Inferential Statistics

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 • 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. One-way ANOVA Table • 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 https://statistics.laerd.com/spss-tutorials/one-way-anova-using-spss-statistics.php http://academic.udayton.edu/gregelvers/psy216/spss/1wayanova.htm

8. 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. = 0.013 < 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) https://statistics.laerd.com/spss-tutorials/two-way-anova-using-spss-statistics.php

16. Example • Determine the effect of major and gender on the total score H0 : μ1 = μ2 = μ3 = μ4 H1 : μ1 != μ2 != μ3 != μ4

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. = 0.024 < α, gender has effect on total score • Major row Sig. = 0.575 > α, major has no effect on total score • Major*Gender row Sig. = 0.298 > 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