1 Completely Randomized Design (CRD) Part I
It is the simplest of designs, but even planning for it requires scientific and statistical decisions (Acid rain
example).
1.1 Exploratory Data Analysis (EDA)
Essential for any statistical analysis. For the CRD we often use boxplots for initial examination of the data.
From the boxplot we can look for evidence of a treatment difference, and possible outliers or problems with
homogeneity of variance.
1.2 ANOVA as a choice of the best fitting model for the mean
Let yij (i= 1, ..., g;j= 1, ..., ni) be the jth observation in group i. In ANOVA from a CRD we consider two
models for yij . The first model, yij =µi+εij , specifies that each group has a different mean value µi. This
is also called the full model for yij . The second model, yij =µ+εij , specifies that all groups have identical
mean values µ. This is also called the reduced model for yij . Note that the reduced model is a special
case (or subset) of the full model. Both models make the assumption that the εij are independent, have
mean zero, and variance σ2. To conduct statistical inference (tests, confidence intervals, etc.) we make the
further assumption that the εij have a normal distribution.
An alternative way to express the models above is by letting µi=µ∗+αi, where µ∗is the overall mean
and αiis the treatment effect of group i. Then the full model is yij =µ∗+αi+εij . This new formulation
of the full model generalizes well to more complicated models, but introduces a complication because there
are now more parameters than groups. For both the full and reduced models, we can develop estimators
for the parameters µand σ2(reduced) or µ, µi, αi,and σ2(full), shown in Display 3.1 in the text. We can
also calculate confidence intervals for our parameters.
To choose between the full and reduced models, we compare their sum of squared residuals (SSR). A
residual ris the error in predicting an observation, r=y−by, where byis the predicted value of y. For the full
model, byij =yi·(the sample group mean), and for the reduced model, byij =y·· (the sample overall mean).
SSR for the full model can never exceed SSR for the reduced model, so we wish to decide if SSR for the full
model has been reduced enough to account for the extra parameters (µi) in the full model.
1.3 Analysis of Variance mechanics
Analysis of variance involves a partition of the total sum of squares for the observations yij. Using our
notation from above, yij −y·· = (yij −yi·)+(yi·
−y··), which equals the residual (from the full model) plus
the ith treatment effect, or rij +bαi.By squaring and summing these terms and cancelling the cross product
we obtain:
g
X
i=1
ni
X
j=1
(yij −y··)2=
g
X
i=1
ni
X
j=1
(yij −yi·)2+
g
X
i=1
ni
X
j=1
(yi·
−y··)2,or SST= SSE+SST rt .
As an example, consider three groups with the following data: Group 1 has y1jvalues of 1, 2, and 3,
Group 2 has y2jvalues of 5, 3, and 4, and Group 3 has y3jvalues of 6, 7, and 5. The overall sample mean
is y·· = (Pg
i=1 P3
j=1 yij)/3g= 36/9 = 4.Then SSTis
1
