Chapter 7. Comparing Two Population Means. Comparing two population means. T-tests: Independent samples and paired variables.

Transcription

1 Chapter 7 Comparing Two Population Means Comparing two population means T-tests: Independent samples and paired variables. Usually subjects are assigned to treatment or control groups and one or more variables which are hypothesized to be affected by the treatment are measured. Chapter 7 1

2 The following null and alternative hypotheses are formulated: Ho: mean for treatment group is equalto that for the control group. Ha: means for treatment and control groups are different. OR Ha: mean for treatment is larger (smaller) than that for the control group. Example: Does increasing calcium intake reduce blood pressure? Observational studies suggest there is a link, and that is strongest in African-American men. Experiment: 21 African-American men participated 10 were assigned to treatment group. 11 were assigned to placebo group. Researchers measured the blood pressure of each subject before treatment & after 12 weeks. LINK: Example1.SAS Chapter 7 2

3 DATA CALCIUM; INFILE 'C:\CLASSES\STA5206\NOTES\chapter7\SAS_ FILES\CALCIUM.DAT'; INPUT GROUP$ BEFORE AFTER; PROC PRINT; The SAS System 02:26 Thursday, October 12, Obs GROUP BEFORE AFTER 1 Calcium Calcium Calcium Calcium Calcium Calcium Calcium Calcium Calcium Calcium Placebo Placebo Placebo Placebo Placebo Placebo Placebo Placebo Placebo Placebo Placebo Chapter 7 3

4 Independent T-test Observation for the two groups are independent. When sample size for each group is small (say<30), population need to be normally distributed. Need to check equality of variances for both population. LINK: Example 1A.SAS DATA CALCIUM; INFILE 'C:\CLASSES\STA5206\NOTES\chapter7\SAS _FILES\CALCIUM.DAT'; INPUT GROUP$ BEFORE AFTER; CHANGE = AFTER - BEFORE; PROC UNIVARIATE NORMAL; run; CLASS GROUP; VAR CHANGE; Chapter 7 4

5 The UNIVARIATE Procedure Variable: CHANGE GROUP = Calcium Tests for Normality Test --Statistic p Value Shapiro-Wilk W Pr < W Kolmogorov-Smirnov D Pr > D > Cramer-von Mises W-Sq Pr > W-Sq > Anderson-Darling A-Sq Pr > A-Sq > The UNIVARIATE Procedure Variable: CHANGE GROUP = Placebo Tests for Normality Test --Statistic p Value Shapiro-Wilk W Pr < W Kolmogorov-Smirnov D Pr > D Cramer-von Mises W-Sq Pr > W-Sq Anderson-Darling A-Sq Pr > A-Sq For the Calcium example, we are interested to find out if Change = After Before 0 Ho: Change = 0 Ha: Change 0 LINK: Example1B.SAS Chapter 7 5

6 DATA CALCIUM; INFILE 'C:\CLASSES\STA5206\NOTES\chapter7\SAS_FILES \CALCIUM.DAT'; INPUT GROUP$ BEFORE AFTER; CHANGE = AFTER - BEFORE; PROC TTEST DATA=CALCIUM; run; CLASS GROUP; VAR CHANGE; The TTEST Procedure Statistics Lower CL Upper CL Lower CL Variable Class N Mean Mean Mean Std Dev Std Dev CHANGE Calcium CHANGE Placebo CHANGE Diff (1-2) Statistics Upper CL Variable Class Std Dev Std Err Minimum Maximum CHANGE Calcium CHANGE Placebo CHANGE Diff (1-2) Chapter 7 6

7 T-Tests Variable Method Variances DF t Value Pr > t CHANGE Pooled Equal CHANGE Satterthwaite Unequal Equality of Variances Variable Method Num DF Den DF F Value Pr > F CHANGE Folded F Example: Clouds were randomly selected or not with silver nitrate.rainfalls were recorded from the clouds. Goal: Determine if cloud seeding increases rainfall LINK: Example2.SAS Chapter 7 7

8 DATA CLOUDS; INFILE 'C:\CLASSES\STA5206\NOTES\chapter7 \SAS_FILES\CLOUDS.DAT'; INPUT GROUP$ RAIN; PROC PRINT; run; The SAS System 02:26 Thursday, October 12, Obs GROUP RAIN 1 Unseeded Unseeded Unseeded Unseeded Unseeded Unseeded Unseeded Unseeded Unseeded Unseeded Unseeded Unseeded Unseeded Unseeded Unseeded Unseeded Unseeded Unseeded Unseeded Unseeded Unseeded Unseeded Unseeded Unseeded Unseeded Unseeded 1.0 Chapter 7 8

9 Obs GROUP RAIN 25 Unseeded Unseeded Seeded Seeded Seeded Seeded Seeded Seeded Seeded Seeded Seeded Seeded Seeded Seeded Seeded Seeded Seeded Seeded Seeded Seeded Seeded Seeded Seeded Seeded Seeded Seeded Seeded Seeded 4.1 LINK: Example2A.SAS Chapter 7 9

10 DATA CLOUDS; INFILE 'C:\CLASSES\STA5206\NOTES\chapter7\ SAS_FILES\CLOUDS.DAT'; INPUT GROUP$ RAIN; PROC UNIVARIATE NORMAL; CLASS GROUP; VAR RAIN; run; The UNIVARIATE Procedure Variable: RAIN GROUP = Seeded Tests for Normality Test --Statistic p Value Shapiro-Wilk W Pr < W < Kolmogorov-Smirnov D Pr > D < Cramer-von Mises W-Sq Pr > W-Sq < Anderson-Darling A-Sq Pr > A-Sq < The UNIVARIATE Procedure Variable: RAIN GROUP = Unseeded Tests for Normality Test --Statistic p Value Shapiro-Wilk W Pr < W < Kolmogorov-Smirnov D Pr > D < Cramer-von Mises W-Sq Pr > W-Sq < Anderson-Darling A-Sq Pr > A-Sq < Chapter 7 10

11 Comparing means when data are non-normal (Distribution free test) When data are nonnormal and sample size is small. We should use nonparametric methods. We also use nonparametric methods when the variables are ordinal. Example: 1 = very mild 2 = mild 3 = moderate 4 = strong Example: Suppose there are two groups: Group A: Control Group B: Treated with a drug to prevent tumor formation. Both Groups are exposed to a chemical that encourages tumor growth. The following values are masses of tumors in groups A and B. A: 3.1, 2.2, 1.7, 2.7, 2.5 B: 0, 0, 1, 2.3 Chapter 7 11

12 We combine bothsets together and sort the data from small to large. Mass Group B B B A A B A A A Rank Sum Ranks A = =33 Sum Ranks B = =12 The smaller the sum of ranks for B, the smaller we expect the tumors to be for group B. Question: Is the sum rank for group A sufficiently larger than the sum rank for group B? The Wilcoxon Test gives us the probability that the difference in rank sums observed is not just by chance alone. LINK: Example2B.SAS Chapter 7 12

13 DATA CLOUDS; INFILE 'C:\CLASSES\STA5206\NOTES\chapter7\ SAS_FILES\CLOUDS.DAT'; INPUT GROUP$ RAIN; PROC NPAR1WAY DATA = CLOUDS WILCOXON; run; CLASS GROUP; VAR RAIN; EXACT WILCOXON; The SAS System 02:26 Thursday, October 12, The NPAR1WAY Procedure Wilcoxon Scores (Rank Sums) for Variable RAIN Classified by Variable GROUP Sum of Expected Std Dev Mean GROUP N Scores Under H0 Under H0 Score ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ Unseeded Seeded Average scores were used for ties. Chapter 7 13

14 Wilcoxon Two-Sample Test Statistic (S) Normal Approximation Z One-Sided Pr < Z Two-Sided Pr > Z t Approximation One-Sided Pr < Z Two-Sided Pr > Z Exact Test One-Sided Pr <= S Two-Sided Pr >= S - Mean Z includes a continuity correction of 0.5. Kruskal-Wallis Test Chi-Square DF 1 Pr > Chi-Square Paired T-test To compare means of two populations, sometimes it s useful to pair data. Examples: Measuring tire wear of two brands of bicycle tires use pair on one bike Measuring effect of a drug on reducing blood pressure Before-After Comparing two methods of teaching a certain math concept. match based on IQ Chapter 7 14

15 Example: We are interested to measure the growing presence of women in the labor force over the period 1968 to A data set called Labor Force contains the labor force participation rate (LFPR) of women in 19 cities in the United States. LINK: Example3.SAS DATA Labor_Force; infile 'C:/Classes/STA5206/Notes/Chapter7/SAS _FILES/labor.dat'; input City $ YR72 YR68; proc print; run; Chapter 7 15

16 The SAS System 23:06 Monday, October 16, Obs City YR72 YR68 1 N.Y L.A Chicago Philadel Detroit San_Fran Boston Pitt St._Loui Connecti Wash._D Cinn Baltimor Newark Minn/St Buffalo Houston Patterso Dallas Since the United States did not change much from 1968 to 1972, a pooled t-test of LFPR rates may be considered. However, the data are naturally paired because the measurements were made in the same cities for each of the two years. If the sample size is small, it is required that the difference between the two variables come from a normally distributed population. LINK: Example3A.SAS Chapter 7 16

17 DATA Labor_Force; infile 'C:/Classes/STA5206/Notes/Chapter7/SAS _FILES/labor.dat'; input City $ YR72 YR68; DIFF = YR72 - YR68; proc univariate normal plot; run; var diff; id city; The SAS System 23:06 Monday, October 16, The UNIVARIATE Procedure Variable: DIFF Tests for Normality Test --Statistic p Value Shapiro-Wilk W Pr < W Kolmogorov-Smirnov D Pr > D Cramer-von Mises W-Sq Pr > W-Sq Anderson-Darling A-Sq Pr > A-Sq Chapter 7 17

18 Extreme Observations Lowest Highest Value City Obs Value City Obs St._Loui Baltimor Newark Minn/St San_Fran Wash._D Philadel Boston Chicago Pitt. 8 Stem Leaf # Boxplot *--+--* Multiply Stem.Leaf by 10**-1 To perform T test use: Proc Means T PRT ; LINK: Example3b.SAS The paired T-test is significant at 5% level. Interestingly, the pooled t-test is not significant at 5% level. Chapter 7 18

19 DATA Labor_Force; infile 'C:/Classes/STA5206/Notes/Chapter7/SAS_ FILES/labor.dat'; input City $ YR72 YR68; diff = YR72 - YR68; proc ttest; run; var diff; The TTEST Procedure Statistics Lower CL Upper CL Lower CL Upper CL Variable N Mean Mean Mean Std Dev Std Dev Std Dev diff Statistics Variable Std Err Minimum Maximum diff T-Tests Variable DF t Value Pr > t diff Chapter 7 19