We are most grateful to Matt Coates, Isaac Dialsingh, Carl Donovan,
Richard Gerlach, Kriss Harris, Gareth Janacek, Rajeev Kumar, Chris
Lloyd, Zoann Nugent, Paul Swank, Ben Torsney and Michael Tsagris for
helpful replies, which have given the student plenty to think about over
the Christmas break. The original question is copied below, followed by
an edited version of the replies (mainly omitting repeated mention of
Kruskal-Wallis).
Thanks and best wishes DFG
__________________
> I'll be grateful for hints on the following, which has arisen in the
> course of a student project using real data from her industrial
> placement. The samples are large and the distributions reasonably
> normal.
>
> She wants to test differences between the means of several groups, but
> the group SDs vary considerably - much more than the means and
> without any clear pattern, along the following (simplified) lines:
>
> A mean 10.2 SD 2.0
> B mean 11.5 SD 0.4
> C mean 10.4 SD 0.6
> D mean 10.9 SD 4.5
> E mean 10.7 SD 3.0
>
> Obviously she cannot use standard ANOVA, which requires equal
> variances, nor is any transformation available to stabilise the
> variances.
__________________
A general linear mixed model will allow modeling different variance for
different groups.
------
For large samples, you can just use weight least squares. To do this,
set the model up as a generalised linear model i.e. with dummies for the
6 groups, and take the weights to be the inverse sample variances. There
is an issue with the degrees of freedom of the T-test, but for large
samples the df will be large anyway, and T is close to normal.
------
Have you got SAS? If so this can be done easily in Proc Mixed.
------
What about doing pairwise two-sample t-tests without assuming equal
variances (default in minitab) and with a bonferroni or other
correction?
------
I assume you do use SPSS. First she can use non parametric test
(Kruskal-Wallis). Second she can use the Brown-Forsythe or the Welch
test, which are robust to deviations from the homoscedasticity
assumptions. These two tests come with follow-up tests as well. Check
the attached file also. Keep in mind that it is not a big problem.
------
The Welch variant is commonly used in this case (and is often an option
in an analysis package) - or you might consider some sort of bootstrap.
------
A slightly adhoc or perhaps preliminary analysis is as follows:
Under the null of no differences (constant mean across groups) each
xbar(i) ~ N(mu, sigma^2(i))
Estimate mu as the weighted average of each group's sample mean
(weighted by sample size in each group). Estimate sigma^2(i) by the
sample variance in each group (square the SDs you gave below). Then,
under normality,
SUM(i=1,5) (xbar(i) - mu)^2/sigma^2(i) ~ Chi-squared(with 4 degrees
of freedom)
since mu was estimated using the 5 sample means.
Of course technically the df of 4 might be adjusted slightly more, since
all sample means are estimates, but if you have large sample sizes in
each group and normality, then this result should hold I guess.
Either way, I doubt you'll find significant differences based on the
numbers you show below.
If you want to get more technical and less artistic or adhoc :) you
could fit a hierarchical model that assumed Y(i,j) ~ N(mu(i),
sigma^2(i)) where i represents each group and j represents each
individual observation in group i. I have code to run a Bayesian MCMC
analysis of this model that can return posterior probabilities for
certain relevant hypotheses (that you may have formed before seeing the
data :)). But I'm sure some software must run a frequentist analysis of
this hierarchical model if that's what you wish for.
------
Try a nonparametric test for rquality of medians.
------
Have you thought of using a non-parametric test such as Kruskal-Wallis
ANOVA or a Median Test?
If the variability is itself of interest, you could test for differences
in in variance using Bartlett's test or something similar.
Just one other thought ... Is it possible that there are other factors
varying within the groups that might be causing the different variances?
If so, adjusting for these might just take away the problem.
------
As per my knowledge If the variance are unequal, the F-test for equality
of mean for fixed fixed is only slightly effected if the sample size of
all groups are equal. If the sample size among groups are equal than
F-test can be used and after that for post hoc pairwise comparison you
can apply the unequal variance test like Tamhame's, Dunnett's, Gamer
Howell. These test are available at the SPSS-13.
> -----------------------------
> David Goda
> SCIT, Univ. of Wolverhampton Phone (01902)321444
> Wulfruna Street Email [log in to unmask]
> Wolverhampton WV1 1SB
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