| Familial correlations measure the resemblance between family members and are used in many fields of study including epidemiology, genetics, heredity, and psychology. Here, an analysis of familial correlations where male and female children of the same family can have different correlations in the unequal family size case is presented. First, three likelihood based tests, namely the likelihood ratio test, Rao score test, and Wald test, and two more asymptotic tests which use Srivastava's estimator of the intraclass correlation coefficient are considered to test the null hypothesis of equality of the intraclass correlation coefficients when families have unequal numbers of children. These methods are implemented on Galton's data set on human stature and a simulation study is conducted to compare the different tests. The simulations show the alternative tests to be better or comparable to the likelihood based tests in certain situations. Additionally, testing the equality of interclass correlations from g independent populations is considered where male and female children of the same family can have different correlations and the family sizes within populations are unequal. For this problem, the likelihood ratio test is compared with two asymptotic alternative tests using Srivastava's estimator of the interclass correlation coefficient that are easier to compute. Simulations are used to study the size and power of these tests. Based on the simulation study, the alternative tests perform well when compared to the likelihood ratio test. Finally, the likelihood ratio test is compared with an asymptotic alternative test of interclass correlation for testing the equality of two parent interclass correlations coefficients, namely, parent-son and parent-daughter interclass correlation coefficients, within families from a single population with unequal family sizes. Both tests are illustrated on Galton's data set on human stature and the results of a simulation study are shown. The results show the alternative test to perform better for certain cases. |
