Developed by Fisher while studying
agriculture and crop output with different fertilizer treatments, needed test
that could evaluate differences between three or more means, developed the
F distribution
Why - problems with applying the
Student's t test to more than 2 sample/population means
number of comparisons between
all means increases with an increase in the number of samples/populations
(k)
alpha level becomes unstable
with addition of more means, chances of making a Type I error increase with
an increase in the number of means being compared
Types of ANOVAs
One-way ANOVA
Examines one factor at a time,
tests for differences among levels of the factor, Example from Zar (1999)
Question - is there a difference
in the average weight of pigs fed different types of feed?
Factor - pig feed
Levels - different feeds (feed
type 1, feed type 2, feed type 3, feed type 4)
Fixed effects or Model I One-way
ANOVA
the levels of the factor are
specifically chosen by the investigator
example - 4 feed types were
specifically chosen
Random effects or Model II One-way
ANOVA
the levels of the factor are
randomly chosen by the investigator
example - 4 feed types were
randomly chosen from many different types of feeds
This course addresses the One-way
ANOVA, can be applied to fixed effects or random effects models
Others
Two-way ANOVA - examines effects
of two factors simultaneously, beyond scope of this course
Mechanics of One-way ANOVA
Focus in on analysis of variance
- comparison between 2 types of variance
among group variance (variation
among the grand mean and the sample means)
note that this is often referred
to the group mean square (MSg) as well as the among group variance
grand mean is the mean for all
data
within group variance (sum of
the variation within each sample - around each sample mean)
note that this is often referred
to the error variance or error mean square (MSe) as well as
the within group variance (MSw)
evaluation uses a one-tailed
F test to determine if the among group variance is greater than the within
group variance
Assumptions of the ANOVA
homogeneity of variance, sample
variances are equal
Bartlett's test for homogeneity
of variance
samples come from normal populations
ANOVA is robust enough to handle
departures from normality and unequal variances
Problems occur when
heterogeneity of variances
is combined with unequal sample sizes
skewness/kurtosis is combined
with smaller sample sizes
Results of ANOVA
no difference between among group
variance and within group variance
this tells us there is no difference
among means
difference between among group
variance and within group variance
this tells us there is a difference
among means but not which ones are different
How we conduct an ANOVA
Mechanics
- Null hypotheses, formulae, interpretation, etc.
Multiple Comparisons Tests - only
used if the results of an ANOVA yield a significant difference between among
group variance and within group variance
ANOVA results only indicate that
a difference exists among means, not where the difference(s) are - which means
are different, have to use multiple comparisons test to detect where the differences
lie
Referred to as ad hoc or a posteriori
tests because they are used after you know there is a significant difference
from the ANOVA
Several types of multiple comparisons
tests - can be combined into three broad categories
Generic multiple comparisons
tests - evaluate all possible pairs/combinations of means
Tukey's HSD test
Student-Newman-Keuls (SNK)
test
unique in that the critical
value of q changes with the range of means being considered, sometimes
referred to as the multiple range test
Control group test - evaluate
differences between experimental group versus the control group (not
differences among experimental groups), also employs one-tailed tests which
is less common practice with other multiple comparisons tests
Dunnett's test
Multiple contrasts tests - can
be used like the traditional tests mentioned above to evaluate differences
among pairs of mean but is better used to evaluate homogeneous groups of
means against other such groups or individual means,
