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Psy 521/621 Univariate Quantitative Methods, Fall 2020 1
ANOVA Example
Below is the output for the SPSS ONEWAY procedure to compare the means of three school types in the
hypothetical teacher satisfaction example.^1 This is a pretty small sample size per group and such a
small sample is not necessarily recommended. It is certainly legitimate to do an ANOVA with this size
sample, but one should be particularly conscious of unequal variances.
Syntax
means tables=satisfaction by school.
oneway satisfaction by school
/statistics=welch.
Menus
Analyze—Compare Means—One-way ANOVA
Oneway
ANOVA
satisfaction
df Mean Square F Sig.
Between Groups
Within Groups
Total
Robust Tests of Equality of Means
satisfaction
Statistic a^ df1 df2 Sig.
Welch 14.259 2 7.211.
a.
Unequal group sizes, very different variances, and small sample sizes can increase Type I error rates (or
sometimes increase Type II error rates; Clinch & Keselman, 1982; Tomarken & Serlin, 1986). If the ratio
of variances in different groups is less than 4 to 1 and the group sizes are over 5, there is generally a
very minor impact on Type I error rates. If data are very non-normal, variances are very unequal, and/or
the group sample sizes are small and unequal, the Welch's robust test is an alternative that will generally
have a more accurate Type I error rate.
(^1) It is also possible to do the same one-way ANOVA using several other procedures in SPSS, including the GLM through the menus or GLM,
ANOVA, or MANOVA (syntax only) commands. Use of the GLM procedure, glm satisfaction by school., will give the eta-squared
(shown as R -squared in output), but the Welch's test is not an option.
Psy 521/621 Univariate Quantitative Methods, Fall 2020 2
Although the group sizes are equal, the small overall sample size and the relatively substantial
differences in the standard deviations (and therefore larger differences in the variances) suggest that it
may be wise to consider the Welch's test in this case (Welch, 1951). The statistical conclusion does not
differ from the standard F -test, however (this could be noted in a write-up, where either or both is
reported, if this comes up). With larger N s, as in most of the applications you will encounter, the
correction would likely be unnecessary.
R
library(lessR)
#do this at the beginning of your run and always run it before you open a data set rm(mydata)
library(haven) mydata = read_sav("c:/jason/spsswin/uvclass/school anova.sav")
#note: you may need to add data=mydata after brief=TRUE library(lessR) ANOVA(satisfaction ~ school, brief=TRUE)
BACKGROUND
Response Variable: satisfaction
Factor Variable: school Levels: 1 2 3
Number of cases (rows) of data: 15 Number of cases retained for analysis: 15
DESCRIPTIVE STATISTICS
n mean sd min max 1 5 6.00 2.00 4.00 8. 2 5 9.00 1.00 8.00 10. 3 5 6.00 0.71 5.00 7.
Grand Mean: 7
BASIC ANALYSIS
Summary Table df Sum Sq Mean Sq F-value p-value school 2 30.00 15.00 8.18 0. Residuals 12 22.00 1.
Association and Effect Size
R Squared: 0. R Sq Adjusted: 0. Omega Squared: 0. Cohen's f: 0.
Write-up
Teacher satisfaction differed significantly among the three groups, F (2,12) = 8.18, p = .006. Because of
the substantial differences in variances among groups and the small sample size, the Welch's test was
conducted. The results from the Welch's test also indicted a significant difference among the groups, F W
(2,12) = 14. 26, p = .003.^2 Teachers were most satisfied in charter schools (M = 9.00, SD = 1.00), and
less satisfied in public (M = 6.00, SD=2.00) and private schools (M = 6.00, SD = .71). Approximately 58%
of the variance in satisfaction was accounted for by school type, η^2 = .58.
References
Welch, B. L. (1951). On the comparison of several mean values: An alternative approach. Biometrika, 38, 330-336.
(^2) The Welch test sometimes may be referred to as W. It is not technically and F , but the F distribution is used as the referent distribution when
determining significance.
Psy 521/621 Univariate Quantitative Methods, Fall 2020 4
Family-wise Confidence Level: 0.
diff lwr upr p adj
2-1 3.00 0.72 5.28 0.
3-1 0.00 -2.28 2.28 1.
3-2 -3.00 -5.28 -0.72 0.
Write-up
The Tukey post hoc tests indicated that teacher satisfaction in the charter schools differed significantly
from teacher satisfaction in public, p = .01, and private schools, p = .01. The Games-Howell test,
however, suggested that the difference between public and charter schools was only marginally
significant, p = .056.
