Pooling time points to perform a Fisher test?


I am currently trying to analyse an experiment to assess if rats with cerebral malformations have a sensitivity deficit on one of their front paws. I put 2 sticky dots on their front paws then look which side they turn to first. I repeated this over 4 trials.

So my data look like that (with "n" being the number of rats contacting a dot on the left side first or on the right side first)

What I want to know is if the proportion of rats turning to the left side first is different in groups A and B. In that case, could I pool all time points and perform a Fisher test or would this be incorrect?

I have tried performing 2-way ANOVA on Prism 6 but realized this was not appropriate as it would average both "left" and "right" subcolums.

If anyone knows how I could treat these data properly, it would be really helpful.
Thanks to everyone who will have taken time to read this :)
Does each rat do four trials or do you use new rats each time?
If each rat does four trials does your data identify the response for each individual rat in each trial?
Each rat does 4 trials and I have 10 rats/group. So my understanding is that I am not supposed to pool the trials because they are not independant events, right? But then, I have no idea what the proper test to use would be. Any idea?

Edit: Sorry I did not see the second part of your post. Yes, the data identify the response for each individual rat in each trial, i.e n1 would be the number of rats which turned left first in trial 1 and n2 would be the number of rats which turned right first in trial 1, with n1+n2 = 10 (the total number of rats in Group A)
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Yes. You're right. You can't just pool the groups because they are not independent.
The structure of your data suggests using a General Linear Model with a logistic response. You would need to code each of your 80 results as 0 for Left and 1 for Right and do a repeated measures logistic GLM Group x Trial x Rats with Rats nested within Group. This is quite complicated.
However, no matter what you do, the nesting nature of Rats in Group will mean that you will always be stuck with effectively one number per rat, or 10 numbers per group. You probably aren't particularly interested in the Trial effect, so I suggest you work out the percentage of Rights for each rat, giving 10 numbers per group. These numbers will range through 0, 25%, 50% to 100% for each rat. These numbers won't be normal so a t test is probably out (although you try it and look at the residuals). Try a Mann-Whitney test between the groups, or perhaps a permutation test.
This will be almost as good as the logistic GLM, and it is much easier to do, interpret and explain.