I think you can still use a t-test, but you have to keep in mind that what you call "an effect" can also appear in the variance. As a consequence, I think you should add a test to your t-test to measure difference in the variances.
Simple example, consider the case when $X$ is a $mathcal{N}(0,1)$ and $Y$ is a $mathcal{N}(mu,sigma)$ and you want to say if $Y$ is the result of "an effect" or it is not.
You can look at a t-test like that:
$frac{|bar{X}-bar{Y}|}{sqrt{var(X)+var(Y)}}$. It works fine even with different variances but it will only check for differences in the mean (i.e $muneq 1$). If you want to test something on the variances also, you will have to test simultaneously (take care of the level of your test) the equality of the mean and the equality of the variances.
More General setting If an effect can appear in a more complex way, you may be obliged to concider a non parametric godness of fit testing procedure to which belong (I guess) the rank test.
Hop this help !
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