6 Exercises

1. Change the example program to obtain the expected $\chi^2$ for the test for additive genetic effects. Find out how many pairs are needed to obtain significant estimates of $a^2$ in 80% of all possible samples.
2. Explore the effect of power of a particular test of altering the proportion of MZ and DZ twins in the sample.
3. Show that the change in expected $\chi^2$ is proportional to the change in sample size.
4. Obtain and tabulate the sample sizes necessary to detect a significant $a^2$ when the population parameter values are as follows:
   

   $a^2$	$c^2$
   0.10	0.00
   0.30	0.00
   0.60	0.00
   0.90	0.00

   In what way do these values change if there are shared environmental effects?
5. Show that with small sample sizes for the number of pairs in each group, some bias in the chi-squared is introduced. Consider whether or not this may be due to the $n - 1$ part of the maximum likelihood loss function (Equation  on page ).