Combining Biological Replicates I am doing an experiment where I track the behavior of larval zebrafish of 3 genotypes during a photomotor assay. The larvae are a result of a heterozygous cross, so following the experiment I genotype each fish to determine whether that data came from a wildtype, het or mutant. The tracking software provides measurements including distance moved, velocity and well location.
Due to constraints with our tracking system, I can only run between 24-48 fish at a time. Because of this, I perform the experiment several times to increase n. When I repeat the experiment, it is done with different larvae on a different day - trying to keep as many factors the same (including time of day, amount of liquid in the wells etc.)
Previously, our lab has normalized the data to the controls of each replicate and then combined the data. To do this, total distance moved by mutant larvae was divided by the average of total distance moved for the controls of that experiment, this was also done to the controls. The "normalized" values from multiple replicates were then combined. I don't think this is the most rigorous way to analyze these data.
I am playing with lots of different ideas, including linear mixed models with Experiment # as a random effect (I recognize this may be an issue as I usually only have 3 replicates).
Any suggestions? Note: I am new to statistics and have been feeling very lost while trying to balance all of the considerations of various models.
 A: It's typically best to stay as close as possible to the original measurements. In this case, the primary original measurement seems to be the distance moved over a fixed period of time. That distance is what I'd suggest modeling.
The distance moved by any fish in the study is then a function of genotype and the experiment number. Your simplest model then might be something like:
distance ~ genotype + exptNo

With only 3 different experiments, the 3-level exptNo categorical term accounts for baseline differences in distance among experiments. That fixed-effect predictor serves a role similar to what a random effect for exptNo would provide if you had more experiments.
You then would evaluate differences among genotypes via the 2 regression coefficients reported for genotype. If the reference level is wildtype, then there would be coefficients for heterozygous and homozygous, each representing the corresponding difference from wildtype. There are standard tests for whether genotype has any overall association with distance and for comparing heterozygous vs homozygous.
There might be a question whether the untransformed distance values are what should be used as outcomes in your model. The usual assumption for a regression model is that the errors around the predictions are uncorrelated, with a distribution independent of the predicted outcome value. My guess is that those errors in distance outcomes will tend to be proportional to distance rather than uncorrelated. In that case you might consider modeling the logarithms of distance instead:
logDistance ~ genotype + exptNo

Then the genotype coefficients would represent differences in logDistance from wildtype, and the exptNo would adjust for proportional rather than additive differences among experiments.
If the fractional differences associated with genotype aren't large, then this second model might not lead to very different results from the way your group has been proceeding. Remember that a difference between two logarithmic values is equal to the logarithm of the corresponding ratio.
The within-experiment normalization your group has been doing is thus similar to what the exptNo term in the second model based on logDistance provides. The ratios of heterozygous and homozygous to wildtype are similar to what the genotype coefficients in the second model provide. If the fractional differences $x$ are small, $\log(1+x) \approx x$ and ratios might not lead to a large practical difference from the log-scale analysis.
The models I suggest have some advantages beyond a stronger theoretical basis.
First, you can test whether the models' underlying assumptions about error distributions (additive or proportional) hold. That's harder to do when you effectively ignore the issue by first taking a wildtype average, then take ratios of individual observations to that average, averaging those ratios for each genotype within each experiment, and then further averaging over the 3 experiments.
Second, it gets around a problem with using the average of the wildtype control group as the denominator for the ratios. The models draw on information from all of the genotypes together, not just from the wildtype control group. That will help minimize the influence of an experiment in which there was some anomaly in that control group.
Third, they provide information about differences among experiments, rather than hiding them in the initial ratios.
Fourth, you can extend the models. If you think that the differences among genotypes (not just baseline values of distance) might differ among experiments, you can include a genotype:exptNo interaction term in the model. You can account for potential confounders. For example, as you seem to have an array of wells each containing a single fish, there might be systematic differences among those wells. That could be handled by an extension of the models (e.g., random effects for well) in a way that the simple ratio analysis can't. Also, depending on how zebrafish breeding is done, you might need to account for the specific brood from which each fish was taken. That can also be accomplished with a regression model.
