Analysis of Variance( ANOVA )

What is the F-test

Another parametric test used to compare the means of two or more groups of independent samples.

It is also known as the analysis of variance (ANOVA)

Three kinds of ANOVA

One-way anova

Two-way anova

Three-way anova

One-way ANOVA

Used when there is only one variable involved

Two-way ANOVA

Used when two variables are involved: the column and the row variables

Why do we use the F-test?

To find out if there is a significant difference between and among the means of the two or more independent groups.

When do we use the F-test?

When there is normal distribution and when the level of measurement is expressed in interval or ratio data just like the t-test and the z-test

How do we use the F-test?

 

Sources of variation

df SS MS

Computed Tabular

Between Groups

K-1 BSS BSS df

MSB = F MSW

See the table at .05

Within Group

(N-1)-(K-1) WSS WSS df

w/ df between and w/in group

Total N-1 TSS

ANOVA Table

* MSB – mean squares between* MSW – mean square within

F - Value

FCv FTv ; disconfirm H₀FCv confirm H₀

A sari-sari store is selling 4 brands of shampoo. The owner is interested if there is a significant difference in the average sales of the four brands of shampoo for one week. The following data are recorded.

A B C D

7 9 2 4

3 8 3 5

5 8 4 7

6 7 5 8

9 6 6 3

4 9 4 4

3 10 2 5

Brand

Example :

F-test One-way ANOVA

7 9 2 4

3 8 3 5

5 8 4 7

6 7 5 8

9 6 6 3

4 9 4 4

3 10 2 5

A B C D

Brand

∑= n₁= =

∑= n ₄ = =

∑= n ₃ = =

∑= n ₂ = =

∑= ∑= ∑= ∑=

ANOVA table

Sources of variation

df SS MS

Computed Tabular

Between Groups (K-1)

Within Group(N-1)-(K-1)

Total (N-1)

F - Value

Scheffé’s Test To find out where the differences lies, another test must

be used, the Scheffé’s Test.

The F-test tells us there is a significant difference in the average sales of the 4 brands of shampoo but as to where the differemce lies, it has to be tested further by another test

(-)²F’ =SW ² (n₁ + n) n₁ n

Where: F’ = Scheffé’s test = mean of group 1₂ = mean of group 2 n₁ = number samples in group 1 n= number samples in group 2 Sw² = within mean squares

Between Brand

F’ (F .05)(K-1)(3.01)(3)

Interpretation

A vs B 9.51 significant

A vs C

A vs D

B vs C

B vs D

C vs D

Example : A vs BF’ =

=

=

F’ = 9.51

Example 2 :

The following data represent the operating time in hours of the 3 types of scientific pocket calculators before a recharge is required. Determine the difference in the operating time of the three calculators. Do the ANOVA at .05 level of significance.

F

4.9 6.4 4.8

5.3 6.8 5.4

4.6 5.6 6.7

6.1 6.5 7.9

4.3 6.3 6.2

6.9 6.7 5.3

5.3 5.9

4.1

4.3

Brand

∑= n₁= =

∑= n ₃ = =

∑= n ₂ = =

∑= ∑= ∑=

F-test two-way ANOVA with interaction effect

Involves 2 variables; column and row variables

Used to find out if there is an interaction effect betwwen 2 variables.

Example 1.

Forty-five language students were randomly assigned to one of three instructors and to one of the three methods of teaching. Achievement was measured on a test administered at the end of the term. Use the two-way AVOVA with interaction effect at .05 LoS to test the ff. hypothesis.

Two-factor ANOVA w/ Significant interaction

TEACHER FACTORA B C40 50 4041 50 4140 48 4039 48 3838 45 38

Total40 45 5041 42 4639 42 4338 41 4338 40 42

Total40 40 4043 45 4141 44 4139 44 3938 43 38

TotalGrand Total

Method of Teaching 1

Method of Teaching 2

Method of Teaching 3

Stepwise method:

Statistics

F-test Two-way ANOVA with interaction effect

TEACHER FACTOR(column)A B C40 50 4041 50 4140 48 4039 48 3838 45 38

Total 198 241 19740 45 5041 42 4639 42 4338 41 4338 40 42

Total 196 210 22440 40 4043 45 4141 44 4139 44 3938 43 38

Total 201 216 199Grand Total 595 667 620

Method of Teaching 1

Method of Teaching 2

Method of Teaching 3

(row)

(row)

(row)

∑= 636

∑= 616

∑= 630

1,882

 

II.

dftotal = N-1dfwithin = k(n-1)dfcolumn = c-1dfrow = r-1dfc r = (c-1)(r-1)

Sources of

variation

df SS MS

Computed Tabular Interpretation

Between Columns 2 178.18

89.09

Rows 2 14.05 7.02

Interaction

4 187.15 46.75

Within 36 129.20 3.59

Total 44 508.58

ANOVA Table

F - Value

FVc: Columns =

Row =

Interaction =

FVt: Columns df =

Row df =

Interaction df =

Sources of

variation

df SS MS

Computed Tabular Interpretation

Between Columns 2 178.18

89.09 24.82 3.26 S

Rows 2 14.05 7.02 1.95 3.26 NS

Interaction

4 187.15 46.75 13.03 2.63 S

Within 36 129.20 3.59

Total 44 508.58

ANOVA Table

F - Value

Decision Rule : FCvFTv; disconfirm

Conclusion :

With the FCv (column) of 24.82 compared to the FTv of 3.26 @ .05 LoS with 2 and 36 df, the is disconfirmed in favor of the research hypothesis which means that there is significant difference in the performance of the 3 groups of students under the 3 different instructors. It implies that instructor B is better than instructor A.With regard to the FCv (row) of 1.95, which it is the FTv of 3.26 @ .05 LoS with 2 and 36 df, the hypothesis of no significant differences in the performance of the students under the 3 different methods of teaching is confirmedHowever, the FCv (interaction) of 13.03 is the FTv of 2.63 @ .05 LoS with $ and 26 df. Thus, the research hypothesis is confirmed which means that there is an interaction effect is present between the instructors and their methods of teaching. Students under instructor B have better performance under methods of teachin 1 and 3, while students under instructor C have better performance under method 2.

FIN..

Thank you !!!