# UNIT 9

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### INTRODUCTION

In Unit 9, we will study the theory and logic of analysis of variance (ANOVA). Recall that a t test requires a predictor variable that is dichotomous (it has only two levels or groups). The advantage of ANOVA over a t test is that the categorical predictor variable can have two or more groups. Just like a t test, the outcome variable in ANOVA is continuous and requires the calculation of group means.TOGGLE DRAWERHIDE FULL INTRODUCTION

#### 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. 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 social sciences research. In non-experimental designs, the one-way ANOVA compares group means between naturally existing groups, such as political affiliation (Democrat, Independent, Republican). In experimental designs, the one-way ANOVA compares group means for participants randomly assigned to different 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 pairwise comparisons determined prior to running the F test or (b) follow-up tests of pairwise comparisons, also referred to as post-hoc tests, to determine exactly which pairwise comparisons are significant.

#### Hypothesis Testing in a One-Way ANOVA

The null hypothesis of the omnibus test is that all k (group) population means are equal, or H0: µ1 = µ2 = … µk. By contrast, the alternative hypothesis is usually articulated by stipulating that “at least one” pairwise comparison of population means is unequal. Keep in mind that this prediction does not imply that all groups must significantly differ from one another on the outcome variable.

#### Assumptions of a 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 (or 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) from Unit 2. Eta squared for the one-way ANOVA is calculated by dividing the sum of squares of between-group differences ( SSbetween) by the total sums of squares in the model ( SStotal), which is reported in SPSS output for the F test. Eta squared for the one-way ANOVA is interpreted by referring to Table 5.2 in the Warner text (p. 208).

#### Journal Article Assignment

By the conclusion of Unit 9, you will have studied three fundamental statistics used in research, including correlation, t tests, and one-way analysis of variance (ANOVA). You are now prepared to analyze a journal article in your career specialization that reports one of these statistical tests. In the journal article assignment, you will follow the general process used in completing DAA assignments:

1. Provide a brief summary of the research study and, in this assignment, why it is relevant to your career.
2. Identify the predictor variables and outcome variables including scales of measurement.
3. Articulate the research question, null hypothesis, and alternative hypothesis.
4. Report the test statistic and interpret it.
5. Provide conclusions as well as the strengths and limitations of the study.
##### Reference

Warner, R. M. (2013). Applied statistics: From bivariate through multivariate techniques (2nd ed.). Thousand Oaks, CA: Sage.

### OBJECTIVES

To successfully complete this learning unit, you will be expected to:

1. Analyze the use of one-way ANOVA in research situations.
2. Describe how ANOVA protects against the risk of inflated Type I error.
3. Analyze the assumptions, calculation, and effect size of one-way ANOVA
4. Studies

Use your Warner text, Applied Statistics: From Bivariate Through Multivariate Techniques, to complete the following:

• Read Chapter 6, “One-Way Between-Subjects Analysis of Variance,” pages 219–260. This reading addresses the following topics:
• Research situations using one-way ANOVA.
• Assumptions of one-way ANOVA.
• Calculation of one-way ANOVA.
• Effect size.
• Planned contrasts and post-hoc tests.
• Reporting and interpreting SPSS output.

In addition to the other required study activities for this unit, PSY learners are required to read the following:Wang, P., Rau, P. P., & Salvendy, G. (2015). Effect of information sharing and communication on driver’s risk taking. Safety Science, 77, 123–132. doi:10.1016/j.ssci.2015.03.013

#### SOE Learners – Suggested Readings

Some programs have opted to provide program-specific content designed to help you better understand how the subject matter in this study is incorporated into your particular field of study. The following readings are suggested for SOE learners.Wabed, A., & Tang, X. (2010). Analysis of variance (ANOVA). In N. J. Salkind (Ed.), Encyclopedia of research design (pp. 27–29). Thousand Oaks, CA: Sage. doi:10.4135/9781412961288.n11

### Journal Article Summary

#### Resources

For this assignment, you will identify a published research article either in the print literature or online in the Capella Library. Your article must be based on empirical (data-based) research; qualitative or purely descriptive research is not appropriate. Select a journal article in your career specialization that reports a correlation, a t test, a one-way ANOVA, or some combination of these test statistics. The library guides listed in the Resources area can help you to locate appropriate articles.The intent of this assignment is to:

• Expose you to professional literature in your discipline.
• Provide practice in the interpretation of statistical results contained in an empirical (data-based) journal article.
• Provide practice in writing and thinking in a concise and economical manner that is typical of scientific discourse.

You will summarize the article in a maximum of 600 words using the DAA Template located in the Resources area. Specific instructions for completing each section of the DAA Template are listed below.You may use some of the author’s own words to summarize the article with proper citation, but avoid lengthy direct quotes (such as copying multiple sentences or paragraphs verbatim). You should not exceed the limit of 600 words. This is a situation where less is better.

#### Step 1: Write Section 1 of the DAA.

• Provide a brief summary of the journal article.
• Include a definition of the specified variables (predictor, outcome) and corresponding scales of measurement (nominal, continuous).
• Specify the sample size of the data set.
• Discuss why the journal article is relevant to your career specialization.

#### Step 2: Write Section 2 of the DAA.

• Discuss the assumptions of the statistical test used in the journal article.
• If possible, identify information in the article about how these assumptions were tested.
• If no information on assumptions is provided, consider this as a limitation of the reported study.

#### Step 3: Write Section 3 of the DAA.

• Specify the research question from the journal article.
• Articulate the null hypothesis and alternative hypothesis.

#### Step 4: Write Section 4 of the DAA.

• Report the results of the statistical test using proper APA guidelines. This includes:
• The statistical notation (such as r, t, or F).
• The degrees of freedom.
• The statistical value of r, t, or F, and the p value.
• Report the effect size and interpretation if one is provided.
• Interpret the test statistic with regard to the null hypothesis.

#### Step 5: Write Section 5 of the DAA.

• Discuss the conclusions of the statistical test as it relates to the research question.
• Conclude with an analysis of the strengths and limitations of the study reported in the journal article.

Submit your DAA Template as an attached Word document in the assignment area.

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