# Inference for quasibinomial GLM with LASSO penalty using selectiveInference package

I would like to carry out inference on a binomial LASSO model, but take into account the fact that my data are overdispersed and use the quasibinomial family instead.

R package selectiveInference, which does inference for LASSO models, only seems to support the binomial family though and not quasibinomial.

To get around this, I was wondering if it would be correct to adjust the z scores and p values returned by fixedLassoInf called using family="binomial" for overdispersion by dividing the z scores by the square root of the estimated dispersion coefficient of a quasibinomial GLM with the selected variables included? (or perhaps all variables included??) Any thoughts if this would be a correct procedure? If it is, I was also wondering then how I should recalculate/adjust the returned confidence intervals? Any thoughts?

[BTW, package hdi, which has a similar aim, also doesn't support quasibinomial, and I also couldn't readily see how that package could be interfaced with package glmmLasso - if that would be possible then overdispersion could perhaps be taken into account using an observation-level random effect; if anyone would know how to do this then let me know too]

The output I had for my data right now was

fixedLassoInf(x, y, beta, lambda, family = "binomial",
intercept=TRUE, alpha=0.1, type="partial")
# Var   Coef Z-score P-value LowConfPt UpConfPt LowTailArea UpTailArea
# 2  2.596  10.710       0     2.194    2.995       0.048      0.050
# 3  1.224  16.400       0     1.101    1.348       0.049      0.050
# 5  2.608  17.219       0     2.356    2.857       0.049      0.050
# 7  0.776  10.588       0     0.655    0.897       0.048      0.050
# 8 -1.857  -5.103       0     1.229    2.462       0.050      0.048

• This is an interesting question, and your approach seems reasonable, but as far as "correctness", you'd have to read the paper to see if their arguments for the Gaussian with unknown variance can be pushed through to a quasi-binomial. Or just run a simulation study... – Andrew M Apr 13 '17 at 19:11
• Can you please add a tag wiki for your new tag selectiveinference? – kjetil b halvorsen Apr 15 '17 at 16:37
• When you remove the penalty and fit a quasibinomial GLM with the selected variables, your point estimates will change. Without the penalty, the fitted values will lie closer to the observed responses, and it seems like your dispersion estimate might end up too low. – eric_kernfeld Apr 23 '17 at 20:43