Section 11 Part 1. Confidence Intervals and Hypothesis Tests for Means from Paired data

Transcription

1 Section 11 Part 1 Confidence Intervals and Hypothesis Tests for Means from Paired data

2 Section 11 Overview Statistical inference from a sample to the population has been limited to data from one measure on one sample. Section 8: One sample confidence intervals and hypothesis tests of the population mean Section 9: One sample confidence intervals and hypothesis tests of the population proportion Section 10: Review for Exam II Section 11 extends inference methods about the mean to paired data (two measures on one sample) and data from two samples (one measure on two samples). Part 1: Paired Data estimation and tests of mean difference Part 2: Two Samples estimation and tests of difference of means. PubH 6414 Section 11 Part 1 2

3 Paired Data Paired data can occur when we have: Two outcome measures on one individual in time (before/after) Two outcome measures on one individual in location (right eye/left eye). Two outcome measures on twins or people carefully matched to be similar. The research question is whether or not the intervention made a difference Is there a change cholesterol from before treatment to after treatment? Is clearing of infection in the right eye better than the left eye? Did the twin who did not received therapy score higher on a depression scale than the other? The parameter of interest is the mean of the differences. PubH 6414 Section 11 Part 1 3

4 Hypothesis Test of the Mean Difference The hypothesis test of the Mean difference is called a Paired t- test The null hypothesis is that there is no change: H o : the mean difference = 0 The alternative hypothesis is that there is a change H A : the mean difference 0 (two-tailed) H A : the mean difference < 0 or >0 (one-tailed) PubH 6414 Section 11 Part 1 4

5 Paired t-test Procedure 1. State the hypotheses 2. Calculate test statistic 2. Calculate the p-value 3. State the conclusion of the test PubH 6414 Section 11 Part 1 5

6 COPD Example The 6 minute walk test can be used to measure physical stamina: How far (in ft) can the patient walk in 6 minutes? A study was designed to evaluate whether a new respiratory therapy intervention for COPD (Chronic Obstructive Pulmonary Disease) patients resulted in a change in 6 minute distance walked PubH 6414 Section 11 Part 1 6

7 COPD Example Distance walked in 6 minutes was measured for a random sample of 16 COPD patients before (Pre) and after (Post) 90 days on the new therapy. Assume that distribution of the mean difference of distance walked is approximately normally distributed Conduct a Paired t-test to test whether there was a significant change in 6 min. distance PubH 6414 Section 11 Part 1 7

8 Pre Post DIFF d s d = = PubH 6414 Section 11 Part 1 8

9 1. State the hypotheses The null and alternative hypotheses are about the population mean difference notated with the Greek letter delta: δ H 0 : δ = 0 H A : δ 0 (Note: Use a one-sided test if you know before collecting data that change can only be in one direction or if interest is in one direction only). PubH 6414 Section 11 Part 1 9

10 Null hypothesis for paired t-test The null hypothesis that δ = 0 is the same as stating that, on average, there is no change. Why? PubH 6414 Section 11 Part 1 10

11 Step 2. Calculate test statistic The test statistic for a paired t-test is a t-statistic t d = is 0 is d 0 SE( diff ) the sample mean difference the hypothesized difference Sd SE( diff ) = n n = number of subjects PubH 6414 Section 11 Part 1 11

12 Step 2. Calculate the test statistic d t = = = S d n PubH 6414 Section 11 Part 1 12

13 Step 3: P-value The p-value of the test statistic is the area beyond ± under the t-distribution with 15 df P-value = 2 P(t>2.905) = > 2 * pt(-2.905, 15) [1] PubH 6414 Section 11 Part 1 13

14 Step 4. State the Conclusion of the test Reject the null hypothesis that the mean difference is zero The data provide sufficient evidence that there was a significant mean change in the 6 minute distance walked after 90 days of therapy for COPD patients at alpha=05. PubH 6414 Section 11 Part 1 14

15 Inference for Paired Data In addition to the paired t-test, Confidence intervals of the population mean difference can be constructed The confidence interval provides information about the precision of the estimate of the mean difference. The confidence interval can also be used to evaluate change. If the 95% confidence interval does not contain the value 0, the change is significant at the alpha = 0.05 level. PubH 6414 Section 11 Part 1 15

16 Confidence Interval of Mean Difference Mean difference ± Confidence Coefficient *SE (diff) d ± t * n 1 S ( d n ) PubH 6414 Section 11 Part 1 16

17 Confidence Interval of Mean Difference 95% confidence interval for mean difference in 6 minute walk distance: d ± t * n 1 S ( d n ) ± 2.13*6.78 = (5.25, 34.13) PubH 6414 Section 11 Part 1 17

18 CI Interpretation The 95% confidence interval does not contain the value 0 (the null value). Same conclusion! The data provide sufficient evidence that there was a significant mean change in the 6 minute distance walked after 90 days of therapy for COPD patients at alpha = PubH 6414 Section 11 Part 1 18

19 Requirements for Paired t-test The sample data consist of two measurements on each subject The mean is an appropriate summary statistic of the data (data are quantitative) The subjects are randomly selected from the population of interest Data are approximately normally distributed or sample size is large. PubH 6414 Section 11 Part 1 19