Single Factor Repeated Measures ANOVA
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Joseph Hilbe, STB/CRC  9-25-91


            ^ranova^ <varlist> [^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
___________________________________________________________________
     Subjects | 13459.9300     14        -         -         -
   Treatments |  284.5800      3       94.8600     7.70    0.0003
        Error |  517.6700      42      12.3255     -         -
___________________________________________________________________
        Total | 14262.1800     59



Help
- ----
Joseph Hilbe
hilbe@asu.edu