Anova

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2. ANOVA and linear regression. We begin by reviewing the beneﬁts and
limitations of classical nonhierarchical regression for ANOVA problems.ANOVA—WHY IT IS MORE IMPORTANT THAN EVER 3
2.1. ANOVAand classical regression: good news. It is well known that many
ANOVA computations can be performed using linear regression computations,
with each row of the ANOVA table corresponding to thevariance of a correspond-
ing set of regression coefﬁcients.
2.1.1. Latin square. For a simple example, consider a Latin square with ﬁve
treatments randomized to a 5×5 array of plots. The ANOVAregression has 25 data
points and the following predictors: one constant, four rows, four columns and
four treatments, with only four in each batch because, if all ﬁve were included,
the predictors wouldbe collinear. (Although not necessary for understanding the
mathematical structure of the model, the details of counting the predictors and
checking for collinearity are important in actuallyimplementing the regression
computation and are relevant to the question of whether ANOVA can be computed
simply using classical regression. As we shall discuss in Section 3.1, we ultimately
will ﬁndit more helpful to include all ﬁve predictors in each batch using a
hierarchical regression framework.)
For each of the three batches of variables in the Latin square problem,
the variance of the J= 5 underlying coefﬁcients can be estimated using the
basic variance decomposition formula, where we use the notation var
J
j=1 for
thesamplevarianceof J items:
E(variance between the ˆ βj ’s)= variance between the true βj ’s
+estimation variance,
(1)
E(var
J
j=1
ˆ βj ) = var
J
j=1 βj + E

var( ˆ βj |βj )

,
E(V ( ˆ β)) = V(β) +Vestimation.
One can compute V( ˆ β) and anestimate of Vestimation directly from the coefﬁcient
estimates and standard errors, respectively, in the linear regression output, and then
use the simple unbiased estimate,
 V(β) = V( ˆ β) − ...