9.6 - Step 3: Test for the main effects of treatments
9.6 - Step 3: Test for the main effects of treatmentsBecause the results are deemed to be not significant then the next step is to test for the main effects of the treatment.
We now define a new variable equal to the sum of the observations for each animal. To test for the main treatment effect, consider the following linear combination of the observations for each dog; that is, the sum of all the data points collected for animal j receiving treatment i.
\(Z_{ij} = Y_{ij1}+Y_{ij2}+\dots + Y_{ijt}\)
This is going to be a random variable and a scalar quantity. We could then define the mean as:
\(E(Z_{ij}) = \mu_{Z_i} \)
Consider testing the following hypothesis that all of these means are equal to one another against the alternative that at least two of them are different, or:
\(H_0\colon \mathbf{\mu}_{Z_1} =\mathbf{\mu}_{Z_2} = \dots = \mathbf{\mu}_{Z_a} \)
ANOVA on the data Zij is carried out using the following MANOVA statement in the SAS program as shown below:
Note: In the upper right-hand corner of the code block you will have the option of copying ( ) the code to your clipboard or downloading ( ) the file to your computer.
options ls=78;
title "Repeated Measures - Coronary Sinus Potassium in Dogs";
data dogs;
infile "D:\Statistics\STAT 505\data\dog1.csv" firstobs=2 delimiter=',';
input treat dog p1 p2 p3 p4;
run;
proc print data=dogs;
run;
/* The class statement specifies treat as a categorical variable.
* The model statement specifies p1 through p4 as the responses
* and treat as the factor.
* The h= option in the manova statement is used to specify over
* which groups the mean response vectors are to be compared.
* The m= option specifies the transformation (if any) to be
* applied to the responses before the means are calculated.
*/
proc glm data=dogs;
class treat;
model p1 p2 p3 p4=treat;
manova h=treat / printe;
manova h=treat m=p1+p2+p3+p4;
manova h=treat m=p2-p1,p3-p2,p4-p3;
run;
h=treat sets the hypothesis test about treatments.
Then we set m = p1+p2+p3+p4 to define the random variable Z as in the above.
Now, we must make sure that we are looking at the correct part of the output! We have defined a new variable MVAR in this case, a single variable that indicates that we are summing these four.
Results for Wilks Lambda:
| Statistic | Value | F Value | Num DF | Den DF | Pr > F |
|---|---|---|---|---|---|
| Wilks' Lambda | 0.63985247 | 6.00 | 3 | 32 | 0.0023 |
| Pillai's Trace | 0.3601453 | 6.00 | 3 | 32 | 0.0023 |
| Hotelling-Lawley Trace | 0.56286025 | 6.00 | 3 | 32 | 0.0023 |
| Roy's Greatest Root | 0.56286025 | 6.00 | 3 | 32 | 0.0023 |
This indicates that there is a significant main effect of treatment. That is that the mean response of our four-time variables differs significantly among treatments.
To fit the MANOVA model and test for treatment main effect
- Open the ‘dog1’ data set in a new worksheet
- Rename the columns treat, dog, p1, p2, p3, and p4, from left to right.
- Name a new column in the worksheet sum.
- Calc > Calculator
- Highlight and select sum for the Store result window
- In the expression window, enter p1+p2+p3+p4, and choose OK. The sum of the responses appears in the sum column in the worksheet.
- Stat > ANOVA > One-way
- Choose Response data are in one column.
- Highlight and select sum to move this to the Responses window.
- Highlight and select treat to move it to the Factor window.
- Choose OK. The results are shown in the results area.
Conclusion
Treatments have a significant effect on the average coronary sinus potassium over the first four time points following occlusion \( \left( \Lambda = 0.640; F = 6.00; d. f. = 3, 32; p = 0.0023 \right) \).
In comparing this result with the results obtained from the split-plot ANOVA, we find that they are identical. The F-value, p-value, and degrees of freedom are all identical. This is not an accident! This is mathematical equality.