Unformatted text preview: )2
¯ TSS =
i
2 j Sum of Square Within (SSW): is a measure of variability
within samples
(yij − yi . )2
¯ SSW =
i
3 j Sum of Squares Between (SSB): is a measure of variability
between samples
ni (¯i . − y.. )2
y
¯ SSB =
i
Chapter 12: Multisample Inference Stat 491: Biostatistics Analysis of Variance (ANOVA)
Multiple Comparisons
The KruskalWallis Test
TwoWay ANOVA Measuring Variabilities It can be proved that TSS can be partitioned as
TSS = SSB + SSW
Degrees of Freedom: is the amount of information
(independent observations) used to estimate variability.
Source of Variation Sum of Square Degrees of Freedom
Between
SSB
k −1
Within
SSW
n−k
Total
TSS
n−1 Chapter 12: Multisample Inference Stat 491: Biostatistics Analysis of Variance (ANOVA)
Multiple Comparisons
The KruskalWallis Test
TwoWay ANOVA Measuring Variabilities
Mean Square: is a sum of square divided by its degrees of
freedom.
Mean Square adjusts a sum of squares for the number of
independent observations used (degrees of freedom).
Mean Squares:
Mean Square
Mean Square Between
Mean Square Within 2
sB
2
sW Formula
= SSB /(k − 1)
= SSW /(n − k ) 2
2
sW is always an estimate of σ 2 . Further, sB can be an
estimate of σ 2 when H0 is true.
2
2
However, sB tends to be bigger than sW when H0 is not true.
2
2
Therefore, we can compare sB and sW to test our hypothesis.
Chapter 12: Multisample Inference Stat 491: Biostatistics Analysis of Variance (ANOVA)
Multiple Comparisons
The KruskalWallis Test
TwoWay ANOVA Test Statistic
The test statistic for testing our hypothesis is:
F= 2
sB H0
∼ Fk −1,n−k
2
sW Our decision rule is to reject H0 when
F > F1−α,k −1,n−k .
The upper 1 − α percentile F1−α,k −1,n−k of can be computed
in R using the command qf (1 − α, k − 1, n − k ).
The pvalue is computed as
p − value = P (Fk −1,n−k > Fcomputed ).
In R, 1 − pf (Fcomputed , k − 1, n − k ).
Chapter 12: Multisample Inference Stat 491: Biostatistics Analysis of Variance (ANOVA)
Multiple Comparisons
The KruskalWallis Test
TwoWay ANOVA ANOVA Table The calculations involved in ANOVA is summarized in the
following table known as the ANOVA table.
Source
Between Samples
Within Samples
Total Sum of
Squares
SSB
SSW
TSS Degrees
of Freedom
k −1
n−k
n−1 Mean Square
2
sB =
2
sW = SSB
k −1
SSW
n −k The R command for ANOVA (in its simplest use) is
aov(formula,data=NULL) Chapter 12: Multisample Inference Stat 491: Biostatistics F
2
2
sB / sW Analysis of Variance (ANOVA)
Multiple Comparisons
The KruskalWallis Test
TwoWay ANOVA An Example: Pulmonary Disease
Twentytwo young asthmatic volunteers were studied to assess the
shortterm eﬀects of sulfurdioxide(SO2 ) exposure under various
conditions. Bronchial reactivity to SO2 (cm H2 O /s ) grouped by
lung function (as deﬁned by forced expiratory volume divided by
forced vital capacity) at screening among 22 asthmatic volunteers
is given below.
FEV/FVC ≤ 74% (Group A): 20.8, 4.1, 30.0, 24.7, 13.8,
FEV/FVC 75 − 84% (Group...
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 Fall '12
 SolomonHarrar
 Statistics, Biostatistics

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