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--- tips:models_with_or_without_intercept [2021/02/12 15:53] – Wolfgang Viechtbauer
+++ tips:models_with_or_without_intercept [2021/02/12 16:09] – Wolfgang Viechtbauer
@@ Line 106: / Line 106: @@
 &\beta_2 = \mu_s - \mu_a.
 \end{align}
-But what about the contrast between random and systematic allocation? It turns out that we can obtain this from the model as the difference between the $\beta_1$ and $\beta_2$ coefficients. In particular, if we subtract $\beta_1$ from $\beta_2$, then
+But what about the contrast between systematic and random allocation? It turns out that we can obtain this from the model as the difference between the $\beta_1$ and $\beta_2$ coefficients. In particular, if we subtract $\beta_1$ from $\beta_2$, then
 $$
 \beta_2 - \beta_1 = (\mu_r - \mu_a) - (\mu_s - \mu_a) = \mu_r - \mu_s
 $$
-so this difference reflects how different random allocation is compared to systematic allocation. Using the ''anova()'' function, we can obtain this contrast with
+so this contrast reflects how different systematic allocation is compared to random allocation. Using the ''anova()'' function, we can obtain this contrast with
 <code rsplus>
 anova(res, L=c(0,-1,1))
@@ Line 224: / Line 224: @@
 </code>
-It is important to realize that this does not test whether there are differences between the different forms of allocation (this is what we tested earlier in the model that included the intercept term). However, we can again use contrasts of the model coefficients to test differences between the levels. For example, let's test the difference between alternating and random allocation and the difference between systematic allocation and random allocation:
+It is important to realize that this does not test whether there are differences between the different forms of allocation (this is what we tested earlier in the model that included the intercept term). However, we can again use contrasts of the model coefficients to test differences between the levels. Let's test all pairwise differences (i.e., between random and alternating allocation, between systematic and alternating allocation, and between systematic and random allocation):
 <code rsplus>
-anova(res, L=rbind(c(-1,1,0),c(-1,0,1)))
+anova(res, L=rbind(c(-1,1,0),c(-1,0,1), c(0,-1,1)))
 </code>
 <code output>
 Hypotheses:
 :     -factor(alloc)alternate + factor(alloc)random = 0
-: -factor(alloc)alternate + factor(alloc)systematic = 0
+: -factor(alloc)alternate + factor(alloc)systematic = 0
+:    -factor(alloc)random + factor(alloc)systematic = 0
 Results:
    estimate     se    zval   pval
 :  -0.4478 0.5158 -0.8682 0.3853
 :   0.0890 0.5600  0.1590 0.8737
+:   0.5369 0.4364  1.2303 0.2186
+</code>
+These are now the exact same results we obtained earlier for the model that included the intercept term.
+Note that the output does not contain an omnibus test for the three contrasts because the matrix with the contrast coefficients (''L'' above) is not of full rank (i.e., one of the three contrasts is redundant). If we only include two of the three contrasts (again, it does not matter which two), then we also get the omnibus test (rest of the output omitted):
+<code rsplus>
+anova(res, L=rbind(c(-1,1,0),c(-1,0,1)))
+</code>
+<code output>
 Omnibus Test of Hypotheses:
 QM(df = 2) = 1.7675, p-val = 0.4132
 </code>
-These are now the exact same results we obtained earlier for the model that included the intercept term.
 ==== Parameterization ====