Analysis of Variance
ANOVA is a technique to differentiate between sample means to draw inferences about
the presence or absence of variations between populations means.
•
•
•
Samples from different populations (treatment groups)
Any difference among the population means?
Null hypothesis: no difference among the means
Examples
•
•
•
•
Effect of different lots of vaccine on antibody titer
Effect of different measurement techniques on serum cholesterol determination
from the same pool of serum
Water samples drawn at various location in a city
Effect of antihypertensive drugs and placebo on mean systolic blood pressure
The key statistic in ANOVA is the F-test of difference of group means, testing if the
means of the groups formed by values of the independent variable (or combinations of
values for multiple independent variables) are different enough not to have occurred by
chance
Assumptions of ANOVA
–
–
–
Observations normally distributed within each population
Population (treatment) variances are equal (Homogeneity of variance or
homoscedasticity)
Observations are independent
Step 1 Formulation of Hypothesis
𝐻0 : 𝜇1 = 𝜇2 = 𝜇3 = 𝜇4 = ⋯ = 𝜇𝑘
𝐻1 : atleast two means are significantly different
Step-2
Choose a level of significance 𝛼 = 0.01, 0.05
Step-3
Test statistic
𝐹=
𝑀𝑆𝑡𝑟
𝑀𝑆𝐸
=
𝑠𝑡2
𝑠𝑒2
Step-4
Computation
In this section we calculate the value of F statistic
Where MStr = Mean-square treatment (Variance of the Treatment)
MSE = Mean-square Error (Variance of the error term)
Sstotal = SSTr + SSE
First of all we calculate the correction factor ie is C.F =
(𝑇..)2
𝑟𝑘
= T..= grand total
And r = no of observation in the treatment (sample) and k = no of treatments
Sstoal = ∑∑(𝑥𝑖𝑗 − 𝑥̅ )
2
2
or ∑∑𝑥𝑖𝑗
− 𝐶. 𝐹
( we square the each value and then add
and minus the correction factor)
SStr =
2
∑ 𝑥.𝑗
𝑘
− 𝐶. 𝐹 and SSE = Sstotal – SStr
Finally after the calculation of these above sum of squares then make an ANOVA table
ANOVA Table
Source of
variation
Between
groups
Within
groups
Total
Df
Sum of squares
Mean square
k-1
Result of SStr
SStr/ k-1
n-k
Result of SSE
SSE/n-k
n-1
Add above two
total
No need
Step-5 Critical Region
𝐹 ≥ 𝐹𝛼 ,(𝑣1,𝑣2)
Where v1 = k-1 and v2 = n-k these are the df.
Step-6
Conclusion
In this section we decide hypothesis would be accepted or rejected.
F-Ratio
𝐹=
𝑆𝑡2
𝑆𝑒2
No need