Assessment 4 Context
Recall that null hypothesis tests are of two types: (1) differences between group means and (2)
association between variables. In both cases there is a null hypothesis and an alternative
hypothesis. In the group means test, the null hypothesis is that the two groups have equal
means, and the alternative hypothesis is that the two groups do not have equal means. In the
association between variables type of test, the null hypothesis is that the correlation coefficient
between the two variables is zero, and the alternative hypothesis is that the correlation
coefficient is not zero.
Notice in each case that the hypotheses are mutually exclusive. If the null is false, the
alternative must be true. The purpose of null hypothesis statistical tests is generally to show
that the null has a low probability of being true (the p value is less than .05) – low enough that
the researcher can legitimately claim it is false. The reason this is done is to support the
allegation that the alternative hypothesis is true.
In this context you will be studying the details of the first type of test again, with the added
capability of comparing the means among more than two group at a time. This is the same type
of test of difference between group means. In variations on this model, the groups can actually
be the same people under different conditions. The main idea is that several group mean values
are being compared. The groups each have an average score or mean on some variable. The
null hypothesis is that the difference between all the group means is zero. The alternative
hypothesis is that the difference between the means is not zero. Notice that if the null is false,
the alternative must be true. It is first instructive to consider some of the details of groups.
One might ask why we would not use multiple t tests in this situation. For instance, with three
groups, why would I not compare groups one and two with a t test, then compare groups one
and three, and then compare groups two and three?
The answer can be found in our basic probability review. We are concerned with the probability
of a TYPE I error (rejecting a true null hypothesis). We generally set an alpha level of .05, which
is the probability of making a TYPE I error. Now consider what happens when we do three t
tests. There is .05 probability of making a TYPE I error on the first test, .05 probability of the
same error on the second test, and .05 probability on the third test. What happens is that these
errors are essentially additive, in that the chances of at least one TYPE I error among the three
tests much greater than .05. It is like the increased probability of drawing an ace from a deck of
cards when we can make multiple draws.
ANOVA allows us do an "overall" test of multiple groups to determine if there are any differences
among groups within the set. Notice that ANOVA does not tell us which groups among the three
groups are different from each other. The primary test in ANOVA is only to determine if there is
a significant difference among the groups somewhere.
You will study the theory and logic of analysis of variance (ANOVA). Recall that a t-test requires
a predictor variable that is dichotomous. The advantage of ANOVA over a t-test is that the
categorical predictor variable includes 3+ values (groups). Just like a t-test, the outcome
variable in ANOVA is quantitative and requires the calculation of group means.
In ANOVA, there are two levels of hypotheses. There is first the overall question of whether all
the group means are equal, or if there are some differences among the means somewhere. This
is called the omnibus null hypothesis test. The test is designed to show that the probability that
the group means are all equal is very low, leading to the researcher being able to legitimately
claim there are differences. This is done with the F test. In ANOVA, once the omnibus null
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hypothesis is rejected, then one may legitimately use special tests, called post hoc tests, to
examine each of the pairs of groups in the set to determine which ones differ and which do not.
For instance, if an ANOVA is performed for three groups, the omnibus null hypothesis is that the
three groups have equal means. If that null is rejected, then the researcher may use special
post hoc tests to compare groups 1 & 2, groups 1 & 3, and groups 2 & 3. Each of these post
hoc tests are themselves null hypothesis tests, similar to the t tests which were studied
previously.
They are designed to control for multiple comparisons, or an inflation of the Type I error rate that
is a result of doing many tests with some fixed probability of error on each test. Most are based
on the assumption that the omnibus null has been rejected.
The Logic of a One-Way ANOVA
The ANOVA, or F-test, relies on predictor variables referred to as factors. A factor is a
categorical (nominal) predictor variable. The term "one-way" is applied to an ANOVA with only
one factor that is defined by two or more mutually exclusive groups. Technically, an ANOVA can
be calculated with only two groups, but the t-test is usually used instead. The one-way ANOVA
is usually calculated with three or more groups, which are often referred to as levels of the
factor.
If the ANOVA includes multiple factors, it is referred to as a factorial ANOVA. An ANOVA with
two factors is referred to as a "two-way" ANOVA; an ANOVA with three factors is referred to as
a "three-way" ANOVA, and so on. Factorial ANOVA is studied in Advanced Inferential Statistics.
In this course, we will focus on the theory and logic of the one-way ANOVA.
ANOVA is one of the most popular statistics used in psychological research. In nonexperimental
designs, the one-way ANOVA compares group means across naturally existing characteristics
of groups, such as political affiliation. In experimental designs, the one-way ANOVA compares
group means for participants randomly assigned to treatment conditions (for example, high
caffeine dose; low caffeine dose; control group).
Avoiding Inflated Type I Error
You may wonder why a one-way ANOVA is necessary. For example, if a factor has four groups
( k = 4), why not just run independent sample t tests for all pairwise comparisons (for example,
Group A versus Group B, Group A versus Group C, Group B versus Group C, et cetera)?
Warner (2013) points out that a factor with four groups involves six pairwise comparisons. The
issue is that conducting multiple pairwise comparisons with the same data leads to inflated risk
of a Type I error (incorrectly rejecting a true null hypothesis—getting a false positive). The
ANOVA protects the researcher from inflated Type I error by calculating a single omnibus test
that assumes all k population means are equal.
Although the advantage of the omnibus test is that it helps protect researchers from inflated
Type I error, the limitation is that a significant omnibus test does not specify exactly which group
means differ, just that there is a difference "somewhere" among the group means. A researcher
therefore relies on either (a) planned contrasts of specific pair wise comparisons determined
prior to running the F-test, or, (b) follow-up tests of pair wise comparisons, also referred to as
post-hoc tests, to determine exactly which pair wise comparisons are significant. Usually, if
planned contrasts are designed correctly, there is no need to perform the omnibus null test, and
the overall ANOVA is not necessary.
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Hypothesis Testing in One-Way ANOVA
The null hypothesis of the omnibus test is that all k population means are equal, or H0: µ1 = µ2
=…µk. By contrast, the alternative hypothesis is usually articulated by stipulating that H0 is not
true. Keep in mind that this prediction does not imply that all groups must significantly differ from
one another on the outcome variable. In fact, for reasons beyond the scope of our present
discussion, it is not even strictly necessary that any two groups differ even if the omnibus null is
rejected.
Assumptions of One-Way ANOVA
The assumptions of ANOVA reflect assumptions of the t-test. ANOVA assumes independence
of observations. ANOVA assumes that outcome variable Y is normally distributed. ANOVA
assumes that the variance of Y scores is equal across all levels (groups) of the factor. These
ANOVA assumptions are checked in the same process used to check assumptions for the t-test
discussed earlier in the course—using the Shapiro-Wilk test and the Levene test.
Effect Size for a One-Way ANOVA
The effect size for a one-way ANOVA is eta squared (η2). It represents the amount of variance
in Y that is attributable to group differences. Recall the concept of sum of squares ( SS). Eta
squared for the one-way ANOVA is calculated by dividing the sum of squares of between-group
differences (SS-between) by the total sums of squares in the model (SS-total), which is reported
in SPSS output for the F-test. Eta squared for the one-way ANOVA is interpreted with .06 as "large."
References
Lane, D. M. (2013). HyperStat online statistics textbook. Retrieved from
http://davidmlane.com/hyperstat/index.html
Warner, R. M. (2013). Applied statistics: From bivariate through multivariate techniques (2nd
ed.). Thousand Oaks, CA: Sage Publications.
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IBM SPSS Step-by-Step Guide: One-Way ANOVA
Note: This guide is an example of creating ANOVA output in SPSS with the grades.sav file.
The variables shown in this guide do not correspond with the actual variables assigned in
Assessment 4. Carefully follow the instructions in the assignment for a list of assigned
variables. Screen shots were created with SPSS 21.0.
Creating One-Way ANOVA Output
To complete Section 2 of the DAA for Assessment 4, you will generate SPSS output for a
histogram, descriptive statistics, and the Shapiro-Wilk test, which are covered in previous
step-by-step guides. The Levene test (homogeneity of variance) is covered in the steps below.
Refer to the Assessment 4 instructions for a list of assigned variables. The example variables
year and final are shown below.
Step 1. Open grades.sav in SPSS.
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Step 2. On the Analyze menu, point to Compare Means and click One-Way ANOVA…
Step 3. In the One-Way ANOVA dialog box:
•
Move the assigned dependent variable into the Dependent List box.
•
Move the assigned independent variable into the Factor box. The examples of final
and year are shown below.
•
Click the Options button.
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Step 4. In the One-Way ANOVA: Options dialog box:
•
Select Homogeneity of variance test (for the Levene test for Section 2 of the DAA).
•
Select Descriptive and Means Plot (for Section 4 of the DAA).
•
Click Continue.
•
Return to the One-Way ANOVA dialog box and select the Post Hoc button.
Step 5. In the One-Way ANOVA: Post Hoc Multiple Comparisons dialog box:
•
Check the Tukey option for multiple comparisons.
•
Click Continue and OK.
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Interpreting One-Way ANOVA Output
A string of ANOVA output will appear in SPSS. (The output below is for the example variable
final.)
Step 1. Copy the Levene test output from SPSS and paste it into Section 2 of the DAA
Template. Then interpret it for the homogeneity of variance assumption.
Test of Homogeneity of Variances
final
Levene Statistic
.866
df1
df2
3
Sig.
101
.462
Step 2. Copy the means plot, paste it into Section 4 of the DAA Template, and interpret it.
Step 3. Copy the descriptives output. Paste it into Section 4 along with the report of means
and standard deviations of the dependent variable at each level of the independent variable.
Descriptives
final
N
Mean
Std. Deviation
Std. Error
95% Confidence Interval for
Minimum
Maximum
Mean
Lower Bound
Upper Bound
Frosh
3
59.33
5.859
3.383
44.78
73.89
55
66
Soph
19
62.42
6.628
1.520
59.23
65.62
48
72
Junior
64
61.47
8.478
1.060
59.35
63.59
40
75
Senior
19
60.89
7.951
1.824
57.06
64.73
43
74
105
61.48
7.943
.775
59.94
63.01
40
75
Total
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Step 4. Copy the ANOVA output, paste it into Section 4, and interpret it.
ANOVA
final
Sum of Squares
Between Groups
df
Mean Square
F
37.165
3
12.388
Within Groups
6525.025
101
64.604
Total
6562.190
104
Sig.
.192
.902
Step 5. Finally, if the overall ANOVA is significant, copy the post hoc output, paste it into
Section 4, and interpret it.
Multiple Comparisons
Dependent Variable: final
Tukey HSD
(I) Year in school
(J) Year in school
Mean
Std. Error
Sig.
Difference (I-J)
Frosh
Soph
Junior
Senior
95% Confidence Interval
Lower Bound
Upper Bound
Soph
-3.088
4.993
.926
-16.13
9.96
Junior
-2.135
4.748
.970
-14.54
10.27
Senior
-1.561
4.993
.989
-14.61
11.48
Frosh
3.088
4.993
.926
-9.96
16.13
Junior
.952
2.100
.969
-4.53
6.44
Senior
1.526
2.608
.936
-5.29
8.34
Frosh
2.135
4.748
.970
-10.27
14.54
Soph
-.952
2.100
.969
-6.44
4.53
Senior
.574
2.100
.993
-4.91
6.06
Frosh
1.561
4.993
.989
-11.48
14.61
Soph
-1.526
2.608
.936
-8.34
5.29
Junior
-.574
2.100
.993
-6.06
4.91
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Running head: DATA ANALYSIS AND APPLICATION TEMPLATE
Data Analysis and Application (DAA) Template
Learner Name
Capella University
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DATA ANALYSIS AND APPLICATION TEMPLATE
Data Analysis and Application (DAA) Template
Use this file for all assignments that require the DAA Template. Although the statistical
tests will change from week to week, the basic organization and structure of the DAA remains
the same. Update the title of the template. Remove this text and provide a brief introduction.
Section 1: Data File Description
1. Describe the context of the data set. You may cite your previous description if the same
data set is used from a previous assignment.
2. Specify the variables used in this DAA and the scale of measurement of each variable.
3. Specify sample size (N).
Section 2: Testing Assumptions
1. Articulate the assumptions of the statistical test.
2. Paste SPSS output that tests those assumptions and interpret them. Properly integrate
SPSS output where appropriate. Do not string all output together at the beginning of the
section.
3. Summarize whether or not the assumptions are met. If assumptions are not met, discuss
how to ameliorate violations of the assumptions.
Section 3: Research Question, Hypotheses, and Alpha Level
1. Articulate a research question relevant to the statistical test.
2. Articulate the null hypothesis and alternative hypothesis.
3. Specify the alpha level.
Section 4: Interpretation
1. Paste SPSS output for an inferential statistic. Properly integrate SPSS output where
appropriate. Do not string all output together at the beginning of the section.
2. Report the test statistics.
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DATA ANALYSIS AND APPLICATION TEMPLATE
3. Interpret statistical results against the null hypothesis.
Section 5: Conclusion
1. State your conclusions.
2. Analyze strengths and limitations of the statistical test.
3
DATA ANALYSIS AND APPLICATION TEMPLATE
References
Provide references if necessary.
4
SPSS Data Analysis Report Guidelines
For the SPSS data analysis report assignments in Assessments 2, 3, and 4, you will use the Data
Analysis and Application (DAA) Template with the five sections described below. As shown in
the IBM SPSS step-by-step guides, label all tables and graphs in a manner consistent with
Capella's APA Style and Format guidelines. Citations, if needed, should be included in the text
and references included in a reference section at the end of the report. The organization of the
report should include the following five sections:
Section 1: Data File Description (One Paragraph)
1. Describe the context of the data set. Cite a previous description if the same data set is
used from a previous assignment. To increase the formal tone of the DAA, avoid firstperson perspective "I." For example, do not write, "I ran a scatter plot shown in Figure
1." Instead, write, "Figure 1 shows. . . ."
2. Specify the variables used in this DAA and the scale of measurement of each variable.
3. Specify sample size (N).
Section 2: Testing Assumptions (Multiple Paragraphs)
1. Articulate the assumptions of the statistical test.
2. Paste SPSS output that tests those assumptions and interpret them. Properly embed SPSS
output where appropriate. Do not string all output together at the beginning of the section.
In other words, interpretations of figures and tables should be near (that is, immediately
above or below) where the output appears. Format figures and tables per APA formatting.
Refer to the examples in the IBM SPSS step-by-step guides.
3. Summarize whether or not the assumptions are met. If assumptions are not met, discuss
how to ameliorate violations of the assumptions.
Section 3: Research Question, Hypotheses, and Alpha Level (One Paragraph)
1. Articulate a research question relevant to the statistical test.
2. Articulate the null hypothesis and alternative hypothesis for the research question.
3. Specify the alpha level (.05 unless otherwise specified).
Section 4: Interpretation (Multiple Paragraphs)
1. Paste SPSS output for an inferential statistic and report it. Properly embed SPSS output
where appropriate. Do not string all output together at the beginning of the section. In
other words, interpretations of figures and tables should be near (that is, immediately
above or below) where the output appears. Format figures and tables per APA formatting.
2. Report the test statistics. For guidance, refer to the "Results" examples at the end of the
appropriate chapter of your Warner text.
3. Interpret statistical results against the null hypothesis.
Section 5: Conclusion (Two Paragraphs)
1. Provide a brief summary (one paragraph) of the DAA conclusions.
2. Analyze strengths and limitations of the statistical test.
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