nlme and lme4 Ignoring squared terms

Question

I am trying to build on a standard translog demand function, which is:

lnY = lnP + lnZ + lnY*lnZ + lnY^2 + lnZ^2

Where Y = demand, P = price, and Z = income.

However, when I include the squared terms in nlme or lme4, they just ignore them. I've tried:

model <- lme(asinh(gallons) ~ asinh(rprc) + asinh(rexp) + asinh(rexp)*asinh(rexp) + asinh(rprc)*asinh(rprc) + asinh(rprc)*asinh(rexp), random=~1|cuid, data = data)

and I've tried:

model <- lme(asinh(gallons) ~ asinh(rprc) + asinh(rexp) + asinh(rexp)^2 + asinh(rprc)^2 + asinh(rprc)*asinh(rexp), random=~1|cuid, data = data)

And I've tried the equivalents for lmer.

The squared terms just don't show up in summary(model), and I know they're being ignored because I've created separate vectors with the squared terms and passed those in and the estimates are different.

Anybody have any advice?

Answer 1

In formulas, the term ^2 is used to create interactions of variables. For example, (x+y+z)^2 creates the main effects and all possible interactions with two variables, i.e., x + y + z + x:y + x:z + y:z. Hence, x^2 inside a formula is the same as x.

Furthermore, also * is used to create interactions, For example, x*y creates x + y + x:y. Hence, x*x inside a formula is the same as x.

To create the squared value inside a formula, you have to use the function I, i.e., I(x^2) or I(x*x).

lme(asinh(gallons) ~ asinh(rprc) + asinh(rexp) +
      I(asinh(rexp)^2) + I(asinh(rprc)^2) + asinh(rprc)*asinh(rexp), 
    random=~1|cuid, data = data)