For Type III SS, the sequence effect is determined by the subject, since subject is
nested within sequence Type III gives the additional reduction in the residual SS
after accounting for the other model terms. For Type I SS, you get the reduction in
the residual SS after accounting for the model terms before the term in question.
Since subject within sequence comes after subject, you get subject Type I SS. Note
that you can omit the sequence effect entirely
lm(outcome~treatment+period+subject, data=example)
The contrast of interest (on treatment) will not be affected. One should also note
that sequence is confounded with period by treatment interaction, so beware. Further,
the subject that has an observation missing is essentially removed from the analysis
(doesn't affect the results). But that is not true if you use a mixed effect modeling
engine, i.e.,
library(nlme)
my.lme <- lme(outcome~treatment+period, data=example, random=~1|subject)
In this model, all 8+7 observations affect the likelihood, and are used in fitting the
fixed effects: The mixed effect model and the purely fixed effect model will give
difference results for unbalanced data, same results for balanced data.
summary(my.lme) give the output:
Linear mixed-effects model fit by REML
Data: example
AIC BIC logLik
265.1293 270.8068 -127.5647
Random effects:
Formula: ~1 | subject
(Intercept) Residual
StdDev: 66.52344 27.39351
Fixed effects: outcome ~ treatment + period
Value Std.Error DF t-value p-value
(Intercept) 333.8187 20.56392 12 16.233219 <.0001
treatmentS -46.6071 10.77655 11 -4.324868 0.0012
period2 15.8929 10.77655 11 1.474763 0.1683
Correlation:
(Intr) trtmnS
treatmentS -0.242
period2 -0.242 -0.077
Standardized Within-Group Residuals:
Min Q1 Med Q3 Max
-1.69842428 -0.41583473 0.06268074 0.52314988 1.28230120
Number of Observations: 26
Number of Groups: 13
The difference in means of treatment levels is S - F = -46.6, with SE 10.78.
If you add the sequence term to test for period by treatment interaction, you get
fixed effects
Fixed effects: outcome ~ treatment + period + sequence
Value Std.Error DF t-value p-value
(Intercept) 337.1429 28.27660 11 11.923035 <.0001
treatmentS -46.6071 10.77654 11 -4.324871 0.0012
period2 15.8929 10.77654 11 1.474765 0.1683
sequence2 -7.2024 40.20272 11 -0.179152 0.8611
The sequence effect is not statistically significant (P=0.8611), and so one would not
worry about treatment by period interaction here. If one had observed significant
sequence effect, then one would need to either (a) analyze only the first period data,
or (b) explain why the interaction is not real and can be ignored.
The advantages to the lme() analysis over the lm() analysis are: (1) sequence effect
automatically gets appropriate denominator (not so for the lm version) (2) all data
are actually used in the analysis, (3) We are trying to infer to the population of
subjects, hence they should be thought of as random effects, not fixed effects (unless
you really are interested in only those particular subjects).
If any of this is confusing, let me know.
Russell Reeve, Ph.D.
Dir of Experimental Design, Analysis, and Quality
[EMAIL PROTECTED]
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