LECTURE 8

1. LECTURE 8 One Way ANOVA

2. One-way ANOVA • In the previous lecture, we looked at using the two sample t-test when you have data from two sample and wish to know whether the mean of the two population from which the samples are drawn are the same or different. • What happen when we have more than two sample and want to compare more than 2 means? • A one-way analysis of variance (ANOVA) is appropriate when you wish to compare the means of more than two sample.

3. The Hypothesis • ANOVA is one of the test used in hypothesis testing. • In ANOVA the F-ratio is calculated to obtain the p-value and a significant p-value (< level of significant ) tells us that the population means are probably not all equal. • The hypothesis for ANOVA in general is • H0 : The means are all equal. • H1 : At least one of the means are not equal.

4. One-way between groups ANOVA with post-hoc comparisons • Used when samples are independent. • If we reject the null hypothesis, we need to identify which ones are significantly different and this requires a post-hoc analysis. • There are a number of post hoc test available for example Scheffe test which allows you to perform every possible comparison but is tough on rejecting the null hypothesis. • In contrast, Tukey’s honestly significant difference (HSD) test is more lenient, but you are restricted in terms of the types of comparison that you can make. For this course, we will be using the HSD post-hoc test.

5. Assumption testing • Before you can conduct the ANOVA you must ensure that the necessary assumptions are met. The assumptions for ANOVA are • Population normality – population from which the samples have been drawn should be normal. • Homogeneity of variance – the scores in each group should have homogeneous (equal) variances. • This assumption can be tested using the Levene’s test where the H0 is variances is equal and H1 is variances are unequal.

6. Example • An economist wished to compare household expenditure on electricity and gas in 4 major towns (KK, tawau, Lahad Datu, Sandakan) in Sabah. • She obtained random samples of 25 households from each town and asked them to keep records of their expenditure over a 6 month period. • This is an independent group design because different households are in different towns. • Since we are comparing more than 2 means and the samples are independent we will conduct a one-way between groups ANOVA with post-hoc comparisons. • The hypothesis • H0: The mean expenditure for the 4 towns are equal • H1: At least one of the mean expenditure for 4 towns is unequal

7. Output Variances are equal H0 is rejected

8. Output These 2 towns have significantly different means

9. Conclusion • Based on the Levene’s test, the p-value is 0.488 that is greater than significant level 0.05, therefore we do not reject H0 and conclude that the assumption of homogeneity of variance is not violated. • Based on the ANOVA, the p-value is 0.013 that is less than significant level 0.05, therefore we reject H0 and conclude that at leastone of the mean expenditure for 4 towns are unequal. • Based on the Tukey’s HSD, the only significant p-value is the one between Kota Kinabalu and Lahad Datu, therefore we concluded that there is a significant difference between the mean expenditure between Kota Kinabalu and Lahad Datu while there is no significant difference for the other combinations.

10. One-way repeated measures ANOVA • Used when the samples are dependent, particularly when the same participants perform under different conditions. • There are 3 assumptions underlying the repeated measure ANOVA • Normality – populations should be normal • Homogeneity of Variance – Assessed by obtaining the F-max which is the largest variance divided by the smallest variance. If this ratio is greater than 3, then the assumption has been violated. • Sphericity – Assessed by the Mauchy’s Test of Spericity. If the p-value is significant (< 0.05), then this this assumption has been violated.

11. Example • You wish to determine whether practice enhances ability to solve anagrams. 8 participants were asked to solve as many anagrams as possible in 10 minutes. • They were then allowed to practice for an hour before being asked to complete another 10 minute timed task. • Participants were then given another practice session and another timed task. The number of anagrams correctly solved was recorded. • Since the same participants were used under 3 different condition and we want to see if there is any significant difference between the 3 sample means, a one-way repeated measure ANOVA is used. • The hypothesis • H0: The mean number of anagrams solved are equal for all 3 sessions • H1: At least 1 of the 3 sessions have unequal mean number of anagrams solved.

12. Output Reject H0, Spericity is violated

13. Output Reject H0

14. Conclusion • The F-max is less than 3 (14.01472/6.36402), therefore the assumption of homogeneity of variances is not violated. • Based on the Mauchy test of Sphericity, the p-value (0.016<0.05 )is significant, therefore the assumption of sphericity is violated. • Since sphericity is not assumed we will base our test on the Huynh-Feldt correction. • The p-value for the F-ratio is 0.006 that is less than significant level 0.05, therefore we reject H0 and conclude that at least 1 of the 3 sessions have unequal mean number of anagrams solved. • A further post-hoc analysis maybe conducted to determine which mean is different but this will not be discussed in this course.