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Version info: Code for this page was tested in Stata 12.
Page 312 shows the analysis of covariance. In Stata you use the contin( ) option to specify continuous variables. In the example below, x is the covariate and c.x indicates to treat this as a continuous variable.
use http://www.ats.ucla.edu/stat/stata/examples/da/chap14, clear
anova y a c.x
Number of obs = 24 R-squared = 0.4522
Root MSE = 4.08761 Adj R-squared = 0.3700
Source | Partial SS df MS F Prob > F
-----------+----------------------------------------------------
Model | 275.828162 3 91.9427208 5.50 0.0064
|
a | 165.793075 2 82.8965373 4.96 0.0178
x | 163.828162 1 163.828162 9.81 0.0053
|
Residual | 334.171838 20 16.7085919
-----------+----------------------------------------------------
Total | 610.00 23 26.5217391
We should note that we have used c. in order to perform contrasts, but such use does not make those variables covariates. This is just a programming trick in Stata to accomplish the contrasts of interest. In general, the c. is used when you have a covariate and you want to tell Stata to treat that variable as a continuous variable (as opposed to a categorical variable).
Below we show how to compute the adjusted means shown on page 314. The adjust command adjusts the means for x and lists them separately for each level of a.
adjust x, by(a)
-------------------------------------------------------------------------------
Dependent variable: y Command: anova
Covariate set to mean: x = 7.5
-------------------------------------------------------------------------------
----------+-----------
a | xb
----------+-----------
1 | 6.53103
2 | 10.3747
3 | 13.0943
----------+-----------
Key: xb = Linear Prediction
On pages 314-316 Keppel shows how to compare group 1 with 2 and 3. We create acomp1 that shows the comparison of interest, and add acomp2 that is orthogonal to acomp1.
generate acomp1=a recode acomp1 1=1 2=-.5 3=-.5 generate acomp2=a recode acomp2 1=0 2=-1 3=1
We now replace a with c.acomp1 c.acomp2. These results are slightly different from Keppel's on page 316, but do match results from other packages. We think the differences may be simply due to rounding error.
anova y c.acomp1 c.acomp2 c.x
Number of obs = 24 R-squared = 0.4522
Root MSE = 4.08761 Adj R-squared = 0.3700
Source | Partial SS df MS F Prob > F
-----------+----------------------------------------------------
Model | 275.828162 3 91.9427208 5.50 0.0064
|
acomp1 | 142.11589 1 142.11589 8.51 0.0085
acomp2 | 27.5922135 1 27.5922135 1.65 0.2135
x | 163.828162 1 163.828162 9.81 0.0053
|
Residual | 334.171838 20 16.7085919
-----------+----------------------------------------------------
Total | 610.00 23 26.5217391
At the bottom of page 320 Keppel shows how to test for homogeneity of regression across groups. This is equivalent to testing the interaction of a*x, where an interaction would indicate that the regression coefficient for x is significantly different across the levels of a. We see that the test of a*x corresponds to that shown at the bottom of page 320. Note that we get the interaction and main effects using the ## notation in Stata 12 which expands to the main effects plus their interaction.
anova y a##c.x
Number of obs = 24 R-squared = 0.5027
Root MSE = 4.10535 Adj R-squared = 0.3645
Source | Partial SS df MS F Prob > F
-----------+----------------------------------------------------
Model | 306.630482 5 61.3260963 3.64 0.0189
|
a | 74.2381202 2 37.1190601 2.20 0.1394
x | 107.524781 1 107.524781 6.38 0.0211
a#x | 30.8023193 2 15.4011596 0.91 0.4188
|
Residual | 303.369518 18 16.8538621
-----------+----------------------------------------------------
Total | 610 23 26.5217391
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