1. Copyright (c) Bani K. Mallick1 STAT 651 Lecture 6

2. Copyright (c) Bani K. Mallick2 Topics in Lecture #6 The language of hypothesis testing Hypothesis tests are carried out using confidence intervals Z-tests are also possible P-values as a way of not having to do the mechanics of many hypothesis tests Statistical power

3. Copyright (c) Bani K. Mallick3 Book Sections Covered in Lecture #6 Chapter 5.4 Chapter 5.6

4. Copyright (c) Bani K. Mallick4 Lecture 5 Review: Confidence Interval for a Population Mean  when  is Known Want 90%, 95% and 99% chance of interval including . 90% 95% 99%

5. Copyright (c) Bani K. Mallick5 Lecture 5 Review: Confidence Intervals There is a general formula given on page 200 If you want a (1-  )100% confidence interval for the population mean  when the population s.d.  is known, use the formula The term z  is the value in Table 1 that gives probability 1 -  /2.  = 0.10, z  = 1.645:  = 0.05, z  = 1.96,  = 0.01, z  = 2.58

6. Copyright (c) Bani K. Mallick6 Lecture 5 Review: Sample Size Determination I want the length of a confidence interval to be 2 x E then the sample size I need is

7. Copyright (c) Bani K. Mallick7 Lecture 5 Review Which make the lengths of CI’s become longer? Sample sizes? Population standard deviation? Degree of confidence?

8. Copyright (c) Bani K. Mallick8 Hypothesis Testing: Beginings Suppose you want to know whether the population mean change in reported caloric intake equals zero Note the emphasis on population There is a reasonably elaborate structure to test such a hypothesis Computers make this relatively simple, for simple problems, but you have to understand the language

9. Copyright (c) Bani K. Mallick9 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 what?

10. Copyright (c) Bani K. Mallick10 Hypothesis Testing Can be thought of as a framework for decision making I like to emphasize confidence intervals, since they give more information The books talks about 1-tailed and 2-tailed tests. We will do only 2-tailed tests.

11. Copyright (c) Bani K. Mallick11 The Null Hypothesis Begin with a hypothesis: hypothesized value (say 0) This is called the null hypothesis It is always of the form given above.

12. Copyright (c) Bani K. Mallick12 The Alternative Hypothesis Write down possible alternatives: hypothesized value The is the alternative hypothesis For our course, it is always two-sided hypothesized value

13. Copyright (c) Bani K. Mallick13 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

14. Copyright (c) Bani K. Mallick14 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%

15. Copyright (c) Bani K. Mallick15 Choose a Decision Rule You can reject Or you can NOT reject For reasons described later, we never accept That’s a trick question on an exam!

16. Copyright (c) Bani K. Mallick16 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 your 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

17. Copyright (c) Bani K. Mallick17 Making Your Decision You have selected your level of confidence 1- . This means your Type I error rate is  You have a null hypothesis You form a confidence interval If the hypothesized value is not in the confidence interval, you reject and say it is false Otherwise, you cannot reject

18. Copyright (c) Bani K. Mallick18 The Z-Test I like confidence intervals because they tell you two things They tell you where the population mean is The length of the confidence interval tells you if your study sample size is too small to be useful

19. Copyright (c) Bani K. Mallick19 The Z-Test There is an equivalent procedure for performing the hypothesis test, called the Z- test It is not as useful as a confidence interval, because it does not give you the confidence interval You actually rarely see it anymore, but sometimes you do

20. Copyright (c) Bani K. Mallick20 The Z-Test Form the Z-statistic: Reject the null hypothesis if  = 0  01, confidence = 99%) |Z| > 2.58  = 0  05, confidence = 95%) |Z| > 1.96  = 0  10, confidence = 90%) |Z| > 1.645

21. Copyright (c) Bani K. Mallick21 WISH Data Again We already know that the population mean change in reported caloric intake is not zero with 99% confidence  = 600, n = 271 = -180, |Z| = 4.9 > 2.58 Reject the null hypothesis!

22. Copyright (c) Bani K. Mallick22 P-values Ubiquitous in journals You need to know what they are They are simply bookkeeping devices to save journal space, since you cannot do 3 confidence intervals for each hypothesis Small p-values indicate that you have rejected the null hypothesis Simple stuff!!!

23. Copyright (c) Bani K. Mallick23 P-values Small p-values indicate that you have rejected the null hypothesis If p < 0.10, this means that you have rejected the null hypothesis with a confidence interval of 90% or a Type I error rate of 0.10 If p > 0.10, you did not reject the null hypothesis at these levels

24. Copyright (c) Bani K. Mallick24 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

25. Copyright (c) Bani K. Mallick25 P-values Small p-values indicate that you have rejected the null hypothesis If p < 0.01, this means that you have rejected the null hypothesis with a confidence interval of 99% or a Type I error rate of 0.01 If p > 0.01, you did not reject the null hypothesis at these levels

26. Copyright (c) Bani K. Mallick26 WISH Data Again Which are true, if any? 99% CI did not include 0 p < 0.01 p < 0.05 p < 0.10 p > 0.01 p > 0.05 p > 0.10

27. Copyright (c) Bani K. Mallick27 A Little Theory Suppose that the p-value = 0.032. Which confidence intervals, if any, include the hypothesized value? 90% 95% 99%

28. Copyright (c) Bani K. Mallick28 A Little Theory Suppose that the p-value = 0.002. Which confidence intervals, if any, include the hypothesized value? 90% 95% 99%

29. Copyright (c) Bani K. Mallick29 A Little Theory Suppose that the p-value = 0.092. Which confidence intervals, if any, include the hypothesized value? 90% 95% 99%

30. Copyright (c) Bani K. Mallick30 A Little Theory Suppose that the p-value = 0.122. Which confidence intervals, if any, include the hypothesized value? 90% 95% 99%

31. Copyright (c) Bani K. Mallick31 Hormone Assay Data Two assay methods for measuring the amount of a hormone Reference Method: the old standby Test Method: A new, cheaper method I was asked by Becton Dickenson Company to say whether the Test method was a reliable substitute for the Reference method

32. Copyright (c) Bani K. Mallick32 Hormone Assay Data Q-Q Plot for differences: Test - Reference

33. Copyright (c) Bani K. Mallick33 Hormone Assay Data Q-Q plot of differences of logarithms: log(test) - log(reference)

34. Copyright (c) Bani K. Mallick34 Hormone Assay Data Q-Q Plot for differences: Test - Reference

35. Copyright (c) Bani K. Mallick35 Hormone Assay Data Q-Q plot of differences of logarithms: log(test) - log(reference)

36. Copyright (c) Bani K. Mallick36 Hormone Assay Data Box Plot for differences: Test - Reference

37. Copyright (c) Bani K. Mallick37 Hormone Assay Data Box plot of differences of logarithms: log(test) - log(reference)

38. Copyright (c) Bani K. Mallick38 Hormone Assay Data It seems to make sense to use the log(Test) and log(Reference) Data, since log(Test) - log(Reference) seems more Gaussian

39. Copyright (c) Bani K. Mallick39 Hormone Assay Data n = 85 For log data, log(test) - log(reference) Sample mean = -0.1083 Sample s.d. = 0.2761 Sample std. error = 0.0299 p-value < 0.01: what is the hypothesis and what is the conclusion?

40. Copyright (c) Bani K. Mallick40 Hormone Assay Data For log data, X = log(test) - log(reference) Sample mean = -0.1083 Sample std. error = 0.0299 Hypothesis: X has population mean = 0 p-value < 0.01: what is the conclusion?

41. Copyright (c) Bani K. Mallick41 Hormone Assay Data For log data, X = log(test) - log(reference) Sample mean = -0.1083 Sample std. error = 0.0299 Hypothesis: X has population mean = 0 p-value < 0.01: what is the conclusion? That a 99% confidence interval did not include zero, and hence you reject the null hypothesis that the population mean = 0 99% CI from -0.0294 to -0.1872

42. Copyright (c) Bani K. Mallick42 Hormone Assay Data For raw data, X = test - reference Sample mean = -0.4424 Sample std. error = 0.3771 Hypothesis: X has population mean = 0 99% CI from -1.44 to 0.55 Is p < 0.01 for this scale? p = 0.244

43. Copyright (c) Bani K. Mallick43 Hormone Assay Data For raw data, X = test - reference, p = 0.244 For log data, X = log(test) - log(reference), p < 0.001 WOW!!! Which is right?

44. Copyright (c) Bani K. Mallick44 Hormone Assay Data For raw data, X = test - reference, p = 0.244 For log data, X = log(test) - log(reference), p < 0.001 WOW!!! Which is right? I believe the log data more: more nearly normal, no outliers, nice histogram Plus, a nonparametric test (later) has p < 0.001

45. Copyright (c) Bani K. Mallick45 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

46. Copyright (c) Bani K. Mallick46 Statistical Power Statistical power is defined as the probability that you will reject the null hypothesis when you should reject it. The power depends crucially on the sample size If you have a very small sample size (n), then you will have low power, i.e., a small chance of finding an effect even if it is there

47. Copyright (c) Bani K. Mallick47 Statistical Power Statistical power is defined as the probability that you will reject the null hypothesis when you should reject it. If you have a very small sample size (n), then you will have low power, i.e., a small chance of finding an effect even if it is there This is why we never accept the null hypothesis: because we can manipulate through n the chance of rejecting it.

48. Copyright (c) Bani K. Mallick48 Numerical Illustration Wish again,  = 600, sample mean = -180 Hypothesized value for pop. Mean = 0 Set  = 0.01, 99% CI n = 1, |Z| = 0.30, NOT Reject n = 10, |Z| = 0.95, NOT Reject n = 20, |Z| = 1.34, NOT Reject n = 100, |Z| = 3.00, Reject n = 271, |Z| = 4.95, Reject