UNDERSTANDING THE INDEPENDENT-SAMPLES t TEST The independent-samples t test evaluates the difference between the means of two independent or unrelated groups. That is, we evaluate whether the means for two independent groups are significantly different from each other. The independent-samples t test is commonly referred to as a between-groups design, and can also be used to analyze a control and experimental group. With an independent-samples t test, each case must have scores on two variables, the grouping (independent) variable and the test (dependent) variable. The grouping variable divides cases into two mutually exclusive groups or categories, such as boys or girls for the grouping variable gender, while the test variable describes each case on some quantitative dimension such as test performance. The t test evaluates whether the mean value of the test variable (e.g., test performance) for one group (e.g., boys) differs significantly from the mean value of the test variable for the second group (e.g., girls). HYPOTHESES FOR THE INDEPENDENT-SAMPLES t TEST Null Hypothesis: H0: µ1 = µ2 where µ1 stands for the mean for the first group and µ2 stands for the mean for the second group. -or- H0: µ1 – µ2 = 0 Alternative (Non-Directional) Hypothesis: Ha: µ1 ≠ µ2 -or- Ha: µ1 – µ2 ≠ 0 Alternative (Directional) Hypothesis: Ha: µ1 < µ2 -or- Ha: µ1 > µ2 (depending on direction) NOTE: the subscripts (1 and 2) can be substituted with the group identifiers For example: H0: µBoys = µGirls Ha: µBoys ≠ µGirls ASSUMPTIONS UNDERLYING THE INDEPENDENT-SAMPLES t TEST 1. The data (scores) are independent of each other (that is, scores of one participant are not systematically related to scores of the other participants). This is commonly referred to as the assumption of independence. 2. The test (dependent) variable is normally distributed within each of the two populations (as defined by the grouping variable). This is commonly referred to as the assumption of normality. 3. The variances of the test (dependent) variable in the two populations are equal. This is commonly referred to as the assumption of homogeneity of variance. Null Hypothesis: H0: 21σ = 22σ (if retained = assumption met) (if rejected = assumption not met) Alternative Hypothesis: Ha: 21σ ≠ 22σTHE INDEPENDENT-SAMPLES t TEST PAGE 2 TESTING THE ASSUMPTION OF INDEPENDENCE One of the first steps in using the independent-samples t test is to test the assumption of independence. Independence is a methodological concern; it is dealt with (or should be dealt with) when a study is set up. Although the independence assumption can ruin a study if it is violated, there is no way to use the study’s sample data to test the validity of this prerequisite condition. It is assessed through an examination of the design of the study. That is, we confirm that the two groups are independent of each other? The assumption of independence is commonly known as the unforgiving assumption (r.e., robustness), which simply means that if the two groups are not independent of each other, one cannot use the independent-samples t test. TESTING THE ASSUMPTION OF NORMALITY Another of the first steps in using the independent-samples t test is to test the assumption of normality, where the Null Hypothesis is that there is no significant departure from normality, as such; retaining the null hypothesis indicates that the assumption of normality has been met for the given sample. The Alternative Hypothesis is that there is a significant departure from normality, as such; rejecting the null hypothesis in favor of the alternative indicates that the assumption of normality has not been met for the given sample. To test the assumption of normality, we can use the Shapiro-Wilks test. From this test, the Sig. (p) value is compared to the a priori alpha level (level of significance for the statistic) – and a determination is made as to reject (p < α) or retain (p > α) the null hypothesis. Tests of Normality.229 15 .033 .917 15 .170.209 15 .076 .888 15 .062Stress ConditionLow StressHigh StressPercentage of time talkingStatistic df Sig. Statistic df Sig.Kolmogorov-SmirnovaShapiro-WilkLilliefors Significance Correctiona. For the above example, where α = .001, given that p = .170 for the Low Stress Group and p = .062 for the High Stress Group – we would conclude that each of the levels of the Independent Variable (Stress Condition) are normally distributed. Therefore, the assumption of normality has been met for this sample. The a priori alpha level is typically based on sample size – where .05 and .01 are commonly used. Tabachnick and Fidell (2007) report that conventional but conservative (.01 and .001) alpha levels are commonly used to evaluate the assumption of normality. NOTE: Most statisticians will agree that the Shapiro-Wilks Test should not be the sole determination of normality. It is common to use this test in conjunction with other measures such as an examination of skewness, kurtosis, histograms, and normal Q-Q plots.THE INDEPENDENT-SAMPLES t TEST PAGE 3 In examining skewness and kurtosis, we divide the skewness (kurtosis) statistic by its standard error. We want to know if this standard score value significantly departs from normality. Concern arises when the skewness (kurtosis) statistic divided by its standard error is greater than z +3.29 (p < .001, two-tailed test) (Tabachnick & Fidell, 2007). We have several options for handling non-normal data, such as deletion and data transformation (based on the type and degree of violation as well as the randomness of the missing data points). Any adjustment to the data should be justified (i.e., referenced) based on solid resources (e.g., prior research or statistical references). As a first step, data should be thoroughly screened to ensure that any issues are not a factor of missing data or data entry errors. Such errors should be resolved prior to any data analyses using acceptable procedures (see for example Howell, 2007 or Tabachnick & Fidell, 2007). TESTING THE ASSUMPTION OF HOMOGENEITY OF VARIANCE Another of the first steps in using the independent-samples t test statistical analysis is to test the assumption of homogeneity of variance, where the null hypothesis