# [R] What kind of test in summary(glm)?

Frank E Harrell Jr fharrell at virginia.edu
Wed Oct 10 14:48:10 CEST 2001

```To go a bit farther, simulations have shown that a t
distribution is inappropriate because it is overly
conservative (due to not having to estimate sigma).
P-values using the normal distribution are more
accurate than those from t.

As Brian noted, the Hauck-Donner effect can ruin
Wald statistics.  When a variable does not have
a dominating effect (i.e., when there is no cell
containing less than perhaps two observations
but this depends on N) Wald stats work reasonably
well.

Frank Harrell

Prof Brian Ripley wrote:
>
> On Wed, 10 Oct 2001, Anne Morgenstern wrote:
>
> > Hello R Users,
> >
> > when I use summary(glm) for a logistic regression model with logit as link
> > function I get one column "z value". What kind of test does R use? (I would
> > have expected a t-test).
>
> It's not doing a test at all.  It is reporting a z value, the coefficient
> divided by the estimated standard error.  Now although that is often
> called a `t value', it is not for a GLM (except in special cases)
> t-distributed under the null hypothesis.  If you mean that you used
> glm(family=binomial), the scale is not estimated and the relevant
> asymptotic theory suggests a normal distribution.  However, that can be
> rather misleading.  Under glm(family=pseudobinomial) the scale is
> estimated, but there is no really relevant theory (as you have a
> quasi-likelihood model) suggesting a t distribution.
>
> If you want to test, use anova.glm not the Wald statistic given by the
> `z value', because of the Hauck-Donner effect.
>
> --
> Brian D. Ripley,                  ripley at stats.ox.ac.uk
> Professor of Applied Statistics,  http://www.stats.ox.ac.uk/~ripley/
> University of Oxford,             Tel:  +44 1865 272861 (self)
> 1 South Parks Road,                     +44 1865 272860 (secr)
> Oxford OX1 3TG, UK                Fax:  +44 1865 272595
>
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--
Frank E Harrell Jr              Prof. of Biostatistics & Statistics
Div. of Biostatistics & Epidem. Dept. of Health Evaluation Sciences
U. Virginia School of Medicine  http://hesweb1.med.virginia.edu/biostat
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