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.










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