1. Copyright (c) Bani Mallick1 STAT 651 Lecture 7

2. Copyright (c) Bani Mallick2 Topics in Lecture #7 Sample size for fixed power Never, ever, accept a null hypothesis Paired comparisons in SPSS Student’s t-distributions Confidence intervals when  is unknown SPSS output on confidence intervals, without formulae

3. Copyright (c) Bani Mallick3 Book Sections Covered in Lecture #7 Chapter 5.5 (sample size) Chapter 6.4 (paired data) Chapter 5.7 (t-distribution) My own screed (never, ever, accept a null hypothesis)

4. Copyright (c) Bani Mallick4 Lecture 6 Review: Hypothesis Testing Suppose you want to know whether the population mean change in reported caloric intake equals zero We have already done this!!!!! Confidence intervals tell you where the population mean  is, with specified probability If zero is not in the confidence interval, then you can reject the hypothesis

5. Copyright (c) Bani Mallick5 Lecture 6 Review: Type I Error (False Reject) A Type I error occurs when you say that the null hypothesis is false when in fact it is true You can never know for certain whether or not you have made such an error You can only control the probability that you make such an error  t is convention to make the probability of a Type I error 5%, although 1% and 10% are also used

6. Copyright (c) Bani Mallick6 Lecture 6 Review: Type I Error Rates Choose a confidence level, call it 1 -  The Type I error rate is   confidence interval:  = 10%  confidence interval:  = 5%  confidence interval:  = 1%

7. Copyright (c) Bani Mallick7 Lecture 6 Review: Type II: The Other Kind of Error The other type of error occurs when you do NOT reject even though it is false This often occurs because you study sample size is too small to detect meaningful departures from Statisticians spend a lot of time trying to figure out a priori if a study is large enough to detect meaningful departures from a null hypothesis

8. Copyright (c) Bani Mallick8 Lecture 6 Review: P-values Small p-values indicate that you have rejected the null hypothesis If p < 0.05, this means that you have rejected the null hypothesis with a confidence interval of 95% or a Type I error rate of 0.05 If p > 0.05, you did not reject the null hypothesis at these levels

9. Copyright (c) Bani Mallick9 Lecture 6 Review: Statistical Power Statistical power is defined as the probability that you will reject the null hypothesis when you should reject it. If  is the Type II error, power = 1 -  The Type I error (test level) does NOT depend on the sample size: you chose it (5%?) The power depends crucially on the sample size

10. Copyright (c) Bani Mallick10 Sample Size Calculations You want to test at level (Type I error)  the null hypothesis that the mean = 0 You want power 1 -  to detect a change of from the hypothesized mean by the amount  or more, i.e., the mean is greater than  or the mean is less than -  There is a formula for this!!

11. Copyright (c) Bani Mallick11 Sample Size Calculations  Look up z  and z  Remember what they are? Find the values in Table 1 which give you readings of 1-  and 1-  Required sample size is

12. Copyright (c) Bani Mallick12 Sample Size Calculations  =0.01  =0.90  =180  =600 Look up z  =2.58 and z  =1.28 (Check this) = 166  =0.01  =0.80  =180  =600, z  =0.84 (Check this) n = 130: the less power you want, the smaller the sample size

13. Copyright (c) Bani Mallick13 More on Sample Size Calculations Most often, sample sizes are done by convention or convenience: Your professor has used 5 rats/group before successfully You have time only to interview 50 subjects in total

14. Copyright (c) Bani Mallick14 More on Sample Size Calculations More often, sample sizes are done by convention or convenience: In this case, the sample size calculations can be used after a study if you find no statistically significant effect You can then guess how large a study you would have needed to detect the effect you have just seen but which was not statistically significant

15. Copyright (c) Bani Mallick15 Never Accept a Null Hypothesis Suppose we use a 95% confidence interval, it includes zero. Why do I say : with 95% confidence, I cannot reject that the population mean is zero. I never, ever say: I can therefore conclude that the population mean is zero. Why is this? Are statisticians just weird? (maybe so, but not in this case)

16. Copyright (c) Bani Mallick16 Never Accept a Null Hypothesis: Reason 1 Suppose we use a 95% confidence interval, it includes zero: [-3,6]. Why do I say : with 95% confidence, I cannot reject that the population mean is zero. Remember the definition of a confidence interval: the chance is 95% that the true population mean is between -3 and 6: hence, the true population mean could be 5, and is not necessarily = 0.

17. Copyright (c) Bani Mallick17 Never Accept a Null Hypothesis: Reason 2 Suppose we use a 95% confidence interval, it includes zero: [-3,6]. Why do I say : with 95% confidence, I cannot reject that the population mean is zero. Potential for chicanery: if you want to accept the null hypothesis, how can you best insure it?

18. Copyright (c) Bani Mallick18 Never Accept a Null Hypothesis: Reason 2 An example of chicanery: generic drugs In the pharmaceutical industry, all the expense involves getting a drug approved by the FDA After a drug goes off-patent, generic drugs can be marketed The main regulation is that the generic must be shown to be “bioeqiuvalent” to the patent drug

19. Copyright (c) Bani Mallick19 Never Accept a Null Hypothesis: Reason 2 The generic must be shown to be “bioeqiuvalent” to the patent drug One way would be to run a study and do a statistical test to see whether the drugs have the same effects/actions: the null hypothesis is that the patent and generic are the same The alternative is that they are not If the null is rejected, the generic is rejected, and $$$ issues arise

20. Copyright (c) Bani Mallick20 Never Accept a Null Hypothesis: Reason 2 Test to see whether the drugs have the same effects/actions: the null hypothesis is that the patent and generic are the same If the null is rejected, the generic is rejected, and $$$ issues arise If you pick a tiny sample size, there is no statistical power to reject the null hypothesis

21. Copyright (c) Bani Mallick21 Never Accept a Null Hypothesis: Reason 2 If you pick a tiny sample size, there is no statistical power to reject the null hypothesis The FDA is not stupid: they insist that the sample size be large enough that any medically important differences can be detected with 80% (1 -  ) statistical power

22. Copyright (c) Bani Mallick22 Never Accept a Null Hypothesis p-values are not the probability that the null hypothesis is true. For example, suppose you have a vested interest in not rejecting the null hypothesis. Small sample sizes have the least power for detecting effects. Small sample sizes imply large p-values. Large p-values can be due to a lack of power, or a lack of an effect.

23. Copyright (c) Bani Mallick23 Paired Comparisons: Count you Number of Populations! The hormone assay data illustrate an important point. Sometimes, we measure 2 variables on the same individuals Reference Method and Test Method There is only 1 population. How do we compare the two variables to see if they have the same mean?

24. Copyright (c) Bani Mallick24 Paired Comparisons: Count you Number of Populations! There is only 1 population. How do we compare the two variables to see if they have the same mean? Answer (Ott & Longnecker, Chapter 6.4): do what we did and first compute the difference of the variables and make inference on this difference: now have 1 variable In making inference, match the number of variables to the number of populations!

25. Copyright (c) Bani Mallick25 Paired Comparisons in SPSS SPSS has a nice routine way of doing a paired comparison analysis, providing confidence intervals and p-values “Analyze” “Compare Means” “Paired Samples t-test” Highlight the variables that are paired and select: use “options” to get other than 95% CI

26. Copyright (c) Bani Mallick26 Paired Comparisons in SPSS Demo using computer comes next

27. Copyright (c) Bani Mallick27 Boxplots and Histograms for Paired data For paired data, SPSS makes it easy to automatically get confidence intervals: it takes the difference of the paired variables for you However, for boxplots, qq-plots, etc., you have to do this manually. Here is how you can define a new variable, called “differen”, in the armspan data for males.

28. Copyright (c) Bani Mallick28 Computing the Difference in Paired Comparisons Click on “Transform” Click on “Compute” New window shows up, in “Target Variable” type in differen Click on “Type & Label” and type in your label (Height - Armspan in Inches) click on “Continue”

29. Copyright (c) Bani Mallick29 Computing the Difference in Paired Comparisons Highlight height and move over by clicking the mover button In “Numeric Expression”, type in the minus sign - Highlight armspan and move over Click on “OK” You are done!

30. Copyright (c) Bani Mallick30 Selecting Cases in SPSS “Data” “Select Cases” Push button of “If condition is satisfied” Select “If” Select “Gender” and move over Then type = ‘Female’ and “Continue” “OK” --> all analyses will be on Females

31. Copyright (c) Bani Mallick31 Student’s t-Distribution In real life, the population standard deviation  is never known We estimate it by the sample standard deviation s To account for this estimation, we have to make our confidence intervals (make a guess): longer or shorter? Stump the experts!

32. Copyright (c) Bani Mallick32 Student’s t-distribution Of course: you have to make the confidence interval longer! This fact was discovered by W. Gossett, the brewmaster of Guinness in Dublin. He wrote it up anonymously under the name “Student”, and his discovery is hence called Students t-distribution because he used the letter t in his paper.

33. Copyright (c) Bani Mallick33 Student’s t-Distribution Effectively, if you want a (1  100% confidence interval, what you do is to replace z  (1.645, 1.96, 2.58) by t  (n-1) found in Table 2 of the book. n-1 is called the degrees of freedom The increase in length of the confidence interval depends on n. If n gets larger, does the CI get larger or smaller?

34. Copyright (c) Bani Mallick34 Student’s t-Distribution The (1  100% CI when  was known was The (1  100% CI when is  unknown is You replace  by s and by t  (n-1)

35. Copyright (c) Bani Mallick35 Student’s t-Distribution Take 95% confidence,  = 0.05 z  = 1.96 n = 3, n-1 = 2, t  (n-1) = 4.303 n = 10, n-1 = 9, t  (n-1) = 2.262 n = 30, n-1 = 29, t  (n-1) = 2.045 n = 121, n-1 = 120, t  (n-1) = 1.98

36. Copyright (c) Bani Mallick36 Student’s t-Distribution Luckily, SPSS is smart. It automatically uses Student’s t- distribution in constructing confidence intervals and p-values! So, all the output you will see in SPSS has this correction built in

37. Copyright (c) Bani Mallick37 Student’s t-Distribution In the old days, people used the t-test to decide whether the hypothesize value is in the CI. If your hypothesis is that  = 0, then you reject the hypothesis if You learn nothing from this not available in a CI, but its value is in SPSS

38. Copyright (c) Bani Mallick38 WISH Numerical Illustration s = 613, Xbar = -180 n = 3, s.e. = 613 / 3 1/2 = 354, t  (n- 1) = 4.303, CI is -180 plus and minus 1523, hence the interval is [-1703, 1343] n = 121, s.e. = 613 / 121 1/2 = 59, t  (n-1) = 1.98, CI is -180 plus and minus 118, hence the interval is [-298,-62] Note change in conclusions!

39. Copyright (c) Bani Mallick39 Armspan Data for Males Outcome is height – armspan in inches In SPSS, “Analyze”, “Descriptives”, “Explore” will get you to the right analysis Illustrate how to do this in SPSS

40. Copyright (c) Bani Mallick40 Armspan Data for Males Sample mean = -0.26 Sample standard error = 0.2391 Lower bound of 95% CI = -0.7406 Upper bound of 95% CI = 0.2206 Is there evidence with 95% confidence that armspans for males differ systematically from heights?

41. Copyright (c) Bani Mallick41 Armspan Data for Males Might ask: what about with 90% confidence Illustrate how to do this in SPSS

42. Copyright (c) Bani Mallick42 Armspan Data for Males Sample mean = -0.26 Sample standard error = 0.2391 Lower bound of 90% CI = -0.6609 Upper bound of 90% CI = 0.1409 Is there evidence with 90% confidence that armspans for males differ systematically from heights?

43. Copyright (c) Bani Mallick43 Armspan Data for Males SPSS will compute the p-value for you as well as confidence intervals. For paired comparisons, “Analyze”, “Compare Means”, “Paired Sample”. Highlight the paired variables. It computes the difference of the first named variable in the list minus the second Illustration in SPSS

44. Copyright (c) Bani Mallick44 Armspan Data for Males t = -1.087 p-value (significance level) = 0.282 SPSS also automatically does a 95% confidence interval for the population mean difference between heights and armspans