#### To the Editor:

Hwang et al^{1} and Foppa and Spiegelman^{2} have presented sample size and power calculations for gene-environment interaction in case-control studies. These calculations have been criticized by Garcia-Closas and Lubin^{3} for relying on a variance estimator under the null (the “null-variance” formula^{4}), rather than under the alternative. The issue is not that the null-variance sample-size calculations are incorrect, but simply that they do not correspond to the test statistics that are generally used. In practice, the variance is most often evaluated under the alternative, which requires different formulas. Relying on the null variance for interaction in logistic regression models (when the variance for the test statistic is in fact estimated under the alternative) tends to underestimate the sample sizes required, especially when interaction odds ratios are relatively large.^{3,5}

Similar issues pertain to case-only designs. Under the assumption of gene-environment independence, Yang et al^{6} presented expressions for calculating required sample size for the case-only estimator of interaction that rely on the null-variance formula. The authors provide the following formula for the sample size required to detect a case-only interaction ratio of magnitude *Ri* with significance level α and power β:

where *Z*_{1−α/2} and *Z*_{β} are the (1−α/2)th and βth quantiles, respectively, of the standard normal distribution, and where *vN* and *vA* are the variance of the test statistic under the null and the alternative, respectively. Yang et al^{6} described how *vN* and *vA* can be computed for various values of the prevalence (*g*) of the genetic factor, the prevalence (*e*) of the environmental factor, the relative risk for the genetic factor alone (*Rg*), and the relative risk for the environmental factor alone (*Re*). However, when case-only estimators are actually employed, the variance for test statistics for the interaction parameter is generally estimated under the alternative.

Here, we give sample-size and power formulas for the case-only interaction estimator that can be used when the variance under the alternative is employed (as is often done in practice). The sample size required to detect an interaction ratio *Ri* using a Wald test statistic with variance evaluated under the alternative and with significance level α and power β is given by Demidenko^{5} as follows:

where for the case-only estimator, *vA* is given by Yang et al^{6} as follows:

The discrepancies between the sample-size formula in Eq. (1) from Yang et al^{6} and that given in Eq. (2) can lead to meaningful differences in the estimated requirement for sample size. For example, in a case-only study with α = 5%, β = 80%, *g* = *e* = 0.1, *Rg* = 1, *Re* = 2, and *Ri* = 2, Yang et al^{6} reported a required sample size of 939; in contrast, the required sample size calculated from Eq. (2) is 785. For larger interaction ratios, the use of null variance employed by Yang et al^{6} can underestimate the required sample size. In a case-only study with α = 5%, β = 80%, *g* = 0.1, *e* = 0.3, *Rg* = 1, *Re* = 2, and *Ri* = 10, Yang et al^{6} reported a required sample size of 34, whereas Eq. (2) gives a sample size of 55. This mirrors the finding of Garcia-Closas and Lubin^{1} for case-control studies, where using the null variance with larger interaction parameters can substantially underestimate the required sample size.

Equation 1 Image Tools |
Equation 2 Image Tools |
Equation (Uncited) Image Tools |

Equation (Uncited) Image Tools |

If power for a fixed sample size *n* is of interest, then this can be calculated by

where Φ^{−1} is the inverse cumulative distribution function for a standard normal random variable. The power and sample-size calculations of Yang et al^{6} should be used only if the investigator intends to calculate the variance of the test statistic under the null (which is generally not done); otherwise the power and sample size calculations given here should be used.

In some cases, an investigator might be interested in testing log(*Ri*)> log(2) or log(*Ri*)>log(3) to detect “sufficient cause” or “epistatic” interactions.^{7,8} Sample size calculations could then proceed as mentioned earlier, using the formula in equation (2), but replacing the denominator (log(*Ri*))^{2} with either [log(*Ri*)− log(2)]^{2} or [log(*Ri*)−log(3)]^{2}, respectively.

Tyler J. VanderWeele

Departments of Epidemiology and Biostatistics

Harvard School of Public Health

Boston, MA

tvanderw@hsph.harvard.edu