4 Power for the continuous case

A common question in genetic research concerns the ability of a study of twins reared together to detect the effects of the shared environment. Let us investigate this issue using Mx. Following the steps outlined above, we start by stipulating that we are going to explore the power of a classical twin study -- that is, one in which we measure MZ and DZ twins reared together. We shall assume that 50% of the variation in the population is due to the unique environmental experiences of individuals ($e^2 =0.5$). The expected MZ twin correlation is therefore 0.50. This intermediate value is chosen to be typical of many of the less-familial traits. Anthropometric traits, and many cognitive traits, tend to have higher MZ correlations than this, so the power calculations should be conservative as far as such variables are concerned. We assume further that the additive genetic component explains 30% of the total variation ($a^2 = 0.30$) and that the shared family environment accounts for the remaining 20% ($c^2=0.20$). We now substitute these parameter values into the algebraic expectations for the variances and covariances of MZ and DZ twins:

Total variance	=	$a^2 + c^2 + e^2$	=	$0.30+0.2+0.5$	=	$1.00$
MZ covariance	=	$a^2 + c^2$	=	$0.30+0.2$	=	$0.50$
DZ covariance	=	$.5a^2 + c^2$	=	$0.15+0.2$	=	$0.35$

In Appendix  we show a version of the Mx code for fitting the ACE model to the simulated covariance matrices. In addition to the expected covariances we must assign an arbitrary sample size and structure. Initially, we shall assume the study involves equal numbers, 1000 each, of MZ and DZ pairs. In order to conduct the power calculations for the $c^2$ component, we can run the job for the full (ACE) model first and then the AE model, obtaining the expected difference in $\chi^2$ under the full and reduced models just as we did earlier for testing the significance of the shared environment in real data. Notice that fitting the full ACE model yields a goodness-of-fit $\chi^2$ of zero. This should always be the case when we use Mx to solve for all the parameters of the model we used to generate the expected covariance matrices because, since there is no sampling error attached to the simulated covariance matrices, there is perfect agreement between the matrices supplied as ``data" and the expected values under the model. In addition, the parameter estimates obtained should agree precisely with those used to simulate the data; if they are not, but the fit is still perfect, it suggests that the model is not identified (see Section 5.7) Therefore, as long as we are confident that we have specified the structural model correctly and that the full model is identified, there is really no need to fit the full model to the simulated covariances matrices since we know in advance that the ``$\chi^2$" is expected to be zero. In practice it is often helpful to recover this known result to increase our confidence that both we and the software are doing the right thing. For our specific case, with samples of 1000 MZ and DZ pairs, we obtain a goodness-of-fit $\chi^2_4$ of 11.35 for the AE model. Since the full model yields a perfect fit ($\chi^2_3=0$), the expected difference in $\chi^2$ for 1 df -- testing for the effect of the shared environment -- is 11.35. Such a value is well in excess of the 3.84 necessary to conclude that $c^2$ is significant at the 5% level. However, this is only the value expected in the ideal situation. With real data, individual $\chi^2$ values will vary greatly as a function of sampling variance. We need to choose the sample sizes to give an expected value of $\chi^2$ such that observed values exceed 3.84 in a specified proportion of cases corresponding to the desired power of the test. It turns out that such problems are very familiar to statisticians and that the expected values of $\chi^2$ needed to give different values of the power at specified significance levels for a given df have been tabulated extensively (see Pearson and Hartley, 1972). The expected $\chi^2$ is known as the centrality parameter ($\lambda$) of the non-central $\chi^2$ distribution (i.e., when the null-hypothesis is false). Selected values of the non-centrality parameter are given in Table 7.1 for a $\chi^2$ test with 1 df and a significance level of 0.05.

Table 7.1: Non-centrality parameter, $\lambda$, of non-central $\chi^2$ distribution for 1 df required to give selected values of the power of the test at the 5% significance level (selected from Pearson and Hartley, 1972).

Desired Power	$\lambda$
0.25	1.65
0.50	3.84
0.75	6.94
0.80	7.85
0.90	10.51
0.95	13.00

With 1000 pairs of MZ and DZ twins, we find a non-centrality parameter of 11.35 when we use the $\chi^2$ test to detect $c^2$ which explains 20% of the variation in our hypothetical population. This corresponds to a power somewhere between 90% ( $\lambda =10.51$) and 95% ($\lambda=13.00$). That is, 1000 pairs each of MZ and DZ twins would allow us to detect, at the 5% significance level, a significant shared environmental effect when the true value of $c^2$ was 0.20 in about 90-95% of all possible samples of this size and composition. Conversely, we would only fail to detect this much shared environment in about 5-10% of all possible studies. Suppose now that we want to figure out the sample size needed to give a power of 80%. Let this sample size be $N^*$. Let $N_0$ be the sample size assumed in the initial power analysis (2000 pairs, in our case). Let the expected $\chi^2$ for the particular test being explored with this sample size be $\chi^2_E$ (11.35, in this example). From Table 7.1, we see that the non-centrality parameter, $\lambda$, needs to be 7.85 to give a power of 0.80. Since the value of $\chi^2$ is expected to increase linearly as a function of sample size we can obtain the sample size necessary to give 80% power by solving:

$$N^* = \frac{\lambda}{\chi^2_E}N_0$$ (44)

$$= \frac{7.85}{11.35}\times 2000$$

$$= 1383$$

That is, in a sample comprising 50% MZ and 50% DZ pairs reared together, we would require 1,383 pairs in total, or approximately 692 pairs of each type to be 80% certain of detecting a shared environmental effect explaining 20% of the total variance, when a further 30% is due to additive genetic factors. It must be emphasized again that this particular sample size is specific to the study design, sample structure, parameter values and significance level assumed in the simulation. Smaller samples will be needed to detect larger effects. Greater power requires larger samples. Larger studies can detect smaller effects, and finally, some parameters of the model may be easier to detect than others.