S160 #20. Comparing Two Means Paired Data. JC Wang. April 5, 2016

Transcription

1 S160 #20 Comparing Two Means Paired Data JC Wang April 5, 2016

2 Outline 1 Paired Data Comparing Means in Paired Data (General) Paired Data JC Wang (WMU) S160 #20 S160, Lecture 20 2 / 12

3 Paired Data, Before-and-After Data Data come in pairs, each subject was measured twice, before and after. Note that it s NOT two samples, it s one sample of subjects, each measured twice. Weight Loss Example: Weight in pounds before and after 12 months on diet Subject Before After JC Wang (WMU) S160 #20 S160, Lecture 20 3 / 12

4 Analysis of Paired Data It s WRONG to analyze the data using two-sample method. The proper procedure is laid out below: 1 Calculate the pairwise differences in a consistent manner: d i = Before After, i = 1,, n. 2 Treat the differences as one sample and Mean difference: D = 1 n (d d n ). Standard error (of D): SE = SD n, where SD = SD d = A 95% c.i. for µ d = µ 1 µ 2 : Sum [ (d i D) 2] n 1 D ± ME (where ME = 1.96 SE) If the confidence interval excludes 0, then means (of before and after) are statistically significant. JC Wang (WMU) S160 #20 S160, Lecture 20 4 / 12

5 Analysis of Paired Data It s WRONG to analyze the data using two-sample method. The proper procedure is laid out below: 1 Calculate the pairwise differences in a consistent manner: d i = Before After, i = 1,, n. 2 Treat the differences as one sample and Mean difference: D = 1 n (d d n ). Standard error (of D): SE = SD n, where SD = SD d = A 95% c.i. for µ d = µ 1 µ 2 : Sum [ (d i D) 2] n 1 D ± ME (where ME = 1.96 SE) If the confidence interval excludes 0, then means (of before and after) are statistically significant. JC Wang (WMU) S160 #20 S160, Lecture 20 4 / 12

6 Analysis of Paired Data It s WRONG to analyze the data using two-sample method. The proper procedure is laid out below: 1 Calculate the pairwise differences in a consistent manner: d i = Before After, i = 1,, n. 2 Treat the differences as one sample and Mean difference: D = 1 n (d d n ). Standard error (of D): SE = SD n, where SD = SD d = A 95% c.i. for µ d = µ 1 µ 2 : Sum [ (d i D) 2] n 1 D ± ME (where ME = 1.96 SE) If the confidence interval excludes 0, then means (of before and after) are statistically significant. JC Wang (WMU) S160 #20 S160, Lecture 20 4 / 12

7 Analysis of Paired Data It s WRONG to analyze the data using two-sample method. The proper procedure is laid out below: 1 Calculate the pairwise differences in a consistent manner: d i = Before After, i = 1,, n. 2 Treat the differences as one sample and Mean difference: D = 1 n (d d n ). Standard error (of D): SE = SD n, where SD = SD d = A 95% c.i. for µ d = µ 1 µ 2 : Sum [ (d i D) 2] n 1 D ± ME (where ME = 1.96 SE) If the confidence interval excludes 0, then means (of before and after) are statistically significant. JC Wang (WMU) S160 #20 S160, Lecture 20 4 / 12

8 Analysis of Paired Data It s WRONG to analyze the data using two-sample method. The proper procedure is laid out below: 1 Calculate the pairwise differences in a consistent manner: d i = Before After, i = 1,, n. 2 Treat the differences as one sample and Mean difference: D = 1 n (d d n ). Standard error (of D): SE = SD n, where SD = SD d = A 95% c.i. for µ d = µ 1 µ 2 : Sum [ (d i D) 2] n 1 D ± ME (where ME = 1.96 SE) If the confidence interval excludes 0, then means (of before and after) are statistically significant. JC Wang (WMU) S160 #20 S160, Lecture 20 4 / 12

9 Analysis of Paired Data It s WRONG to analyze the data using two-sample method. The proper procedure is laid out below: 1 Calculate the pairwise differences in a consistent manner: d i = Before After, i = 1,, n. 2 Treat the differences as one sample and Mean difference: D = 1 n (d d n ). Standard error (of D): SE = SD n, where SD = SD d = A 95% c.i. for µ d = µ 1 µ 2 : Sum [ (d i D) 2] n 1 D ± ME (where ME = 1.96 SE) If the confidence interval excludes 0, then means (of before and after) are statistically significant. JC Wang (WMU) S160 #20 S160, Lecture 20 4 / 12

10 SD = Weight Loss Example, revisited Subject Before After Difference Difference D D = ( 4.6) ( 19.6) ( 4.6) = 11.1 and hence SE = SD n = = 4.2 JC Wang (WMU) S160 #20 S160, Lecture 20 5 / 12

11 Weight Loss Example, continued The margin of error is ME = 1.96 SE = = 8.2. Hence a 95% confidence interval for mean weight loss (i.e., mean difference of Before and After) is ( , ) = (1.4, 17.8) which excludes 0 and hence the mean weight loss is statistically significant. JC Wang (WMU) S160 #20 S160, Lecture 20 6 / 12

12 iclicker Question 20.1 Measurements of the left-hand and right-hand gripping strengths of 10 left-handed writers are recorded: Person Left hand Right hand Should this data set be treated as a paired data problem? A. Yes. B. No. C. Cannot determine. JC Wang (WMU) S160 #20 S160, Lecture 20 7 / 12

13 iclicker Question 20.2 Measurements of the left-hand and right-hand gripping strengths of 10 left-handed writers are recorded: Person Left hand Right hand A 95% confidence interval for µ left µ right is calculated and the result is (0.22,6.98). Which statement below is true about the confidence interval? A. We are 95% confident that the difference in sample means is between 0.22 and B. There is a 95% probability that the difference in the true mean gripping strengths of the two hands is between 0.22 and C. We are 95% confident that the difference in the true mean gripping strengths is between 0.22 and D. There is a 95% probability that the difference in the sample mean gripping strengths of the two hands is between 0.22 and E. None of the previous. JC Wang (WMU) S160 #20 S160, Lecture 20 8 / 12

14 iclicker Question 20.3 Measurements of the left-hand and right-hand gripping strengths of 10 left-handed writers are recorded: Person Left hand Right hand A 95% confidence interval for µ left µ right is calculated and the result is (0.22,6.98). Which statement below is true about the confidence interval? A. The average gripping strength of the left hand differs significantly than that of the right hand since the confidence interval includes 0. B. The average gripping strength of the left hand differs significantly than that of the right hand since the confidence interval excludes 0. C. The difference in average gripping strengths of the two hands is insignificantly since the confidence interval excludes 0. D. The difference in average gripping strengths of the two hands is insignificantly since the confidence interval includes 0. E. None of the previous. JC Wang (WMU) S160 #20 S160, Lecture 20 9 / 12

15 (General) Paired Data Paired data are data in which natural matchings occur. Even when we have two samples, if each observation in one sample is uniquely matched to an observation in the other sample, then we have paired data. The analysis of such data should follow that of before-and-after paired data. JC Wang (WMU) S160 #20 S160, Lecture / 12

16 (General) Paired Data Paired data are data in which natural matchings occur. Even when we have two samples, if each observation in one sample is uniquely matched to an observation in the other sample, then we have paired data. The analysis of such data should follow that of before-and-after paired data. JC Wang (WMU) S160 #20 S160, Lecture / 12

17 (General) Paired Data Paired data are data in which natural matchings occur. Even when we have two samples, if each observation in one sample is uniquely matched to an observation in the other sample, then we have paired data. The analysis of such data should follow that of before-and-after paired data. JC Wang (WMU) S160 #20 S160, Lecture / 12

18 (General) Paired Data Paired data are data in which natural matchings occur. Even when we have two samples, if each observation in one sample is uniquely matched to an observation in the other sample, then we have paired data. The analysis of such data should follow that of before-and-after paired data. JC Wang (WMU) S160 #20 S160, Lecture / 12

19 HIC, Head Injury Criterion Example Driver Passenger Acura Integra Audi Chevrolet Camaro Ford Escort Honda Accord LX Toyota Corolla Fx Volvo 740 GLE JC Wang (WMU) S160 #20 S160, Lecture / 12

20 HIC Example, continued The differences of HIC Driver Passenger: Mean difference: D = S = and hence SE = 72.40/ 7 = ME = = A 95% c.i. for mean difference in HICs (Driver Passenger): ( , ) = (19.95, ) which excludes 0 and hence the mean difference in HICs is statistically significant. JC Wang (WMU) S160 #20 S160, Lecture / 12

21 HIC Example, continued The differences of HIC Driver Passenger: Mean difference: D = S = and hence SE = 72.40/ 7 = ME = = A 95% c.i. for mean difference in HICs (Driver Passenger): ( , ) = (19.95, ) which excludes 0 and hence the mean difference in HICs is statistically significant. JC Wang (WMU) S160 #20 S160, Lecture / 12

22 HIC Example, continued The differences of HIC Driver Passenger: Mean difference: D = S = and hence SE = 72.40/ 7 = ME = = A 95% c.i. for mean difference in HICs (Driver Passenger): ( , ) = (19.95, ) which excludes 0 and hence the mean difference in HICs is statistically significant. JC Wang (WMU) S160 #20 S160, Lecture / 12

23 HIC Example, continued The differences of HIC Driver Passenger: Mean difference: D = S = and hence SE = 72.40/ 7 = ME = = A 95% c.i. for mean difference in HICs (Driver Passenger): ( , ) = (19.95, ) which excludes 0 and hence the mean difference in HICs is statistically significant. JC Wang (WMU) S160 #20 S160, Lecture / 12

24 HIC Example, continued The differences of HIC Driver Passenger: Mean difference: D = S = and hence SE = 72.40/ 7 = ME = = A 95% c.i. for mean difference in HICs (Driver Passenger): ( , ) = (19.95, ) which excludes 0 and hence the mean difference in HICs is statistically significant. JC Wang (WMU) S160 #20 S160, Lecture / 12