Tuesday, March 4, 2014

One-Way Analysis of Variance with SPSS

The purpose of this paper is to state the statistical assumptions of the one-way analysis of variance with SPSS, select independent and dependent variables, develop the null and alternative hypotheses, calculate an ANOVA, and include a post hoc test, report on the p value, interpret the confidence interval, and reject or retain the null hypothesis. In addition, syntax and output files will be presented with an analysis of the results.

Statistical Assumptions of this Test

The statistical assumptions of this test are that the observations within each sample must be independent of each other; the populations from which the samples are drawn are normal; and the populations have equal variances (Green & Salkind, 2014).

The independent variable chosen for this assignment was race, which is a variable with three or more levels that include White, Black, and Other Races. The dependent variable is income.

The null hypothesis is:

H0: µ1 = µ2 = µ3; There are no differences in income between groups (different races). Race does not have a significant effect on income.

An alternative hypothesis is:

H0: µ1 ≠: µ2 ≠ µ3; There is a difference among the three groups. Race has an effect on income.

Interpretation of Tests

A one-way analysis of variance was conducted to evaluate the relationship between race and income. The independent variable, race included three levels: White, Black, and Other Races. The dependent variable was the level of income. The ANOVA was significant at the .05 level, F(2, 1338) = 13.75, p = 0. Since p > .05, we cannot reject the null hypothesis, which states that race has no effect on income. In this case, race has an effect on income. The strength of the relationship between income and race as assessed by ɳ2, was weak (.02), with race accounting for 2% of the variance of the dependent variable.

I conducted follow-up tests to evaluate pairwise differences among the means. Because the variances among the three groups ranged from 4.74 to 8.49, I decided not to assume that the variances were homogeneous and conducted post hoc comparisons with the use of the Dunnett's C test, a test that does not assume equal variances among the three groups. There was a significant difference among the means. Therefore, race significantly affects income. According to the Tukey HSD and Dunnett C Post Hoc tests, there was not a significant difference between the incomes of White and Black participants, although there was a significant difference between the incomes of Whites and other races and between the incomes of Blacks and other races.

SPSS Syntax and Output Files

UNIANOVA INCOME BY RACE
/METHOD=SSTYPE(3)
/INTERCEPT=INCLUDE
/POSTHOC=RACE(TUKEY QREGW C)
/EMMEANS=TABLES(RACE)
/PRINT=ETASQ HOMOGENEITY DESCRIPTIVE
/CRITERIA=ALPHA(.05)
/DESIGN=RACE.

Univariate Analysis of Variance
[DataSet1] C:\Users\Deborah\Desktop\Stats\gss04student_corrrected.sav
 Between-Subjects Factors Value Label N RACE OF RESPONDENT 1 WHITE 1052 2 BLACK 194 3 OTHER 95

 Descriptive Statistics Dependent Variable:   TOTAL FAMILY INCOME RACE OF RESPONDENT Mean Std. Deviation N WHITE 11.06 2.177 1052 BLACK 10.12 2.913 194 OTHER 10.74 2.606 95 Total 10.90 2.351 1341 Levene's Test of Equality of Error Variancesa Dependent Variable:   TOTAL FAMILY INCOME F df1 df2 Sig. 15.598 2 1338 .000 Tests the null hypothesis that the error variance of the dependent variable is equal across groups. a. Design: Intercept + RACE

 Tests of Between-Subjects Effects Dependent Variable:   TOTAL FAMILY INCOME Source Type III Sum of Squares df Mean Square F Sig. Partial Eta Squared Corrected Model 149.163a 2 74.582 13.746 .000 .020 Intercept 61258.850 1 61258.850 11290.745 .000 .894 RACE 149.163 2 74.582 13.746 .000 .020 Error 7259.427 1338 5.426 Total 166844.000 1341 Corrected Total 7408.591 1340 a. R Squared = .020 (Adjusted R Squared = .019)

Estimated Marginal Means
 RACE OF RESPONDENT Dependent Variable:   TOTAL FAMILY INCOME RACE OF RESPONDENT Mean Std. Error 95% Confidence Interval Lower Bound Upper Bound WHITE 11.064 .072 10.923 11.205 BLACK 10.119 .167 9.790 10.447 OTHER 10.737 .239 10.268 11.206
Post Hoc Tests

RACE OF RESPONDENT
 Multiple Comparisons Dependent Variable:   TOTAL FAMILY INCOME (I) RACE OF RESPONDENT (J) RACE OF RESPONDENT Mean Difference (I-J) Std. Error Sig. 95% Confidence Interval Lower Bound Upper Bound Tukey HSD WHITE BLACK .95* .182 .000 .52 1.37 OTHER .33 .250 .390 -.26 .91 BLACK WHITE -.95* .182 .000 -1.37 -.52 OTHER -.62 .292 .086 -1.30 .07 OTHER WHITE -.33 .250 .390 -.91 .26 BLACK .62 .292 .086 -.07 1.30 Dunnett C WHITE BLACK .95* .220 .43 1.46 OTHER .33 .276 -.33 .98 BLACK WHITE -.95* .220 -1.46 -.43 OTHER -.62 .339 -1.42 .19 OTHER WHITE -.33 .276 -.98 .33 BLACK .62 .339 -.19 1.42 Based on observed means.  The error term is Mean Square(Error) = 5.426. *. The mean difference is significant at the .05 level.

Homogeneous Subsets
 TOTAL FAMILY INCOME RACE OF RESPONDENT N Subset 1 2 Tukey HSDa,b,c BLACK 194 10.12 OTHER 95 10.74 WHITE 1052 11.06 Sig. 1.000 .377 Ryan-Einot-Gabriel-Welsch Rangec BLACK 194 10.12 OTHER 95 10.74 10.74 WHITE 1052 11.06 Sig. .068 .334 Means for groups in homogeneous subsets are displayed.  Based on observed means.  The error term is Mean Square(Error) = 5.426. a. Uses Harmonic Mean Sample Size = 180.380. b. The group sizes are unequal. The harmonic mean of the group sizes is used. Type I error levels are not guaranteed. c. Alpha = .05.

References

Green, S. B., & Salkind, N. J. (2014). Using SPSS for Windows and Macintosh: Analyzing and understanding data (7th ed.). Upper Saddle River, NJ: Pearson Education.