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.