TWO WAY ANOVA

1. Dr PRIYANKA MISHRA

2. What is Two-Way ANOVA?
Assumptions of Two-Way ANOVA
One Example
Objectives
Hypothesis
Data Analysis
Conclusion

3.  Two- Way ANOVA is an extension of One-Way ANOVA.
 Here we have two independent variables and one dependent variable
 Here two independent variables are studied. That’s why it is called Two –
Way ANOVA.
 The independent variables can have number of categories called levels or
factors.
 Examples of dependent variable in ANOVA are Sales, Performance,
Emotional Intelligence, Health, opinion on any attribute etc.
 Examples of Independent variable are Gender, cities, academic qualification,
colors, position etc.

4.  Scale: The independent variables should be in categorical
scale(nominal or ordinal) and the dependent variable has to be in
continuous scale(interval or ratio).
 Independence: The data should be independent of each other i.e. the
data of one group doesn’t influence the other group
 Normality: The data should be normally distributed
 Homogeneity of variance: variance of all groups should be equal
 Group sizes should be same: each group should have same number
of respondents
 The residuals are also normally distributed

5. A company wants to find out whether the Sales of their
product is influenced by their place of respondents and
the education of the respondents. The company selects a
sample of 27 respondents.
Place: Mumbai, Delhi, Pune
Education: Undergraduate, Graduate, Post Graduate

6. To study whether place influence sales
To study whether education influences sales
To study whether education influences sales in different cities

7.  H01: Sales of product do not differ in different places.
 H11: Sales of product differ in different places.
 H02: Sales of product do not differ based on the education of respondents
 H12: Sales of product differ based on the education of respondents
 H03: Sales of product do not differ with education level of respondents belonging
to different places.
 H13:Sales of product differ with education level of respondents belonging to
different places.

8. Hypothesis 1: Main Effect
Hypothesis 2: Main Effect
Hypothesis 3: Interaction Effect

10. Tests of Normality
Place Kolmogorov-Smirnova Shapiro-Wilk
Statistic df Sig. Statistic df Sig.
Sales
Mumbai .205 9 .200* .955 9 .741
Delhi .188 9 .200* .916 9 .357
Pune .209 9 .200* .820 9 .034
*. This is a lower bound of the true significance.
a. Lilliefors Significance Correction
Tests of Normality
Education Kolmogorov-Smirnova Shapiro-Wilk
Statistic df Sig. Statistic df Sig.
Sales
Undergraduate .250 8 .150 .918 8 .416
Graduate .137 12 .200* .955 12 .716
Post graduate .175 7 .200* .915 7 .429
*. This is a lower bound of the true significance.
a. Lilliefors Significance Correction

11. Tests of Normality
Place Kolmogorov-Smirnova
Shapiro-Wilk
Statistic df Sig. Statistic df Sig.
Residual for Sales
Mumbai .198 9 .200*
.912 9 .332
Delhi .186 9 .200*
.899 9 .244
Pune .211 9 .200*
.895 9 .222
*. This is a lower bound of the true significance.
a. Lilliefors Significance Correction
Tests of Normality
Education Kolmogorov-Smirnova Shapiro-Wilk
Statistic df Sig. Statistic df Sig.
Residual for Sales
Undergraduate .147 8 .200* .931 8 .521
Graduate .248 12 .039 .855 12 .042
Post graduate .137 7 .200* .985 7 .981
*. This is a lower bound of the true significance.
a. Lilliefors Significance Correction

12. The Shapiro-Wilk significant value is >0.05. which proves
that the data is normal.
The Shapiro-Wilk significant value is >0.05 for residuals
also. Hence the residuals are also normally distributed.

13.  The Levene’s Test shows that the significant value is 0.084
which implies that homogeneity of variance is not
significant(p>0.05).
 That means that the error variance across all groups are equal.
 Hence the assumption of homogeneity of variance is
satisfying.
Levene's Test of Equality of Error Variancesa
Dependent Variable: Sales
F df1 df2 Sig.
2.152 8 18 .084
Tests the null hypothesis that the error variance of the dependent variable is equal across groups.
a. Design: Intercept + Place + Education + Place * Education

14. Since the p>0.05 for place, so the null hypothesis (H01) is accepted,
and p<0.05 for education, the null hypothesis(H02)is rejected and
since the p>0.05 for the interaction effect, the null hypothesis (H03)
is accepted.
Tests of Between-Subjects Effects
Dependent Variable: Sales
Source Type III Sum of
Squares
df Mean Square F Sig. Partial Eta
Squared
Corrected Model 4638.957a
8 579.870 4.000 .007 .640
Intercept 30867.022 1 30867.022 212.948 .000 .922
Place 639.603 2 319.802 2.206 .139 .197
Education 3584.455 2 1792.228 12.364 .000 .579
Place * Education 92.752 4 23.188 .160 .956 .034
Error 2609.117 18 144.951
Total 38801.000 27
Corrected Total 7248.074 26
a. R Squared = .640 (Adjusted R Squared = .480)

15. Multiple Comparisons
Dependent Variable: Sales
Tukey HSD
(I) Education (J) Education Mean Difference
(I-J)
Std. Error Sig. 95% Confidence Interval
Lower Bound Upper Bound
Undergraduate
Graduate -13.17 5.495 .068 -27.19 .86
Post graduate -32.14*
6.231 .000 -48.05 -16.24
Graduate
Undergraduate 13.17 5.495 .068 -.86 27.19
Post graduate -18.98*
5.726 .010 -33.59 -4.36
Post graduate
Undergraduate 32.14*
6.231 .000 16.24 48.05
Graduate 18.98*
5.726 .010 4.36 33.59
Based on observed means.
The error term is Mean Square(Error) = 144.951.
*. The mean difference is significant at the .05 level.

16. Multiple Comparisons
Dependent Variable: Sales
Tukey HSD
(I) Place (J) Place Mean
Difference (I-J)
Std. Error Sig. 95% Confidence Interval
Lower Bound Upper Bound
Mumbai
Delhi 4.89 5.676 .671 -9.60 19.37
Pune 13.56 5.676 .069 -.93 28.04
Delhi
Mumbai -4.89 5.676 .671 -19.37 9.60
Pune 8.67 5.676 .302 -5.82 23.15
Pune
Mumbai -13.56 5.676 .069 -28.04 .93
Delhi -8.67 5.676 .302 -23.15 5.82
Based on observed means.
The error term is Mean Square(Error) = 144.951.

17.  Using the Tukey HSD further, we can conclude that there is no difference
in the responses of respondents with education level of undergraduate
and graduate.
 But there is a significant difference in when the education level of
respondents are (postgraduate and graduate) and (post graduate and
undergraduate).
 In the post hoc test of place, the data is not significant in any combination
of place. That means the responses are same irrespective of cities.

18. From the above data analysis, it is concluded that, there is
a significant difference in the sales of the product with
different education. No effect on sales with respondents
from different cities. Even the interaction effect is also not
significant.
All assumptions of Two-Way ANOVA are being met in the
case.