PSY7625_U04A1_R_Hale

Running head: DATA ANALYSIS AND APPLICATION 1
Data Analysis and Application
Todd Hale
Capella University

Introduction
For this assignment, a two way factorial ANOVA is completed on a set of data that seeks
to answer the question as to whether the mean heart rate of test subjects are moderated by a
150mg dose of caffeine and if the mean heart rates of test subjects are moderated by whether the
individual engages in physical exercise or not. In addition, the two-way factorial ANOVA will
analyze any interaction between the independent variables of exercise and caffeine. The analysis
will include a descriptive overview of the data, methods used to analyze the data, and an
interpretation of the results.
Data File Description
The data file was downloaded from the Capella University student website. The sample
contains 16 entries (N = 16). The data set includes three variables; “caffeine”, “exercise”, and
“heartrate”. The heart rate variable is the outcome or dependant variable and is a scale variable.
The caffeine variable is considered a nominal/categorical variable with “1” representing no
caffeine ingested by the subject and “2” representing a 150 mg dose of caffeine ingested by the
subject. The exercise variable is also a nominal/categorical variable with “1” representing that
the subject was excluded from exercise and “2” representing that the subject engaged in 30
minutes of exercise. In this experiment, caffeine will be factor “A” and exercise will be factor
“B”.
Testing Assumptions

The ANOVA procedure contains four main assumptions that must be met prior to
performing the analysis (George & Mallery, 2012; Howell, 2011; Warner, 2013). Ensuring these
assumptions are met is paramount to achieving a logical result using the ANOVA procedure.
Independence of observations
The first assumption involves the method in which samples were taken for entry into the
database (George & Mallery, 2012; Howell, 2011; Warner, 2013). For results to be accurate, all
samples must be taken independently from other samples to ensure that no sample is in any way
dependent on another sample. This assumption is simply verified by the researcher to ensure that
this assumption is met when samples are selected. In this case, the researcher is confident that
this assumption has not been violated.
Normality of the Distribution
A second assumption for the ANOVA procedure to produce accurate results is that the
dependant variable is at least interval or ratio level and normally distributed (George & Mallery,
2012; Howell, 2011; Warner, 2013). One can see a histogram of the heart rate variable as figure
1. A visual inspection shows that the data appear to be normally distributed and symmetric. The
distribution is unimodal and does not appear to have a negative or positive skew. In addition, the
visual inspection shows that the distribution does not contain any outliers.

Figure #1
In addition to the histogram for the outcome variable heart rate, one can see the
descriptive statistics for this variable on table 1. As can be seen in table 1, the heart rate variable
has a mean (M = 80) as well as median of 80. The standard deviation is 10.954 (SD = 10.954)
and the variance is 120.

Table #1
Statistics
Heartrate
N
Valid 16
Missing 0
Mean 80.00
Median 80.00
Mode 75a
Std. Deviation 10.954
Variance 120.000
Skewness .000
Std. Error of Skewness .564
Kurtosis -.491
Std. Error of Kurtosis 1.091
Minimum 60
Maximum 100
a. Multiple modes exist. The smallest
value is shown
The distribution has no skew as is evidenced by the skewness statistic of .000. In addition, the
distribution is slightly platykurtic, but still within normal range, with a kurtosis statistic of -.491
as can be seen in table 1. One can see an additional confirmation of normality of the distribution
for the heart rate variable with the result of the Shapiro-Wilk test of normality found in table 2,
W(16) = .983, p < .983.
Table #2
Tests of Normality
Kolmogorov-Smirnova
Shapiro-Wilk
Statistic df Sig. Statistic df Sig.
Heartrate .113 16 .200*
.983 16 .983
*. This is a lower bound of the true significance.

a. Lilliefors Significance Correction
Homogeneity of Variance
The final major assumption when using the ANOVA procedure is the homogeneity of
variance assumption (Howell, 2011; Warner, 2013). This assumption involves ensuring that there
is no significant difference in the variances in the data should be relatively the same. To test this
assumption, the Levene’s homogeneity test is applied to the data. The test result demonstrates
that the heart rate variable does not violate this assumption, F(3,12) = 0, p < 1.00 as can be seen
in table 3.
Table #3
Levene's Test of Equality of Error Variancesa
Dependent Variable: Heartrate
F df1 df2 Sig.
.000 3 12 1.000
Tests the null hypothesis that the error variance of
the dependent variable is equal across groups.
a. Design: Intercept + Caffeine + Exercise +
Caffeine * Exercise
Research Question, Hypotheses, and Alpha Level
In this study, the main research question is “Does caffeine, at a dose of 150 mg or
physical exercise moderate one’s heart rate?” A second research question would be “Is there any
interaction between the two independent variables of caffeine level and exercise on an
individual’s heart rate?” There are three hypothesis which are: H0: µA1 = µA2, H0: µB1 = µB2, and
finally, H0: No A X B Interaction. The alternative hypothesis will: H1: µA1 ≠ µA2, H1: µB1 ≠ µB2 and
finally, H1: There is an A X B Interaction. The alpha level for all testing will be set at .05.

Interpretation
As can be seen in table 6, the grand mean for all data is 80. This is interpreted to be the
average number for all data included in the analysis.
Table 6
1. Grand Mean
Mean Std. Error 95% Confidence Interval
Lower Bound Upper Bound
80.000 1.614 76.484 83.516
The mean estimates for the “A” factor caffeine, include 72.5 for the mean of non-caffeine
participants in the study and a mean of 87.5 for those that did ingest caffeine during the study.
This information can be seen in table 7.
Table 7
Estimates
Caffeine Mean Std. Error 95% Confidence Interval
No Caffeine 72.500 2.282 67.528 77.472
150 mg Caffeine 87.500 2.282 82.528 92.472

Additionally, the means for the “no exercise” respondant’s heart rate was 75 while the mean
heart rate for those that did exercise was 85 as can be seen in table 8.
Table 8
Estimates
Exercise Mean Std. Error 95% Confidence Interval
No Exercise 75.000 2.282 70.028 79.972
Half Hour of Exercise 85.000 2.282 80.028 89.972
The between subject affects descriptive statistics can be seen on table 4. As can be seen
on table 4, the mean heart rate for those subjects with no caffeine and no exercise, according to
the data was 67.50. In addition, the mean hear rate for subjects that consumed 150 mg of caffeine
and exercised regularly is 85. This already seem to imply that as one increases caffeine intake or
one increases one’s exercise regimen, one’s heart rate increases.

Table #4
Descriptive Statistics
Caffeine Exercise Mean Std. Deviation N
No Caffeine
No Exercise 67.50 6.455 4
Half Hour of Exercise 77.50 6.455 4
Total 72.50 8.018 8
150 mg Caffeine
No Exercise 82.50 6.455 4
Total 87.50 8.018 8
Total
No Exercise 75.00 10.000 8
Total 80.00 10.954 16
Additional results from the ANOVA analysis can be viewed in table 5. The interpretation
of the result of the factorial ANOVA is as follows: the H0 for the “A” factor, caffeine, is rejected
in favor of H1, meaning that there is a difference in the mean heart rate of subjects based on
whether the test subject consumed a 150 mm dose of caffeine, FA(1, 12) = 21.6, p < .001 as can
be seen on table 5. In addition, H0 will be rejected in favor of H1 for the “B” factor, exercise,
indicating that the mean heart rate was different based on whether or not the test subject engaged
in exercise, FB(1,12) = 9.6, p < .009 as can be seen on table 5. Finally, with respect to any
interaction between factor A and factor B, the null hypothesis H0 will not be rejected based on
the results of the ANOVA, FA*B(1,12) = 0, p < 1.00 as can be seen on table 5. Thus this can be
interpreted to mean that the ANOVA failed to find any evidence that would support an
interaction between the caffeine intake and exercise.

Table #5
Tests of Between-Subjects Effects
Source Type III Sum
of Squares
df Mean Square F Sig. Partial Eta
Squared
Noncent.
Parameter
Observed
Powerb
Corrected Model 1300.000a
3 433.333 10.400 .001 .722 31.200 .985
Intercept 102400.000 1 102400.000 2457.600 .000 .995 2457.600 1.000
Caffeine 900.000 1 900.000 21.600 .001 .643 21.600 .989
Exercise 400.000 1 400.000 9.600 .009 .444 9.600 .811
Caffeine *
Exercise
.000 1 .000 .000 1.000 .000 .000 .050
Error 500.000 12 41.667
Total 104200.000 16
Corrected Total 1800.000 15
a. R Squared = .722 (Adjusted R Squared = .653)
b. Computed using alpha = .05
One can visually see the trajectory plots for the caffeine by exercise interaction on image
1. The image demonstrates that the lines are parallel to one another further suggesting that there
is no interaction between the variables of caffeine and exercise.

Image #1
An effect size can be calculated for the “A” factor, “B” factor, and the interaction of “A” and
“B” using the formula ƞ2
A = SSA / SStotal, ƞ2
B = SSB / SStotal, and ƞ2
A*B = SSA*B = SSA*B / SStotal
(Howell, 2011; Warner, 2013). Completing the appropriate equation with the output data from
SPSS results in an effect size for factor “A” of .008 which would be considered a small effect
size. In addition, the calculated effect size for factor “B” is .003 which would also be considered
a small effect size. Finally the “AXB” interaction effect size would be 0 as there was no
significant interaction.

Conclusion
The results of the factorial ANOVA clearly demonstrate a relationship between the
independent variable of caffeine and its affect on one’s heart rate. In addition the test
demonstrated a relationship between exercise and one’s heart rate. However, the test found no
interaction between caffeine intake and exercise as those variables relate to one’s heart rate. The
test does have its limitations including the rather small sample size of only 16 respondents. A
larger sample size might reveal additional insights.

References
George, D., & Mallery, P. (2012). IMB SPSS statistics 19 step by step: A simple guide and
reference (12th. ed.). Boston, MA: Pearson Publishing.
Howell, C. (2011). Fundamental statistics for the behavioral sciences (7th ed. ed.). Belmont,
CA: Wadsworth, Cengage Learning.
Warner, R. (2013). Applied statistics: From bivariate through multivariate techniques (2nd. ed.).
Thousand Oaks, CA.: Sage Publications, Inc.

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