The way ANOVA works -- well, I don't know how it works, but what it does is: It gives you a percentage for every variable that shows you how much of your hypothesis is directly attributable to that variable. For example, (and these numbers are made up) let's say that you're trying to figure out why some people do well on a test. Your variables might be Time Studied, IQ, GPA, etc.... ANOVA goes through each variable and tells you 1) if it actually affects the outcome and 2) how much it affects it. It might say that yes, Time spent studying affects your grade on a test, and it accounts for 45% of how well you do, or something like that.
In this situation, there would certainly be a pile of variables tossed in, and all that it would take for them to make their claim is for ANOVA to tell them that gender of children IS statistically significant. It might only account for 5% of the reason parents get divorced, but it does have an effect.