1. Inferential Statistics
Analysis of Variance – ANOVA
Faculty of Information Technology
King Mongkut’s University of Technology North Bangkok

2. Content Estimation Hypothesis testing Forming hypothesis
Testing population means
Testing population variances
Testing categorical data / proportion
Hypothesis about many population means
One-way ANOVA
Two-way ANOVA

3. Analysis of Variance (ANOVA)
Test if any of multiple means are different from each other
One-way ANOVA: 1 variables – 3 or more groups
Dependent variable is assumed is of interval or ratio scale
Also used with ordinal scale data
Can describe the effect of independent variable on dependent variable
Two-way ANOVA: two independent, one dependent variables
MANOVA: Two or more dependent variables
Can describe interaction between two independent variables

4. One-way ANOVA
Test the means (of dependent variable) between groups as specified by an independent variable that are organized in 3 or more groups (dichotomous)
Occupation: Student, Lecturer, Doctor (1 var - 3 groups)
Salary: dependent variable
Assumptions
Dependent variable is either an interval or ratio (continuous)
Dependent variable is approximately normally distributed for each category of the independent variable
There is equality of variances between the independent groups (homogeneity of variances).
Independence of cases.

5. One-way ANOVA Concept Between-Group Variance Within-Group Variance
Total Variance = Between-Group Variance + Within-Group Variance
Between-Group Variance
Describe the difference of means between groups, which is the effect on variable of interest
Within-Group Variance
Describe the difference of means within each group, which is the effect caused by other factors, called Error
H0 : μ1 = μ2 = μ3 = … = μn
H1 : μ1 != μ2 != μ3 != … != μn (at least one different pair)

6. k: number of groups n: number of samples
One-way ANOVA Table
Source of Variance
Degree of Freedom (df)
Sum Square (SS)
Mean Square (MS)
F-ratio
Between Groups
(Treatment)
k-1
Within Groups
(Error)
n-k
Total
n-1
SST = SSB + SSW
k: number of groups n: number of samples
df: degree of freedom

7. One-way ANOVA: SPSS Analyze -> Compare Means -> One-way ANOVA
Option -> Tick…
Homogeneity of variance test
Descriptive (optional)
Welch
Post Hoc - used when the result is significant (at least one of the means is different) to find the group with the different mean

8. At least one pair is different
Example
Determine if the means of total score are different in the 5 Sections
H0 : μ1 = μ2 = μ3 = μ4 = μ5
H1 : μ1 != μ2 != μ3 != μ4 != μ5
At least one pair is different

9. Result: Descriptives and Variances
Check Levene test
“Sig.” > = 0.05, thus variances are equal in all groups
If not, need to refer to the Robust Tests of Equality of Means Table (Welch) instead of the ANOVA Table

10. Result: ANOVA Table
Sig. = < α, thus at least one of the group has different means
Use Post-Hoc tests To find the pair with different mean

11. Result: Post Hoc Tests The pair that Sig. < α has different mean
Section 1 and 4
Section 2 and 4
Section 2 and 5
Section 3 and 4
Section 4 and 5

12. Two-way ANOVA
Use to determine the effect of 2 or more factors (independent variables) on one dependent variable
Occupation: Student, Lecturer, Doctor
Age: less than 20, 20-30, 31-40, 41 or older
Salary: dependent variable
Assumptions
Dependent variable is either interval or ratio (continuous)
The dependent variable is approximately normally distributed for each combination of levels of the two independent variables
Homogeneity of variances of the groups formed by the different combinations of levels of the two independent variables.
Independence of cases

13. Two-way ANOVA Concept Two-way ANOVA compares
Means between columns
Means between rows
Means from the interaction of factors
Sum Square Row (SSR): variation effect of the 1st factor
Sum Square Column (SSC): variation effect of the 2nd factor
Sum Square Row Column (SSRC): variation effect of the interaction of the two factors
Sum Square Error (SSE): Error caused by external factors
Sum Square Total (SST) = SSR + SSC + SSRC + SSE

14. Two-way ANOVA Table r: number of rows c: number of columns
n: number of samples
df: degree of freedom

15. Two-way ANOVA: SPSS
Analyze -> General Linear Model -> Univariate
Multivariate is MANOVA
Add dependent variable and two or more factors (independent variables)
Option -> tick “Homogeneity tests” (optional “Descriptive”)
Plot -> add one factor (containing more groups) to “Horizontal Axis” and other to “Separate Lines” then click “Add”
To obtain profile plot
Post Hoc to find pair that has different means (similar to One-way ANOVA, optional)

16. Example Determine the effect of major and gender on the total score

17. Result Compare Error to Corrected Total
Error should be less than 20% of corrected total
Error is very large compared to corrected total
Total score is effected by other external factors
Gender row Sig. = < α, gender has effect on total score
Major row Sig. = > α, major has no effect on total score
Major*Gender row Sig. = > α, the interaction between two factors has no effect on total score

18. Result: Profile Plot

19. Example
Determine the effect of section and gender on the total score