Analysis of Variance

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13 Statistics for Business and Management

Two-way ANOVA

In a factorial design we examine the influence of two independent variables (factors) on a dependent variable.

SST is the total sum of square difference for each observation from the total mean.

One-way ANOVA: SST= SSW+SSB

Two-way ANOVA: SST= SS(A)+ SS(B) + SS(AB) + SSE

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Two-way ANOVA

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Sum of squared difference for each observation from the total mean

Sum of squared difference for each cell from the expected mean under the assumption of no interaction (multiplied by the number of observations in each cell)

Sum of squared difference for each row's mean from the total mean (multiplied by the number of observations in each row)

Sum of squared difference for each column's mean from the total mean (multiplied by the number of observations in each column)

Sum of squared

difference for each observation from its cell mean

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15 םיקסע להנמב םיטסיטטס םילדומ

Two-way ANOVA

SSA:

I

2

nJ ∑(xi∙∙ − x∙∙∙ )

i=1

SSB:

J

2

nI ∑(x∙ j∙ − x∙∙∙ )

j =1

I J

2

SS(AB): n∑∑(xij∙ − xi∙∙ − x∙ j∙ + x∙∙∙ )

i=1 j =1

2

I J nij

SSE:

∑∑∑(xijk − xij∙ )

i=1

I J

j =1 k =1

nij 2

SST:

∑∑∑(xijk − x∙∙∙ )

i=1 j =1 k =1

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Two-way ANOVA

Source of Variation

Sum of Squares SS

Degrees of Freedom df

Mean Sum of Squares MS

Ratio of MSS F score

Significance test – P-value

Factor A

SSBA

a-1

SSBA/(a-1)=MSBA

MSBA/MSW

Prob(F(a-1),(N-ab)>Fc)

Factor B

SSBB

b-1

SSBB/(b-1)=MSBB

MSBB/MSW

Prob(F(b-1),(N-ab)>Fc)

Interaction

SS(AB)

(a-1)(b-1)

SS(AB)/(a-1)(b-1)=MS(AB)

MS(AB)/MSW

Prob(F(a-1)(b-1),(N-ab)>Fc)

Error

SSW

N- (ab)

SSW/N- (ab)=MSW

Total

SST

N-1

SST/N-1=MST

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17 םיקסע להנמב םיטסיטטס םילדומ

Example 3

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In a study that aims to examine the influence of gender and education on the number of times people use facebook a day collocated the following data:[pic 36]

Use 0.05 significance level to test whether gender, education, and the interaction between them influence the number of times people use facebook a day.

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18 Statistics for Business and Management

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Example 3

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19 Statistics for Business and Management

Example 3

Source of Variation

Sum of Squares SS

Degrees of Freedom df

Mean Sum of Squares

MS