Section 14.1 – How Can We Compare
Several Means? One-Way ANOVA
Overview
When a quantitative response variable has a
categorical explanatory variable with several
categories, we use analysis of variance or
ANOVA, to compare the means of the
response variable across the different
categories.
Overview
—  Let
g denote the # of different categories or
groups for the explanatory variable.
—  Each group has a corresponding population
of subjects.
—  Each population has a mean: µ1, µ2, …, µg.
Overview
The ANOVA test is a significance test of the
hypotheses:
◦  H0: µ1=µ2=…= µg.
◦  Ha: at least two of the population means are not
equal
—  The test analyzes whether the differences
observed in the sample means could have
reasonable occurred by chance, if the null
hypothesis is true.
— 
Assumptions
—  The
population distribution of the response
variable for the g groups are normal, with
the same standard deviation for each group.
—  Randomization
◦ Independent random samples are selected
from the g populations for a survey.
◦ Subjects are randomly assigned to the g
groups in an experiment.
Variability
The evidence against the null hypothesis is stronger when
the variability within each group sample is smaller.
Variability
—  The
evidence against the null hypothesis is
also stronger when the sample means for the
groups are farther apart.
—  Lastly, the evidence against the null
hypothesis is stronger for larger sample sizes.
Variability
—  The ANOVA
F test statistic summarizes:
◦  F = Between groups variability
Within-groups variability
—  The
test statistic for comparing means has
the F-distribution.
F-distribution
F-distribution
The F distribution has two degrees of
freedom.
◦  N = total sample size over all groups
◦  g = # of groups
◦  df1 = g – 1
◦  df2 = N - g
Assumptions Revisited
Moderate violations of the normality
assumption are not serious, especially when
the sample sizes are rather large.
Assumptions Revisited
Moderate violations of the same standard
deviation assumption are also not serious.
◦  If the groups have the same sample size, this
violation can be more severe.
◦  If the groups do not have the same size, the
largest standard deviation should be no more
than twice the smallest standard deviation.
Example 1 – Red Dye Number 40
S.W. Laagakos and F. Mosteller of Harvard
University fed mice different doses of red
dye number 40 and recorded the time of
death in weeks. Results for female mice,
dosage and time of death are shown in the
data
X1
X2
X3
X4
70
49
30
34
77
60
37
36
83
63
56
48
87
67
65
48
92
70
76
65
93
74
83
91
◦  X3 = time of death for group with
medium dosage
100
77
87
98
◦  X4 = time of death for group with
high dosage
102
80
90
102
102
89
94
103
97
◦  X1 = time of death for control group
◦  X2 = time of death for group with
low dosage
96
Reference: Journal Natl. Cancer
Inst., Vol. 66, p 197-212
Example 1
Example 1
A P-value of 0.029 is small enough to
—  reject the null hypothesis; and thus,
—  conclude that “we have significant
evidence to claim that different dosages of
red dye are associated with different
average lifespan for mice.”
Analyzing Data
—  The
MS values represent the mean
squares estimates of the variance.
◦  We have an estimate between groups
(FACTOR) and within groups (ERROR).
—  The
SS values represent the total sum of
squares before dividing by the degrees of
freedom.