After using Lasso regression to get the best reduced model, can we apply General Linear Hypothesis Test (GLHT) to test whether the reduced model is enough to replace the full model or not?
The purpose of LASSO is to make tests like this unnecessary.
However, I would advise against using any automated procedure blindly. LASSO is a very good method of getting an appropriately penalized regression using only statistical criteria. But there are other reasons to include a variable in a model:
1) A small effect is interesting - if the literature or theory suggests a large effect and you find a small one, that is interesting. But a variable with a small effect will be removed by any automated program.
2) The variable is of great theoretical interest - if your study is about the effect of X on Y then you should often include X in the model and report on that model, even if standard methods say to remove it.
3) The effect is in a higher level interaction. - It is very rare that you want to include an interaction but not the main effects
4) The variable has a large effect on other parameters - such things are not always found by LASSO or any other method.
The purpose of LASSO is to make tests like this unnecessary.
However, I would advise against using any automated procedure blindly. LASSO is a very good method of getting an appropriately penalized regression using only statistical criteria. But there are other reasons to include a variable in a model:
1) A small effect is interesting - if the literature or theory suggests a large effect and you find a small one, that is interesting. But a variable with a small effect will be removed by any automated program.
2) The variable is of great theoretical interest - if your study is about the effect of X on Y then you should often include X in the model and report on that model, even if standard methods say to remove it.
3) The effect is in a higher level interaction. - It is very rare that you want to include an interaction but not the main effects
4) The variable has a large effect on other parameters - such things are not always found by LASSO or any other method.
I also agree with Peter. One of the advantage of Lasso is that it leads to variable selection, ends to a simple and interpretable model. However, the choice of tuning parameters should be made with care.
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