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Inferential StatisticsAnalysis of Variance – ANOVA
Faculty of Information TechnologyKing Mongkut’s University of Technology North Bangkok
ContentEstimationHypothesis testing
Forming hypothesisTesting population meansTesting population variancesTesting categorical data / proportionHypothesis about many population means
One-way ANOVA Two-way ANOVA
Analysis of Variance (ANOVA)Test if any of multiple means are different from each other
One-way ANOVA: 1 variables – 3 or more groupsDependent variable is assumed is of interval or ratio scale
Also used with ordinal scale dataCan describe the effect of independent variable on
dependent variableTwo-way ANOVA: two independent, one dependent variablesMANOVA: Two or more dependent variables
Can describe interaction between two independent variables
One-way ANOVATest 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
AssumptionsDependent variable is either an interval or ratio (continuous)Dependent variable is approximately normally distributed for each
category of the independent variableThere is equality of variances between the independent groups
(homogeneity of variances).Independence of cases.
One-way ANOVA ConceptTotal Variance = Between-Group Variance + Within-Group VarianceBetween-Group Variance
Describe the difference of means between groups, which is the effect on variable of interest
Within-Group VarianceDescribe the difference of means within each group, which is
the effect caused by other factors, called ErrorH0 : μ1 = μ2 = μ3 = … = μn
H1 : μ1 != μ2 != μ3 != … != μn (at least one different pair)
One-way ANOVA Table
SST = SSB + SSW
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
k: number of groups n: number of samplesdf: degree of freedom
MSW
MSBF
1k
SSBMSB
kn
SSWMSW
n
TXSSTij
n
i
K
j
22
11
K
j j
jn
iij
K
j n
TXSSW
j
1
2
1
2
1
One-way ANOVA: SPSSAnalyze -> Compare Means -> One-way ANOVAOption -> Tick…
Homogeneity of variance testDescriptive (optional)WelchPost 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.phphttp://academic.udayton.edu/gregelvers/psy216/spss/1wayanova.htm
ExampleDetermine if the means of total score are
different in the 5 SectionsH0 : μ1 = μ2 = μ3 = μ4 = μ5
H1 : μ1 != μ2 != μ3 != μ4 != μ5
At least one pair is different
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
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
Result: Post Hoc TestsThe pair that Sig. < α has
different meanSection 1 and 4Section 2 and 4Section 2 and 5Section 3 and 4Section 4 and 5
Two-way ANOVAUse to determine the effect of 2 or more factors (independent
variables) on one dependent variableOccupation: Student, Lecturer, DoctorAge: less than 20, 20-30, 31-40, 41 or olderSalary: dependent variable
AssumptionsDependent variable is either interval or ratio (continuous)The dependent variable is approximately normally distributed for
each combination of levels of the two independent variablesHomogeneity of variances of the groups formed by the different
combinations of levels of the two independent variables.Independence of cases
Two-way ANOVA ConceptTwo-way ANOVA compares
Means between columnsMeans between rowsMeans from the interaction of factors
Sum Square Row (SSR): variation effect of the 1st factorSum Square Column (SSC): variation effect of the 2nd factorSum Square Row Column (SSRC): variation effect of the
interaction of the two factorsSum Square Error (SSE): Error caused by external factorsSum Square Total (SST) = SSR + SSC + SSRC + SSE
Two-way ANOVA Table r: number of rows c: number of columnsn: number of samplesdf: degree of freedom
Two-way ANOVA: SPSSAnalyze -> General Linear Model -> Univariate
Multivariate is MANOVAAdd 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
ExampleDetermine the effect of major and gender on the total
score H0 : μ1 = μ2 = μ3 = μ4
H1 : μ1 != μ2 != μ3 != μ4
Result
Compare Error to Corrected TotalError should be less than 20% of corrected totalError is very large compared to corrected totalTotal score is effected by other external factors
Gender row Sig. = 0.024 < α, gender has effect on total scoreMajor row Sig. = 0.575 > α, major has no effect on total scoreMajor*Gender row Sig. = 0.298 > α, the interaction between
two factors has no effect on total score
Result: Profile Plot
ExampleDetermine the effect of section and gender on the total
score