Unbalanced design effect

Question

In my project, I am looking at the relationship between X and Y by the grouping factor "gender".

This is at two stages. The first is to see how XY correlate in each subgroup (Pearsons r) and the second is to see if there is a gender difference in the XY correlation (general linear model).

I am using SPSS and the scatter plot shows a linear relationship.

My sample comprises 60 males and 95 females.

Should I be correcting the effect of unbalanced design (i.e. different numbers in each subgroup) for my analyses?

I know Sum of Squares Type III corrects it for GLM but I am unsure about the bivariate correlation. (Does it matter for a correlation analysis?)

EDIT

In response to the comments below:

1. I have used Probability Proportional to Size (PPS) cluster sampling and have taken into consideration the population distribution. For example, if the chosen entity has 700 members and half of them are females, I have strived to have this reflected in my sample.

2. I have inadvertently simplified the question above. I am testing two hypotheses. The first is to ascertain the relationship between X and Y. It is a simple bivariate correlation analysis. (Please ignore the first stage mentioned in the question above!). The second is to ascertain the effect of gender on the X-Y relationship.

3. I have created the interaction term and I have included the main effects in my model.

My question again:

1. Should I 'correct for' (unsure if this is the right terminology) for the different subgroup sizes in the X-Y correlation analysis? This is a simple bivariate correlation analysis. (i.e it has nothing to do with subgroup analysis by gender)
2. Should I correct for the different subgroup sample sizes when doing the subgroup analysis by gender?

Answer 1

@whuber raises an important distinction--you should think about that. It might also help if I clarify what appear to be some misunderstandings in the question.

First, I wouldn't say that type III SS correct for an unbalanced design. I have discussed what the different types of Sums of Squares are and how they work, mostly here, but with some supplementary information here. It may be worth your time to read them. Briefly, if your covariates are correlated, then when you perform hypothesis tests, type III SS does not use all of the information you have. Type I SS allows you to use all of the SS. In addition, the fact that you have differing numbers of men and women does not mean that sex and your predictor, $X$, are correlated. Since it sounds like you have observational data, they likely are correlated, but they just as likely would be if the $n_g$'s were equal. As a result, I would recommend you use type I SS, which is implemented in SPSS by using the feature that lets you enter your predictors by block, and then checking the 'F-change' test.

The effect of unequal $n_g$'s is to reduce the power of the hypothesis test comparing the groups. For a given $N$, and effect size, $d$, you maximize your power by having each $n_g=\frac{1}{2}N$. (An analogy that helps me is to think about the areas of rectangles, if the length of the perimeter of a rectangle is fixed, you maximize the area by making the sides equal--i.e., square.)

Finally, it is not generally the best strategy to conduct your analysis in stages as you describe. I would recommend that you just fit one full model with $Y$, $X$, and sex. Since it appears that your goal is to determine if sex makes a difference in the $X$ - $Y$ relationship, you should include a sex *$X$ interaction term, and keep it in the final model whether it is significant or not.