Loss of predictive power of polygenic risk score when dataset contains missing variants
Which software package around you using, PRSice? Generally missing values comprising half the total observations are extremely difficult to calculate using regression analysis (or many statistical analyses for that matter.
One way forward is to include (an) example(s) of all 40 variants into your data set and perform the analysis with and without the added data, then compare this against a data set where all 40 variants are observed and your samples are excluded. You might find the missing values have low probability of PRS and this would then explain your results.
Missing analysis in regression is a specialist area of statistics and is not for the faint hearted, so a work-around is the best way forward.