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Note the ANCOVA is a combination of ANOVA and Regression.

I read that Quade's test assumes an equal distribution of the covariate amongst the groups. How strict is this assumption? I'm confused because my covariates are significantly associated with the independent variable (group), therefore violating the homogeneity of regression slopes, which is another reason why I thought I should not use ANCOVA in the first place. The second reason is I have a slight violation of homogeneity of variance, which I cannot get around with very unequal sample sizes.

Can you offer any advice on what test is appropriate? In short, I have three covariates (two continuous, one categorical), and the two continuous covariates are very significantly associated with the grouping. TIA!

I am looking at the variability of thickness measurements on a retinal scan. Two measurements are made on each scan and I am using the absolute difference of the measurements as the outcome variable. Each scan also has an associated disease/diagnosis, as well as what type of fluid is present. The predictors would be 1) baseline thickness of tissue A, 2) baseline thickness of tissue B, 3) type of fluid present, and 4) diagnosis/disease.

My primary interest is looking at type of fluid as the predictor, and baseline thickness and diagnosis as covariates, but my data violate the assumptions of homogeneity of variances and regression slopes.

Any advice would be greatly appreciated!

Based on the statistical properties of your outcome, we can advise you on the proper estimator. Note, heteroskedasticity shouldn't be of great concern (could be "fixed" with robust SEs). However, there are other important assumptions -- e.g., multicollinearity and normality of residuals -- that could present barriers for unbiased estimation.

P.S. If you come across the notion of over-dispersion and its "easy fix" with negative-binomial model -- don't believe that. Just estimate a few and compare the performance.