3a.4 - The Effects Model

In effects model we are no longer estimating means, but instead are estimating the deviations of treatment means from the overall mean (the \(tau_i\)). The model must include the intercept, so we have the following design matrix for IML:

/* The Effects Model */
x={
1	1	0,
1	1	0,
1	0	1,
1	0	1,
1	-1	-1,
1	-1	-1};
 

Here we have another omission of a treatment level, but for a different reason. In the effects model, we have the constraint \(\Sigma \tau_i =0\). As a result, we need to remove one treatment level. Again, the coding for treatment level3 has been omitted.

Re-running with this design matrix yields:

Regression Coefficients
Beta_0 3.5
Beta_1 -2
Beta_2 0
ANOVA
  df SS MS F
Treatment 2 16 8 16
Error 3 1.5 0.5  
Total 5 17.5    
 

The regression coefficient \(\beta_0\) is the overall mean and the coefficients \(\beta_1\) and \(\beta_2\) are \(\tau_1\) and \(\tau_2\) respectively. The estimate for \(\tau_3\) is obtained as \(– (\tau_1)-( \tau_2) = 2.0\).

If we change the coding in Minitab now to be Effect coding (1,0,-1), the default setting, we get the following:

Regression Equation

y = 3.500 - 2.000 trt_level1 - 0.000 trt_level2 + 2.000 trt_level3

The ANOVA table is the same as for the dummy-variable regression model above.

The intermediates were:

xprimex
6 0 0
0 4 2
0 2 4
check
1 0 0
0 1 0
0 0 1
xprimey
21
-8
-4
SumY2
89.5
CF
73.5
xprimexinv
0.1666667 0 0
0 0.3333333 -0.166667
0 -0.166667 0.3333333