© 2008 McGraw-Hill Higher Education The Statistical Imagination Chapter 11: Bivariate Relationships: t-test for Comparing the Means of Two Groups

© 2008 McGraw-Hill Higher Education Bivariate Analysis Bivariate – or “two variable” – analysis involves searching for statistical relationships between two variables A statistical relationship between two variables asserts that the measurements of one variable tend to consistently change with the measurements of the other, making one variable a good predicator of the other

© 2008 McGraw-Hill Higher Education Independent and Dependent Variables The predictor variable is the independent variable The predicted variable is the dependent variable

© 2008 McGraw-Hill Higher Education Three Approaches to Measuring Statistical Relationships 1. Difference of means testing (Ch. 11 & 12) 2. Counting the frequencies of joint occurrences of attributes of two nominal/ordinal variables (Ch. 13) 3. Measuring the correlation between two interval/ratio variables (Ch. 14 & 15)

© 2008 McGraw-Hill Higher Education Difference of Means Testing Compares means of an interval/ratio variable among the categories or groups of a nominal/ordinal variable Chapter 11. The two-group difference of means test – for a dependent interval/ratio and an independent dichotomous nominal/ordinal variable Chapter 12. Analysis of variance – to test for a difference among three or more group means

© 2008 McGraw-Hill Higher Education Frequencies of Joint Occurrences of Two Nominal Variables Chapter 13. Chi-square test – to determine a relationship between two nominal variables Web site Chapter Extensions to Chapter 13: Gamma test – to determine a relationship between two ordinal variables

© 2008 McGraw-Hill Higher Education Measuring Correlation Chapter Correlation – to determine a relationship between two interval/ratio variables Web site Extensions to Chapter 15: Rank-order correlation test – to determine a relationship between two numbered ordinal level variables

© 2008 McGraw-Hill Higher Education 2-Group Difference of Means Test: Independent Samples (t-test) Useful for testing a hypothesis that the means of a variable differ between two populations comprised of different groups of individuals

© 2008 McGraw-Hill Higher Education When to Use an Independent Samples t-test Two variables from one population and sample, one interval/ratio and one dichotomous nominal/ordinal Or: There are two populations and samples and one interval/ratio variable; the samples are representative of their population The interval/ratio variable is typically the dependent variable The groups do not consist of same subjects Population variances are assumed equal

© 2008 McGraw-Hill Higher Education Features of an Independent Samples t-test The t-test focuses on the computed difference between two sample means and addresses the question of whether the observed difference between the sample means reflects a real difference in the population means or is simply due to sampling error

© 2008 McGraw-Hill Higher Education Features of an Independent Samples t-test (cont.) Step 1. Stating the H 0 : The mean of population 1 equals the mean of population 2 That is, there is no difference in the means of the interval/ratio variable, X, for the two populations

© 2008 McGraw-Hill Higher Education Features of an Independent Samples t-test (cont.) Step 2. The sampling distribution is the approximately normal t-distribution The pooled variance formula for the standard error is used when we can assume that population variances are equal The separate variance formula for the standard error is used when we cannot assume that population variances are equal

© 2008 McGraw-Hill Higher Education Features of an Independent Samples t-test (cont.) Step 4. The effect is the difference between the sample means The test statistic is the effect divided by the standard error The p-value is estimated using the t-distribution table

© 2008 McGraw-Hill Higher Education Assumption of Equality of Population Variances When one sample variance is not larger than twice the size of the other, this suggests that the two population variances are equal and we assume equality of variances We may use the pooled variance estimate of the standard error Equality of variances is also termed homogeneity of variances or homoscedasticity

© 2008 McGraw-Hill Higher Education Assumption of Equality of Population Variances (cont.) Heterogeneity of variances, or heteroscedasticity, is when variances of the two populations appear unequal Here we use the separate variance estimate of the standard error and calculate degrees of freedom differently

© 2008 McGraw-Hill Higher Education Test for Nonindependent or Matched-Pair Samples This is a test of the difference of means between two sets of scores of the same research subjects, such as two questionnaire items or scores measured at two points in time This test is especially useful for before-after or test-retest experimental designs

© 2008 McGraw-Hill Higher Education When to Use a Nonindependent Samples t-test There is one population with a representative sample from it There are two interval/ratio variables with the same score design Or: There is a single variable measured twice for the same sample subjects There is a target value of the variable (usually zero) to which we may compare the mean of the differences between the two sets of scores

© 2008 McGraw-Hill Higher Education Features of a Nonindependent Samples or Matched-Pair t-test Step 1. Stating the H 0 : The mean of differences between the scores in a population is equal to zero

© 2008 McGraw-Hill Higher Education Nonindependent Samples or Matched-Pair t-test (cont.) Step 2. The sampling distribution is the approximately normal t-distribution The standard error is calculated as the standard deviation of differences between scores divided by the square root of n - 1

© 2008 McGraw-Hill Higher Education Nonindependent Samples or Matched-Pair t-test (cont.) Step 4. The effect is the mean of differences between scores The test statistic is the effect divided by the standard error The p-value is estimated using the t-distribution table

© 2008 McGraw-Hill Higher Education Distinguishing Between Practical and Statistical Significance A hypothesis test determines significance in terms of likely sampling error – whether a sample difference is so large that there probably is a difference in the populations Practical significance is an issue of substance. A statistically significant difference may not be practically significant

© 2008 McGraw-Hill Higher Education Practical and Statistical Significance (cont.) E.g., a hypothesis test reveals a statistically significant difference in the mean number of personal holidays of men and women in a corporation: women average 0.1 days per year more. The test tells us with 95% confidence that the 0.1 day difference in the samples truly exists in the populations However, is one-tenth day per year meaningful? Might such a small statistical effect be accounted for by some other variable?

© 2008 McGraw-Hill Higher Education Four Aspects of Statistical Relationships When examining a relationship between two variables, we can address four things: existence, direction, strength, and practical applications These four aspects provide a checklist for what to say in writing up the results of a hypothesis test

© 2008 McGraw-Hill Higher Education Existence of a Relationship Existence: On the basis of statistical analysis of a sample, can we conclude that a relationship exists between two variables among all subjects in the population? Established by rejection of the H 0 Testing for the existence of a relationship is the first step in any analysis. If a relationship is found not to exist, the other three aspects of a relationship are irrelevant

© 2008 McGraw-Hill Higher Education Direction of a Relationship Direction: Can the dependent variable be expected to increase or decrease as the independent variable increases? Direction is stated in the alternative hypothesis ( H A ) of step 1 of the six steps of statistical inference

© 2008 McGraw-Hill Higher Education Strength of a Relationship Strength: To what extent are errors reduced in predicting the scores of a dependent variable when an independent variable is used as a predictor?

© 2008 McGraw-Hill Higher Education Practical Applications of a Relationship Practical Applications: In practical, everyday terms, how does knowledge of a relationship between two variables help us understand and predict outcomes of the dependent variable?

© 2008 McGraw-Hill Higher Education Existence of a Relationship for 2-Group Difference of Means Test Existence: Established by using independent samples or nonindependent samples t-test When the H 0 is rejected, a relationship exists

© 2008 McGraw-Hill Higher Education Direction of a Relationship for 2-Group Difference of Means Test For the two group tests, direction and strength are not relevant Direction: Not relevant Strength: Not relevant

© 2008 McGraw-Hill Higher Education Practical Applications of Relationship for a 2-Group Difference of Means Test Practical Applications: Describe the effect of the test in everyday terms, where the effect of the independent variable on the dependent variable is the difference between sample means

© 2008 McGraw-Hill Higher Education Statistical Follies Avoid a common tendency: Difference in means testing is so widely used that researchers often focus too heavily on mean differences while ignoring the differences in variances (or standard deviations)