Testing for the significance of treatment on intervention/target group compared to control group when group sizes are different?

Question

So I have have been conducting an experiment of an interesting feed on livestock. So essentially what I have done is, divided the total livestock $N$ in two different sized groups $n_1 $ and $ n_2$. I am not going to name them control or intervention, because of reasons you will see below.

Now, Baselining
I've conducted baselining, that is for 14 days sampled $N$, that is both $n_1$ as well as $n_2$ twice everyday to get their volume and fat % data points without any sorts of intervention for each livestock.

Stage 1
I fed the $n_1$ group the new food, by replacing a fixed portion of their regular food. I didn't change anything with the $n_2$ group and fed it the regular food. For 14 days I sampled $N$, that is both $n_1$ as well as $n_2$ twice everyday to get their volume and fat % data points with this setup for each livestock.

Stage 3
I reverted back to completely regular food for the $n_1$ group. I now fed the $n_2$ group the new food, by replacing the same fixed portion of their regular food. For 14 days I sampled $N$, that is both $n_1$ as well as $n_2$ twice everyday to get their volume and fat % data points with this setup.

How do I go ahead with understanding statistically whether there was an impact of my new feed on livestock on the fat and/or volume or just total fat production $ Total\ fat = Volume *fat \ \% $ ?

A good point to mention would be that typical values of each livestock are different than other: for example one member of $n_1$ would have volumes somewhere around 800 ml per sampling whereas as another from $n_1$ would have volumes somewhere around 2000 ml. So I guess its random in a way.

Right of the mind I can think of applying Student's T paired test for each group individually between two stages. But how ? Should it be between baselining and stage 1 for group $n_1$ for example ? Or between stage 1 and stage 2 for $n_1$ ? Similarly what about $n_2$ ? And how do I compare between the groups ?

I just need to (dis)prove the hypothesis that my new food increases the volume or fat % or total fat. I've seen the best practices post but couldnt find much whether the methods there apply to my trials.

Answer 1

Your terminology is a little confusing, so let me summarize: you took all your $N$ cows, divided them into 2 groups and

phase 1: fed both groups normally as you would for 14 days

phase 2: after phase 1 you gave a different feed (a portion of total feed) to the first group of cows, the second group was fed normally, again for 14 days

phase 3: after phase 2 you gave a different feed (a portion of total feed) to the second group of cows, the first group was fed normally, again for 14 days

Each cow in each group was measured for fat / volume twice a day, for a total of 2*14*3=84 measurements for each cow. Is this correct?

If so this is a case for a linear mixed model. The cows are your observations of which you took repeated measures from. With a mixed model you can treat each cow as a random effect, meaning each cow has a certain random effect associated with it, which is unique to it.