Hello,
First, lets rephrase the initial problem to capture its main points:
There are N_f farms. For each farm, there is a measure of its thermal environment HLI. HLI is continuous. There is one value of HLI per farm. There are N_a animals at each farm. In total, there are N_f*N_a animals. The individual animal ID is unique. In total, there are N_f*N_a unique IDs in the data set. Animals at each farm are of two genotypes. The respiration rate mean_rr is an outcome. It is measured (observed) once for each animal. In total, there are N_f*N_a observations in the data set.
The HLI is measured at the farm level. ***If HLI is different for all N_f farms (there are N_f distinct values of HLI)***, then HLI and farm are confounded thus no random farm effect should be specified.
Assuming the *** statement is correct, we fit the following model to test if slopes for HLI are the same between the two genotypes:
proc mixed data=x;
class genotype;
model mean_rr = genotype HLI genotype*HLI /solution;
run;
Or, in Proc GLM:
proc glm data=x;
class genotype;
model mean_rr = genotype HLI genotype*HLI;
run;
There is no random coefficient for HLI. HLI is a covariate in your problem, a continuous variable we adjust for (analysis of covariance chapter in "SAS System For Mixed Models"). Moreover, if only main effect for HLI was included in the model it would have no effect on significance testing of genotype as genotype is the animal level effect thus within subject variance (unaffected by having only main effect for HLI) is used in test of significance.
UPD. Now, suppose the observed HLI levels came from the population of the HLI levels (HLI is random). We can fit such model as follows:
proc mixed data=x;
class genotype;
model mean_rr = genotype /solution;
random HLI genotype*HLI;
run;
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