WOMBAT – A program for Mixed Model Analyses by Restricted Maximum Likelihood

FAQ: Polynomial regression

Adjustment factors

The default form of regression equation in WOMBAT is chosen to account for rank deficiencies in the fixed effects part of the coefficient matrix in the MME when estimating variance components. If your aim is to obtain `adjustment factors' for other analyses, it is best to fit the appropriate model from the start.

  • Intercepts are readily fitted, but may cause additional dependencies among fixed effects, in particular for nested covariables. If so, you will need to identify suitable fixed effects levels to be `zero-ed out' to account for these dependencies, and specify them in the parameter file via the ZEROUT statement.
  • An alternative way to fit a higher order polynomial regression is to include appropriate powers of the covariable as additional columns in the data file and fit these as additional, linear covariables. This will directly yield the βi type regression coefficients.

When fitting a complicated fixed effects model, care is needed in the interpretation of results, paying particular attention to the estimability of individual effects!

  • Q: I am fitting age as a quadratic covariable in the model, using the default polynomial regression. WOMBAT gives me estimates of regression coefficients in FixSolutions.out, but:
    1. I can't find the intercept, and
    2. these estimates don't make sense, i.e. when I plug them into the regression equation, I don't get a curve which fits the data.
      What is wrong?
  • A: When you say “regression equation”, you are thinking of an equation of the form:
    y = β0 + β1 x + β2 x2
    with intercept, linear and quadratic regression coefficients given by β0, β1, and β2, respectively. Correct?
    If so, this is the problem:
    1. WOMBAT deviates both records and covariables from their respective means prior to analysis, and then fits an “intercept-free” regression equation of form:
      y - μy = α1 ( x - μx) + α2 ( x - μx)2
    2. This means that you can not use the estimates αi from WOMBAT directly in your regression equation (blue) above!
    3. Fortunately, there is a straightforward relationship between the αi coefficients from WOMBAT, and the coefficients βi you want. You need the means, μy and μx, for records and covariables (you find these in file SumModel.out). You can then calculate your coefficients as:
      • β0 = μy - α1 μx + α2 μx2
      • β1 = α1 - 2 α2 μx and
      • β2 = α2.
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