This is the simplest repeated measures ANOVA model if we treat it as a univariate model:
$$y_{it} = a_{i} + b_{t} + \epsilon_{it}$$
where $i$ represents each case and $t$ the times we measured them (so the data are in long form). $y_{it}$ represents the outcomes stacked one on top of the other, $a_{i}$ represents the mean of each case, $b_{t}$ represents the mean of each time point and $\epsilon_{it}$ represents the deviations of the individual measurements from the case and time point means. You can include additional between-factors as predictors in this setup.
We do not need to make distributional assumptions about $a_{i}$, as they can go into the model as fixed effects, dummy variables (contrary to what we do with linear mixed models). Same happens for the time dummies. For this model, you simply regress the outcome in long form against the person dummies and the time dummies. The effect of interest is the time dummies, the $F$-test that tests the null hypothesis that $b_{1}=...=b_{t}=0$ is the major test in the univariate repeated measures ANOVA.
What are the required assumptions for the $F$-test to behave appropriately? The one relevant to your question is:
\begin{equation}
\epsilon_{it}\sim\mathcal{N}(0,\sigma)\quad\text{these errors are normally distributed and homoskedastic}
\end{equation}
There are additional (more consequential) assumptions for the $F$-test to be valid, as one can see that the data are not independent of each other since the individuals repeat across rows.
If you want to treat the repeated measures ANOVA as a multivariate model, the normality assumptions may be different, and I cannot expand on them beyond what you and I have seen on Wikipedia.
The linear regression model gives us a prediction that theoretically can take on any value in a continuous range when the regressor input is changed. In your data with the two categorical regressors, each of which can take on only two values, the two-way ANOVA (reaped measures or not) will give us only four predictions depending on the input combination of values of these two categories. These four predictions won't match the observed values in your sample. The differences between those four predictions and the observed data are basically the residuals.
Yes, for the two-way repeated measures ANOVA (like for the regular two-way ANOVA), those residuals must be normally distributed. In other words, the dependent variable must be normally distributed within the categorical combo "bins" (there are four of them in your dataset A∩G1, B∩G1, A∩G2, B∩G2).
Best Answer
They do not generalize each other.
The repeated measures ANOVA is a general data analysis technique for longitudinal or panel data where many observations are drawn within individuals or clusters. A random effect, usually an intercept, accounts for the correlated errors so that appropriate estimation and inference can be achieved. The random intercept can be thought of as handling hundreds of unknown covariates common in clusters in an empirical way.
On the other hand, the paired t-test is actually just a t-test, where you exploit the cleverness of the design, having two observations within each cluster, and subtract one from the other, again so that the mean differences are independent and heteroscedastic.