Using lme to perform a nested ANCOVA in r

Question

I want to perform a nested ANCOVA using the function lme in r. My data concerns the growth rate (sgr) of fish at different temperatures (temp). In the trial there were 3 replicate tanks each housing 8 fish at each level of temperature (15,18,21 and 24 degrees Celsius) so that a total of 12 tanks were included in the trial. I want to control for the effects that individual tanks may have had on growth and since the initial mass of the fish also effects growth rate I want to include it as a covariate.

These are the variables I plan to include in the model:

sgr= continuous dependent variable

temp= fixed main effect

mass= covariate

tank= random factor (tank coded 1-12)

Using lme I have coded my model as such:

m1=lme(sgr~temp+mass,random=~1|tank)

I want to know two things:

1. If the code I have used to include tank as a random factor has been done correctly?

2. Can a covariate (mass) be added this way using lme (just like a regular ANCOVA) when the model also includes a random effect?

Thank you in advance for any assistance provided?

Answer 1

If you use the following notation: $sgr_{ft}$ is the growth rate for fish $f$ in tank $t$, $temp_t$ temperature in tank $t$ and $mass_f$ the mass of fish $f$ then you estimate a regression of the form:

$sgr_{ft} = \beta_{0t} + \beta_1 temp_t + \beta_2 mass_f$ using the R-code lmeModel<-lme(sgr ~ mass + temp , random=~1|tank). Note that the intercept can be different for each tank ($\beta_{0t}$ has $t$ as subscript).

The $\beta_{0t}$ can be extracted from lmeModel using the extraction functions fixef and ranef. In fact, $\beta_{0t}$ has two components, i.e. $\beta_{0t}=\beta_0+b_{0t}$, where $\beta_0$ is a fixed effect and $b_{0t}$ a random effect. The function fixef gived you the maximum likelihood estimator of the fixed effects, the function ranef gived the best linear unbiased predictors (BLUP) of the random effects, you will see that ranef yields one value for each tank.

You may also try a model like $sgr_{ft} = \beta_{0t} + \beta_{1t} temp_t + \beta_2 mass_f$, where the coefficient $\beta_{1t}$ also depends on the tank, using the code lmeModel<-lme(sgr ~ mass + temp , random=~1+temp|tank)

Using the R-code lmeModel<-lme(sgr ~ mass + temp , random=~1|tank/temp) is similar to the first model, only that in the the first case you assume a different intercept for each tank while in the model with random=~1|tank/temp you assume a different intercept for each temperature within each tank. So it is something like $sgr_{ft} = \beta_{0t,T} + \beta_1 temp_t + \beta_2 mass_f$, where T in $\beta_{0t,T}$ stands for temperature. However, if you treat temperature as a factor (i.e. a categorical variable) then I think it does not make much difference.

EDIT 27-07-2016:

About the question in your comment; if temperature is categorical, having four values, then this categorical variable is replaces by three (the number of categories minus 1) dummy variables, $D_1, D_2, D_3$.

In that case lmeModel<-lme(sgr ~ mass + temp , random=~1|tank) estimates $sgr_{ft} = \beta_{0t} + \beta_{11} D_{1t} + \beta_{12} D_{2t} + \beta_{13} D_{3t}+ \beta_2 mass_f$ while lmeModel<-lme(sgr ~ mass + temp , random=~1+temp|tank) estimates $sgr_{ft} = \beta_{0t} + \beta_{11t} D_{1t} + \beta_{12t} D_{2t} + \beta_{13t} D_{3t}+ \beta_2 mass_f$