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I am having difficulty interpreting the output for a quasibinomial model.

My first issue is that I have used the function 'autoplot' to test assumptions, and the normal Q-Q plot is skewed:

autoplot results

I am unsure whether or not it is okay to proceed with fitting the anova, or how to adjust my data if it is not ok to proceed.

Assuming that it was okay to proceed, when running the anova there is a significant effect of 'Group' (using a critical p-value of 0.05). However, there are no significant effects found in the output when using the summary function. How would I interpret this, or write this result in a report?


Efetphagoquasi <- glm(cbind(NumPhago,NumNotPhago)~Group*Day, data = Efet, family = "quasibinomial")

autoplot(Efetphagoquasi, smooth.colour = NA) 

summary(Efetphagoquasi)

Call:
glm(formula = cbind(NumPhago, NumNotPhago) ~ Group * Day, family = "quasibinomial", 
    data = Efet)

Deviance Residuals: 
     Min        1Q    Median        3Q       Max  
-10.2932   -4.5115   -0.4734    3.9493   10.7405  

Coefficients:
                     Estimate Std. Error t value Pr(>|t|)
(Intercept)          -0.25131    0.35544  -0.707    0.483
GroupTreatment       -0.03054    0.50317  -0.061    0.952
Day14                -0.10749    0.50472  -0.213    0.832
Day28                 0.30633    0.50079   0.612    0.544
GroupTreatment:Day14 -1.30657    0.78674  -1.661    0.104
GroupTreatment:Day28 -0.72701    0.71971  -1.010    0.318

(Dispersion parameter for quasibinomial family taken to be 24.87212)

    Null deviance: 1443.3  on 47  degrees of freedom
Residual deviance: 1157.4  on 42  degrees of freedom
AIC: NA

Number of Fisher Scoring iterations: 5

anova(Efetphagoquasi, test="F")


Analysis of Deviance Table

Model: quasibinomial, link: logit

Response: cbind(NumPhago, NumNotPhago)

Terms added sequentially (first to last)


          Df Deviance Resid. Df Resid. Dev      F  Pr(>F)  
NULL                         47     1443.3                 
Group      1  113.280        46     1330.0 4.5545 0.03871 *
Day        2  100.189        44     1229.8 2.0141 0.14613  
Group:Day  2   72.481        42     1157.4 1.4571 0.24445  
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

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Most of those plots don't make a lick of sense for a GLM, at least not for the binomial (or quasi-binomial) models. The "theoretical" quantiles are still drawn from a normal (0,1) distribution even when the prevalence of the outcome is very low (meaning that errors are likely to be skewed even after standardizing by variance).

Fitting the quasi-likelihood model is extra insurance against some of the more important (not depicted) modeling assumptions. See my post here about quasilikelihood models.

To your question about the ANOVA, it would help to print the actual output. But basically, the ANOVA for Group is a 3 degree of freedom test for the 1 group level effect as well as its interactions with the time fixed effects. It's not entirely uncommon to see non-significant coefficient terms, but significant nested tests (one would need to draw a strange hyperdimensional confidence interval to visualize how the 3 terms do not intersect the "null" 0,0,0 hypothesis.

You can just report them the way you normally would, e.g. "The phagocyte prevalence ratio was exp(\beta_1) at baseline (95% CI...), this ratio decreased by exp(\gamma_1) at day 14 (95% CI...) and by exp(\gamma_2) at day 28 (95% CI). A global trend test for group was performed, the p-value was X.XX see figure X.

Ideally plot the predicted effect over time along with 95% CIs, since the last sentence is too technical for non-statistical reviewers.

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