ANOVA: Analysis of Variance

34 Pages·2017·0.4 MB·English

ANOVA: Analysis of Variance
Marc H. Mehlman
[email protected]
UniversityofNewHaven
“The analysis of variance is (not a mathematical theorem but) a
simple method of arranging arithmetical facts so as to isolate
and display the essential features of a body of data with the
utmost simplicity.” – Sir Ronald A. Fisher
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Table of Contents
1 ANOVA: One Way Layout
2 Comparing Means
3 ANOVA: Two Way Layout
4 Chapter #12 R Assignment
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ANOVA (analysis of variance) is for testing if the means of k different
populations are equal when all the populations are independent, normal
and have the same unknown variance.
An ANOVA test compares the randomness (variance) within groups
(populations) to the randomness between groups. To test if the means of
all the populations are equal, one considers the ratio
variance between groups
variance within groups
as a test statistic. A large ratio would indicate a difference between in
means between the groups.
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ANOVA:OneWayLayout
ANOVA: One Way Layout
ANOVA: One Way Layout
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ANOVA:OneWayLayout
The Idea of ANOVA
 The sample means for the three samples are the same for each set.
 The variation among sample means for (a) is identical to (b).
 The variation among the individuals within the three samples is much less
for (b).
 CONCLUSION: the samples in (b) contain a larger amount of variation
among the sample means relative to the amount of variation within the
samples, so ANOVA will find more significant differences among the
means in (b)
- assuming equal sample sizes here for (a) and (b).
- Note: larger samples will find more significant differences. 7
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ANOVA:OneWayLayout
Note:
When k = 2, one usually uses the two–sample t test. However,
ANOVA will give the same result.
When k > 2, hypothesis testing two populations at a time does not
work well. For instance, if one has four populations and each test is a
significance level 0.05, then the significance level of all (cid:0)4(cid:1) = 6 tests
2
would be 1−(1−0.05)6 = 0.265.
The ANOVA procedure is computationally intense - one usually uses a
computer program.
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ANOVA:OneWayLayout
Assumptions for doing ANOVA
1 the populations are normal.
2 the populations have same (unknown) variance.
The above conditions are robust in the sense one can use ANOVA if the
populations are approximately normal (otherwise the Kruskal–Wallis Test –
a nonparametric test) and the population variances are approximately
equal.
Convention: Rule for establishing equal variance
If the largest sample standard deviation is less than twice the smallest
sample standard deviation, one can use ANOVA techniques under the
assumption the variances are all the same.
Some textbooks use four times the smallest sample variance instead of just
twice.
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ANOVA:OneWayLayout
The Treatment or Factor is what differs between populations.
Example
A Blood pressure drug is administered to k populations in k different
doses. One samples from each of the the k populations.
dosage #1 X ,··· ,X
11 1n1
. .
. .
. .
dosage #k X ,··· ,X
k1 kn
k
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ANOVA:OneWayLayout
Definition
Let
def
k = # of levels (populations)
n d=ef sample size of random sample from jth population
j
def
N = n +n +···+n = total number of random varibles
1 2 k
x¯ d=ef sample mean from jth population
j
s2 d=ef sample variance from jth population
j
def 1 (cid:88)k (cid:88)ni
x¯ = the grand mean = x
ij
N
i=1 j=1
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ANOVA:OneWayLayout
Definition
(cid:88)k (cid:88)ni
SS = (x −x¯)2 = Sum of Squares Total
TOT ij
i=1 j=1
def
SS = Sum of Squares between levels
A
= n (x¯ −x¯)2+n (x¯ −x¯)2+···+n (x¯ −x¯)2
1 1 2 2 k k
def
SS = Sum of Squares within the levels
E
= (n −1)s2+(n −1)s2+···+(n −1)s2
1 1 2 2 k k
Theorem
SS = SS +SS .
TOT A E
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