risk factor–outcome association, or with the with a continuous outcome, as the outcome can only take outcome through a pathway not via the risk factor of inter- two values. As a result, the required sample sizes to obtain est, then the estimate associated with this IV may be 80% power are much larger. biased. If the function and relevance of some variants as For a given applied example, the magnitude of the IVs are uncertain, investigators will have to balance the causal effect of a risk factor is fixed, as is the expected pro- risk of a biased analysis against the risk of an underpow- portion of variance in the risk factor explained by each ered analysis. Sensitivity analysis may be a valuable tool to variant. However, the expected proportion of variance in assess the homogeneity of IV estimates using different sets the risk factor explained by the IV depends on the choice of variants. of IV. The required sample size for a given power level can If there are missing data, this may adversely impact the be reduced (or equivalently the expected power at a given power of an analysis. When there are multiple genetic vari- sample size can be increased) by including more genetic ants, individuals with sporadic missing genetic data can be variants into the IV. This can be achieved by using multiple 13 25 variants as separate IVs, or as a single IV using an allele included in an analysis using an imputation approach. Downloaded from https://academic.oup.com/ije/article/43/3/922/761826 by DeepDyve user on 20 July 2022 928 International Journal of Epidemiology, 2014, Vol. 43, No. 3 This can minimize the impact of missing data on the power 90% of the power of the complete-data analysis can be ob- of the analysis, particularly if the distributions of genetic tained while only measuring the risk factor for 10% of par- variants are correlated (the variants are in linkage ticipants. This means that obtaining measurements of the disequilibrium). risk factor, which may be expensive or impractical for a The calculations in this paper make several assump- large sample, should not be the prohibitive factor for a tions. The distribution of the IV estimator is assumed to be Mendelian randomization investigation. well approximated by a normal distribution. This is known to be a poor approximation when the IV is weak; how- Supplementary Data ever, if the IV is weak, then the power will usually be low. The standard deviation of this normal distribution is Supplementary data are available at IJE online. assumed to be close to the first-order term from the delta Conflict of interest: None declared. expansion. This term only involves the uncertainty in the coefficient from the genetic association with the outcome. References The uncertainty in the estimate of the genetic association 1. Davey Smith G, Ebrahim S. Data dredging, bias, or confounding. with the risk factor is not accounted for. Typically, this un- BMJ 2002;325:1437. certainty will be small in comparison as the genetic associ- 2. Davey Smith G, Ebrahim S. ‘Mendelian randomization’: can ation with the outcome is assumed to be mediated through genetic epidemiology contribute to understanding environmental the risk factor. Again, if this uncertainty is large, then the determinants of disease? Int J Epidemiol 2003;32:1–22. power of the analysis will usually be low. If a more precise 3. 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Sample size and power calculations in Mendelian randomization with a single instrumental variable and a binary outcome

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Oxford University Press
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Copyright © 2022 International Epidemiological Association
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0300-5771
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1464-3685
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10.1093/ije/dyu005
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Abstract

Background: Sample size calculations are an important tool for planning epidemiological studies. Large sample sizes are often required in Mendelian randomization investigations. Methods and results: Resources are provided for investigators to perform sample size and power calculations for Mendelian randomization with a binary outcome. We initially provide formulae for the continuous outcome case, and then analogous formulae for the binary outcome case. The formulae are valid for a single instrumental variable, which may be a single genetic variant or an allele score comprising multiple variants. Graphs are provided to give the required sample size for 80% power for given values of the causal effect of the risk factor on the outcome and of the squared correlation between the risk factor and instrumental variable. R code and an online calculator tool are made available for calculating the sample size needed for a chosen power level given these parameters, as well as the power given the chosen sample size and these parameters. Conclusions: The sample size required for a given power of Mendelian randomization in- vestigation depends greatly on the proportion of variance in the risk factor explained by the instrumental variable. The inclusion of multiple variants into an allele score to explain more of the variance in the risk factor will improve power, however care must be taken not to introduce bias by the inclusion of invalid variants. Key words: Mendelian randomization, sample size, power, binary outcome, allele score Key Messages Resources are provided for investigators to perform sample size and power calculations for Mendelian randomization with a binary outcome. The sample size required for a given power level is greater with a binary outcome than a continuous outcome, and is highly dependent on the proportion of the variance in the risk factor explained by the instrumental variable. V The Author 2014. Published by Oxford University Press on behalf of the International Epidemiological Association 922 This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited. Downloaded from https://academic.oup.com/ije/article/43/3/922/761826 by DeepDyve user on 20 July 2022 International Journal of Epidemiology, 2014, Vol. 43, No. 3 923 Introduction needed in a Mendelian randomization study to obtain a given power level. We initially present formulae with a con- Sample size calculations are an important part of experimen- tinuous outcome (this reviews material previously covered tal design. They inform an investigator of the expected power by Freeman et al. ) and then analogous formulae with a bin- of a given analysis to reject the null hypothesis. If the power ary outcome. We concentrate on estimates from the ratio of an analysis is low, then not only is the probability of re- (or Wald) method, as this method makes few parametric as- jecting the null hypothesis low, but when the null hypothesis sumptions, relying only on a linear relationship between the is rejected, the posterior probability that the rejection of the 1 conditional expectation of the outcome (or in the binary null hypothesis is not simply a chance finding is low. case, the logistic function of the probability of the outcome) Mendelian randomization is the use of genetic variants and the risk factor. If the imprecision in the estimate of as instrumental variables for assessing the causal effect of a 2 the genetic association with the risk factor is negligible, then risk factor on an outcome from observational data. estimates of power and sample size from the ratio method Genetic variants are chosen which are specifically associ- also correspond to those from assessment of the causal rela- ated with a risk factor of interest, and not associated with tionship of the risk factor on the outcome by testing the as- variables which may be confounders of the association be- 3 sociation between the genetic variant and outcome. tween the risk factor and outcome. Such a variant divides Other estimation approaches are possible with a binary the population into groups which are similar to treatment outcome but these either give equivalent estimates to the arms in a randomized controlled trial. Under the instru- 5,6 ratio method with a single IV (the two-stage predictor sub- mental variable assumptions, a statistical association be- stitution method ) or are not recommended for general tween the genetic variant and the outcome implies that the 7 use in applied practice. These include the two-stage re- risk factor has a causal effect on the outcome. However, sidual inclusion method, due to inconsistency for a param- as genetic variants typically explain a small proportion of eter with a natural interpretation, and the generalized the variance in risk factors, the power to detect a signifi- method of moments (GMM) and structural mean models cant association between the variant and outcome in an 8 (SMM) methods, due to potential lack of identifiability of applied Mendelian randomization context can be low. the causal parameter (S Burgess et al., unpublished data). Sample size analysis is particularly important to inform whether a null finding is representative of a true null causal relationship, or simply a lack of power to detect an effect Power with a continuous outcome size of clinical interest. With a single IV and a continuous outcome, the IV esti- Sample size calculations have been previously presented mates from the ratio, two-stage least squares (2SLS) and for Mendelian randomization experiments with continu- limited information maximum likelihood (LIML) methods ous outcomes. Calculations based on asymptotic statistical coincide. The estimator can be expressed as the ratio be- theory have been presented with a single instrumental vari- tween the coefficient from the regression of the outcome able (IV), whether that IV is a single genetic variant or an (Y) on the genetic variant (G), divided by the coefficient allele score. An allele score (also called a genetic risk from the regression of the risk factor (X) on the variant: score) is a single variable summarizing multiple genetic variants as a weighted or unweighted sum of risk factor- GY b ¼ : ð1Þ IV increasing alleles. A simulation study for estimating GX power has also been presented with both single and mul- The asymptotic variance of this IV estimator is given by tiple IVs. These approaches have shown good agreement. the formula: However, in many cases, the outcome in a Mendelian ran- varðR Þ domization experiment is binary (dichotomous), such as varðb Þ¼ ð2Þ IV N varðXÞq GX disease. In this paper, we present power calculations for where R ¼ Y – b . X is the residual of the outcome on sub- Mendelian randomization studies with a binary outcome. Y 1 We assume the context of a case-control study where the traction of the causal effect of the risk factor, and q is GX causal parameter of interest is an odds ratio, although the the square of the correlation between the risk factor X and 17 2 the IV G. The coefficient of determination (R ) in the re- calculations are also valid for other study designs. gression of the risk factor on the IV is an estimate of q . GX The IV in these calculations could either be a single genetic Methods and Results 10 variant or an allele score. We give results for the asymptotic variance of IV estima- The asymptotic variance of the conventional regression tors with a single IV, and for the resulting sample size (ordinary least squares, OLS) estimator of the association Downloaded from https://academic.oup.com/ije/article/43/3/922/761826 by DeepDyve user on 20 July 2022 924 International Journal of Epidemiology, 2014, Vol. 43, No. 3 between the risk factor X and the outcome Y is given by We use these formulae to construct power curves for the formula: Mendelian randomization using a significance level of 0.05. In Figure 1 (left), we fix the squared correlation q varðÞ R GX varðb Þ¼ : ð3Þ OLS at 0.02, meaning the variant explains on average 2% of the N varðÞ X variance of the risk factor, and vary the size of the effect The sample size necessary for an IV analysis to demon- b ¼ 0.05, 0.1, 0.15, 0.2, 0.25, 0.3 and the sample size strate a non-zero association for a given magnitude of causal N ¼ 1000 to 10 000. In Figure 1 (right), we fix the size of effect is therefore approximately equal to that for a conven- the effect at b ¼ 0.2 and vary the squared correlation tional epidemiological analysis to demonstrate the same q ¼ 0.005, 0.01, 0.015, 0.02, 0.025, 0.03 and the sam- GX magnitude of association divided by the q value for the GX ple size as before. In each of the figures, the power to de- IV. If the significance level is a and the power desired to tect a positive causal relationship is displayed; this tends to test the null hypothesis is 1b, then the sample size required 0.025 as the sample size tends to zero. We see that the to test a causal effect of size b using IV analysis is: power increases as the causal effect increases, and as the IV a 2 ðz þz Þ varðÞ R ð1 Þ b Y explains more of the variance in the risk factor (the q GX Sample size ¼ ð4Þ 2 2 varðÞ X b q parameter or the expected value of the R statistic GX increases). where z is a quantile function, so that z is the 100a per- Similar formulae to these have been made available in centile point on the standard normal distribution. If the an online tool for calculating either power for a given sam- significance level is 0.05 and the power is 0.8, then the ple size or sample size needed for a given power, taking the sample size to test for a change of b standard deviations in causal effect (b ) and squared correlation (q ) param- Y per standard deviation increase in X is: GX eters, as well as the variance of the risk factor and out- 7:848 Sample size¼ : ð5Þ come, and the observational (OLS) coefficient of the risk b q 1 GX factor from regression on the outcome. For a given sample size N, the power to detect a causal ef- fect (in the same direction the true effect) can be calculated as: Power with a binary outcome Uðb q N  zÞð6Þ GX 1 ð1 Þ With a single IV and a binary outcome, the same IV esti- mator as in the continuous outcome case can be eval- where U is the cumulative distribution function of the uated, except that a logistic model is typically used in the standard normal distribution. This is the inverse function regression of the outcome on the genetic variant. The of the quantile function (U(z ) ¼ a). asymptotic variance of this estimator can be approximated β = 0.05 ρ = 0.005 GX β = 0.10 ρ = 0.010 β = 0.15 GX β = 0.20 1 ρ = 0.015 GX β = 0.25 1 2 ρ = 0.020 GX β = 0.30 ρ = 0.025 GX ρ = 0.030 GX 0 2000 4000 6000 8000 10000 0 2000 4000 6000 8000 10000 Sample size Sample size Figure 1. Power curves varying the sample size with continuous outcome and a single instrumental variable. Left panel: for a fixed value of the IV strength (q ¼ 0.02) and different values of the size of the causal effect (b ¼ 0.05, 0.1,…, 0.3). Right panel: for a fixed value of the causal effect GX (b ¼ 0.2) and varying the size of the IV strength (q ¼ 0.005, 0.01,…, 0.03) GX Power (%) 0 20 40 60 80 100 Power (%) 0 20 40 60 80 100 Downloaded from https://academic.oup.com/ije/article/43/3/922/761826 by DeepDyve user on 20 July 2022 International Journal of Epidemiology, 2014, Vol. 43, No. 3 925 using the delta method for the ratio of two estimates. We use these approximations to calculate the number The leading term in the expansion is: of cases needed to obtain 80% power in a Mendelian ran- domization analysis with a binary outcome for different varðb Þ GY varðb Þ¼ : ð7Þ IV values of b and q , assuming a 1:1 ratio of cases to con- ^ GX GX trols. The results are displayed in Figure 2. We note that Although further terms from the delta method could be when the genetic variants explain a small proportion of the included, these are usually much smaller in magnitude. In variance in the risk factor, large sample sizes are required the simulation example later in the paper, if the association to detect even moderately large causal effects with reason- between the risk factor and IV is estimated using data on able power. the entire sample of control participants, the second and An R script for performing sample size and power cal- third terms in the expansion are two orders of magnitude culations is provided in the Appendix (available as smaller than the leading term (Figure 1). The asymptotic Supplementary data at IJE online). This code enables the variance of the coefficient b from logistic regression is: GY calculation of the sample size required for a chosen power level given the values of b and q , as well as the power 1 1 GX varðb Þ¼ P ð8Þ GY given the values of b , q and the chosen sample size. E g PðY¼1jG¼g ÞPðY¼0jG¼g Þ 1 GX i i i i A calculator using this code is available online. where i indexes individuals. This expression is obtained by differentiation of the log-likelihood. If the probability of Validation simulation an event does not depend greatly on the value of the gen- etic IV, then P(Y ¼ 1 j G ¼ g )  P(Y ¼ 1) which is the ratio In order to validate the estimates of sample size and power, of cases to participants in the sample. This approximation we simulate data on a genetic variant, a continuous risk will be reasonable if the genetic variant does not explain a factor and an outcome. The data-generating model for in- large proportion of the variance in the risk factor, and/or dividuals indexed by i is: the effect of the risk factor on the outcome is not extreme. We assume (without loss of generality) that the mean of G g  Nð0,1Þð13Þ 2 2 is 0 and the variance is 1, so that E(R g ) ¼ N, where N is x  Nðg q ,1  q Þ i i i GX GX the sample size. The square of the coefficient b is ap- y  Binomialð1, expitðb þ b x ÞÞ GX i i 0 1 proximately equal to var(X) q . This gives: GX where expit(x) ¼ (exp(x)/1 þ exp(x)) is the inverse of the logit function and b is the log odds ratio per unit (which varðb Þ¼ : ð9Þ IV N varðXÞq PðY ¼ 1Þ PðY ¼ 0Þ here equals 1 standard deviation) increase in the risk fac- GX tor. The genetic variant is modelled by a standard normal The sample size required to detect an effect of size b per distribution; it can be regarded as a standardized weighted standard deviation increase in X for 80% power with a sig- allele score. The parametric relationship between X, G and nificance level of 0.05 is therefore q ensures that the proportion of variance in the risk fac- GX tor explained by the instrumental variable in a large sample 7:848 Sample size ¼ ð10Þ is q . We also simulate data with a dichotomous risk fac- 2 GX b q PðY ¼ 1Þ PðY ¼ 0Þ 1 GX tor; details are given in the Web Appendix (available as Supplementary data at IJE online). where the effect b is a log odds ratio. If there are to be We set b ¼3 so that the outcome has a prevalence of an equal number of cases and controls, P(Y ¼ 1) ¼ about 5% in the population from which the case-control P(Y ¼ 0) ¼ 0.5, and: sample is taken. We take three values of b ¼ 0.1, 0.2, 0.3, 31:392 three values of q ¼ 0.01, 0.02, 0.03, three sample sizes Sample size ¼ : ð11Þ GX b q 1 GX (10 000, 20 000, and 30 000 cases), and two values of the The corresponding power to detect a causal effect of size ratio of cases to controls (1:1 and 1:2). For each set of par- b with a significance level of 0.05 is: ameter values, we calculate the estimate of the power from equation (12) using a significance level of 0.05, and com- pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Uðb q ðNPðY¼1ÞPðY¼0ÞÞ1:96Þ: ð12Þ pare this with the number of times the 95% confidence 1 GX interval for the ratio estimate excludes the null based on Similar power curves to Figure 1 in the binary outcome 10 000 simulated datasets. setting are given in the Web Appendix (available as The 95% confidence interval for the ratio method used Supplementary data at IJE online). in calculating the power of the simulation method is Downloaded from https://academic.oup.com/ije/article/43/3/922/761826 by DeepDyve user on 20 July 2022 926 International Journal of Epidemiology, 2014, Vol. 43, No. 3 2 2 ρ =1% ρ =0.5% GX GX 2 2 ρ =2% ρ =1.0% GX GX ρ =3% ρ =1.5% GX GX 2 2 ρ =5% ρ =2.0% GX GX 2 2 ρ =8% ρ =2.5% GX GX ρ =3.0% GX 1.1 1.2 1.3 1.4 1.5 1.1 1.2 1.3 1.4 1.5 Odds ratio per SD increase in risk factor Odds ratio per SD increase in risk factor Figure 2. Number of cases required in a Mendelian randomization analysis with a binary outcome and a single instrumental variable for 80% power with a 5% significance level and 1:1 ratio of cases:controls varying the size of causal effect [odds ratio per standard deviation (SD) increase in risk fac- 2 2 tor, exp(b )] for different values of IV strength. Left panel: q ¼ 1%–8%. Right panel: q ¼ 0.5%–3.0% GX GX constructed using Fieller’s method, and so does not rely factor; details are given in Web Table A1 (available as Supplementary data at IJE online). In comparing estimates on the same asymptotic assumption as the analytical method for estimating the power. Previous simulations of power with equal numbers of cases, greater power is have shown that confidence intervals from Fieller’s method achieved when there is a case:control ratio of 1:2 than with maintain nominal coverage levels even with weak instru- a ratio of 1:1. However, when the total sample size is fixed, ments. To obtain a case-control sample of the necessary the estimate of power is greatest when the numbers of size, we initially simulate data for a large number of indi- cases and controls are equal. This can be seen by compar- viduals, and then take the required number of cases and ing estimates with 30 000 cases and a ratio of 1:1, and controls from this population. with 20 000 cases and a ratio of 1:2. In response to concerns from a reviewer that the power estimates may not be valid with a discrete instrumental Simulation results variable (such as a single nucleotide polymorphism) or Results from the validation simulation are given in when there is confounding, additional validation simula- Table 1. The Monte Carlo standard error (the expected tions were performed in these scenarios. Results are given variation from the true value due to the limited number of in the Web Appendix (Web Tables A2–A4, available as simulations) in the simulation estimates of power is at Supplementary data at IJE online). No substantial differ- most 0.5%. The coverage levels of the 95% confidence ences were observed from the validation simulation in the interval from Fieller’s method are close to 95% throughout main paper when the instrumental variable was discrete. (between 94.8 and 95.9 for the 54 scenarios). When there was confounding, estimates from the analyt- We note that estimates of power from the formula of ical formula slightly overestimated power, particularly equation (12) are similar to those from the simulation ap- when the confounding was in the same direction as the proach. There is no apparent systematic bias in the esti- causal effect. However, this overestimation was slight (on mates from the analytical formula, with simulation average less than 1% when the confounding was in the op- estimates being greater and less than those from the for- posite direction, and less than 2% when the confounding mula a similar number of times (when rounded to nearest was in the same direction). As the magnitude of confound- 0.1%, the estimate from the simulation was less 24 times ing is not possible to estimate in applied practice, conserva- and greater 19 times). Estimates from both approaches are tive estimates of the correlation and causal effect parameters used in power calculations are recommended, no more different than would be expected due to chance alone. Similar results are obtained with a dichotomous risk particularly if confounding is thought to be substantial. Number of cases required for 80% power 2K 10K 20K 30K 40K 50K Number of cases required for 80% power 5K 20K 40K 60K 80K 100K Downloaded from https://academic.oup.com/ije/article/43/3/922/761826 by DeepDyve user on 20 July 2022 International Journal of Epidemiology, 2014, Vol. 43, No. 3 927 Table 1. Validation simulation to compare estimates of power in a Mendelian randomization analysis with a continuous risk fac- tor and a binary outcome from analytical formula and simulation study with a 5% significance level varying the size of causal ef- fect (b ), the IV strength (q ), the sample size and the ratio of cases to controls GX Case:control ratio ¼ 1:1 10000 cases 20000 cases 30000 cases Formula Simulation Formula Simulation Formula Simulation b ¼ 0.1 10.5% 10.2% 16.9% 16.6% 23.1% 22.4% b ¼ 0.2 29.3% 28.4% 51.6% 51.2% 68.8% 69.5% q ¼ 0.01 1 GX b ¼ 0.3 56.4% 56.4% 85.1% 85.0% 95.7% 95.7% b ¼ 0.1 16.9% 17.2% 29.3% 28.9% 41.0% 41.1% b ¼ 0.2 51.6% 51.0% 80.7% 80.2% 93.4% 93.6% q ¼ 0.02 1 GX b ¼ 0.3 85.1% 84.9% 98.9% 98.9% 99.9% 100.0% b ¼ 0.1 23.1% 22.9% 41.0% 40.8% 56.4% 57.0% b ¼ 0.2 68.8% 68.5% 93.4% 93.3% 98.9% 99.0% q ¼ 0.03 1 GX b ¼ 0.3 95.7% 95.5% 99.9% 99.9% 100.0% 100.0% Case:control ratio ¼ 1:2 10000 cases 20000 cases 30000 cases Formula Simulation Formula Simulation Formula Simulation b ¼ 0.1 12.6% 12.9% 21.0% 21.4% 29.3% 28.9% b ¼ 0.2 37.2% 37.5% 63.7% 64.4% 80.7% 81.1% q ¼ 0.01 1 GX b ¼ 0.3 68.8% 68.2% 93.4% 93.3% 98.9% 98.8% b ¼ 0.1 21.0% 21.2% 37.2% 37.8% 51.6% 51.6% b ¼ 0.2 63.7% 63.9% 90.4% 90.7% 97.9% 97.9% q ¼ 0.02 1 GX b ¼ 0.3 93.4% 93.2% 99.8% 99.8% 100.0% 100.0% b ¼ 0.1 29.3% 29.0% 51.6% 51.4% 68.8% 68.8% b ¼ 0.2 80.7% 80.8% 97.9% 97.7% 99.8% 99.9% q ¼ 0.03 1 GX b ¼ 0.3 98.9% 98.9% 100.0% 100.0% 100.0% 100.0% Discussion score approach. With an allele score, power can be further increased by the use of relevant weights for the variants. In this paper, we have provided information on sample Provided that weights are not derived naively from the sizes and power calculations in a Mendelian randomiza- data under analysis, the allele score approach avoids some tion analysis with a single IV and a binary outcome. We of the problems of bias from weak instruments resulting have shown in the continuous setting how the power de- from using many IVs. A disadvantage of the inclusion of pends on the magnitude of causal effect and the proportion many variants in an IV analysis, whether in a multiple IV of variance in the risk factor explained by the IV. With a or an allele score model, is that one or more of the variants binary outcome, the precision of the coefficient in the re- may not be a valid IV. If a variant is associated with a con- gression of the outcome on the IV is reduced compared founder of the risk factor–outcome association, or with the with a continuous outcome, as the outcome can only take outcome through a pathway not via the risk factor of inter- two values. As a result, the required sample sizes to obtain est, then the estimate associated with this IV may be 80% power are much larger. biased. If the function and relevance of some variants as For a given applied example, the magnitude of the IVs are uncertain, investigators will have to balance the causal effect of a risk factor is fixed, as is the expected pro- risk of a biased analysis against the risk of an underpow- portion of variance in the risk factor explained by each ered analysis. Sensitivity analysis may be a valuable tool to variant. However, the expected proportion of variance in assess the homogeneity of IV estimates using different sets the risk factor explained by the IV depends on the choice of variants. of IV. The required sample size for a given power level can If there are missing data, this may adversely impact the be reduced (or equivalently the expected power at a given power of an analysis. When there are multiple genetic vari- sample size can be increased) by including more genetic ants, individuals with sporadic missing genetic data can be variants into the IV. This can be achieved by using multiple 13 25 variants as separate IVs, or as a single IV using an allele included in an analysis using an imputation approach. Downloaded from https://academic.oup.com/ije/article/43/3/922/761826 by DeepDyve user on 20 July 2022 928 International Journal of Epidemiology, 2014, Vol. 43, No. 3 This can minimize the impact of missing data on the power 90% of the power of the complete-data analysis can be ob- of the analysis, particularly if the distributions of genetic tained while only measuring the risk factor for 10% of par- variants are correlated (the variants are in linkage ticipants. This means that obtaining measurements of the disequilibrium). risk factor, which may be expensive or impractical for a The calculations in this paper make several assump- large sample, should not be the prohibitive factor for a tions. The distribution of the IV estimator is assumed to be Mendelian randomization investigation. well approximated by a normal distribution. This is known to be a poor approximation when the IV is weak; how- Supplementary Data ever, if the IV is weak, then the power will usually be low. The standard deviation of this normal distribution is Supplementary data are available at IJE online. assumed to be close to the first-order term from the delta Conflict of interest: None declared. expansion. This term only involves the uncertainty in the coefficient from the genetic association with the outcome. References The uncertainty in the estimate of the genetic association 1. Davey Smith G, Ebrahim S. Data dredging, bias, or confounding. with the risk factor is not accounted for. Typically, this un- BMJ 2002;325:1437. certainty will be small in comparison as the genetic associ- 2. Davey Smith G, Ebrahim S. ‘Mendelian randomization’: can ation with the outcome is assumed to be mediated through genetic epidemiology contribute to understanding environmental the risk factor. Again, if this uncertainty is large, then the determinants of disease? Int J Epidemiol 2003;32:1–22. power of the analysis will usually be low. If a more precise 3. 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Journal

International Journal of Epidemiology – Oxford University Press

Published: Jun 1, 2014

Keywords: instrumental variables

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