Simple Comparative Experiments – Introduction to ANOVA
Read Section 3.4 in the text
Note: These notes were modified from lecture notes created by Tisha Hooks and Christopher Malone.
In the last set of notes we were introduced to the statistical model for the general linear model. Recall, the model
introduced in the last set of notes:
yij = µ + τi + εij
where i = 1, 2, .., a and j = 1, 2, …, n.
This model has certain assumptions that need to hold in order for decisions made using the model to be valid. As
discussed previously, the mean of the responses in the ith group is ______________ and the variance of yij is denoted
by ______. Also, it is assumed that the response vector follows a normal distribution. Therefore, the assumptions
for the model can be summarized as follows:
If the assumptions hold, then all the hypothesis tests and confidence intervals are said to be exact – which is a nice
feature. If these assumptions are grossly violated, then the results from the hypothesis tests and confidence
intervals may be misleading.
Why?
Model Assumptions for the ANOVA
1. Random error terms, εij (and ultimately yij), are ________________________. The lack of independence of
the error terms can have serious effects on inferences.
Note: In a designed experiment, the _______________________________ of experimental units to
treatment groups allows us to assume the error terms are independent.
2. Random error terms, εij (and ultimately yij), have a ____________________ variance (σ2) for all treatment
groups.
Note: If the variance is not constant across all treatment groups, the F-tests and t-tests can be
seriously affected.
3. Random error terms, εij (and ultimately yij), are ______________________ distributed with a mean of _____.
Note: A slight departure from normality doesn’t usually affect the analysis, but a sever departure
from normality affects hypothesis tests and confidence intervals.
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Let’s use the full cement mortar dataset to investigate the model assumptions. One plot that can be used to assess
the validity of the model assumptions is a plot of the ___________________ (or error terms) vs. the ______________
values (page 83 in text).
How do we calculate these?
If the model assumptions have been met, the points should create a random scatter of points above and below the
line at ______. This is usually referred to as displaying a ___________________________________. This plot is used
for checking:

Whether the mean function is appropriate. That is, is the data linear?

Constant variance assumption – Var(yij) = σ2.
The following are bad examples, i.e. scenarios where the model assumptions have been violated (i.e., not met).
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Another plot used to assess the model assumptions is a plot of the ____________________ vs. _________________
or ______________ (page 82 in text). This plot is used for checking:

Independence – We ____________ want to see smooth trends or extreme bouncing back and forth.
The following are bad examples.
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The last plot we are going to use to assess the model assumptions is a ________________________ probability plot.
In this plot, we interested in checking the normality assumption. That is, we’re checking to make sure the following
holds:
In this plot, we’re hoping to see a straight line. If this happens, then the assumption of normality has been satisfied.
Before using this plot, it is helpful to make the plotting window square in order to properly assess the normality
assumption.
Here are some examples of plots that violate the normality assumption.
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Using Minitab to obtain the plots
We can obtain all the plots discussed above using the following process in Minitab. Choose Stat  ANOVA 
General Liner Model. Next, click on the Graphs button indicated below and then click OK twice.
Once you click on this button you’ll get the following options.
Option 1: Obtain each graph individually
Option 2: Obtain all graphs at once
The first option will produce the plots given above. The second option will produce the following output.
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