# Interpretation of a log likelihood function for PROC NLMIXED in SAS

I have a data set of skewed nutrient intake values, from around 7800 individuals, of whom around 3000 had two measures of daily nutrient intake (the others only had one measure), so this is a repeated measures design, very unbalanced. The boxcox transformation in the MASS package in R identified a suitable lambda (0.4) to use for the transformation to linearity, and the nutrient value transformation takes the normal Box Cox form of (x^L-1)/L where x is the nutrient intake value (&response) and L is lambda (0.4 in this case). The purpose of this transformation is to provide a linear dependent variable for a subsequent nlmer analysis in R although the original analysis was performed in SAS.

PROC NLMIXED in SAS, unlike other SAS procedures, requires a user-specified MODEL statement. In the SAS code, a general log likelihood function has been provided, which is (rest of SAS syntax omitted, the & are there as the syntax uses SAS macro language):

ll1=log(1/(sqrt(2*pi*A_VAR_E)));
ll2=(-(boxcoxy-x2b2u2)**2)/(2*A_VAR_E)+(&amtlambda-1)*log(&response);
ll=ll1+ll2;
model &response ~ general(ll);


I can follow some of this log likelihood specification, as I can see some relationship to the probability density function of the normal distribution. But I don't understand the parameterisation to the dataset at hand. The values are:

1. A_VAR_E appears to be the variance of the residuals
2. boxcoxy is the BoxCox-transformed values of &response
3. x2b2u2 is the fitted values based on the fixed and random effects in the model (incorporates age, race, weekend intake effects)
4. &amtlambda is 0.4 in this case
5. &response is the raw nutrient intake value, that was Box Cox transformed to boxcoxy, and is the dependent variable in this analysis.

This likelihood function seems to have deviated from the usual form. In particular, I do not understand why ll2 incorporates (&amtlambda-1)*log(&response) into the denominator. What is this part of the likelihood function doing? Why would the dependent variable &response be on both sides of the equation in the model statement? In ll2 the dependent variable appears to be included twice, once as its transformation boxcoxy in the numerator and once as its raw value in the denominator.

I would appreciate any help with understanding why this likelihood has been parameterised this way.

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