Chapter 5 Analysis of variance
SPSS –Analysis of variance
Data file used: gss.sav
How to get there: Analyze
Compare Means
…
One-way ANOVA …
To test the null hypothesis that several population means are equal, based on the results of several
independent samples. The test variable is measured on an interval- or ratio scale (for example age), and is
grouped by a variable which can be measured on a nominal or discrete ordinal scale (for example life
existing of the categories Dull, Routine and Exciting).
An independent T test and a one-way ANOVA for two independent samples test the same hypothesis.
You must select the dependent variable, and specify the factor to define the different groups. You can move
more than one variable into the Dependent List to test all of them. See following figure.
Button Options …
Here you can choose to get descriptives of the data (Descriptive), and to test for equal variances in the
groups (Homogeneity-of-variance).
Button Post Hoc …
To see if, and if yes which, groups differ among themselves, there are several possibilities. You can use the
Bonferroni procedure (see following figure) when there are equal variances in the groups, which can be
tested with the Homogeneity-of-variance test (Button Options).
Output of running one-way ANOVA
We performed a one-way ANOVA, with age as dependent variable, and life as factor, which exists of the
groups: 0 = “Not applicable” 1 = “Dull”
2 = “Routine” 3 = “Exciting” 8 = “Don’t know”
Oneway
Descriptives
Age of Respondent
Dull
Routine
Exciting
Total
N
Mean
52,62
47,28
44,54
46,33
65
457
471
993
Std. Deviation
20,059
18,191
16,106
17,479
Std. Error
2,488
,851
,742
,555
95% Confidence Interval for
Mean
Lower Bound Upper Bound
47,64
57,59
45,61
48,95
43,08
46,00
45,24
47,42
Minimum
19
19
18
18
Maximum
89
89
87
89
Test of Homogeneity of Variances
Age of Respondent
Levene
Statistic
8,287
df1
2
df2
990
Sig.
,000
ANOVA
Age of Respondent
Between Groups
Within Groups
Total
Sum of
Squares
4492,439
298568,2
303060,6
df
2
990
992
Mean Square
2246,220
301,584
F
7,448
Sig.
,001
The table ‘Descriptives’ speaks for itself.
In the table ‘Test of Homogeneity of Variances’ you find the result of Levene’s Test for Equality of
Variances. It tests the condition that the variances of both samples are equal, indicated by the Levene
Statistic. In this statistic, a high value results normally in a significant difference, in this example that is
Sig. = 0,000. Strictly speaking, the Bonferroni procedure can therefore not be used, as it assumes equal
variances.
However, we are dealing with large samples, which reduces the problem, and the Bonferroni test can be
used and interpreted with care.
In the table ‘ANOVA’ the variation (Sum Of Squares), the degrees of freedom (df), and the variance (Mean
Square) are given for the within and the between groups, as well as the F value (F) and the significance of
the F (Sig.). Sig. indicates whether the null hypothesis – the population means are all equal – has to be
rejected or not.
As you can see, there is much difference between the two Mean Squares (2246,220 and 301,584), resulting
in a significant difference (F = 7,448; Sig. = 0,001). This means that H0 must be rejected. Thus: the average
age of people who find life Dull, Routine, or Exciting are not all equal.
But we don’t know yet which means differ from each other. Rejecting a null-hypothesis means that
NOT ALL population means differ. We don’t know yet whether one or more means vary from each other!
Therefore, we perform the Bonferoni procedure. The output shows us:
Post Hoc Tests
Multiple Comparisons
Dependent Variable: Age of Respondent
Bonferroni
(I) Is life exciting or dull
Dull
Routine
Exciting
(J) Is life exciting or dull
Routine
Exciting
Dull
Exciting
Dull
Routine
Mean
Difference
(I-J)
5,34
8,08*
-5,34
2,74*
-8,08*
-2,74*
Std. Error
2,302
2,298
2,302
1,140
2,298
1,140
Sig.
,062
,001
,062
,049
,001
,049
95% Confidence Interval
Lower Bound Upper Bound
-,18
10,86
2,57
13,59
-10,86
,18
,01
5,48
-13,59
-2,57
-5,48
-,01
*. The mean difference is significant at the .05 level.
The table ‘Multiple Comparisons’ shows that two out of three groups vary:
Dull vs Routine Sig. = 0,062 which is higher than the Sig. level of 0.05. These groups do not vary.
Dull vs Exciting Sig. = 0,001 which is lower than the Sig. level of 0.05. These groups vary.
Exciting vs Routine
Sig. = 0,049 which is lower than the Sig. level of 0.05. These groups vary (although
only just).
Because the Bonferroni test assumes equal variances, which does not hold in this case, you can do a test
that does not assume equal variances, for example the Tamhane’s T2 test. We you perform this test, you
will see that in this case it results in the same conclusions.
How to get there: Analyze
General Linear Model
…
A General Linear Model is, as the name suggest, general in that it incorporates many different models, so
that many different tests can be performed. Among these, are the one- and two-way ANOVA, and
regression analyses. More difficult designs can be analysed as well.
Univariate …
A univariate GLM is a test with only one dependent variable. There can be one or more independent
variable or factors and/or variables.
A one-way ANOVA is a univariate GLM with exactly one independent variable (e.g. fixed factor).
A two-way ANOVA is a univariate GLM with exactly two independent variables (e.g. fixed factors).
You can test null hypotheses about the effects of other variables on the means of various groupings of a
single dependent variable. You can investigate interactions between factors as well as the effects of
individual factors. Also, the effects of covariates and covariate interactions with factors can be included.
In a univariate GLM you must select in the source variable list the variable you want to test and move it
into the Dependent Variable Box. You can select only one dependent variable. Then, select the variables
(factors) whose values define the groups and move them into the Fixed Factor box (or Random Factor(s)
box if appropriate). If you have covariates, you move them into the Covariate(s) box. To obtain the default
univariate GLM that contains all main effects and interactions, click OK. See following figure.
Button Plots …
To obtain plots so that you can examine interaction visually. In the Factors list box, you see all main factors
in your model. To plot the means for the values of a singe factor, move that factor into the Horizontal Axis
box and click Add. If you want to see the means for all combinations of values of two factors, move the
second factor into the Separate Lines Box and click Add. The horizontal-axis factor means will be plotted
separately for each value of the second factor. If you move a third factor into the Separate Plots box,
separate plots for each value of this factor will be produced. See following figure.
Button Post Hoc …
When an overall F test has shown significance, you can use post hoc tests to evaluate differences among
specific means.
To pinpoint differences between all possible pairs of values of a factor variable, select the factors to be
tested. Move them to the ‘Post Hoc Tests for:’ list box. Additionally, select one of the multiple comparison
procedures, for example Bonferroni or Tukey. Many different tests are available. For information about a
test, point to its name in the dialog box and click the right mouse button. See following figure.
Output of running univariate GLM
Univariate Analysis of Variance
Between-Subjects Factors
RS Highest
Degree
0
1
2
Respondent's
Sex
3
4
1
2
Value Label
Less than
HS
N
80
High school
480
Junior
college
Bachelor
Graduate
Male
Female
67
181
92
450
450
Tests of Between-Subjects Effects
Dependent Variable: Number of Hours Worked Last Week
Source
Corrected Model
Intercept
DEGREE
SEX
DEGREE * SEX
Error
Total
Corrected Total
Type III Sum
of Squares
22211,324a
918896,468
7409,113
9296,790
551,792
170221,266
1762191,000
192432,590
df
9
1
4
1
4
890
900
899
Mean Square
2467,925
918896,468
1852,278
9296,790
137,948
191,260
a. R Squared = ,115 (Adjusted R Squared = ,106)
F
12,904
4804,440
9,685
48,608
,721
Sig.
,000
,000
,000
,000
,577
Profile Plots
Estimated Marginal Means of Number of Hours Worked Last Week
60
Estimated Marginal Means
50
40
Respondent's Sex
Male
30
Less than HS High school Junior college Bachelor
Female
Graduate
RS Highest Degree
The ‘Profile Plot’ is a line plot for the average hours worked. Is there an interaction between gender and
education? You see that the two lines don’t cross. The shapes of the lines for males and females are quite
similar. That suggests that there is no interaction. You don’t expect pairs of lines drawn from real data to
have exactly the same shape, even if there is no interaction between gender and education in the population.
Your goal is to determine whether the interaction observed in the sample is large enough to believe that it
also exists in the population.
The table ‘Between-Subjects Factors’ speaks for itself.
The table ‘Tests of Between-Subjects Effects’ is very similar to the one-way analysis of variance table
ANOVA. What has changed is the number of hypotheses you are testing. In one-way analysis of variance,
you tested a single hypotheses. Now you can test three hypotheses: one about the main effect of degree,
one about the main effect of gender, and one about the degree-by-gender interaction.
Mean Square for Degree = the variability of the sample means of the five degree groups
Mean Square for Gender = the variability of the sample means of the two gender groups
Error Mean Square = the variability of the observations within the 10 cell means, that is 5 (Degree) x 2
(Gender). It is a kind of ‘Within Groups Mean Square’.
Remember: If the null hypotheses for an effect is true, then the corresponding F ratio is expected to be 1.
You look at the observed significance level for each observed F ratio to see if you can reject the
corresponding null hypotheses.
Always, you first have to look at possible interaction effect, since it doesn’t make sense to talk about main
effects if there is significant interaction between the factors.
In this example, Sig. = 0.577, so you validate the null hypotheses that there is no interaction between the
two variables. The effect of the type of degree on hours worked seems to be similar for males and females.
The absence of interaction tells you that it’s reasonable to believe that the difference in average hours
worked between males and females is the same for all degree categories.
Since you didn’t find an interaction between degree and gender, it makes sense to test hypotheses about the
main effects of degree and gender.
The F statistic for the degree main effect is 9,685. The observed significance level is 0,000. This means that
H0 must be rejected. The variable Degree has influence on the average hours worked. A posthoc test will
reveal more about the differences in degree.
The F statistic for the Gender main effect is 48,608. The observed significance level is 0,000. This means
that H0 must be rejected. The variable Gender has influence on the average hours worked.