Repeated measure block design got significant block and block interactions. Each block analysis showed 2 out of 5 blocks, not normally distributed

Question

I have Per Capita fecundity of females from two population of Drosophila (evolved and ancestral-Population type) females at 5 different age points (age fixed factor), where same females were used for fecundity measurement. My unit of analysis for per capita fecundity at each age point(1,5,10,15 and20) comes by-counting the number of eggs laid by group of 10 females divided by number of females alive at the start of that day point. So basically its a fraction type data. This experiment was carried out with 5 independent replicate population of the two population type. Thus represents repeated measure (female fecundity measured at different age points)block design with 5 statistical blocks.

However, I got significant block effect and block interactions with other fixed effects, when I ran LMM under lme4 package (lmer) taking block as random factor. So we analysed each block seperately,therefore first checked for normality distribution of each block, 2 out of 5 blocks were not normally distributed (residual distribution was checked S-W test). Here's the qq-plot from one of those blocks (Block 1-W = 0.97244, p-value = 0.005062):

Other blocks showed better qq-plots, although the Shapiro-Wilk test still suggested non-normality (Block 5-W = 0.9795, p-value = 0.02518):

So whether can i still go for parametric test with this much deviation from normality can be accepted or should i go for non-parametric test for these 2 blocks. I thought of doing GLMM (glmer) but I am not aware of what distribution would fit my data type? Is it Poisson or quasipoisson or Gamma?

Answer 1

With positive count data in both the numerator and the denominator of your fecundity measure, it might not be surprising that residuals don't follow a normal distribution. Poisson count data have a variance equal to the mean, necessarily smaller in magnitude at low counts and larger at high counts. That might explain the heavy-tailed nature of your qq plots. Here's a qq plot for a lm() fit of 450 Poisson-distributed Y values versus corresponding X mean values ranging from 1 to 15; code below. It has the same overall shape as yours.

Even if you eventually present the results in terms of that measure, your statistical analysis might best be done directly at the count level. That means starting with a Poisson generalized linear model (log link) and working from there, perhaps moving to a quasi-Poisson or negative binomial model.

You would model the actual egg counts, using the log of the number of females as an offset for this rate-type analysis. Time and population type would still be fixed effects (with their interaction, which seems to be of interest), and the "blocks" (as I understand, 10 total, 5 for each population type) treated as random effects. The R DHARMa package provides useful tools for residual diagnostics.

One potential problem in the design: this assumes that per-capita fecundity is independent of the number of females alive at the start of each time point. If crowding affects fertility, then neither your fecundity index nor using the log of the number of females as an offset (fixed regression coefficient of 1) would be valid. Check that mortality is the same for the 2 population types at a minimum, and see if there's evidence of non-proportionality of egg counts against number of females under otherwise similar circumstances.

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Code for the plot

> myX <- rep(1:15,30)
> length(myX)
[1] 450
> set.seed(1234)
> myY <- rpois(450,myX)
> poisDF <- data.frame(x=myX,y=myY)
> plot(lm(y~x,poisDF))