Whether there is significant difference between two gender groups

Question

A data set contains 5 columns:

There are 16 people in the dataset(using subjectID to identify them). Each year, when these people were ages 11, 12, 13, 14, and 15, they provided us a score $y$. We now want to know whether there is a significant difference between male and female people (in their average $y$ score). How could I do? I found that this dataset does not meet the normality assumption. And in "Male" group, say subject 45 will appear 5 times, subject 268 will appear 5 times as well... Could I just use the mean of this "Male" group to make comparison with the mean of "Female" group?

And I also wonder if I can use a fixed effect model to compare these two groups:

We can see the difference between "Male" group and "Female" group is 0.13521, and it is not significant at 5% level as its p-value=0.228 (cannot reject $H_0$:"genderM"=0). So could I just say the difference is not significant?

Answer 1

Each measurement occasion is nested with an individual. There are a variety of ways to handle this issue: mixed effects models and cluster-robust standard errors are a few of them.

In a mixed effect model (aka multilevel model), you model each outcome value as a function of an intercept, (centered) age, and a measurement residual. You then model the intercept (and maybe also the slope of age) as a function of gender and an intercept (and slope) residual. The coefficient on gender is the effect of gender on the average outcome for an individual. The models look as follows:

$$y_{it} = \beta_{0t}+\beta_{1i}X^{age}_{it}+\epsilon_{it}\\ \beta_{0i}=\gamma_{00}+\gamma_{01}X^{gender}_{i}+u_{0i}\\ \beta_{1i}=\gamma_{10}+\gamma_{11}X^{gender}_{i}+u_{1i} $$ where $i$ indexes unit-ID and $t$ indexes measurement occasion for individual $i$. The third line can be replaced by $\beta_{1i}=\gamma_{10}+u_{1i}$ if you don't think the effect of age depends on gender but varies for each individual or $\beta_{1i}=\gamma_{10}$ if you think the effect of age is constant for all individuals.

To use cluster-robust standard errors, simply run the model $y_j=\beta_0+\beta_1X^{age}_{j}+\beta_2X^{gender}_{j}+\beta_3X^{age}_{j}X^{gender}_{j}+\epsilon_j$ where $j$ indexes each row in the data set, and then request a cluster-robust standard error with unit-ID as the clustering variable. If age is centered, $\beta_2$ will be the effect of gender on the outcome.