confoundingspsst-test
In an article I found recently they were able to compute the difference in means of two groups (presumably with a t-test) while adjusting for confounders; they called this the aDiff (adjusted difference).
Now I understand that you can use linear regression for continuous variables when you want to adjust for confounders, but with linear regression you are not comparing means.
I assume you're trying to estimate the causal effect of $x_1$ on $y$, rather than just trying to predict $y$. In general, to find a correct conditioning set (if one exists), you need to know the causal relationships among the variables. Here are some examples to illustrate why:
In this case, the true effect of $X_1$ on $Y$ is zero. However, there are two unobserved confounders $U_1$ and $U_2$, and one observed confounder $X_2$. If we had observed $U_1$ and $U_2$ we could condition on all three confounders and get an unbiased estimate. However, since we only observed $X_2$ we are stuck. If we don't control for $X_2$ it will confound our estimate of the effet of $X_1$ on $Y$. But if we do control for it, we induce an association between $U_1$ and $U_2$, and therefore between $X_1$ and $Y$.
There are many cases in which you should not control for a particular covariate. If you don't know the causal structure, you may accidentally bias your estimate by controlling for the wrong set of covariates. In this case you can apply a causal structure learning algorithm as a first step, before you try to estimate the causal effect of $x_1$ on $y$.
You could use linear regression. t-tests and anova are simply special cases of the general linear regression model.
You say: " However I thought that linear regression was only for continuous variables .... ". That is not right, you can include categorical predictors, you must just code them correctly, using dummy variables. If you do not know what that means, you can read about them in various posts here.
For instance, the post Coding categorical variables for regression have examples, and there are many others. Just use site search!
Best Answer
Independent t-test is just a special case of linear regression. Given an outcome y and a binary indicator d, where d=0 for group A and d=1 for group B:
$y = \beta_0 + \beta_1 d$
The intercept $\beta_0$ is the mean of group A, the combined term $\beta_0 + \beta_1$ is the mean of group B. The p-value for $\beta_1$ then tests if the difference is different from zero. You'll see the p-value of this regression is identical to the p-value of the t-test variant.
Now, to adjust the mean for another covariate, x, the model simply becomes:
$y = \beta_0 + \beta_1 d + \beta_2x$
Now, $\beta_1$ is the so-called adjusted difference between groups A and B.