Tl;dr: I highlighted my primary questions at the bottom of the post.
I'm having difficulty with the workflow for a multiple logistic regression for my (genomic) data and I'm starting to think this comes from a fundamental misunderstanding of GLMs.
To elaborate, I have a series of proportions for a large number of gene features which can (essentially) be divided into respective "successes" (Cnt1) and "failures" (Cnt2) (see mock dataframe).
Pair Cnt1 Cnt2 Gene Tissue A1:B1 23 201 A T1 A1:B1 43 565 B T1 A1:B1 65 123 A T2 A1:B1 76 678 B T2 A1:B1 63 675 A T3 A1:B1 77 618 B T3 A2:B2 34 143 A T1 ...
Foremost, I'd like to see how pair status (Gene A vs Gene B) influences the proportion of Cnt (e.g. pr(Cnt1)).
For this, I reasoned this model would suffice:
pr(Cnt1) ~ Gene
or in R syntax,
glm(cbind(Cnt2,Cnt1) ~ Gene, family= binomial(link="logit"), data = x)
However, I need to identify pairs that exhibit strong differences in pr(Cnt1) between Gene A and Gene B but also differ between T1, T2, or T3.
I've tried to look at the interaction between Gene:Tissue (see code below) but this where I start to get lost conceptually.
glm(cbind(Cnt2,Cnt1) ~ Gene*Tissue, family= binomial(link="logit"), data = x)
An example summary of the coefficients from one pair:
Estimate Std. Error z value Pr(>|z|) (Intercept) 3.42965051 0.1122062 30.5656166 3.506965e-205 GeneB 0.14472442 0.1356404 1.0669714 2.859848e-01 tissueT2 -0.18315952 0.1963746 -0.9327047 3.509724e-01 tissueT3 0.06658153 0.1481779 0.4493351 6.531900e-01 GeneB:tissueT2 -0.27579716 0.2330609 -1.1833697 2.366627e-01 GeneB:tissueT3 -0.87463736 0.3661340 -2.3888451 1.690143e-02
It was recommended that I run an Anova (i.e. via "car") on the fit for each pair. This is where whatever tenuous understanding I have falls apart. Why do I need to run the anova on these models? I'd guess it's to test the fit of the model but is it truly essential in this scenario?
The anova output reduces all the predictor variable interactions to simply the Gene:Tissue interaction.
Given the fact that I'm (presumably) interested in pairs for which the variation in pr(Cnt1) is explained by an interaction between Gene and at least one of the tissues, could I just use the glm output?
Additionally, why is the intercept so low?
(1) Why should one run an anova after fitting a logistic regression and what is it's purpose in this context?
(2) Can I use the output of the model summary, and additionally, what does the extremely low intercept in this example mean?
I understand that is an incredibly convoluted question but some help and/or (if possible) resources that might answers would be greatly appreciated!