2 Steps in Power Analysis

The basic approach to power analysis is to imagine that we are doing an identical study many times. For example, we pretend that we are trying to estimate $a,c$, and $e$ for a given population by taking samples of a given number of MZ and DZ twins. Each sample would give somewhat different estimates of the parameters, depending on how many twins we study, and how big $a,c$, and $e$ are in the study population. Suppose we did a very large number of studies and tabulated all the estimates of the shared environmental component, $c^2$. In some of the studies, even though there was some shared environment in the population, we would find estimates of $c^2$ that were not significant. In these cases we would commit ``type II errors.'' That is, we would not find a significant effect of the shared environment even though the value of $c^2$ in the population was truly greater than zero. Assuming we were using a $\chi^2$ test for 1 d.f. to test the significance of the shared environment, and we had decided to use the conventional 5% significance level, the probability of Type II error would be the expected proportion of samples in which we mistakenly decided in favor of the null hypothesis that $c^2 = 0$. These cases would be those in which the observed value of $\chi^2$ was less than 3.84, the 5% critical value for 1 d.f. The other samples in which $\chi^2$ was greater than 3.84 are those in which we would decide, correctly, that there was a significant shared environmental effect in the population. The expected proportion of samples in which we decide correctly against the null hypothesis is the power of the test.

Designing a genetic study boils down to deciding on the numbers and types of relationships needed to achieve a given power for the test of potentially important genetic and environmental factors. There is no general solution to the problem of power. The answers will depend on the specific values we contemplate for all the factors listed above. Before doing any power study, therefore, we have to decide the following questions in each specific case:

1. What kinds of relationships are to be considered?
2. What significance level is to be used in hypothesis testing?
3. What values are we assuming for the various effects of interest in the population being studied?
4. What power do we want to strive for in designing the study?

When we have answered these questions exactly, then we can conduct a power analysis for the specified set of conditions by following some basic steps:

1. Obtain expected covariance matrices for each set of relationships by substituting the assumed values of the population parameters in the model for each relationship.
2. Assign some initial arbitrary sample sizes to each separate group of relatives.
3. Use Mx to analyze the expected covariance matrices just as we would to analyze real data and obtain the $\chi^2$ value for testing the specific hypothesis of interest.
4. Find out (from statistical tables) how big that $\chi^2$ has to be to guarantee the power we need.
5. Use a simple formula (given below) to multiply our assumed sample size and solve for the sample size we need.

It is essential to remember that the sample size we obtain in step five only applies to the particular effect, design, sample sizes, and even to the distribution of sample sizes among the different types of relationship assumed in a specific power calculation. To explore the question of power fully, it often will be necessary to consider a number, sometimes a large number, of different designs and population values for the relevant effects of genes and environment.