Re: [R] Helmert contrasts for repeated measures and split-plot expts

From: Spencer Graves <spencer.graves_at_pdf.com>
Date: Fri 13 Oct 2006 - 15:19:58 GMT

<comments in line>

Roy Sanderson wrote:
> Dear R-help
>
> I have two separate experiments, one a repeated-measures design, the other
> a split-plot. In a standard ANOVA I have usually undertaken a
> multiple-comparison test on a significant factor with e.g TukeyHSD, but as
> I understand it such a test is inappropriate for repeated measures or
> split-plot designs.
>
> Is it therefore sensible to use Helmert contrasts for either of these
> designs? Whilst not providing all the pairwise comparisons of TukeyHSD,
> presumably the P-statistic for each Helmert contrast will indicate clearly
> whether that contrast is significant and should be retained in the model.
> (This seems to come with the disadvantage that the parameter values are
> harder to interpret than with Treatment contrasts.) In the
> repeated-measures design the factor in question has three levels, whilst in
> the split-plot design it has four.
>

      You don't need to restrict yourself to Helmert vs. treatment contrasts: You can use any set of "contrasts" that will provide estimates of (k-1) parameters for a factor with k levels and interpret the p values as you suggest. I see two issues with doing this: correlation among parameter estimates and individual vs. group p values.

CORRELATED PARAMETER ESTIMATES: Helmert contrasts are orthogonal for a balanced design but will produce correlated parameter estimates with an unbalanced design. This will generally increase the p values due to "variance inflation" created by the correlation. If one or more correlations are too large, you may wish to try custom contrasts that produce parameter estimates that are essentially uncorrelated; this should give you the smallest p value you could expect for that comparison. If I was interested in, e.g., 2*k comparisons, I might run the same analysis several times with different contrasts, taking the p value for each comparison from an analysis in which the coefficient for that comparison had a low correlation with the remaining (k-2) coefficients for that k-level factor.

INDIVIDUAL VS. GROUP p VALUES: In many but not all cases, under the null hypothesis of no effect, a p value will follow a uniform distribution. Thus, if we compute 1,000 p values using a typical procedure when nothing is going on, we can expect roughly 50 of them to be less than 0.05 by chance alone. The Bonferroni inequality suggests that if we do m comparisons, we should multiply the smallest p value by m to convert it to a family- or group-wise p value. This is known to be conservative, and with more than (k-1) comparisons among k levels of a factor, it is extremely conservative. In that case, I would be inclined to multiple the smallest p value by (k-1), even if I considered many more than (k-1) comparisons among the k levels. I don't know a reference for doing this, and if I were going to do it for a publication, I might do some simulations to check it. Perhaps someone else might enlighten us both on how sensible this might be.

      Hope this helps. 
      Spencer Graves

> Many thanks in advance
> Roy
> ----------------------------------------------------------------------------
> -------
> Roy Sanderson
> Institute for Research on Environment and Sustainability
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> University of Newcastle
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> United Kingdom
>
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>
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>
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