Calculating Analysis of Variance (ANOVA) and Post Hoc Analyses Follo.docx

Calculating Analysis of Variance (ANOVA) and Post Hoc
Analyses Following ANOVA
Analysis of variance (ANOVA)
is a statistical procedure that compares data between two or
more groups or conditions to investigate the presence of
differences between those groups on some continuous dependent
variable (see
Exercise 18
). In this exercise, we will focus on the
one-way ANOVA
, which involves testing one independent variable and one
dependent variable (as opposed to other types of ANOVAs, such
as factorial ANOVAs that incorporate multiple independent
variables).
Why ANOVA and not a
t
-test? Remember that a
t
-test is formulated to compare two sets of data or two groups at
one time (see
Exercise 23
for guidance on selecting appropriate statistics). Thus, data
generated from a clinical trial that involves four experimental
groups, Treatment 1, Treatment 2, Treatments 1 and 2
combined, and a Control, would require 6
t
-tests. Consequently, the chance of making a Type I error (alpha
error) increases substantially (or is inflated) because so many
computations are being performed. Specifically, the chance of
making a Type I error is the number of comparisons multiplied
by the alpha level. Thus, ANOVA is the recommended
statistical technique for examining differences between more
than two groups (
Zar, 2010
).

ANOVA is a procedure that culminates in a statistic called the
F
statistic. It is this value that is compared against an
F
distribution (see
Appendix C
) in order to determine whether the groups significantly differ
from one another on the dependent variable. The formulas for
ANOVA actually compute two estimates of variance: One
estimate represents differences between the groups/conditions,
and the other estimate represents differences among (within) the
data.
Research Designs Appropriate for the One-Way ANOVA
Research designs that may utilize the one-way ANOVA include
the randomized experimental, quasi-experimental, and
comparative designs (
Gliner, Morgan, & Leech, 2009
). The independent variable (the “grouping” variable for the
ANOVA) may be active or attributional. An active independent
variable refers to an intervention, treatment, or program. An
attributional independent variable refers to a characteristic of
the participant, such as gender, diagnosis, or ethnicity. The
ANOVA can compare two groups or more. In the case of a two-
group design, the researcher can either select an independent
samples
t
-test or a one-way ANOVA to answer the research question. The
results will always yield the same conclusion, regardless of
which test is computed; however, when examining differences
between more than two groups, the one-way ANOVA is the
preferred statistical test.
Example 1: A researcher conducts a randomized experimental
study wherein she randomizes participants to receive a high-
dosage weight loss pill, a low-dosage weight loss pill, or a
placebo. She assesses the number of pounds lost from baseline
to post-treatment

378
for the three groups. Her research question is: “Is there a
difference between the three groups in weight lost?” The
independent variables are the treatment conditions (high-dose
weight loss pill, low-dose weight loss pill, and placebo) and the
dependent variable is number of pounds lost over the treatment
span.
Null hypothesis: There is no difference in weight lost among the
high-dose weight loss pill, low-dose weight loss pill, and
placebo groups in a population of overweight adults.
Example 2: A nurse researcher working in dermatology
conducts a retrospective comparative study wherein she
conducts a chart review of patients and divides them into three
groups: psoriasis, psoriatric symptoms, or control. The
dependent variable is health status and the independent variable
is disease group (psoriasis, psoriatic symptoms, and control).
Her research question is: “Is there a difference between the
three groups in levels of health status?”
Null hypothesis: There is no difference between the three
groups in health status.
Statistical Formula and Assumptions
Use of the ANOVA involves the following assumptions (
Zar, 2010
):
1.
Sample means from the population are normally distributed.
2.
The groups are mutually exclusive.
3.
The dependent variable is measured at the interval/ratio level.
4.
The groups should have equal variance, termed “homogeneity of

variance.”
5.
All observations within each sample are independent.
The dependent variable in an ANOVA must be scaled as interval
or ratio. If the dependent variable is measured with a Likert
scale and the frequency distribution is approximately normally
distributed, these data are usually considered interval-level
measurements and are appropriate for an ANOVA (
de Winter & Dodou, 2010
;
Rasmussen, 1989
).
The basic formula for the
F
without numerical symbols is:
F
=
Mean Square Between Groups
Mean Square Within Groups
The term “mean square” (
MS
) is used interchangeably with the word “variance.” The
formulas for ANOVA compute two estimates of variance: the
between groups variance and the within groups variance. The
between groups variance
represents differences between the groups/conditions being
compared, and the
within groups variance
represents differences among (within) each group's data.
Therefore, the formula is
F
= MS between/MS within.
Hand Calculations

Using an example from a study of students enrolled in an RN to
BSN program, a subset of graduates from the program were
examined (
Mancini, Ashwill, & Cipher, 2014
). The data are presented in
Table 33-1
. A simulated subset was selected for this example so that
379
the computations would be small and manageable. In actuality,
studies involving one-way ANOVAs need to be adequately
powered (
Aberson, 2010
;
Cohen, 1988
). See
Exercises 24
and
25
for more information regarding statistical power.
TABLE 33-1
MONTHS FOR COMPLETION OF RN TO BSN PROGRAM BY
HIGHEST DEGREE STATUS
Participant #
Associate's
Participant #
Bachelor's
Participant #
Master's
Degree
Degree
Degree
1
17
10
16
19

17
2
19
11
15
20
21
3
24
12
16
21
20
4
18
13
12
22
21
5
24
14
16
23
12
6
24
15
12
24
16
7
16
16
16
25

20
8
16
17
12
26
18
9
20
18
10
27
12
The independent variable in this example is highest degree
obtained prior to enrollment (Associate's, Bachelor's, or
Master's degree), and the dependent variable was number of
months it took for the student to complete the RN to BSN
program. The null hypothesis is “There is no difference between
the groups (highest degree of Associate's, Bachelor's, or
Master's) in the months these nursing students require to
complete an RN to BSN program.”
The computations for the ANOVA are as follows:
Step 1: Compute correction term,
C
.
Square the grand sum (
G
), and divide by total
N
:
C
=
460
2

27
=
7
,
837.04
Step 2: Compute Total Sum of Squares.
Square every value in dataset, sum, and subtract
C
:
(
17
2
+
19
2
+
24
2
+
18
2
+
24
2

+
16
2
+
16
2
+
…
+
12
2
)
−
7
,
837.04
=
8
,
234
−
7
,
837.04
=
396.96
Step 3: Compute Between Groups Sum of Squares.
Square the sum of each column and divide by

N.
Add each, and then subtract
C
:
178
2
9
+
125
2
9
+
157
2
9
−
7
,
837.04
(
3
,
520.44
+
1
,
736.11

+
2
,
738.78
)
−
7
,
837.04
=
158.29
Step 4: Compute Within Groups Sum of Squares.
Subtract the Between Groups Sum of Squares (Step 3) from
Total Sum of Squares (Step 2):
396.96
−
158.29
=
238.67
380
Step 5: Create ANOVA Summary Table (see
Table 33-2
).
a.
Insert the sum of squares values in the first column.
b.
The degrees of freedom are in the second column. Because the
F
is a ratio of two separate statistics (mean square between
groups and mean square within groups) both have different
df

formulas—one for the “numerator” and one for the
denominator:
Mean square between groups
d
f
=
number of groups
−
1
Mean square within groups
d
f
=
N
-number of groups
For this example
,
the
d
f
for the numerator is
3
−
1
=
2
.
The
d
f
for the denominator is
27
−

3
=
24
.
c.
The mean square between groups and mean square within
groups are in the third column. These values are computed by
dividing the
SS
by the
df
. Therefore, the
MS
between = 158.29 ÷ 2 = 79.15. The
MS
within = 238.67 ÷ 24 = 9.94.
d.
The
F
is the final column and is computed by dividing the
MS
between by the
MS
within. Therefore,
F
= 79.15 ÷ 9.94 = 7.96.
TABLE 33-2
ANOVA SUMMARY TABLE
Source of Variation
SS
df
MS
F

Between Groups
158.29
2
79.15
7.96
Within Groups
238.67
24
9.94
Total
396.96
26
Step 6: Locate the critical
F
value on the
F
distribution table (see
Appendix C
) and compare it to our obtained
F
= 7.96 value. The critical
F
value for 2 and 24
df
at α = 0.05 is 3.40, which indicates the
F
value in this example is statistically significant. Researchers
report ANOVA results in a study report using the following
format:
F
(2,24) = 7.96,
p
< 0.05. Researchers report the exact

p
value instead of “
p
< 0.05,” but this usually requires the use of computer software
due to the tedious nature of
p
value computations.
Our obtained
F
= 7.96 exceeds the critical value in the table, which indicates
that the
F
is statistically significant and that the population means are not
equal. Therefore, we can reject our null hypothesis that the
three groups spent the same amount of time completing the RN
to BSN program. However, the
F
does not indicate which groups differ from one another, and
this
F
value does not identify which groups are significantly different
from one another. Further testing, termed multiple comparison
tests or post hoc tests, is required to complete the ANOVA
process and determine all the significant differences among the
study groups.
Post Hoc Tests
Post hoc tests
have been developed specifically to determine the location of
group differences after ANOVA is performed on data from more
than two groups. These tests were developed to reduce the
incidence of a Type I error. Frequently used post hoc tests are
the Newman-Keuls test, the Tukey Honestly Significant
Difference (HSD) test, the Scheffé test, and the Dunnett test (
Zar, 2010
; see
Exercise 18

for examples). When these tests are
381
calculated, the alpha level is reduced in proportion to the
number of additional tests required to locate statistically
significant differences. For example, for several of the
aforementioned post hoc tests, if many groups' mean values are
being compared, the magnitude of the difference is set higher
than if only two groups are being compared. Thus, post hoc
tests are tedious to perform by hand and are best handled with
statistical computer software programs. Accordingly, the rest of
this example will be presented with the assistance of SPSS.
SPSS Computations
The following screenshot is a replica of what your SPSS
window will look like. The data for ID numbers 24 through 27
are viewable by scrolling down in the SPSS screen.
382
Step 1: From the “Analyze” menu, choose “Compare Means”
and “One-Way ANOVA.” Move the dependent variable, Number
of Months to Complete Program, over to the right, as in the
window below.
Step 2: Move the independent variable, Highest Degree at
Enrollment, to the right in the space labeled “Factor.”
Step 3: Click “Options.” Check the boxes next to “Descriptive”
and “Homogeneity of variance test.” Click “Continue” and
“OK.”
383
Interpretation of SPSS Output
The following tables are generated from SPSS. The first table
contains descriptive statistics for months to completion,
separated by the three groups. The second table contains the
Levene's test of homogeneity of variances. The third table
contains the ANOVA summary table, along with the
F

and
p
values.
The first table displays descriptive statistics that allow us to
observe the means for the three groups. This table is important
because it indicates that the students with an Associate's degree
took an average of 19.78 months to complete the program,
compared to 13.89 months for students with a Bachelor's and
17.44 months for students with a Master's degree.
One Way
The second table contains the Levene's test for equality of
variances. The Levene's test is a statistical test of the equal
variances assumption. The
p
value is 0.488, indicating there was no significant difference
among the three groups' variances; thus, the data have met the
equal variances assumption for ANOVA.
The last table contains the contents of the ANOVA summary
table, which looks much like
Table 33-2
. This table contains an additional value that we did not
compute by hand—the exact
p
value, which is 0.002. Because the SPSS output indicates that
we have a significant ANOVA, post hoc testing must be
performed.
384
Return to the ANOVA window and click “Post Hoc.” You will
see a window similar to the one below. Select the “LSD” and
“Tukey” options. Click “Continue” and “OK.”
The following output is added to the original output. This table
contains post hoc test results for two different tests: the LSD

(Least Significant Difference) test and the Tukey HSD
(Honestly Significant Difference) test. The LSD test, the
original post hoc test, explores all possible pairwise
comparisons of means using the equivalent of multiple
t
-tests. However, the LSD test, in performing a set of multiple
t
-tests, reports inaccurate
p
values that have not been adjusted for multiple computations (
Zar, 2010
). Consequently, researchers should exercise caution when
choosing the LSD post hoc test following an ANOVA.
The Tukey HSD comparison test, on the other hand, is a more
“conservative” test, meaning that it requires a larger difference
between two groups to indicate a significant difference than
some of the other post hoc tests available. By requiring a larger
difference between the groups, the Tukey HSD procedure yields
more accurate
p
values of 0.062 to reflect the multiple comparisons (
Zar, 2010
).
385
Post Hoc Tests
Observe the “Mean Difference” column. Any difference noted
with an asterisk (*) is significant at
p
< 0.05. The
p
values of each comparison are listed in the “Sig.” column, and
values below 0.05 indicate a significant difference between the
pair of groups. Observe the
p
values for the comparison of the Bachelor's degree group

versus the Master's degree group. The Tukey HSD test indicates
no significant difference between the groups, with a
p
of 0.062; however, the LSD test indicates that the groups
significantly differed, with a
p
of 0.025. This example enables you see the difference in results
obtained when calculating a conservative versus a lenient post
hoc test. However, it should be noted that because an
a priori
power analysis was not conducted, there is a possibility that
these analyses are underpowered. See
Exercises 24
and
25
for more information regarding the consequences of low
statistical power.
Final Interpretation in American Psychological Association
(Apa) Format
The following interpretation is written as it might appear in a
research article, formatted according to APA guidelines (
APA, 2010
). A one-way ANOVA performed on months to program
completion revealed significant differences among the three
groups, F(2,24) = 7.96,
p
= 0.002. Post hoc comparisons using the Tukey HSD
comparison test indicated that the students in the Associate's
degree group took significantly longer to complete the program
than the students in the Bachelor's degree group (19.8 versus
13.9 months, respectively) (
APA, 2010
). However, there were no significant differences in program
completion time between the Associate's degree group and the
Master's degree group or between the Bachelor's degree group
and the Master's degree group.

386
Study Questions
1.
Is the dependent variable in the
Mancini et al. (2014)
example normally distributed? Provide a rationale for your
answer.
2.
What are the two instances that must occur to warrant post hoc
testing following an ANOVA?
3.
Do the data in this example meet criteria for homogeneity of
variance? Provide a rationale for your answer.
4.
What is the null hypothesis in the example?
5.
What was the exact likelihood of obtaining an
F
value at least as extreme as or as close to the one that was
actually observed, assuming that the null hypothesis is true?
6.
Do the data meet criteria for “mutual exclusivity”? Provide a
rationale for your answer.
387
7.
What does the numerator of the
F
ratio represent?
8.

What does the denominator of the
F
ratio represent?
9.
How would our final interpretation of the results have changed
if we had chosen to report the LSD post hoc test instead of the
Tukey HSD test?
10.
Was the sample size adequate to detect differences among the
three groups in this example? Provide a rationale for your
answer.
388
Answers to Study Questions
1.
Yes, the data are approximately normally distributed as noted
by the frequency distribution generated from SPSS, below. The
Shapiro-Wilk (covered in
Exercise 26
)
p
value for months to completion was 0.151, indicating that the
frequency distribution did not significantly deviate from
normality.
2.
The two instances that must occur to warrant post hoc testing
following an ANOVA are (1) the ANOVA was performed on
data comparing more than two groups, and (2) the
F
value is statistically significant.
3.

Yes, the data met criteria for homogeneity of variance because
the Levene's test for equality of variances yielded a
p
of 0.488, indicating no significant differences in variance
between the groups.
4.
The null hypothesis is: “There is no difference between groups
(Associate's, Bachelor's, and Master's degree groups) in months
until completion of an RN to BSN program.”
5.
The exact likelihood of obtaining an
F
actually observed, assuming that the null hypothesis is true, was
0.2%.
6.
Yes, the data met criteria for mutual exclusivity because a
student could only belong to one of the three groups of the
highest degree obtained prior to enrollment (Associate,
Bachelor’s, and Master’s degree).
7.
The numerator represents the between groups variance or the
differences between the groups/conditions being compared.
8.
The denominator represents within groups variance or the extent
to which there is dispersion among the dependent variables.
9.
The final interpretation of the results would have changed if we
had chosen to report the LSD post hoc test instead of the Tukey

HSD test. The results of the LSD test indicated that the
389
students in the Master's degree group took significantly longer
to complete the program than the students in the Bachelor's
degree group (
p
= 0.025).
10.
The sample size was most likely adequate to detect differences
among the three groups overall because a significant difference
was found,
p
= 0.002. However, there was a discrepancy between the results
of the LSD post hoc test and the Tukey HSD test. The
difference between the Master's degree group and the Bachelor's
degree group was significant according to the results of the LSD
test but not the Tukey HSD test. Therefore, it is possible that
with only 27 total students in this example, the data were
underpowered for the multiple comparisons following the
ANOVA.
390
Data for Additional Computational Practice for Questions to be
Graded
Using the example from
Ottomanelli and colleagues (2012)
study, participants were randomized to receive Supported
Employment or treatment as usual. A third group, also a
treatment as usual group, consisted of a nonrandomized
observational group of participants. A simulated subset was
selected for this example so that the computations would be
small and manageable. The independent variable in this example
is treatment group (Supported Employment, Treatment as
Usual–Randomized, and Treatment as Usual–Observational/Not
Randomized), and the dependent variable was the number of
hours worked post-treatment. Supported employment refers to a

type of specialized interdisciplinary vocational rehabilitation
designed to help people with disabilities obtain and maintain
community-based competitive employment in their chosen
occupation (
Bond, 2004
).
The null hypothesis is: “There is no difference between the
treatment groups in post-treatment number of hours worked
among veterans with spinal cord injuries.”
Compute the ANOVA on the data in
Table 33-3
below.
TABLE 33-3
POST-TREATMENT HOURS WORKED BY TREATMENT
GROUP
Participant #
Supported Employment
Participant #
TAU Observational
Participant #
TAU Randomized
1
8
6
15
11
25
2
9
7
18
12
28
3
15
8

9
13
35
4
17
9
18
14
30
5
24
10
16
15
15
“TAU” = Treatment as Usual.
391
EXERCISE 33
Questions to Be Graded
Name:
_____________________________________________________
__ Class: _____________________
Date:
_____________________________________________________
______________________________
Follow your instructor's directions to submit your answers to
the following questions for grading. Your instructor may ask
you to write your answers below and submit them as a hard
copy for grading. Alternatively, your instructor may ask you to
use the space below for notes and submit your answers online at
http://evolve.elsevier.com/Grove/statistics/
under “Questions to Be Graded.”
1.
Do the data meet criteria for homogeneity of variance? Provide
a rationale for your answer.

2.
If calculating by hand, draw the frequency distribution of the
dependent variable, hours worked at a job. What is the shape of
the distribution? If using SPSS, what is the result of the
Shapiro-Wilk test of normality for the dependent variable?
3.
What are the means for three groups' hours worked on a job?
4.
What are the
F
value and the group and error
df
for this set of data?
392
5.
Is the
F
significant at α = 0.05? Specify how you arrived at your
answer.
6.
If using SPSS, what is the exact likelihood of obtaining an
F
actually observed, assuming that the null hypothesis is true?
7.
Which group worked the most weekly job hours post-treatment?
Provide a rationale for your answer.
8.
Write your interpretation of the results as you would in an APA-

formatted journal.
9.
Is there a difference in your final interpretation when comparing
the results of the LSD post hoc test versus Tukey HSD test?
Provide a rationale for your answer.
10.
If the researcher decided to combine the two Treatment as Usual
groups to represent an overall “Control” group, then there
would be two groups to compare: Supported Employment versus
Control. What would be the appropriate statistic to address the
difference in hours worked between the two groups? Provide a
rationale for your answer.
I just need help in question 5th and 6th of questions to be
graded, exercise 33

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