According to the health belief model, group 3 would be expected to rate the ben- efit of exercise the highest, followed by group 2 and then group 1.
SPSS Commands for a One-Way ANOVA
The computation for ANOVA can be found in statistical texts such as Kleinbaum, Kupper, Muller, and Nizam (1997). Here we only show the SPSS commands. It is im- portant to remember that the ANOVA procedure requires that the independent vari- able be categorical (that is, nominal or ordinal) and that the dependent variable be continuous (that is, interval or ratio).
Interpreting a One-Way ANOVA
As noted previously, an ANOVA tests the difference between (or among) group means.
The statistic computed and evaluated to determine whether differences exist is the F test.
An F test that is statistically significant indicates that there is a better-than-chance dif- ference between the mean scores of at least two of the groups. If there are more than two groups and the overall F test is statistically significant, a follow-up (or post hoc) test is conducted to identify the specific group differences. Though a number of post hoc test options exist, the choice of a particular test should be based on the characteristics of the distributions under consideration. For simplicity, Tukey’s post hoc test is presented.
To conduct a one-way ANOVA:
From the Data Editor Screen:
√Analyze. . . .Compare Means. . . .√One-Way ANOVA Highlight and transfer Xvariable into the Factor box Highlight and transfer Yvariable into the Dependent List box
√Options (dialog box)
√Descriptive
√Homogeneity of Variance Test
√Continue
√Post-Hoc (dialog box)
√Tukey
√Continue
√OK
Table 8.6 presents the first table from the SPSS output. This table, labeled Descriptives, provides group and overall descriptive statistics for the dependent vari- able (in this case, benefit rating). From a strictly descriptive perspective, the means of the three groups appear to be different. In particular, the mean benefit rating for group 1 is 3.14, that for group 2 is 6.22, and that for group 3 is 8.07.
Table 8.7, labeled Test of Homogeneity of Variances, provides the results of Levene’s test, which can be used to determine whether the homogeneity-of-variance as- sumption is tenable. The null hypothesis for this test is that the variances across the groups are equal. In this situation we want to retain the null hypothesis; therefore, we are look- ing for a p value >.05. Table 8.7 shows that the Levene statistic of .769 is not statistically significant (p = .466), and we can accept the null hypothesis that the variances are equal.
TABLE 8.6 SPSS PRINTOUT USING THE ONE-WAY ANOVA DESCRIPTIVE COMMAND.
Descriptives
BENEFIT
1.00 2.00 3.00 Total
36 36 28 100
3.1389 6.2222 8.0714 5.6300
N Mean Lower Bound Minimum Maximum
95% Confidence Interval for Mean
Upper Bound 2.33180
2.35568 2.17611 3.04065
2.3499 5.4252 7.2276 5.0267
3.9279 7.0193 8.9152 6.2333
.00 2.00 2.00 .00
9.00 10.00 10.00 10.00 .38863
.39261 .41125 .30407 Std.
Deviation Std.
Error Mean scores for benefit (dependent variable) by exercise group
Exercise group (independent variable)
TABLE 8.7. SPSS PRINTOUT FOR TEST OF HOMOGENEITY OF VARIANCES.
Levene Statistic
.769 2 df1
97 df2
Test of Homogeneity of Variances
Look for a p value >.05 .466
Sig.
BENEFIT
Table 8.8, labeled ANOVA, gives the results of the one-way ANOVA test. Re- member, the F test in this case is an omnibus test; that is, the F test provides evidence that there is at least one pairwise difference between the groups under consideration.
The p value for the overall F test is in the last column of the table. A p <.05 indicates that there is a statistically significant difference between at least two of the groups. In Table 8.8, the F statistic is 38.139 and the p value is <.001, indicating that the means for at least two of the groups are statistically different from each other.
Table 8.9, labeled Multiple Comparisons, presents the results of the post hoc com- parison (in this example, using Tukey’s HSD [honestly significantly different] test). The Multiple Comparisons table provides information for all pairwise comparisons (that is, each group mean is compared to every other group mean). If the p value for any row in this table were less than the designated value (for example, .05), the difference between the means for the two groups under consideration would be considered a sta- tistically significant difference (or a better-than-chance difference). As noted in Table 8.9, the mean differences between all comparisons are statistically significant. To find the mean difference, subtract each group listed on the right in the first column from the group listed on the left in this column. So for the first row, subtract 6.222 (the mean of group 2) from 3.1389 (the mean for group 1) to obtain the mean difference
TABLE 8.8. SPSS PRINTOUT OF ANOVA SUMMARY TABLE.
Between Groups
Sum of Squares
Mean Square
Within Groups Total
402.925 512.385 915.310
201.463 5.282 2
97 99 df
ANOVA
38.139 .000 F Sig.
F ratio: calculated by dividing the mean square between by the mean square within
Variation is partitioned into variability due to group differences (between) and variability due to individual differences (within) within each group
Degrees of freedom
p value associated with the observed F statistic (p < .05), indicating a statistically significant result—reject the null that the group means are equal
BENEFIT
TABLE 8.9. SPSS PRINTOUT SHOWING COMPARISONS AMONG THE GROUPS.
*The mean difference is significant at the .05 level.
(I) 1=little, 2=
some, 3 = a lot
(J) 1=little, 2=
some, 3 = a lot Lower
Bound
Upper Bound 95% Confidence Interval
Multiple Comparisons
p values identify the statistically significant pairwise group differences
To obtain results in column 2 subtract the mean of the group listed to the right from that listed on the left Dependent Variable: BENEFIT Tukey HSD
Mean Difference
(I-J) Std. Error
1.00 2.00
3.00
2.00 1.00
3.00
3.00 1.00
2.00
.54172 .57912 .54172 .57912 .57912 .57912
.000 .000 .000 .005 .000 .005
−3.0833*
−4.9325*
3.0833*
−1.8492*
4.9325*
1.8492*
−4.3728
−6.3110 1.7939
−3.2277 3.5541 .4708
−1.7939
−3.5541 4.3728
−.4708 6.3110 3.2277 Sig.
of –3.0833. By examining the direction of the mean differences as noted in the first column, we see that group 2 (some exercise) and group 3 (a lot of exercise) have higher mean scores than group 1 (little exercise) and that group 3 has a higher mean score than group 2. These results suggest that participants who exercise the most rate the most benefit of exercise. Those who exercise a moderate amount fall somewhere in between.
In summary, then, our researcher can say that there is a difference in perceived benefit of exercise among participants based on the number of hours they exercise each week. Those participants who exercise the most report the most benefit. Those participants who report exercising a moderate amount each week rate the benefit of exercise lower than do those reporting more than seven hours of exercise per week, but higher than do those reporting only a few hours of exercise per week.
Summary
In this chapter, we have presented a review of some statistical concepts that are im- portant to understand in order to conduct assessments for reliability and validity of scales. Many of the tests used to evaluate reliability and validity are based on corre- lation. Some validity assessments also require the use of analysis of variance. The information presented in this chapter should be sufficient to understand the analyses presented in later chapters. However, students who wish to read more about correla- tion and about ANOVA and other statistical tests are referred to the texts by Klein- baum, Kupper, Muller, and Nizam (1997), Norman and Streiner (2000), and Tabachnick and Fidell (2001). Students who wish to read more about SPSS can con- sult the SPSS manual or a how-to text such as that by Cronk (1999).
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LEARNING OBJECTIVES
At the end of this chapter, the student should be able to
1. Understand the relationship between random error and reliability.
2. Define the terms observed score, true score, and error score.
3. Define the term reliability.
4. Describe the theoretical equation representing the relationship among the observed, true, and error scores.
5. Use the theoretical equation to compute observed-score, true-score, and error-score variance.
6. Use the theoretical equation to compute the reliability coefficient.
7. Understand the relationship between the observed score and true score and the reliability coefficient.
8. State the relationship between the correlation of two parallel tests and reliability.
9. Use the reliability coefficient to determine the proportion of observed-score vari- ance due to true-score and error-score variance.
10. Use the reliability coefficient to determine the reliability index.