[u07d2] Unit 7 Discussion 2 – Two Versions of Independent Samples t Test

Analyze why there are two different versions ("Equal variances assumed" and "Equal variances not assumed") of the t test on the SPSS printout and how you decide which one is more appropriate.

INTRODUCTION

Units 7 and 8 review the theory, logic, and application of t tests. The t test is a basic inferential statistic often reported in research. You will discover that t tests, as well as analysis of variance (ANOVA) studied in Units 9 and 10, compare group means on some quantitative outcome variable.

Logic of the t Test

Imagine that a researcher compares the mean IQ scores of Class A versus Class B. The mean IQ for Class A is 102 and the mean IQ for Class B is 105. Is there a significant difference in mean IQ between Class A and Class B?

To answer this question, the researcher conducts an independent samples t test. The independent samples t test compares two group means in a between-subjects (between- S) design. In this between- S design, participants in two independent groups are measured only once on some outcome variable.

By contrast, a paired samples t test compares group means in a within-subjects (within- S) design for a single group. Each participant is measured twice on some outcome variable, such as a pretest-posttest design. For example, a researcher could measure self-esteem for a class of students prior to taking a public speaking course (pretest) and then measure self-esteem again after completing the public speaking course (posttest). The paired samples t test determines if there is a significant difference in mean scores from the pretest to the posttest.

Units 7 and 8 focus on the logic and application of the independent samples t test. There are two variables in an independent samples t test: the predictor variable ( X) and the outcome variable ( Y). The predictor variable must be dichotomous, meaning that it can only have two values or groups (for example, male = 1; female = 2).

Group membership must be mutually exclusive. In non-experimental designs, group membership is based on some naturally occurring characteristic of a group (such as gender). In experimental designs, participants are randomly assigned to one of two group conditions (for example, treatment group = 1; control group = 2). In contrast to the dichotomous (nominal) predictor variable, the outcome variable must be continuous to calculate a group mean (for example, IQ scores, self-esteem scores).

Assumptions of the t Test

All quantitative statistics, including the independent samples t test, operate under assumptions checked prior to calculating the t test in SPSS. Violations of assumptions can lead to erroneous inferences regarding a null hypothesis. The first assumption is independence of observations. For predictor variable X in an independent samples t test, participants are assigned to one and only one "condition" or "level," such as a treatment group or control group. This assumption is not statistical in nature; it is controlled by proper research procedures that maintain independence of observations.

The second assumption is that outcome variable Y is continuous and normally distributed. This assumption is checked by a visual inspection of the Y histogram and calculation of skewness and kurtosis values. A researcher may also conduct a Shapiro-Wilk test in SPSS to check whether a distribution is significantly different from normal. The null hypothesis of the Shapiro-Wilk test is that the distribution is normal. If the Shapiro-Wilk test is significant, then the normality assumption is violated. In other words, a researcher wants the Shapiro-Wilk test to not be significant at p < .05. 

Unit 7 -t Tests: Theory and Logic

The third assumption is referred to as the homogeneity of variance assumption. Ideally, the amount of variance in Y scores is approximately equal for Group 1 and Group 2. This assumption is checked in SPSS with the Levene test. The null hypothesis of the Levene test is that group variances are equal. If the Levene test is significant, then the homogeneity assumption is violated. In other words, a researcher wants the Levene test to not be significant at p < .05. SPSS output for the t test provides two versions of the t test: "Equal variances assumed" and "Equal variances not assumed." The statistics you present depend on the outcome of the Levene test. If the Levene test is not significant, researchers report the "Equal variances assumed" version of the t test. If the Levene test is significant, researchers report the more conservative "Equal variances not assumed" calculation of the t test.

Hypothesis Testing for a t Test

The null hypothesis for a t test predicts no significant difference in population means, or H0: μ1 = μ2. A directional alternative hypothesis for a t test is that the population means differ in a specific direction, such as H1: μ1 > μ2 or H1: μ1 < μ2. A non-directional alternative hypothesis simply predicts that the population means differ, but it does not stipulate which population mean is significantly greater ( H1: μ1 ≠ μ2). For t tests, the standard alpha level for rejecting the null hypothesis is set to .05. SPSS output for a t test showing a p value of less than .05 indicates that the null hypothesis should be rejected; there is a significant difference in population means. A p value greater than .05 indicates that the null hypothesis should not be rejected; there is not a significant difference in population means.

Effect Size for a t Test

There are two commonly reported estimates of effect size for the independent samples t test, eta squared (η2) and Cohen's d. Eta squared is analogous to r2 studied in Units 5 and 6. It estimates the amount of variance in Y that is attributable to group differences in X. Eta squared ranges from 0 to 1.00, and it is

interpreted similarly to r2 in terms of small (.02), medium (.13), and large (.26) effect sizes. Eta squared is calculated as a function of an obtained t value and the study degrees of freedom.

Cohen's d is an alternate effect size representing the difference between the means of the two groups divided by the standard deviation of the variable of interest for both groups combined (pooled standard deviation). A small Cohen's d indicates a high degree of overlap in population means (see Figure 3.4 on p.

106 of the Warner text). A large Cohen's d indicates a low degree of overlap in population means (see Figure 3.5 on p. 106 of the Warner text). A Cohen's d of .20 is considered a small effect, .50 is a medium effect, and .80 is a large effect. Larger effect sizes indicate findings of greater practical or clinical significance.

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. Evaluate research situations using the t test.

2. Identify the assumptions of the independent samples t test.

3. Analyze hypothesis testing for the t test.

4. Understand effect sizes for the t test.

5. Analyze two calculations of the t test and when they are reported.

[u07s1] Unit 7 Study 1 – Readings

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

• Read Chapter 5, "Comparing Group Means Using the Independent Samples t Test," pages 185–218.

This reading addresses the following topics:

◦ Research situations using the independent samples t test.

◦ t-test assumptions.

◦ Factors affecting the size of the t ratio.

◦ Effect sizes.

Suggested Readings

Stone, E. (2010).t test, independent samples. In N. J. Salkind (Ed.),E ncyclopedia of research design (pp. 1552–1556). Thousand Oaks, CA: Sage. doi:10.4135/9781412961288.n475