ANOVA - STATISTICAL TOOL

2. ANALYSIS OF VARIANCE
(ANOVA)
Presented by,
Neethu Asokan

3. • Biostatistics is the science of collection, analysis
and interpretation of facts and numbers connected
with biology.
• It is also called biometrics.
STUDENT’S-t TEST
CHI-SQUARE TEST
FISHER’S TEST(F)
ANOVA
BIOSTATISTICS

4. WHAT IS ANOVA?
ANOVA refers to the examination of differences
among the samples. It is an extremely useful
technique concerning research in biology.
The term ANOVA was first proposed by
R.A.Fisher.
It is a different way to summarize the differences
between several means and comparing these means in
one step. This method is called ANOVA or one-way
ANOVA

5. WHY ANOVA?
• In real life things do not result in two groups being
compared.
• Two-sample t-tests are problematic
– Increasing the risk of an error
– At .05 level of significance, with 100 comparisons, 5 will show
a difference when none exists (experiment wise error)
– So the more t-tests you run, the greater the risk of an error
(rejecting the null when there is no difference)
• ANOVA allows us to see if there are differences
between means with an OMNIBUS test
A STATISTICAL TEST

6. HYPOTHESIS OF ANOVA
H0: The (population) means of all groups under
consideration are equal.
Ha: The (pop.) means are not all equal.

7. ASSUMPTIONS IN ANALYSIS
OF VARIANCE
The samples are independently drawn.
The population are normally distributed,
with common variance.
They occur at random and independent of
each other in the groups.
The effects of various components are
additive.

8. WORKING PROCEDURE
The procedure of calculation in direct
method are lengthy as well as time
consuming and this is not popular in
practice for all purposes.
Therefore a short-cut method based on the
squares of the individual values are usually
used.
This method is more convenient.

9. TECHNIQUES OF ANOVA
The analysis of variance has been classified
into
a. One-Way classification
b. Two-Way classification

10. One-Way ANOVA:
Single independent variable is involved.
Example: Effect of pesticide(independent
variable) on the oxygen consumption (dependent
variable) in a sample of insect.
Two-Way ANOVA:
 Two independent variable is involved.
Example: Effects of different levels of
combination of a pesticide(independent variable)
and on insect hormone (dependent variable) on
the oxygen consumption of a sample of insect.

11. ILLUSTRATION
A certain manure was used on four plots of land A,B,C and D. Four beds were
prepared in each plot and manure used. The output of the crop in the beds of plots
A,B,C and D is given below.
Using ANOVA find out whether the difference in the means of the production of
crops of the plots is significant or not.
A B C D
6 15 9 8
8 10 3 12
10 4 7 1
8 7 1 3
ONE-WAY ANOVA

12. • FIND OUT THE MEAN OF EACH SAMPLE
SUM OF THE SAMPLES ÷ NUMBER OF
SAMPLE

13. • FIND COMBINED MEAN

14. • FIND THE SUM OF SQUARES BETWEEN THE
SAMPLES or SS BETWEEN

15. • FIND MS BETWEEN OR MEAN SQUARE
BETWEEN THE SAMPLES
How to find the degree of freedom??

16. FIND SUM OF SQUARE WITHIN THE
SAMPLE OR SS WITHIN

17. SAMPLE X1
X1 X1- X1¯ (X1- X1¯) 2
∑ (X1- X1¯) 2

20. FIND MS WITHIN
DEGREES OF FREEDOM

21. SOURCE OF
VARIATION
SUM OF SQUARE DEGREES OF
FREEDOM
MEAN SQUARE
BETWEEN
SAMPLES
WITHIN SAMPLES
TOTAL

22. MAKING ANOVA TABLE

23. FIND F VALUE

26. INFERENCE

27. TWO-WAY ANOVA
It is used when the data are classified on the basis of two
factors. It is also called two factor analysis of variance.
ILLUSTRATION:
Set up two-way ANOVA table for the following
results. Per acre production data for sorghum.
NAME OF FERTILIZERS VARIETY OF SORGHUM SEEDS
CO.1 CO.5 CO.9
UREA 6 5 5
AMMONIUM SULPHATE 7 5 4
ZINC SULPHATE 3 3 3
POTASH 8 7 4

32. STEP 8
DEGREE OF FREEDOM
c =number of item column, r= number of item row

35. SETTING A TWO WAY ANOVA TABLE

36. Source of
variation
Sum of
square
Df Mean
square
F
calculate
d value
F table
value at
5%
Between
columns
Between
rows
Error
total

40. WHEN ANOVA?
• Data must be experimental
• If you do not have access to statistical software, an
ANOVA can be computed by hand
• With many experimental designs, the sample sizes
must be equal for the various factor level combinations.
• ANOVA formulas change from one experimental design
to another

41. 3 WAY ANOVA
• The three-way ANOVA is used to determine if
there is an interaction effect between three
independent variables on a continuous
dependent variable.
• It is only appropriate to use a three-way ANOVA
if your data "passes" six assumptions that are
required for a three-way ANOVA to give you a
valid result.

42. ASSUMPTIONS
• Assumption #1: Your dependent variable should
be measured at the continuous level (i.e., it is
an interval or ratio variable).
• Assumption #2: Your three independent
variables should each consist of two or more
categorical, independent groups.
• Assumption #3: You should have independence
of observations, which means that there is no
relationship between the observations in each
group or between the groups themselves.

43. • Assumption #4: There should be no significant
outliers.
• Assumption #5: Your dependent variable should
be approximately normally distributed for each
combination of the groups of the three
independent variables.
• Assumption #6: There needs to be homogeneity
of variances for each combination of the groups
of the three independent variables.
• You can check assumptions #4, #5 and #6 using SPSS
Statistics.

44. REFERENCE
• Fundamentals of Mathematical
Statistics Paperback – 2014; S.C. Gupta
• Research Methodology And Statistical
Techniques; Santhosh gupta,2002
• Research methodology- tools and technique; C .
R Kothari