1. Analysis and Interpretation Inferential Statistics ANOVA
EDRS6208
Analysis and Interpretation
Inferential Statistics
ANOVA
Madgerie Jameson, UWI SOE

2. OUTLINE Definition of Analysis of Variance
Logic of ANOVA ( the Theory behind ANOVA)
The F test
One way ANOVA
Post Hoc Tests
Interpreting the results

3. Analysis of Variance
Suppose the Ministry of Education decides to test three different methods of teaching Mathematics. After teachers implemented the different methods for a term, the testing and measurement unit wanted to know if the mean scores of students taught with the three different methods are the same.
Questions:
What data would they require?
How will they test for this equity of means?

4. Analysis of Variance Definition
The Analysis of Variance (ANOVA) is statistical model that is used to analyse situations in which we want to compare more than two conditions.
It is used to test the null hypothesis that the mean of three or more populations are equal.

5. Recall opening example
The ministry developed three different methods to teach Mathematics. They want to determine whether the three methods produce different mean scores.
So we test the null Hypothesis
H0 : µ1 = µ2 = µ3 ( all three population means are equal)
H1 : Not all three population means are equal.

6. You Ask Is there an overall average difference?
Is this difference statistically significant?
If so, is the size of the difference managerially significant?
The three methods M2 M1 M3

7. You can Test the three hypotheses
H0: µ1 = µ2 or Ho: µ1 = µ3 or Ho: µ2 =µ3 (using t test)
If you reject even one of the three hypothesis, then you must reject the null Hypothesis “H0 : µ1 = µ2 = µ3”
Combining the type ! Error probabilities for the three tests will give a large type 1 error probability test for H0 : µ1 = µ2 = µ3
The procedure that can test the equality of three means in one test is the ANOVA

8. One Way ANOVA
A procedure to make tests by comparing the means of several population.
In one way ANOVA, we analyse one factor or variable.
Testing the equality of the mean of the Mathematics scores of students who are taught using the three different methods.
One factor is considered the effect size of the different teaching methods.

9. Assumptions of One-Way ANOVA
The following assumptions must hold true to use one-way ANOVA.
The populations from which the samples are drawn are (approximately) normally distributed.
The populations from which the samples are drawn have the same variance (or standard deviation).
The samples drawn from different populations are random and independent.
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10. Using the example of the three teaching methods we must assume:
The scores of all the students taught by each method are ( approximately) normally distributed.
The means of the all three distributions of scores for the three teaching methods may or may not be the same, but all three distributions have the same variance
When we take samples from an ANOVA test these samples are drawn independently and randomly from three different populations.

11. The variance between samples ( mean square between samples MSB).
ANOVA is applied by Calculating two estimates of the variance , of the population distribution
The variance between samples ( mean square between samples MSB).
It gives an estimate of the variance based on the samples taken from different populations e.g. the three teaching methods.
MSB is based on the values of the mean scores of the three samples of students taught by the three different methods.
If the mean of all populations under consideration are equal , the means of the prospective samples will still be different. but the variations among them is expected to be small. However if the means of the population under consideration are not all equal, the variation among the means of respective samples is expected to be large, and consequently, the value of MSB is expected to be large.

12. The variance within samples ( mean square within samples MSW).
It gives an estimate of the variance within the data of different samples.
MSW is based on the scores of individual students included in the three samples taken from the three population.
The concept of MSW is similar to the concept of the pooled standard deviation, Sp

13. Note
The one-way ANOVA test is always right-tailed with the rejection region in the right tail of the F distribution curve.

14. THE F DISTRIBUTION Definition
The F distribution is a continuous curve skewed to the right.
The F distribution has two numbers of degrees of freedom: df for the numerator and df for the denominator.
The units of an F distribution, denoted F, are nonnegative.
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15. For an F distribution, degrees of freedom for the numerator and degrees of freedom for the denominator are usually written as follows.
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16. Three F distribution curves.
The figure shows three f distribution curves for three sets of degrees of freedom for the numerator and denominator. The fist number gives the degrees of freedom associated with the numerator, and the second number gives the degrees of freedom associated with the denominator. Notice as the degrees of freedom increase, the peak of the curve moves to the right, that is, skewness decreases.
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17. Exercise
Find the F value for 8 degrees of freedom for the numerator, 14 degrees of freedom for the denominator, and .05 area in the right tail of the F distribution curve.
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18. Obtaining the F Value using the statistical table
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19. The critical value of F for 8 df for the numerator, 14 df for the denominator, and .05 area in the right tail.
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20. Calculating the Value of the Test Statistic
Test Statistic F for a One-Way ANOVA Test
The value of the test statistic F for an ANOVA test is calculated as
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21. Example
Fifteen form one students were randomly assigned to three groups to experiment with three different methods of teaching Mathematics. At the end of the term, the same test was given to all 15 students. The table gives the scores of students in the three groups.
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22. Calculate the value of the test statistic F
Calculate the value of the test statistic F. Assume that all the required assumptions for ANOVA are assumed to hold true.

23. Solution Let x = the score of a student
k = the number of different samples (or treatments)
ni = the size of sample i
Ti = the sum of the values in sample i
n = the number of values in all samples
= n1 + n2 + n
Σx = the sum of the values in all samples = T1 + T2 + T
Σx² = the sum of the squares of the values in all samples
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24. Calculate MSB and MSW To calculate MSB and MSW, we
first compute the between-samples sum of squares, denoted by SSB and
the within-samples sum of squares, denoted by SSW.
The sum of SSB and SSW is called the
total sum of squares
and is denoted by SST; that is,
SST = SSB + SSW
The values of SSB and SSW are calculated using the following formulas.
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25. Between- and Within-Samples Sums of Squares
The between-samples sum of squares, denoted by SSB, is calculates as
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26. Between- and Within-Samples Sums of Squares
The within-samples sum of squares, denoted by SSW, is calculated as
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27. Let us return to the example
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28. ∑x = T1 + T2 + T3 = 324+369+388 = 1081 n = n1 + n2 + n3 = 5+5+5 = 15
= 80,709
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29. Substitute all the values in the formula for SSB, SSW and SST
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30. Calculating the Values of MSB and MSW
MSB and MSW are calculated as
where k – 1 and n – k are, respectively, the df for the numerator and the df for the denominator for the F distribution. Remember, k is the number of different samples.
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32. Draw the ANOVA Table
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33. ANOVA Table for the Example
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34. Back to the question
The scores of 15 form one students who were randomly assigned to three groups in order to experiment with three different methods of teaching Mathematics.
At the 1% significance level, can we reject the null hypothesis that the mean Mathematics score of all fourth-grade students taught by each of these three methods is the same?
Assume that all the assumptions required to apply the one-way ANOVA procedure hold true.

35. Solution
Step 1:
H0: μ1 = μ2 = μ3 (The mean scores of the three groups are all equal)
H1: Not all three means are equal
Step 2: Because we are comparing the means for three normally distributed populations, we use the F distribution to make this test.

36. A one-way ANOVA test is always right-tailed
Step 3:
α = .01
A one-way ANOVA test is always right-tailed
Area in the right tail is .01
df for the numerator = k – 1 = 3 – 1 = 2
df for the denominator = n – k = 15 – = 12
The required value of F is 6.93
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37. Critical value of F for df = (2,12) and α = .01.
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38. The value of the test statistic F = 1.09
Steps 4 & 5:
The value of the test statistic F = 1.09
It is less than the critical value of F = 6.93
It falls in the nonrejection region
Hence, we fail to reject the null hypothesis
We conclude that the means of the three population are equal.
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39. Example 2
From time to time, unknown to its employees, the research department at Post Bank observes various employees for their work productivity. Recently this department wanted to check whether the four tellers at a branch of this bank serve, on average, the same number of customers per hour. The research manager observed each of the four tellers for a certain number of hours. The following table gives the number of customers served by the four tellers during each of the observed hours.
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40. Result
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41. Question
At the 5% significance level, test the null hypothesis that the mean number of customers served per hour by each of these four tellers is the same. Assume that all the assumptions required to apply the one-way ANOVA procedure hold true.
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42. Solution
Step 1:
H0: μ1 = μ2 = μ3 = μ4 (The mean number of customers served per hour by each of the four tellers is the same)
H1: Not all four population means are equal
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43. Step 2:
Because we are testing for the equality of four means for four normally distributed populations, we use the F distribution to make the test.
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44. A one-way ANOVA test is always right-tailed.
Step 3:
α = .05.
A one-way ANOVA test is always right-tailed.
Area in the right tail is .05.
df for the numerator = k – 1 = 4 – 1 = 3
df for the denominator = n – k = 22 – = 18
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45. Critical value of F for df = (3, 18) and α = .05.
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47. Step 4: Σx = T1 + T2 + T3 + T4 =108 + 87 + 93 + 110 = 398
n = n1 + n2 + n3 + n4 = = 22
Σx² = (19)² + (21)² + (26)² + (24)² + (18)² + (14)² + (16)² + (14)² + (13)² + (17)² + (13)² + (11)² + (14)² + (21)² + (13)² + (16)² + (18)² + (24)² + (19)² + (21)² + (26)² + (20)² = 7614
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48. Substitute all the values for formulas SSB,SSW
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50. ANOVA Table
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51. The value for the test statistic F = 9.69
Step 5:
The value for the test statistic F = 9.69
It is greater than the critical value of F = 3.16
It falls in the rejection region
Consequently, we reject the null hypothesis
We conclude that the mean number of customers served per hour by each of the four tellers is not the same.
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52. Significance of mean effect
When there is a significant difference a post hoc statistic is performed. “post hoc” is a short version of the Latin phrase that translates to “ after this, therefore because of this.” The post hoc test consist of pair wise comparisons that are designed to compare all different combinations of the treatment groups. It takes every pair of groups and perform a t test on each pair of groups.

53. Post hoc results in SPSS
SPSS was used to perform a post hoc test on the results of the previous example.
The F test revealed difference among the four groups.
The results of the post hoc are as follows.

54. Teller
Mean Difference
Std. Error
Sig
95% confidence level
Lower Bound
Upper Bound
Teller A Teller B
Teller C
Teller D
7.100*
6.100*
-.400
1.795
1.895
.005
.015
.995
2.03
1.03
-5.70
12.17
11.17
4.90
Teller B Teller A
-7.100*
-1.000
-7.500*
1.712
.936
.003
-12.17
-5.84
-12.57
-2.03
3.84
-2.43
Teller C Teller A
Teller B
-6.100*
1.000
-6.500*
1.995
.010
-11.17
-3.84
-11.57
-1.03
5.84
-1.43
Teller D Teller A
.400
7.500*
6.500*
1.875
.996
-4.90
2.43
1.43
5.79
12.57
11.57
* Mean difference is significant at the .05 level

55. Tukey HSD
This test display subsets of groups that have the same means.
The Tukey test creates two subsets of groups with statistically similar means.
Teller N
Subset 1 2 A B C D
Sig 6 5
14.50
15.50
.943
21.60
22.00
.996

56. In Class exercise