COURSE: JUST 3900 INTRODUCTORY STATISTICS FOR CRIMINAL JUSTICE Chapter 10: t Test for Two Independent Samples Instructor: Dr. John J. Kerbs, Associate Professor Joint Ph.D. in Social Work and Sociology © 2013 - - DO NOT CITE, QUOTE, REPRODUCE, OR DISSEMINATE WITHOUT WRITTEN PERMISSION FROM THE AUTHOR: Dr. John J. Kerbs can be emailed for permission at [email protected] Independent-Measures Designs Allows researchers to evaluate the mean difference between two populations using data from two separate samples. The identifying characteristic of the independentmeasures or between-subjects design is the existence of two separate or independent samples. Thus, an independent-measures design can be used to test for mean differences between two distinct populations (such as men versus women) or between two different treatment conditions (such as drug versus no-drug). Teaching Method A vs. B Independent-Measures Designs (cont'd.) The independent-measures design is used in situations where a researcher has no prior knowledge about either of the two populations (or treatments) being compared. In particular, the population means and standard deviations are all unknown. Because the population variances are not known, these values must be estimated from the sample data. The t Statistic for an IndependentMeasures Research Design As with all hypothesis tests, the general purpose of the independent-measures t test is to determine whether the sample mean difference obtained in a research study indicates a real mean difference between the two populations (or treatments) or whether the obtained difference is simply the result of sampling error. Remember, if two samples are taken from the same population and are given exactly the same treatment, there will still be some difference between the sample means (i.e., this difference is called sampling error). The hypothesis test provides a standardized, formal procedure for determining whether the mean difference obtained in a research study is significantly greater than can be explained by sampling error Two Population Distributions Hypothesis Tests and Effect Size with the Independent Measures t Statistic To prepare the data for analysis, the first step is to compute the sample mean and SS (or s, or s2) for each of the two samples. The hypothesis test follows the same four-step procedure outlined in Chapters 8 and 9. Elements of a t statistics: Single-Sample & Independent Measures NOTE: This alternative formula on the bottom left for pooled variance works when you have sample variances for the first and second samples (s2) and not sum of squares (SS) as noted here Hypothesis Testing with the Independent-Measures t Statistic Book Example: For 10 students who watched Sesame Street (group 1) and 10 students who did not watch the show (group 2), was there a difference in their average high-school grades. Key Information to Run t-tests for independentmeasures n1 = 10 n2 = 10 M1 = 93 M2 = 85 SS1 = 200 SS2 = 160 Hypothesis Testing with the Independent-Measures t Statistic Step 1: State the hypotheses and select the α level. For the independent-measures test, H0 states that there is no difference between the two population means. For a two-tailed test, the hypotheses are as follows: Example: H0: µ1 - µ2 = 0 H1: µ1 - µ2 ≠ 0 Select α: α = 0.01 (two tail) For a one-tail test, H0: µ1 - µ2 ≤ 0 H1: µ1 - µ2 > 0 Hypothesis Testing with the Independent-Measures t Statistic Step 2: Locate the critical region. The critical values for the t statistic are obtained using degrees of freedom that are determined by adding together the df value for the first sample and the df value for the second sample. For two samples with n = 10 in each sample, df is calculated as follows: df = df1 + df2 df = (n1 - 1) + (n2 -1) df = 9 + 9 = 18 To find the critical t value for 18df at α = 0.01, we look at the t-distribution table: t = +/- 2.878 Example: Hypothesis Testing with the Independent-Measures t Statistic Step 3: Compute the test statistic. The t statistic for the independent-measures design has the same structure as the single sample t introduced in Chapter 9. However, in the independent-measures situation, all components of the t formula are doubled: there are two sample means, two population means, and two sources of error contributing to the standard error in the denominator. Three key parts to the t-statistic calculation: Part A) Calculate the pooled variance Part B) Use the pooled variance to calculate the estimated standard error Part C) Compute t-statistic Hypothesis Testing with the Independent-Measures t Statistic Step 3 (Part A): Pooled Variance Step 3 (Part B): Estimated Standard Error Step 3 (Part C): Compute t-Statistic t= 1 − 2 −(µ1 − µ2 ) (1 − 2 ) = 1 − 2 −(µ1 − µ2 ) (1 − 2 ) = 93−85 −0 2 8 2 = =4 Hypothesis Testing with the Independent-Measures t Statistic Step 4: Make a decision. If the t statistic ratio indicates that the obtained difference between sample means (numerator) is substantially greater than the difference expected by chance (denominator), we reject H0 and conclude that there is a real mean difference between the two populations or treatments. T = 4.00>+2.878, thus reject H0 and conclude that there is a significant difference in high school grades for those who watched Sesame Street as compared to those who did not watch this show. Measuring Effect Size for the Independent-Measures t Effect size for the independent-measures t is measured in the same way that we measured effect size for the single-sample t in Chapter 9. Specifically, you can compute an estimate of Cohen’s d or you can compute r2 to obtain a measure of the percentage of variance accounted for by the treatment effect. Measuring Effect Size for the Independent-Measures t Compute an Estimated Cohen’s d Est Cohen’s d = Magnitude of d 1 − 2 2 = 93−85 20 = 8 4.7 = 1.79 Evaluation of Effect Size d = 0.2 Small effect (mean difference around 0.2 standard deviations) d = 0.5 Medium effect (mean difference around 0.5 standard deviations) d = 0.8 Large effect (mean difference around 0.8 standard deviations) 2 2 4 16 2 Compute r = 2 = 2 = = 16 = 0.47 or 47% 34 + 4 +18 Percent of Variance Explained as Measured by r2 16+18 Evaluation of Effect Size r2 = 0.01 (0.01*100 = 1%) Small effect r2 = 0.09 (0.09*100 = 9%) Medium effect r2 = 0.25 (0.25*100 = 25%) Large effect Removing Treatment Effects NOTE: We added 4 points to those students who did not watch Sesame Street NOTE: We subtracted 4 points to those students who watched Sesame Street Confidence Intervals and Hypothesis Tests Calculate a 95% Confidence Interval for the Sesame Street example. For a two-tail critical value with df=18 and α = 0.05, tcrit = +/- 2.101. NOTE: Because 0 is not in the 95% C Interval, we can conclude that the value of 0 is not in µ1 - µ2 = M – M +/-t(1 − 2 ) the 95% Confidence Interval. Alternatively, the = 93 – 85 +/- 2.101(2) = 8 +/- 4.202 = 3.798 value of 0 is rejected with 95% confidence. This is the same as rejecting H0 with α = 0.05 to 12.202 Sample Variance and Sample Size Standard error is positively related to sample variance (larger variance leads to larger sample error). Standard error is inversely related to sample size (large sample sizes lead to smaller standard error). Larger variance produces smaller t-statistic and reduces the likelihood of a significant finding. Larger variance also produces smaller measures of effect size Larger samples produces larger values for t-statistic and increases the likelihood of rejecting H0. Sample size has no effect on Cohen’s d and only a small influence on r2 Homogeneity of Variance Assumption Most hypothesis tests usually work reasonably well even if the set of underlying assumptions are violated. The one notable exception is the assumption of homogeneity of variance for the independentmeasures t test. Requires that the two populations from which the samples are obtained have equal variances Necessary in order to justify pooling the two sample variances and using the pooled variance in the calculation of the t statistic Homogeneity of Variance Assumption If the assumption is violated, then the t statistic contains two questionable values: (1) the value for the population mean difference which comes from the null hypothesis, and (2) the value for the pooled variance. The problem is that you cannot determine which of these two values is responsible for a t statistic that falls in the critical region. In particular, you cannot be certain that rejecting the null hypothesis is correct when you obtain an extreme value for t. Homogeneity of Variance Assumption If the two sample variances appear to be substantially different, you should use Hartley’s F-max test to determine whether or not the homogeneity assumption is satisfied. If homogeneity of variance is violated, Box 10.2 presents an alternative procedure for computing the t statistic that does not involve pooling the two sample variances. Homogeneity of Variance Assumption Hartley’s F-max test Step 1: Compute Sample Variance for each separate sample NOTE: If the F-max test rejects the (s2 = ). 2() hypothesis of equal variances, or if you suspect that the homogeneity of variance assumption is not justified, you should not compute an independent-measures t-statistic using pooled variance. In such cases, use the alternative formula for the t-statistic on the next slide. Step 2: Compute F-Max = Step 3: Find critical F-Max value k = number of separate samples (2) df = n – 1 for each sample (Hartley test assumes all samples are the same size) Select an alpha (α) level for critical F-Max value - homogeneity tests use larger alpha levels at 0.05 or 0.01 Step 4: If computed F-Max < F-crit, then no evidence that the homogeneity of variance assumption has been violated. 2() Homogeneity of Variance Assumption Alternative formula for t-statistic Step 1: Calculate the standard error using the two separate sample variances as in equation 10.1 NOTE: The following alternative formula for the t-statistic does not pool sample variances and does not require the homogeneity of variance assumption. Step 2: The value of degrees of freedom for the t-statistic is adjusted using the following equation: Decimal values for df should be rounded down to the next lower integer. By lowering the df, you push the boundaries of the critical region farther out. This makes the test more demanding.