- The document discusses two-way analysis of variance (ANOVA), which analyzes data with two independent variables. - It explains the assumptions, factors, hypotheses, and calculations involved in a two-way ANOVA. Main effects and interactions between the independent variables are examined. - An example is provided to illustrate a two-way ANOVA comparing two teaching methods (A and B) across different school types. The analysis tests for main effects of method and school type, and their interaction.

1. ACHARYA NARENDRA DEVA UNIVERSITY OF AGRICULTURE &
TECHNOLOGY, KUMARGANJ, AYODHYA (U.P.) 224229
Assignment
on
ANOVA Two way
Course No : STAT-502 4(3+1)
Course name : Statistical methods for applied sciences
Presented to : Presentedby :
Dr. Vishal Mehta Dev Narayan Yadav
Assistant Professor Id. No. A-11144/19/22
Department of Agril. Statistics Ph. D. 1st Semester
Soil Science and Agril. Chemistry

2. ANOVA
 What is ANOVA ?
 The ANOVA investigates independent measurement from several
treatment or levels of one or more than one factors.
 The techniques of ANOVA consists of proportion the total sum of
squares due to different factor and the error.
 It is basically of two types ;
1.ONE WAY ANOVA
2. TWO WAY ANOVA

3. ASSUMPTIONS;
 The two or more categories ,
 Independent group are founded .
 The dependent variable should be measured the
continuous level.
 There should be no significant outliers.
 There need to be homogeneity of variance of each
combination of the groups of the two independent
variable.

4. TWO WAY ANOVA
 Two way ANOVA is used for data analysis when
you have two independent variable (Two –way) and
2 or more levels of either or both independent
variable.
 The two way ANOVA compares the mean difference
between group that mean difference between
groups that have been split into one two
independent variable.

5.  The primary purpose of a two way ANOVA is
to understand if there is an interaction between
the two independent variable on the dependent
variable
 EXAMPLE : We could use a two way ANOVA two
understand whether there is an interaction
between gender and education level on test ,are
your independent variable ,and text anxiety is
your dependent variable

6. The Two-way ANOVA
 Main effect
 A main effect is the effect on performance
of one treatment variable considered in
isolation (ignoring other variables in the
study)
 Interaction
 An interaction effect occurs when the
effect of one variable is different
across levels of one or more other
variables

7. Assumptions for the Two
factor ANOVA
1. Observation within each sample are
independent
2. Populations are normally or approximation
normally distribution
3. Population from which the samples are
selected must have equal variance
(homogeneity of variance )
4. The groups must have the same sample size

8. FACTORS
 The two independent variable in a two-way
ANOVA are called factor
 The idea is that there are two variable factor,
which affect the dependent variable
 Each factor will have two or more levels within it
 The degrees of freedom for each factor is one
less than the number of levels

9. Hypotheses
 There are three sets of hypotheses with the two
way ANOVA#
 The null hypotheses for each of the sets are given
below
 The population mean of the first factor are equal .
This is like the one way ANOVA for the row factor.
 The population mean of the second factor are
equal . This is like the one way ANOVA for the
column factor
 There is no interaction between the two factors

10. TWO- WAY ANOVA HYPOTHESIS
TEST FOR A
 H0: No difference among means for levels of A
 HA: At least two A means differ significantly
 Test statistic: F =
𝑴𝑺𝑨
𝑴𝑺𝑬
 Rej. region: Fobt< F(2, 12, .05) = 3.89
 Decision: Reject H0 – variable A has an effect.

11. Two-way ANOVA – hypothesis
test for B
 H0: No difference among means for levels of B
 HA: At least two B means differ significantly
 Test statistic: F =
𝑴𝑺𝑩
𝑴𝑺𝑬
 Rej. region: Fobt< F(1, 12, .05) = 4.75
 Decision: Reject H0 – variable B has an effect

12. Two-way ANOVA – hypothesis
for AB
 H0: A & B do not interact to affect mean response
 HA: A & B do interact to affect mean response
 Test statistic: F =
𝑴𝑺𝑨𝑩
𝑴𝑺𝑬
 Rej. region: Fobt< F(2, 12, .05) = 3.89
 Decision: Reject H0 – A & B do interact...

13. Two-way ANOVA – Example
 (a) Do the appropriate analysis to answer
the questions posed by the researcher
(all αs = .05)
 (b) The London School Board is currently
using Method B and, prior to this
experiment, had been thinking of
changing to Method A because they
believed that A would be better. At α =
.01, determine whether this belief is
supported by these data

14. Two-way ANOVA – Example
 HO: μA – μB = 0
 HA: μA – μB > 0
 Test statistic: t = 𝑿𝑨−𝑿𝑩 −𝟎
𝑴𝑺
𝑬
𝟏
𝒏𝟏
+
𝟏
𝒏𝟐
 𝒕𝒄𝒓𝒊𝒕=𝒕 𝟐𝟒,𝟎𝟏 =2.492

15. Two-way ANOVA – Example

16. Two-way ANOVA – Example
𝑆𝑆𝐵=
40252+43252+43852
10
-CM =7440.2
∑ 𝑇𝑖𝑗
2
𝑛𝑖𝑗
=
19102
5
+
43252
10
+ …………+
21802
5
+
22052
5
=5380634.2

17. Solution the Example -
 = 1896𝑺𝑺𝑨𝑩 = 5380634.2 – 5409528.33
– 5413447.5 + 5406007.5
=1646.667
 𝑺𝑺𝒕𝒐𝒕𝒂𝒍 = ΣX2 – CM = 5437581.0 –
5406007.5
 = 31573.5
 𝑺𝑺𝑬 = 𝑺𝑺𝒕𝒐𝒕𝒂𝒍 – 𝑺𝑺𝑨 – 𝑺𝑺𝑩 – 𝑺𝑺𝑨𝑩6.0

18. Two-way ANOVA TABLE–
Example
 Source d.f. SS MS F
 A 1 3520.83 3520.83 4.46*
 B 2 7440.00 3720.00 4.71*
 AB 2 1646.67 823.33 1.04
 Error 24 18966.0 790.25
 Total 29 31573.5
 * Reject H0.