RCH 8303, Quantitative Data Analysis 1
Course Learning Outcomes for Unit VI
Upon completion of this unit, students should be able to:
1. Perform statistical tests using software tools.
1.1 Perform analysis of variance (ANOVA) using appropriate data file and menu options.
2. Explain results of statistical tests.
2.1 Describe the selection process of the variables in the data file.
2.2 Discuss the differences between alternative hypotheses.
2.3 Elaborate on options available for missing or incomplete data.
2.4 Describe the assumptions for an ANOVA.
2.5 Contrast the differences between an ANOVA and an independent sample t-test.
3. Judge whether null hypotheses should be rejected or maintained.
3.1 Explain the differences between the null and alternative hypotheses, and perform option
selection.
3.2 Perform an ANOVA option selection.
Course/Unit
Learning Outcomes
Learning Activity
1.1
Unit Lesson
Chapter 6
Unit VI Assignment 2
2.1, 2.2, 2.3, 2.4, 2.5 Unit Lesson
Unit VI Assignment 1
3.1, 3.2 Unit Lesson
Unit VI Assignment 2
Required Unit Resources
Chapter 6: Simple Statistical Tests
Unit Lesson
Unit VI Plan
The Unit VI assignment will be in two parts. Part 1 of your assignment requires you to complete modules of
the Collaborative Institutional Training Initiative (CITI) Program Essentials of Statistical Analysis (EOSA) that
relate directly to the reading in this unit. Each of the modules has a final quiz that must be completed and
successfully passed to demonstrate your knowledge of basic statistics and the research process.
For Part 2, you will review how to conduct an analysis of variance (ANOVA) to determine whether differences
in means are statistically significant between three or more groups.
These are the topics of the Unit VI CITI EOSA course.
UNIT VI STUDY GUIDE
ANOVA
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Analysis of Variance (ID 17625): This module describes and explains conducting analyses comparing three or
more groups and testing for equality of variances. The module describes the assumptions of the ANOVA,
which are the same basic assumptions of the independent-samples t-test and what to do if the data violates
one or more of the assumptions. The module concludes with testing homogeneity of variances.
Following Up Significant ANOVA (ID 17626): This module describes and explains conducting pairwise
comparisons and planned contrast ANOVAs. The module explains what to do if the ANOVA is significant and
how to determine which groups mean are different from the others.
Unit VI Lesson
Unit VI builds on the Unit IV t-test with an ANOVA being used to test three or more groups. Many people think
simply doing a t-test with three or four groups is an option; however, once you exceed two groups you need to
use an ANOVA with more than two groups.
The t-test discussed in Unit V is used to compare two means (e.g., training and no training runtimes), while
Unit VI focuses on the ANOVA, which is used to compare means between more than two groups (e.g.,
runtimes between runners in groups A, B, and C). For this unit, we will be asking the question; Are there
differences in runtimes between runners in A, B, and C? The example in the lesson will work through an
ANOVA and answer this question.
As noted in Unit III, once data are collected, a researcher needs to be able to describe, summarize, and,
potentially, detect patterns in the data they have recorded with meaningful numerical scales such as
histogram. After reviewing the data, decisions must be made regarding whether the assumptions of the test
have been met. If they have, then conducting of the test can proceed. Refer to these tutorials on the two
assumptions associated with these tests: Homogeneity of Variances and Testing for Normality.
For an example of an ANOVA, make sure when you access R that you also load R Commander. Type in
library(Rcmdr) or see Unit I for a refresher on how to gain access to R Commander. Once R and R
Commander have been loaded the next step is to load the data set threeruntime that will be used (Figure 1).
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Figure 1
Data Set Threeruntime Successfully Uploaded
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Viewing the updated data set allows a user to examine the type of category information and numeric values
(Figure 2).
Figure 2
Visual Representation of Threeruntime Data Set
Note that in Figure 2, the variable category has three groups, and the variable runtimes contains continuous
or interval information. This allows the data in runtimes to be measured by the grouping and an ANOVA to be
performed.
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The next step is to plot the data set in groups. This will allow us the opportunity to view the data set in
graphical display. Select Graphs and then Histogram (Figure 3).
Figure 3
Histogram Selection Menu
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Since our data has three grouping variables, selecting the Plot by groups option will allow us to view the data
set by groupings (Figure 4).
Figure 4
Plot by Groups Selection Menu
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Selecting Plot by groups offers our grouping variable to select. Since we only have one, we select category
and then press “OK.” This will produce the histogram of our data set (Figure 5).
Figure 5
Groups Variable Selection Menu
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Data display of our grouping variable (Figure 6) is shown below.
Figure 6
Plot by Groups Data Display
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Following the data plot by groups, we need to check homogeneity of variances. Please review the
Homogeneity of Variances tutorial discussed earlier. For our threeruntime data set, select Statistics,
Variances, and Bartlett’s test (Figure 7)
Figure 7
Homogeneity of Variances selection menu
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Selecting “OK” will run the Bartlett’s Test (Figure 8).
Figure 8
Bartlett’s Test Selection Menu
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Bartlett’s Test output results indicating homogeneity of variances (Figure 9) are shown below.
Figure 9
Bartlett’s Test Output Results
A Bartlett’s test of the homogeneity of variance was performed on the runtime variable based on category.
Since the p-value of this test was greater than .05, indicating normality, the null hypothesis would not be
rejected. If a researcher wanted to include this result in a paper, the following narrative could be used:
A Bartlett’s test was performed to examine the runtime variable’s variance across the three groups.
The result of the test was not significant, K2
(2) = 5.67, p = .059.
If the Bartlett test was significant, departures from normality may be causing the test to report a significant
value. One option is to perform the Levene test, which has been shown to be less sensitive to discrepancies
to normality. Another option would be to transform the data so that if follows a normal distribution (which is
outside the scope of this lesson).
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We are now ready to conduct the ANOVA. The menu options are shown below (Figure 10).
Figure 10
ANOVA Selection Menu
Once a one-way ANOVA is selected, a new menu option is provided. First, a default name of the model is
provided (AnovaModel.1). Second, a user must select the grouping variable. Third, a user selects the
response variable.
Since an ANOVA will only show whether there is a difference between the groups, a pairwise comparison
between groups must be examined with the p-value is < .05. Thus, the Pairwise comparison of means
checkbox should be activated. Finally, if a user wants to perform the Welch version of test, which assumes
unequal variances, the other checkbox should be activated (Figure 11).
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Figure 11
ANOVA Options Selection Menu With Pairwise Comparisons
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Selecting “Ok” will perform the test and display results of the ANOVA (Figure 12).
Figure 12
Output of ANOVA for AnovaModel.1
As indicated in Figure 12, the result of the test was significant (p < .05).
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As displayed below (Figure 13), the differences between the three groups are significant.
Figure 13
Pairwise Comparison of Means Using Tukey’s (1949) Contrasts
In addition to simultaneous tests for the general linear hypotheses, simultaneous confidence intervals can be
obtained (Figure 14).
Figure 14
Pairwise Confidence Intervals Using Tukey’s Contrasts
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Many researchers also display the differences of means using graphical representation. Viewing the means in
a linear fashion and a plot of means aids in visual representation of the data means (Figure 15), as shown
below.
Figure 15
Plot of Family-Wise 95% Confidence Intervals
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In addition, one could plot the means and confidence intervals a different way (Figure 16).
Figure 16
Plot of Means
This concludes Unit V. The ANOVA used to compare more than two means (e.g., runtimes between runners
in groups A, B, and C) was discussed. An ANOVA works with more than two groups. In Unit VI, we will be
asking these questions: Are there differences in runtimes between runners in A, B, and C? Is there a positive
or negative relationship between two variables?
Unit VI starts on a different type of outcome form of testing. Units IV and V conducted tests that compared the
means, and in some cases, a causation could be determined. The focus of Unit VI is Correlation, which is a
methodology that tests to determine whether there is a relationship between variables.
Reference
Tukey, J. W. (1949, June). Comparing individual means in the analysis of variance. Biometrics, 5(2), 99–114.
https://doi.org/10.2307/3001913
Learning Activities (Nongraded)
Nongraded Learning Activities are provided to aid students in their course of study. You do not have to submit
them. If you have questions, contact your instructor for further guidance and information.
When studying APA formatting, pay particular attention to the sections that pertain to formatting for research
and statistics. Review these sections as needed.
