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Unformatted text preview: EXST 7025 – Biological Population Statistics II
Analysis of Covariance and Lack of Fit Symmary Page 1 Lack of Fit and Model adequacy
We have had manuscripts returned with the opinion expressed by editor or referee that means should
be fitted instead of raw data (e.g. growth data). It is never WRONG to fit data to the full data set.
LOF is a test of “model adequacy”.
If the MSLoF is significantly larger than the MSPE, then the means are not being adequately
described by the model. There is some information in the means that is not adequately covered by
If the test of MSLoF not rejected this suggests the model is adequate in the sense that the means
are adequately described by the regression line.
Models can be fitted to means and will give the same estimates if properly weighted.
Although the model fit is the same the R2 value may appear to be much improved. It does in fact
reflect the fact that the regression is fitting the means and not the individual observations.
Models fitted to means will NOT necessarily give the correct variance. The test of LoF will determine
if there is a significant difference. However, even if not “significant difference” they are not
necessarily “significantly the same”. We cannot test that (remember type II error).
Still, it is never WRONG to fit data to the full data set. It just may not look as good.
The test of MSLoF can be done in SAS (GLM or MIXED) using TYPE I SS by fitting the model with
its independent variable and then fitting the independent variable as a categorical variable. This fits
the remaining variation (SSE regression – SSE ANOVA) which is equal to SSLoF. Also, the error
term will appropriately be the full model error term.
Analysis of Covariance
The TYPE I “sequential tests” can be used to test a series of hypotheses where at each step we want to
determine if the addition of some component to a model improves the model or not.
In Analysis of Covariance (AnCoVa) we want to test the following series of hypotheses. Given a
regression and a desire to determine if some several groups can be fitted by the same regression we
can test the following hypotheses.
Is fitting a slope an improvement over a flat line through the mean? [test slope]
Is fitting a separate intercept to each group better than a single line? [test separate intercepts]
Is fitting a separate slope to each group better than parallel lines? [test separate slope]
These are the “usual” tests for multisource regression. Other tests may be of interest and a
different order can be used.
If the order of tests is clear, the tests can be viewed as a series of Full and Reduced models. The
appropriate tests can be given by the TYPE I SS if the desired tests can be obtained from a given
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This note was uploaded on 12/29/2011 for the course EXST 7025 taught by Professor Geaghan,j during the Spring '08 term at LSU.
- Spring '08