EXSC 520
CASE STUDY: FACTORIAL ANALYSIS OF VARIANCE TEMPLATE
Instructions: To successfully complete this case study, all of the following instructions must be
adhered to. Only paste tables/figures that contain pertinent information. Only paste tables/
figures where you are instructed to do so. HIGHLIGHT (in yellow) only relevant the data from
tables; as well as your response to each question. In the write-up section, all information must be
provided in paragraph format. When statistical values are discussed, the numerical value must be
given and a reference must be made to the table(s)/figure(s) where the value(s) can be found. An
example of this is “heart rate was statistically greater during running versus cycling (189 vs. 143
bpm) as indicated in Table 1.
I. Factorial Analysis of Variance
Research question: “is there a statistical difference for skin temperature across time
while taking into account the “factor” “environment” (hot versus cold)?”
Assumptions Testing
1. Data level of measurement- what is the level of measurement for the data used in this
case study? _ The level of measurement for this case study is also interval since it is
referring to the temperature of the skin which is equal distance from one temperature to
the next and there is no absolute zero since zero does not mean the absence of heat.
_____________________________________________________________
2. Would the amount of skewness and kurtosis in these affect the analysis? How do you
know? Give numerical values to support your conclusion. __
The skewness for time 0 in hot is 1.897 and the kurtosis is 1.525 so neither would affect
the analysis as they are within the parameters of +-1.96. The skewness for time 0 in the
cold is 4.534 and the kurtosis is 7.434 so they are both well out of the acceptable range
and would have an effect on the analysis.
The skewness for time 15 in the hot is .604 and the kurtosis is .743 so they would not
affect the analysis. The skewness for time 15 in the cold is .835 and the kurtosis is .549
so they also wouldn’t have an effect.
The skewness for time 30 in the hot is .104 and the kurtosis is .318 so neither would
affect the analysis. The skewness for time 30 in the cold is 1.045 and the kurtosis is .723
so they would not affect the analysis.
________________________________________________________________________
______________________________________________________________________
Table 1. PASTE TABLE with skewness and kurtosis (EACH FACTOR) BELOW
Descriptives
Environment Statistic Std. Error
Time_0 Hot Mean 33.6909 .22298
95% Confidence Interval
for Mean
Lower
Bound
33.1941
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Upper
Bound
34.1877
5% Trimmed Mean 33.6510
Median 33.6000
Variance .547
Std. Deviation .73953
Minimum 32.70
Maximum 35.40
Range 2.70
Interquartile Range 1.00
Skewness 1.254 .661
Kurtosis 1.950 1.279
Cold Mean 31.7636 1.03399
95% Confidence Interval
for Mean
Lower
Bound
29.4598
Upper
Bound
34.0675
5% Trimmed Mean 32.1929
Median 32.5000
Variance 11.761
Std. Deviation 3.42937
Minimum 21.70
Maximum 34.10
Range 12.40
Interquartile Range 1.80
Skewness -2.997 .661
Kurtosis 9.508 1.279
Time_15 Hot Mean 37.5191 .18748
95% Confidence Interval
for Mean
Lower
Bound
37.1014
Upper
Bound
37.9368
5% Trimmed Mean 37.5101
Median 37.4100
Variance .387
Std. Deviation .62180
Minimum 36.70
Maximum 38.50
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Range 1.80
Interquartile Range 1.31
Skewness .399 .661
Kurtosis -.950 1.279
Cold Mean 33.1909 .24614
95% Confidence Interval
for Mean
Lower
Bound
32.6425
Upper
Bound
33.7393
5% Trimmed Mean 33.2066
Median 33.2000
Variance .666
Std. Deviation .81636
Minimum 31.80
Maximum 34.30
Range 2.50
Interquartile Range 1.23
Skewness -.552 .661
Kurtosis -.702 1.279
Time_30 Hot Mean 35.7545 .21882
95% Confidence Interval
for Mean
Lower
Bound
35.2670
Upper
Bound
36.2421
5% Trimmed Mean 35.7551
Median 35.8000
Variance .527
Std. Deviation .72576
Minimum 34.50
Maximum 37.00
Range 2.50
Interquartile Range 1.10
Skewness .069 .661
Kurtosis -.407 1.279
Cold Mean 35.9000 .25726
95% Confidence Interval
for Mean
Lower
Bound
35.3268
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Upper
Bound
36.4732
5% Trimmed Mean 35.9278
Median 36.1000
Variance .728
Std. Deviation .85323
Minimum 34.40
Maximum 36.90
Range 2.50
Interquartile Range 1.70
Skewness -.691 .661
Kurtosis -.925 1.279
3. Was the assumption of normality met and how do you know? _The assumption of
normality was met for almost all of the times except for time 0 cold where the sig value
was less than .05 under the Shapiro-Wilk section at <.001.
________________________________________________________________________
_______________________________________________________________________
Table 2. PASTE Tests of NORMALITY table BELOW
Tests of Normality
Environmen
t
Kolmogorov-SmirnovaShapiro-Wilk
Statistic df Sig. Statistic df Sig.
Time_0 Hot .222 11 .135 .896 11 .167
Cold .402 11 <.001 .569 11 <.001
Time_15 Hot .182 11 .200*.917 11 .295
Cold .176 11 .200*.939 11 .510
Time_30 Hot .142 11 .200*.978 11 .954
Cold .176 11 .200*.903 11 .200
*. This is a lower bound of the true significance.
a. Lilliefors Significance Correction
Figure 1. PASTE Histogram with normal curve (EACH FACTOR) BELOW
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FrequencyFrequency
Histogram
Normal
for
Environment=
Hot
20.00
22.00
24.00
26.00
Mean
= 33.63
Std:
Dev.=
74
New
34.00
Time_o
Histogram
Normal
for
Environment=
Cold
Mean
31.73
Std.
Dev.
nem
28.00 30.00 32.00
Time_o
EXSC 520
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Frequency
Frequency
Histogram
Nome
for
Environment=
Hot
Mean
=
37.52
Std.
Dev.
nem
Time_15
Normal
Histogram
for
Environment=
Cold
Mean
=
33.13
Sid,
Dev.
=
316
nem
Time_15
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Frequency
Frequency
Normal
Histogram
for
Environment=
Hot
Mean
=
35.75
Std.
Dev.
nem
36.00
Time_30
Normal
Histogram
for
Environment=
Cold
Mean
=
35.99
Std,
Dev.
=
353
nem
Time_30
EXSC 520
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Expected
Normal
Expected
Normal
Normal
Q-Q
Plot
of
Time_0O
for
Environment=
Hot
-1
-2
a2
33 a4
35 36
Observed
Value
Normal
Q-Q
Plot
of
Time_0O
for
Environment=
Cold
“1
-2
-4
20
22
24
26
28
30
32
34
Observed
Value
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Figure 2. PASTE Normal Q-Q Plot (EACH FACTOR) BELOW
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Expected
Normal
Expected
Normal
4
2
4
2
Normal
Q-Q
Plot
of
Time_15
for
Environment=
Hot
365
370)
75
380)
Observed
Value
Normal
Q-Q
Plot
of
Time_15
for
Environment=
Cold
385
390)
Py
Ey
Observed
Value
4
38
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Expected
Normal
Expected
Normal
4
2
4
2
Normal
Q-Q
Plot
of
Time_30
for
Environment=
Hot
34 35 36
Observed
Value
Normal
Q-Q
Plot
of
Time_30
for
Environment=
Cold
7
38
4
35 36
Observed
Value
7 8
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4. Were there any outliers; how do you know? _Time 0 had two outliers one in the hot
environment and one in the cold environment which can be seen as there are data points
outside of the box and whiskers plot in the first table below.
________________________________________________________________________
_______________________________________________________________________
Table 3. PASTE Box-and-Whiskers Plot (EACH FACTOR) BELOW
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Time_15.
Time_30
38.00
36.00
34.00
32.00
37.00
Hot
Cold
Environment
36.50
36.00
3550
35.00
3450
34.00
Hot
Cold
Environment
EXSC 520
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Factorial ANOVA- Between-Within (Split-Plot) Design
Table 4. PASTE Within-Subjects Factors table BELOW
Within-Subjects
Factors
Measure:
MEASURE_1
Time
Dependent
Variable
1Time_0
2Time_15
3Time_30
Table 5. PASTE Between-Subjects Factors table BELOW
Between-Subjects Factors
Value
Label N
Environmen
t
1Hot 11
2Cold 11
Table 6. PASTE Descriptive Statistics table BELOW
Descriptive Statistics
Environmen
t Mean
Std.
Deviation N
Time_0 Hot 33.6909 .73953 11
Cold 31.7636 3.42937 11
Total 32.7273 2.61410 22
Time_15 Hot 37.5191 .62180 11
Cold 33.1909 .81636 11
Total 35.3550 2.32546 22
Time_30 Hot 35.7545 .72576 11
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Cold 35.9000 .85323 11
Total 35.8273 .77655 22
Table 7. PASTE Box’s Test of Equality of Matrices table BELOW
Box's Test of
Equality of
Covariance
Matricesa
Box's M 20.361
F2.835
df1 6
df2 2898.113
Sig. .009
Tests the null
hypothesis that the
observed
covariance
matrices of the
dependent
variables are equal
across groups.
a. Design: Intercept
+ Environment
Within Subjects
Design: Time
Table 8. PASTE Mauchley’s Test of Sphericity table BELOW
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Table 9. PASTE Tests of Within-Subjects Effects table(s) BELOW
Table 10. PASTE Levene’s Test of Equality of Error Variances table BELOW
Levene's Test of Equality of Error Variancesa
Levene
Statistic df1 df2 Sig.
Time_0 Based on Mean 2.275 1 20 .147
Based on Median 1.361 1 20 .257
Based on Median and
with adjusted df
1.361 1 10.569 .269
Based on trimmed mean 1.482 1 20 .238
Time_15 Based on Mean .864 1 20 .364
Based on Median .902 1 20 .354
Based on Median and
with adjusted df
.902 1 19.621 .354
Based on trimmed mean .856 1 20 .366
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Time_30 Based on Mean .394 1 20 .538
Based on Median .249 1 20 .623
Based on Median and
with adjusted df
.249 1 18.914 .624
Based on trimmed mean .372 1 20 .549
Tests the null hypothesis that the error variance of the dependent variable is equal
across groups.
a. Design: Intercept + Environment
Within Subjects Design: Time
Table 11. PASTE Tests of Between-Subjects Effects table BELOW
Table 12. PASTE Pairwise Comparisons table(s) (FOR EACH FACTOR) BELOW – This
could include the table for the interaction comparison which (may) require the use of the
Syntax calculation.
Pairwise Comparisons
Measure: MEASURE_1
(I)
Environment
(J)
Environment
Mean
Difference (I-
J)
Std.
Error Sig.b
95% Confidence Interval
for Differenceb
Lower
Bound
Upper
Bound
Hot Cold 2.037*.482 <.001 1.032 3.041
Cold Hot -2.037*.482 <.001 -3.041 -1.032
Based on estimated marginal means
*. The mean difference is significant at the .05 level.
b. Adjustment for multiple comparisons: Bonferroni.
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Pairwise Comparisons
Measure: MEASURE_1
(I) Time (J) Time
Mean
Difference (I-
J) Std. Error Sig.b
95% Confidence Interval for
Differenceb
Lower Bound Upper Bound
1 2 -2.628*.463 <.001 -3.838 -1.417
3-3.100*.475 <.001 -4.341 -1.859
2 1 2.628*.463 <.001 1.417 3.838
3-.472 .185 .057 -.955 .011
3 1 3.100*.475 <.001 1.859 4.341
2.472 .185 .057 -.011 .955
Based on estimated marginal means
*. The mean difference is significant at the .05 level.
b. Adjustment for multiple comparisons: Bonferroni.
Table 12. PASTE Univariate Tests table(s) (FOR EACH FACTOR) BELOW
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Figure 3. PASTE Profile Plots figure BELOW
Write-up- Use these questions as a guide to write an abstract (paragraph). Use past tense.
The write-up must include the following. Use these questions as a guide.
1. What is the research question?
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2. What is the mean and standard deviation for skin temperature at minute 0, 15, and 30
while working in either a hot or cold environment?
3. Was Levene’s test significant? How do you know?
4. What Mauchley’s Test of Sphericity met, how do you know?
5. Was the ANOVA statistically significant? (Hint: look at the p sig value in Test of Within-
Sujects table)
6. List all (if any) significant post-hoc comparisons including the interaction comparison
7. What do the statistical findings of these data suggest about the research question?
8. What are your thoughts on the finding of this analysis, and what are the practical
application(s) of this finding? Elaborate on what the study means, why, how you know,
and relate the findings to exercise physiology knowledge. Use references if needed.
(Hint- does this study “make sense?” What could influence results?)
The research question was “is there a statistical difference for skin temperature across time while
taking into account the “factor” “environment” (hot versus cold)?”. The mean and standard
deviation for skin temperature at minute 0 in the hot environment was 33.691 and .740 as seen in
table 6 above. For cold at minute 0 the mean and standard deviation was 31.763 and 3.429. The
mean for skin temperature at minute 15 in the hot was 37.519 and the standard deviation
was .622. The mean for skin temperature in the cold for minute 15 was 33.191 and the standard
deviation was .816. The mean and standard deviation for temperature at minute 30 in the hot was
35.755 and .726. For cold at minute 30 the mean was 35.9 and the standard deviation was .853.
Levene’s test was not significant since the sig value in table 10 was larger than .05 for all values.
Mauchley’s test of sphericity was met since the Greenhouse-Geisser was larger than .05 in table
8 and the sig value was less than .05 in table 9 for the Greenhouse-Geisser row. The ANOVA
was statistically significant since the p value was less than .05 at .001 in table 11. All of the
comparisons were found to be statistically significant except for the comparison between point 2
and 3 which was the time points 15 and 30 as it had a sig value of .057. These statistical findings
suggest that there would be a significant difference in skin temperature for both hot and cold
environments for up to the first 15 minutes but not between 15 and 30 minutes. From a
physiological perspective this does make sense when again, homeostasis is taken into
consideration. When a body initially enters a new different environment from what it was just in,
there needs to be an adjustment period so that the body can either warm or cool itself down.
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