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f-testnova.htmlPost-Hoc Tests
Topic 3 of 5
Learning Objective:
Interpret the results from a one-way ANOVA.
Post-Hoc Tests
If the overall F-test leads the researcher to reject the null, belief in the null hypothesis is transferred to the alternative. But the alternative merely states that some of the means may not be equal to the others. How can the researcher decide which population means are likely to differ from the others?
A tempting, but ill-advised, approach is to conduct a t-test involving every possible pair of means using a .05 alpha probability. In general, you would conduct a t-test comparing groups 1 and 2, 1 and 3, 1 and 4, 2 and 3, 2 and 4, and 3 and 4. The problem with doing the multiple t-tests is controlling the probability of a type I error. With six independent hypothesis tests and alpha equal to .05 for each individual test, the probability of one or more type I errors for the six tests is actually about 0.26. Hence, if all four population means are equal, the probability of deciding at least two of the means differ from each other is about one in four—certainly much greater than one in twenty, as would be the case with a .05 alpha level.
The Bonferroni Approach
To avoid what is called the inflated probability of a type I error, there are a number of approaches that can be used after an overall F-test is found to be significant. One of the easiest ways of approaching the problem is to use a Bonferroni approach. The approach is rather simple and involves dividing alpha for the family or set of comparisons by the number of hypothesis tests within the set. Thus, if the set of comparisons involves 10 hypothesis tests, divide .05, the probability for one or more type I errors in the set of 10 hypothesis tests, by 10 to get .005. Perform the individual tests using alpha equal to.005 for each test.
The Bonferroni approach is the basis for one type of post-hoc analysis offered in many statistical packages like SPSS. Post-hoc analyses are examined only if the initial F-test for a one-way ANOVA is found to be statistically significant (i.e., the null is rejected). Post-hoc analyses allow researchers to see which means are significantly different from one another. Some statisticians, however, feel that the Bonferroni approach overcorrects for the inflated risk of type I error and therefore is somewhat lacking in statistical power. There are several alternatives to the Bonferroni approach listed in SPSS. They apply different models to control the probability of a type I error for the set of comparisons. The models vary in statistical power and make different assumptions. The focus here will be on interpreting the SPSS output rather than an in-depth discussion of the various types of post-hoc tests.
The results displayed here are based on the same data used in the activity above in which the null hypothesis that four population means are equal was rejected. The results shown in Table 1 are sample means for each of the four cultures. Table 2 shows the results for Turkey’s Honestly Significant Difference Test (HSD), one type of post-hoc analysis.
Table 1: Sample means for each of the four cultures.
| Dependent | |||
|---|---|---|---|
| Cultural Group | Mean | N | Std. Deviation |
| American Indians | 53.8180 | 25 | 8.59169 |
| Inner City | 58.6210 | 25 | 10.93985 |
| Suburban | 62.7753 | 25 | 9.81548 |
| Undocumented | 68.5096 | 25 | 11.69701 |
| Total | 60.9310 | 100 | 11.52910 |
Table 2: Results for Turkey's Honestly Significant Difference Test (HSD).
| Multiple Comparisons | |||||||
| Dependent Variable: Dependent | |||||||
| (I) Culture | (J) Culture | Mean Difference (I-J) | Std. Error | Sig. | 95% Confidence Interval | ||
| Lower Bound | Upper Bound | ||||||
| Tukey HSD | American Indians | Inner City | -4.80308 | 2.92117 | 359 | -12.4408 | 2.8346 |
| Suburban | -8.95739* | 2.92117 | .015 | -16.5951 | -1.3197 | ||
| Undocumented | -14.69168* | 2.92117 | .000 | -22.3294 | -7.0540 | ||
| Inner City | American Indians | 4.80308 | 2.92117 | .359 | -2.8346 | 12.4408 | |
| Suburban | -4.15431 | 2.92117 | .489 | -11.7920 | 3.4834 | ||
| Undocumented | -9.88860* | 2.92117 | .006 | -17.5263 | -2.2509 | ||
| Suburban | American Indians | 8.95739* | 2.92117 | .015 | 1.3197 | 16.5951 | |
| Inner City | 4.15431 | 2.92117 | .489 | -3.4834 | 11.7920 | ||
| Undocumented | -5.73429 | 2.92117 | .209 | -13.3720 | 1.9034 | ||
| Undocumented | American Indians | 14.69168* | 2.92117 | .000 | 7.0540 | 22.3294 | |
| Inner City | 9.88860* | 2.92117 | .006 | 2.2509 | 17.5263 | ||
| Suburban | 5.73429 | 2.92117 | .209 | -1.9034 | 13.3720 | ||
| *. The mean difference is significant at the 0.05 level. |
To interpret the results in Table 2, first look at the number -4.80308. This is the difference in sample means for the American Indians sample and the Inner City sample (remember, the data are fictitious and only being used to illustrate how SPSS results are used). To verify the calculation, subtract 58.6210 (the Inner City sample mean) from 53.6210 (the American Indian mean). Note the result is negative because the Inner City mean is greater than is the American Indian mean.
Now, look at the number 359. This value is the p-value for the individual comparison. Recall that the HSD procedure is controlling alpha (.05) for the set of all comparisons, but you can reject the null for this individual comparison if the p-value (Sig) is less than .05. Hence the decision in the first row is to retain the null that the population mean for American Indians and Inner City groups are equal; that is, you would not reject the null hypothesis and do not have evidence that the means for those two groups are different from one another.
Finally, look at the first row in the columns labeled 95% confidence interval. You can see that the value of 0, which would indicate no mean difference between the two groups, is contained within the confidence interval. This is another indication that there is no evidence that the two means differ from one another.
Decide whether each statement below is consistent with the post-hoc results from SPSS.
Hint: See the “sig.” value that corresponds to the mean comparison being discussed and note whether the “sig.” value (the p-value) is less than .05.
The American Indian and suburban groups show statistically significant different attitudes toward elderly individuals.
Consistent
Not Consistent
SUBMIT Incorrect TAKE AGAINThe inner city and suburban groups show statistically significant different attitudes toward elderly individuals.
Consistent
Not Consistent
SUBMIT Incorrect TAKE AGAINUndocumented individuals have higher scores on the measure used to assess the dependent variable than American Indians.
Consistent
Not Consistent
SUBMIT Incorrect TAKE AGAIN CONTINUE
