Top variables from LASSO not significant in regular regression I have a table with 65 records and 1000 variables for which I use LASSO to perform feature selection.  Then, in order to quantify the relative impact of the variables, I regress the response only on these variables using traditional linear regression.  
Out of the 25 variables, only about 5 are statistically significant.  How would I interpret the fact that LASSO tells us that 25 of these variables out of 1000 are important, but linear regression then indicates that only 5 of these are significant? 
 A: Kodiologist rightfully points out that there is no reason whatsoever to believe these concepts to be related.
Richard Hardy, in the comments, points out the major statistical flaw in your procedure.  To elaborate a bit, the cross validation procedure you did to select the optimal value of the regularization parameter is itself also subject to noise.  If you bootstrap your data and do the entire procedure many times, you will find that your choice of the regularization parameter is not consistent.  The rub is that your confidence intervals for your estimated parameters need to reflect this variation as well, and it is this source of variation that Richard points out is causing your intervals to be too small.
One simple way to convince yourself of this is to consider the predictors that LASSO did not include in the model.  In your subsequent linear regression you left them out.  You have essentially then said, with 100% certainty, that these parameters are zero.  The others that you left in the model you have ascribed some variation to.  Why the difference?  Do you really believe that the true parameters for the selected away variables are zero, with 100% confidence?
Thankfully, this thought experiment also points to a solution.  If you would like to use LASSO and also estimate the standard errors of the final parameters, you can use a bootstrap procedure.
for each bootstrap sample B from your training data
    split B into cross validation folds
    for each cross validation fold
        fit LASSO for each considered regularization parameter
        get estimate of out of sample error for each regularization parameter
    find optimal regularization parameter for training data B
    fit LASSO model with the optimal parameter on the entire sample B
    record the estimated parameters from the full LASSO model on B
return the variance of the estimated parameters over the bootstrap samples

This gives you a fair record of the variance in your estimated parameters.
A: Although hasty glosses of lasso regularization and statistical significance might suggest that they do the same thing, they do very different things. The statistical significance of a coefficient tests a null hypothesis about that coefficient given a smaller model as a background assumption. Lasso regularization, on the other hand, estimates coefficient values while minimizing a penalty term based on the sum of the absolute values of the coefficients. These very different goals mean that there's no reason these methods should agree about anything.

in order to quantify the relative impact of the variables, I regress the response only on these variables using traditional linear regression.

Why not use the coefficients from the lasso model that you already fit? Isn't that the model you're trying to assess?
