Mediation has been thought of in terms of the proportion of effect explained, or the relative attenuation of b1, the coefficient for the primary predictor X1, when the mediator, X2, is added to the model. The goal is to show that b1*, the coefficient for X1 in the reduced model (i.e., the model with only X1, differs from b1, its coefficient in the full model (i.e., the model with both X1 and the mediator X2. If X1 and X2 are correlated, then showing that b2, the coefficient for X2, differs from zero is equivalent to showing b1* differs from b1. Thus the problem reduces to detecting an effect of X2, controlling for X1. In short, it amounts to the more familiar problem of inflating sample size to account for loss of precision due to adjustment for X1.
The approach here is to approximate the expected information matrix from the regression model including both X1 and X2, to obtain the expected standard error of the estimate of b2, evaluated at the MLE. The sample size follows from comparing the Wald test statistic (i.e., the ratio of the estimate of b2 to its SE) to the standard normal distribution, with the expected value of the numerator and denominator of the statistic computed under the alternative hypothesis. This reflects the Wald test for the statistical significance of a coefficient implemented in most regression packages.
The function provides methods to calculate sample sizes for the mediation problem for linear, logistic, Poisson, and Cox regression models in four cases for each model:
The function is also generally applicable to the analogous problem of calculating sample size adequate to detect the effect of a primary predictor in the presence of confounding. Simply treat X2 as the primary predictor and consider X1 the confounder.
masize(model,opts,alpha=0.025,gamma=0.2)
Schmoor C, Sauerbrer W, Schumacher M. Sample size considerations for the evaluation of prognostic factors in survival analysis. Stat Med 2000; 19:441-52.
Vittinghoff E, Sen S, McCulloch CE. Sample size calculations for evaluating mediation. Stat Med 2009; 28:541-57.
For logistic model, the approximate solution due to Hsieh is implemented in the function logistic.approx, and can be used for all four cases. Arguments are p, b2, rho, sdx2, alpha, and gamma. For a binary mediator with prevalence f2, sdx2 should be reset to $\sqrt{f2(1-f2)}$. Simulating the information matrix of the logistic model provides somewhat more accurate sample size estimates than the Hsieh approximation. The functions for cases CpCm, BpCm, CpBm, and BpBm are respectively logistic.ccs, logistic.bcs, logistic.cbs, and logistic.bbs, as for the Poisson and Cox models. Arguments for these functions include p, b1, sdx1 or f1, b2, sdx2 or f2, rho, alpha, gamma, and ns. As in other functions, sdx1, sdx2, alpha, and gamma are set to the defaults listed above. These four functions call two utility functions, getb0 (to calculate the intercept parameter from the others) and antilogit, which are supplied.
For Poisson model, The function implementing the approximate solution based on the variance inflation factor is poisson.approx, and can be used for all four cases. Arguments are EY (the marginal mean of the Poisson outcome), b2, sdx2, rho, alpha and gamma, with sdx2, alpha and gamma set to the usual defaults; use sdx2=$\sqrt{f2(1-f2)}$ for a binary mediator with prevalence f2 (cases CpBm and BpBm). For cases CpCm and BpCm (continuous mediators), the approximate formula is also the analytic solution. For these cases, we supply redundant functions poisson.cc and poisson.bc, with the same arguments and defaults as for poisson.approx (it's the same function). For the two cases with binary mediators, the functions are poisson.cb and poisson.bb. In addition to m, b2, f2, rho, alpha, and gamma, b1 and sdx1 or f1 must be specified. Defaults are as usual. Functions using simulation for the Poisson model are available: poisson.ccs, poisson.bcs, poisson.cbs, and poisson.bbs. As in the logistic case, these require arguments b1 and sdx1 or f1. For this case, however, the analytic functions are faster, avoid simulation error, and should be used. We include these functions as templates that could be adapted to other joint predictor distributions.
For Cox model, the function implementing the approximate solution, using the variance inflation factor and derived by Schmoor et al., is cox.approx, and can be used for all four cases. Arguments are b2, sdx2, rho, alpha, gamma, and f. For binary X2 set sdx2 = $\sqrt{f2(1-f2)}$. The approximation works very well for cases CpCm and BpCm (continuous mediators), but is a bit less accurate for cases CpBm and BpBm (binary mediators). We get some improvement for those cases using the simulation approach. This approach is implemented for all four, as functions cox.ccs, cox.bcs, cox.cbs, and cox.bbs. Arguments are b1, sdx1 or f1, b2, sdx2 or f2, rho, alpha, gamma, f, and ns, with defaults as described above. Slight variants of these functions, cox.ccs2, cox.bcs2, cox.cbs2, and cox.bbs2, make it possible to allow for early censoring of a fraction fc of observations; but in our experience this has virtually no effect, even with values of fc of 0.5. The default for fc is 0.
A summary of the argumentss is as follows, noting that additional parameter seed can be supplied for simulation-based method.
ab## linear model
# CpCm
opts <- list(b2=0.5, rho=0.3, sdx2=1, sdy=1)
masize("linear1",opts)
# BpBm
opts <- list(b2=0.75, rho=0.3, f2=0.25, sdx2=sqrt(0.25*0.75), sdy=3)
masize("linear1",opts,gamma=0.1)
## logistic model
# CpBm
opts <- list(p=0.25, b2=log(0.5), rho=0.5, sdx2=0.5)
masize("logistic1",opts)
opts <- list(p=0.25, b1=log(1.5), sdx1=1, b2=log(0.5), f2=0.5, rho=0.5, ns=10000, seed=1234)
masize("logistic4",opts)
opts <- list(p=0.25, b1=log(1.5), sdx1=1, b2=log(0.5), f2=0.5, rho=0.5, ns=10000, seed=1234)
masize("logistic4",opts)
opts <- list(p=0.25, b1=log(1.5), sdx1=4.5, b2=log(0.5), f2=0.5, rho=0.5, ns=50000, seed=1234)
masize("logistic4",opts)
## Poisson model
# BpBm
opts <- list(m=0.5, b2=log(1.25), rho=0.3, sdx2=sqrt(0.25*0.75))
masize("poisson1",opts)
opts <- list(m=0.5, b1=log(1.4), f1=0.25, b2=log(1.25), f2=0.25, rho=0.3)
masize("poisson5",opts)
opts <- c(opts,ns=10000, seed=1234)
masize("poisson9",opts)
## Cox model
# BpBm
opts <- list(b2=log(1.5), rho=0.45, f=0.2, sdx2=sqrt(0.25*0.75))
masize("cox1",opts)
opts <- list(b1=log(2), f1=0.5, b2=log(1.5), f2=0.25, rho=0.45, f=0.2, seed=1234)
masize("cox5",c(opts, ns=10000))
masize("cox5",c(opts, ns=50000))Run the code above in your browser using DataLab