While demonstrates the intuitively compelling PRE principle, there are problems with it. There is no way to test whether any value of lambda shows a particularly strong or weak relationship between variables; it can take different values depending on whether you set up the dependent variable in the rows or the columns; and it can be very low, even when there is an intuitively clear association between nominal variables. Lambda for table 21.4, for example, is just 0.10—that is, if you guess that all white families with children in the United States have two parents and that all black families have one parent, you make 10% fewer errors than if you guess that all families have two parents.

With bivariate data on nominal variables, many researchers use x^{2} (chi-square). Chi- square tells you whether or not a relation exists between or among variables and it tells you the probability that a relation is the result of chance. But it is not a PRE measure of correlation, so it doesn’t tell you the strength of association among variables.

The principal use of x^{2}, then, is for testing the null hypothesis that there is no relation between two nominal variables. If, after a really good faith effort, we fail to accept the null hypothesis, we can reject it. Using this approach, we never prove anything using statistical tests like x^{2}. We just fail to disprove things. As it turns out, that’s quite a lot.