Single Factor Repeated Measures ANOVA
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Joseph Hilbe, STB/CRC 9-25-91
^ranova^ [^if^ exp][^in^ range]
^ranova^ automatically checks for missing values across variables listed on
the command line. When a missing value is found in any variable, it
deletes the variable from active memory. However, the original data set is
restored to memory upon completition of the analysis. The program provides
information regarding excluded observations.
The statistical design permits analysis of repeated (treatment) measures
on the same individuals. Total model variability is divided into:
^SS Treatment^ - the variability resulting from the independent variable;
i.e., the levels or categories of response
^SS Within^ - the variablity that cannot be accounted for by the
independent variable
a. ^SS Subjects^ - the variability resulting from individual differences.
b. ^SS Error^ - the variability resulting from random factors
Since each individual provides a value for each level or category of the in-
dependent variable, it is possible to measure the individual difference var-
iability. This is not possible in randomized designs. However, a warning
must be given regarding the use of the test. In many instances when
individuals arebeing measured over time there may be a carry-over effect from early measure-
ments to later ones. This will bias the experiment. Moreover, when there
are more than two measurements, the model has the assumption of homogeneity
of covariance. This assumption is violated, for example, when one pair of
levels are fairly close in time whereas another pair are more distant.
Violations of this sort affect Type I error rate. This problem is ameliorated by using the
Huynh-Feldt correction or by transforming the repetitions of dependent var-
iables into separate dependant variables and analyzing the model by
profile analysis or MANOVA.
If a significant difference between levels of the independent variable
has been determined, Tukey HSD tests may be calculated to ascertain which
level(s)are significantly different. The formula is:
^HSD = q * sqrt((MS error)/N)^
where the appropriate ^q^ value is found on a Studentized Range Table.
Output of a ^ranova^ run with one missing value appears as:
.
.
^. ranova Var1-Var4 in 5/20^
Mean Standard Deviation
Var1 101.2667 14.9450
Var2 103.4667 15.1510
Var3 104.4000 17.0955
Var4 107.3333 15.9135
Single Factor Repeated Measures ANOVA
Number of obs in model = 15 Number of vars = 4
Number of obs dropped = 5
Source | SS df MS F Prob > F
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Subjects | 13459.9300 14 - - -
Treatments | 284.5800 3 94.8600 7.70 0.0003
Error | 517.6700 42 12.3255 - -
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Total | 14262.1800 59
Help
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Joseph Hilbe
hilbe@asu.edu