# Compare result from two different set of observations with different sampling

An experiment is to be performed for the one month. However due to some problem it could be performed only for certain number of days in that month. Out of these days when experiment was performed, some days are less reliable. The outcome of the experiment is an event which can be true or false. During each month of experiments, certain number of events are detected. Some of these events are detected during the perfect data conditions while the others are observed when data is less reliable. (total m1+n1 events recorded where m1 is during perfect conditions and n1 is for less reliable conditions) Similar experiment is performed for another month. Experiment for second month also results in a certain number of events. However, for this month the days when data is available and days when the available data is perfect is different. (total m2+n2 events recorded where m2 is during perfect conditions and n2 is for less reliable conditions) My question is how do I compare between the event count for these two months. What sort of normalization I need to perform to make these two results comparable. The idea is depicted in the following figure:

Would be thankful if you can point me in the right direction.

## 1 Answer

I am assuming that the experiment results between different days is independent, but that the experiment within the same day is dependent.

Variability due to the above is variability you probably want to "normalize" out. However, I think there is no need for normalization per se. What you want is Generalized Linear Mixed Effect Model (GLMM). The mixed model factors in correlation within the same day, if you include a random intercept for "Day" in the model.

I think you can probably ignore the month of collection and just factor in the Day. Give an # to all the different days and include that as a categorical factor variable with a random intercept.

Also, it is a GLMM instead of just an LMM because it sounds like you are looking at event counts. Counts are typically not normally distributed but follow the Poisson distribution. So you want a Poisson GLMM probably. If you have excessive "0" counts, then maybe Zero-Inflated Poisson GLMM. Sometimes, data can also be overdispersed than the Poisson can handle so people go to Negative Binomial GLMM or Zero-Inflated Negative Binomial GLMM respectively.

If you use R, look into glmer() for Poisson and glmer.nb() for negative binomial. These are part of the lme4 package in R which can do GLMMs.