How to interpret two way ANOVA repeated measurements results?

I measured the percentage of individuals that crossed a barrier at six different time points for four different genotypes (e.g for replicate 1 of genotype 1: at 0h 0/20, at 0.5h 19/20 etc. animals crossed the barrier). I have a different number of repeats between the genotypes (for demonstration the screenshots only shows data from two genotypes, 8 replicates). I want to analyze statistically if there is a difference between the two genotypes in general (so over all six time points together) meaning if one genotype is more or less likely to cross the barrier than the other in the span of the six times (so not at individual times).

For more context, the barrier is made out of an aversive substance that the control individuals do not like. With increased time the control individuals get hungry and start searching for food so the response to this aversive substance reduces and they cross the barrier.

The time points represent hours, so I measured the number of individuals that crossed this barrier at 0.5, 1, 2, 3 and 4 hours but for the same individuals.

So for example if 1 individual out of 20 crossed the barrier at 0.5 h, 5% crossed the border. One time point later I measured the same worms again. So now the individual that crossed the barrier at 0.5 h and 3 more individuals are on the other side, so 4 individuals were counted (20%) etc.

I want to investigate other individuals with different genotypes than that of the control and see if their behaviour to this aversive barrier differs, so perhaps they can't sense it at all so it would be expected that more individuals cross the barrier or start to cross it earlier than control or the opposite effect is seen that fewer individuals cross the barrier or they cross it later showing increased sensitivity to the barrier.

Individuals did not go back to the other side so this movement is only in one direction.

The data provided is only imaginary.

I performed a two way ANOVA repeated measurements but I'm not sure which p value answers my question. Can I just take the p value for Genotype? (my column value)? Or should I perform another statistical test?

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• Welcome to Cross Validated! Please say more about the "barrier" whose crossing you are using as an outcome, and what the time points represent. In particular: Are the same animals evaluated at all time points? if an animal has crossed the barrier at one time point, is it counted as crossed at all later times? With that information provided by editing the question (comments are easy to overlook and can be deleted) you should be able to get some advice on a better way to proceed than ANOVA. With fractions of successes as your outcome, something like binomial regression will be more reliable.
– EdM
Commented Jul 10 at 14:47
• I edited my post, thank you for your input. Commented Jul 10 at 15:50

One problem in working with percentages as your outcome is that it doesn't take into account the number of observations. After all, you would believe more strongly that 10% is closer to the true proportion if it was based on 100 out of 1000 individuals instead of just 1 out of 10.

Another is that percentages often don't meet the assumptions needed for ANOVA. The errors in estimates are a good deal different at the extremes near 0% and 100% than they are around 50%. ANOVA assumes that errors are constant across the range of predictions.

A third potential problem here is that you are evaluating the total percentage that have crossed as a function of time. That value necessarily increases with time, and is limited at 100%. It can be more informative to evaluate the probability of crossing within each time interval, based on the number of individuals who hadn't yet crossed at the beginning of the interval and the number of those who had crossed by the end.*

It's best to use a binomial regression, a type of generalized linear model, for this type of data. That takes into account the actual numbers involved and the way that the underlying variance changes as a function of the proportions. Logistic regression is one type of binomial model, and should be available in your software. You would set up the model similarly to your ANOVA, with time (as a factor variable), genotype, and their interaction as predictors. You also might include replicates (included as "Subject" in your ANOVA summary) if you expect batch effects, but that can be tricky. You might use a "mixed model" that includes replicate as a "random effect" to adjust for possible baseline differences beyond those due to the combination of time and genotype.

Depending on your software, you might have to re-format the data. You might need to have 1 row for each combination of time interval and genotype, with one column indicating the time interval and other columns for: the number who hadn't yet crossed the barrier at the beginning of the interval, the number who crossed during that interval, and the genotype. If you use a mixed model, you would have such separate rows for each replicate and a column indicating the replicate.

The interaction coefficient (similar to Time x Genotype in your ANOVA summary) would provide a test of whether the changes in crossing probability over time differed among the genotypes. You would then use post-modeling tools to evaluate the differences further. That's not really different from ANOVA: the ANOVA only tells you that there are some differences, but with more than 2 genotypes it doesn't tell you which particular differences are significant.

*What you have can be thought of as a simple survival model. You start with a number of individuals, none of whom have experienced the event of interest (crossing from the starting side to the other side of the barrier). Time progresses, and some individuals experience the event. Once an individual has experienced the event, that individual is no longer "at risk" for that event. That's conceptually the same as evaluating deaths over time.