7. Comparing Two GroupsExample: Does cell phone use while driving impair reaction times?Outcome measure: mean response time for a subject over a large number of trialsSlide 4Types of variables and samplesSlide 6se for difference between two estimates (independent samples)Slide 8CI comparing two proportionsExample: College Alcohol Study conducted by Harvard School of Public Health (http://www.hsph.harvard.edu/cas/)Slide 11Comments about CIs for difference between two population proportionsSlide 13Slide 14Quantitative Responses: Comparing MeansExample: GSS data on “number of close friends”Slide 17Significance Tests for m2 - m1Slide 19Slide 20Equivalence of CI and Significance TestAlternative inference comparing means assumes equal population standard deviationsSlide 23Slide 24Test of H0: m1 = m2 Ha: m1  m2How does software get df for “unequal variance” method?Some comments about comparing meansSlide 28This effect size measure is sometimes called “Cohen’s d.” He considered d = 0.2 = weak, d = 0.5 = medium, d > 0.8 large. Example: Which study showed the largest effect?Comparing Means with Dependent SamplesSlide 31Slide 32Slide 33Slide 34Slide 35Some commentsSlide 37Slide 38Slide 39A few summary questionsSlide 417. Comparing Two GroupsGoal: Use CI and/or significance test to compare means (quantitative variable) proportions (categorical variable) Group 1 Group 2 EstimatePopulation mean Population proportionWe conduct inference about the difference between the means or difference between the proportions (order irrelevant). 1 2 2 11 2 2 1 ˆ ˆ y ym mp p p p--Example: Does cell phone use while driving impair reaction times?•Article in Psych. Science (2001, p. 462) describes experiment that randomly assigned 64 Univ. of Utah students to cell phone group or control group (32 each). Driving simulating machine flashed red or green at irregular periods. Instructions: Press brake pedal as soon as possible when detect red light.See http://www.psych.utah.edu/AppliedCognitionLab/•Cell phone group: Carried out conversation about a political issue with someone in separate room.•Control group: Listened to radio broadcastOutcome measure: mean response time for a subject over a large number of trials•Purpose of study: Analyze whether (conceptual) population mean response time differs significantly for the two groups, and if so, by how much.•DataCell-phone group: = 585.2 milliseconds, s1 = 89.6Control group: = 533.7, s2 = 65.3. Shape? Outliers?1y2yTypes of variables and samples•The outcome variable on which comparisons are made is the response variable.•The variable that defines the groups to be compared is the explanatory variable.Example: Reaction time is response variableExperimental group is explanatory variable -- a categorical var. with categories: (cell-phone, control)Or, could express experimental group as “cell-phone use” with categories (yes, no)•Different methods apply for independent samples -- different samples, no matching, as in this example and in “cross-sectional studies”dependent samples -- natural matching between each subject in one sample and a subject in other sample, such as in “longitudinal studies,” which observe subjects repeatedly over timeExample: We later consider a separate experiment in which the same subjects formed the control group at one time and the cell-phone group at another time.se for difference between two estimates (independent samples)•The sampling distribution of the difference between two estimates is approximately normal (large n1 and n2) and has estimatedExample: Data on “Response times” has 32 using cell phone with sample mean 585.2, s = 89.6 32 in control group with sample mean 533.7, s = 65.3What is se for difference between sample means of 585.2 – 533.7 = 51.4?2 21 2( ) ( )se se se= +(Note this is larger than each separate se. Why?)So, the estimated difference of 51.4 has a margin of error of about 2(19.6) = 39.2 (more precise details later using t distribution)95% CI for  –  is about 51.4 ± 39.2, or (12, 91). Interpretation: We are 95% confident that population mean  for cell phone group is between 12 milliseconds higher and 91 milliseconds higher than population mean  for control group.(In practice, good idea to re-do analysis without outlier, to check its influence. What do you think would happen?)1 1 12 2 22 2 2 21 2/ 89.6 / 32 15.84/ 65.3/ 32 11.54( ) ( ) (15.84) (11.54) 19.6se s nse s nse se se= = == = == + = + =CI comparing two proportions•Recall se for a sample proportion used in a CI is•So, the se for the difference between two sample proportions for independent samples is •A CI for the difference between population proportions isAs usual, z depends on confidence level, 1.96 for 95% confidenceˆ ˆ(1 ) /se np p= -2 21 1 2 21 21 2ˆ ˆ ˆ ˆ(1 ) (1 )( ) ( )se se sen np p p p- -= + = +1 1 2 22 11 2ˆ ˆ ˆ ˆ(1 ) (1 )ˆ ˆ( ) zn np p p pp p- -- � +Example: College Alcohol Study conducted by Harvard School of Public Health (http://www.hsph.harvard.edu/cas/)Trends over time in percentage of binge drinking (consumption of 5 or more drinks in a row for men and 4 or more for women, at least once in past two weeks) and of activities perhaps influenced by it?“Have you engaged in unplanned sexual activities because of drinking alcohol?”1993: 19.2% yes of n = 12,7082001: 21.3% yes of n = 8783What is 95% CI for change saying “yes”?•Estimated change in proportion saying “yes” is 0.213 – 0.192 = 0.021.95% CI for change in population proportion is 0.021 ± 1.96(0.0056) = 0.021 ± 0.011, or roughly (0.01, 0.03)We can be 95% confident that the population proportion saying “yes” was between about 0.01 larger and 0.03 larger in 2001 than in 1993.1 1 2 21 2ˆ ˆ ˆ ˆ(1 ) (1 ) (.192)(.808) (.213)(.787)0.005612,708 8783sen np p p p- -= + = + =Comments about CIs for difference between two population proportions•If 95% CI for is (0.01, 0.03), then 95% CI for is (-0.03, -0.01). It is arbitrary what we call Group 1 and Group 2 and what the order is for comparing the proportions.•When 0 is not in the CI, we can conclude that one population proportion is higher than the other. (e.g., if all positive values for Group 2 – Group 1, then conclude population proportion higher for Group 2 than Group 1)2 1p p-1 2p p-•When 0 is in the CI, it is