Analysis of Variance (ANOVA) - University of North

29 Pages·2005·0.7 MB·English

by  David Tenenbaum

Analysis of Variance (ANOVA)
• Up to this point in the course, we have examined test
statistics that are designed to allow us to compare two
samples in terms of some descriptive measure of those
samples (i.e. usually comparing the mean as a measure
of central tendency, divided by a standard error, which
incorporates some information about dispersion)
• In some instances, we wish to compare more than two
samples, in order to see if there are significant
differences between them: Suppose we have soil
moisture samples from 4 or 5 different catchments on
the same day, and we want to determine whether or not
any of them are significantly more wet or more dry
than the others
David Tenenbaum –GEOG 090 –UNC-CH Spring 2005
Analysis of Variance (ANOVA)
• While we could simply do a large number of two-
sample t-tests to compare the mean soil moistures of
each pair of catchments, this would be a very tedious
procedure (if there were 4 catchments this would
require 6 t-tests, for 5 catchments – 10 t-tests, etc.)
• As an alternative, we can make use of statistical
procedure that, instead of comparing a pair of samples to
one another in isolation, can detect significant
differences between more than two samples at once
by considering the samples simultaneously
• This procedure is known as analysis of variance (usually
referred to as ANOVA), and makes it comparisons based
on the partitioning of variance
David Tenenbaum –GEOG 090 –UNC-CH Spring 2005
When to Use ANOVA
• ANOVA can be applied when we have more than two
samples of the some sort of interval or ratio data, and
we can distinguish the samples from one another in a
categorical fashion, i.e.
• I have soil moisture samples from 4 catchments
• Each sample contains the same kind of data (several
measurements in each catchment describing
volumetric water content per volume of soil – ratio
data)
• The samples are from 4 different catchments (which
can be considered to be in groups or categories)
• The question I want to ask is if the mean of any group is
significantly different from the others
David Tenenbaum –GEOG 090 –UNC-CH Spring 2005
How ANOVA Works
• Formally, the null hypothesis in an ANOVA states that
all the sample means are equal, i.e.
H : x = x = x … = x
0 1 2 3 j
• The alternate hypothesis is that any of the sample
means is not the same as all the others, i.e.
H : At least one x is different
A j
• However, the ANOVA approach does not test these
hypotheses by operating directly upon the means
• Rather, it uses the computation of variances to compare
the means of the groups, such that the test examines the
contribution to total variance made within the groups
versus the contribution made between the groups
David Tenenbaum –GEOG 090 –UNC-CH Spring 2005
How ANOVA Works
• While the ANOVA tests for the equality of the group
means, it does so by using two independent estimates
of the common variance σ2:
• One estimate of the overall variance is created by
calculating a pooled estimate of each of the within-
group variances
• The other estimate of the overall variance consists of
the between-group variance
• The concept here is to compare the variation within
groups to the variation between groups
• If the variation within groups is larger than the
variation between groups, then we should reject H
0
David Tenenbaum –GEOG 090 –UNC-CH Spring 2005
How ANOVA Works
• This makes sense, as based on the following reasoning:
• If the dispersion of values of observations within
groups is greater than the dispersion of values
between groups, clearly the groups are not dividing
up the observations in a way that would lead us to
believe that the values of observations in a
particular group are significantly different from
the values in another group
• On the other hand, if the difference in dispersion
between groups is greater than that within each
group, clearly the values in one group or another
represent some sort of significantly different
condition
David Tenenbaum –GEOG 090 –UNC-CH Spring 2005
Components of Variance
• We make the following assumptions using ANOVA:
• All of the observations are drawn from the same
population (therefore the population variances for
each of the groups are assumed to be equal)
• The observations between and within samples are
random and independent (i.e. this cannot be a
situation where the same set of objects is being
sampled on multiple occasions and groups each
represent an occasion -- ANOVA is analogous to a
multi-sample t-test NOT a multi-sample paired
values t-test
• The observations in each group are normally
distributed
David Tenenbaum –GEOG 090 –UNC-CH Spring 2005
Components of Variance
• With these assumptions in mind, we can think of the
variation in the observations as consisting of two parts:
• The first component of variance is random or
experimental error, that arises whenever we conduct a
sample from any distribution that has a range of values;
by pure chance we will find variation in the values
observed (the random part), and our method of
collecting observations may also introduce some non-
systemic error (the experimental part)
• The second component is systematic variance, that is
due to some known or unknown influences that explain
differences in the phenomenon at various times and
places (i.e. our samples or groups)
David Tenenbaum –GEOG 090 –UNC-CH Spring 2005
Components of Variance
• Thus, we expect to find random or experimental error
expressing itself in terms of the variance that we
calculate using a pooled estimate of the variation
within groups, and systematic variance to express
itself in terms of the variance we calculate to express the
variation between groups
• The test statistic is formulated to test whether or not the
ratio of the between-group variance to the within-group
variance is large enough that we can consider the
between-group variance to be significantly large with
respect to the within-group variance, such that we can
state that we have found systematic variance that
indicates a significant difference between the groups
David Tenenbaum –GEOG 090 –UNC-CH Spring 2005
ANOVA - An F-test
• ANOVA uses an F-test to test the hypotheses, with an
expression of the variation within groups in the
numerator and an expression of the variation between
groups in the denominator
• While conceptually, this is reasonably easy to explain,
most of the difficulty in using ANOVA lies in
performing the calculations to develop those two
measures of the variation of the observation values
• There is nothing particularly complex about these
calculations; they consist of the usual computation of
means, statistical distances, squaring those distances,
summing etc. but with a two-dimensional array of
values, there is more computation required here
David Tenenbaum –GEOG 090 –UNC-CH Spring 2005