I'm analyzing an experiment using linear mixed models but am not sure whether my model is appropriate or whether I'm violating the assumption of conditional independence. I've asked a statistician at my department and he wasn't sure either.

The problem can be understood with this example:

There is a puppy sitting in the middle of a room and I'm feeding it 7 different types of food. The variable I'm interested in is how the loudness of its barking changes from before to after getting the food. In other words, I want to know how the nominal variable "type of food" affects the continuous variable "difference in loudness". Crucially, I am measuring this loudness difference with 16 different microphones located at different points in the room and I am measuring several technical replicates (e.g. I'm averaging across 100 barks before feeding and 100 barks after feeding for each experiment). I'm repeating this experiment multiple times, each time with a different puppy and a different set of microphones (not always located exactly at the same place).

To summarise, after averaging across technical replicates my data is structured like this: deltaLoudness = nPuppies x nTypesOfFood x nMicrophones

My question is whether I can treat each microphone as a conditionally independent observation. I.e. can I ask: deltaLoudness ~ 1+ food + (1|puppy)

My thinking is the following:

1) The measurements from the microphones are clearly not independent, however, they are also not identical and may get different levels of environmental noise (say my experimental chamber has a busy street on one side and a lunch area on the other side)

2)As I gather from here (https://stats.stackexchange.com/q/352884), technical replicates are fine to be included. But I'm not sure I would consider this a technical replication.

3) I could rearrange the data to include true technical replicates. This would, however, force me to become mainly interested in an interaction term. I.e. loudness ~ 1+ beforeAfter + food +beforeAfter*food + (1|puppy) + (1|puppy:microphone)


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