Analysis of variance

ONE-WAY ANALYSIS OF VARIANCE (ANOVA)
The one-way analysis of variance is used to compare the means of several
groups, e.g. the mean Hb levels of patients with different types of sickle cell
disease. The analysis is called one-way as the data are classified just one
way, in this case by the type of sickle cell disease. The method is based on
assessing how much of the overall variation in the data is attributable to
differences between the group means, and comparing this with the amount
attributable to differences between individuals in the same group. Hence the
name analysis of variance.
R
Type of sickle cell disease
Hb SS Hb S / beta thalasemia Hb SC
1 7.2 8.1 10.7
2 7.7 9.2 11.3
3 8 10 11.5
4 8.1 10.4 11.6
5 8.3 10.6 11.7
6 8.4 10.9 11.8
7 8.4 11.1 12
8 8.5 11.9 12.1
9 8.6 12 12.3
10 8.7 12.1 12.6
11 9.1 12.6
12 9.1 13.3
13 9.1 13.3
14 9.8 13.8
15 10.1 13.9
16 10.3
∑X = 430.2
∑X2
= 4651.8
K = No. of groups N = Total No. of observations.
H0 : mean1=mean2=mean3
1

Steps to calculate F statistic
1. Total SS = N
X
X
2
2 )(∑
−∑
2. Between groups SS = n
X
n
X
n
X
n
X 2222
)(
3
)3(
2
)2(
1
)1( ∑
−
∑
+
∑
+
∑
3. Within groups SS (residual) (error) = Total SS – between groups
SS
4. Between groups mean square(MS) = Between SS / K-1
5. Within groups (MS) = Within groups SS / N-K
6. F = )(_
)(_
MSgroupsWithin
MSgroupsBetween
ANOVA table
Source of
variation
SS d.f. MS=SS/d.f.
F=between
groups
MS /
Within
groups MS
Between
groups
99.89 2 49.94 49.99
P<0.001
Within
groups
37.96 38 0.999
Total 137.85 40
2

Assumptions
1.The data are normally distributed.
If the observations are extremely skewed, especially for small samples, the
Kruskal-Wallis nonparametric procedure should be used.
2.The population value for the standard deviation between individuals is the
same for each group. This is estimated by the square root of the within
groups mean square. Moderate departure from normality may be safely
ignored, but the effect of unequal standard deviations may be serious.
Levene’s test of homogeneity of variances will help.
Relationship with the two sample t test
One way ANOVA is an extension of the two sample t test. When there are
only 2 groups, it gives exactly the same results as the t test. The F value
equals the square of the corresponding t value and the percentage points of
the F distribution with (1, N-2) degrees of freedom are the same as the
square of the t distribution with N-2 degrees of freedom.
3

Analysis of variance

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