1. HSRP 734: Advanced Statistical Methods May 22, 2008

2. Course Website Course site in Public Health Sciences (PHS) website: http://www.phs.wfubmc.edu/publi c/edu_statMeth.cfm http://www.phs.wfubmc.edu/publi c/edu_statMeth.cfm

3. Course Syllabus

4. HSRP 734: Advanced Statistical Methods Categorical Data Analysis Logistic Regression Survival analysis Cox PH regression

5. What is Categorical Data Analysis? Statistical analysis of data that are non- continuous Includes dichotomous, ordinal, nominal and count outcomes Examples: Disease incidence, Tumor response

6. What is Logistic Regression? A statistical method used to model dichotomous or binary outcomes (but not limited to) using predictor variables.

7. What is Logistic Regression? Used when the research method is focused on whether or not an event occurred, rather than when it occurred Time course information is not used

8. Logistic Regression quantifies “effects” using Odds Ratios Does not model the outcome directly, which leads to effect estimates quantified by means (i.e., differences in means) Estimates of effect are instead quantified by “Odds Ratios”

9. The Logistic Regression Model predictor variables is the log(odds) of the outcome. dichotomous outcome

10. The Logistic Regression Model intercept is the log(odds) of the outcome. model coefficients

11. A Short Review

12. Philosophy of Science Idea: We posit a paradigm and attempt to falsify that paradigm. Science progresses faster via attempting to falsify a paradigm than attempting to corroborate a paradigm. (Thomas S. Kuhn. 1970. The Structure of Scientific Revolutions. University of Chicago Press.)

13. Philosophy of Science The fastest way to progress in science under this paradigm of falsification is through perturbation experiments. In epidemiology, –often unable to do perturbation experiments –it becomes a process of accumulating evidence Statistical testing provides a rigorous data- driven framework for falsifying hypothesis

14. The P-Value What is the probability of having gotten a sample mean as extreme as 4.8 if the null hypothesis was true (H 0 :  = 0)? P-value = probability of obtaining a result as or more “extreme” than observed if H 0 was true. Consider for the above example, if p = 0.0089 (less than a 9 out of 1,000 chance) What if p = 0.0501 (5 out of 100 chance) ?

15. Hypothesis Testing 1.Set up a null and alternative hypothesis 2.Calculate test statistic 3.Calculate the p-value for the test statistic 4.Based on p-value make a decision to reject or fail to reject the null hypothesis 5.Make your conclusion

16. Hypothesis Testing Your decision vs. Truth Truth: H 0 TrueTruth: H 0 False Decision: Fail to reject H 0 Correct DecisionIncorrect Decision Type II Error (  ) Decision: Reject H 0 Incorrect Decision Type I Error (  ) Correct Decision (Power)

17. Hypothesis Testing Type I error (  ) = the probability of rejecting the null hypothesis given that H 0 is true (the significance level of a test). Type II error (  ): the probability of not rejecting the null hypothesis given that H 0 is false (not rejecting when you should have). Power = 1 - 

18. Power The power of a test is: The probability of rejecting a false null hypothesis under certain assumed differences between the populations. We like a study that has “high” power (usually at least 80%).

19. Any difference can become significant if N is large enough Even if there is statistical significance is there clinical significance?

20. Controversy around HT and p-value “A methodological culprit responsible for spurious theoretical conclusions” (Meehl, 1967; see Greenwald et al, 1996) “The p-value is a measure of the credibility of the null hypothesis. The smaller the p-value is, the less likely one feels the null hypothesis can be true.”

21. HT and p-value “It cannot be denied that many journal editors and investigators use p-value < 0.05 as a yardstick for the publishability of a result.” “This is unfortunate because not only p-value, but also the sample size and magnitude of a physically important difference determine the quality of an experimental finding.”

22. HT and p-value Consider a new cancer drug that possibly shows significant improvements. Should we consider a p = 0.01 the same as a p = 0.00001 ?

23. HT and p-value “[We] endorse the reporting of estimation statistics (such as effect sizes, variabilities, and confidence intervals) for all important hypothesis tests.” – Greenwald et al (1996)

24. Reporting Statistics Reporting I. Statistical Methods The changes in blood pressure after oral contraceptive use were calculated for 10 women. A paired t-test was used to determine if there was a significant change in blood pressure and a 95% confidence was calculated for the mean blood pressure change (after-before).

25. Reporting Statistics Reporting II. Results Blood pressure measurements increased on average 4.8 mmHg with standard deviation of 4.57. The 95% confidence interval for the mean change was (1.53, 8.07). There was evidence that blood pressure measurements after oral contraceptive use were significantly higher than before oral contraceptive use (p = 0.009).

26. HSRP 734 Lecture 1: Measures of Disease Occurrence and Association

27. Objectives: 1.Define and compute the measures of disease occurrence and association 2.Discuss differences in study design and their implications for inference

28. Example CT images rated by radiologist (Rosner p.65)

29. Rated as normal Rated as questionable Rated as abnormal Normal 39613 Abnormal 5244

30. (Cell %) Row % Col % Rated as normal Rated as questionable Rated as abnormal Normal 39 (35.8%) 67% 88.6% 6 (5.5%) 10.3% 75% 13 (11.9%) 22.4% 22.8% 58 Abnormal 5 (4.6%) 9.8% 11.4% 2 (1.8%) 3.9% 25% 44 (40.4%) 86.3% 77.2% 51 44857109

31. Basic Probability Conditional probability –Restrict yourself to a “subspace” of the sample space MaleFemale Young20%10% Old35%

32. Conditional probabilities Probability that something occurs (event B), given that event A has occurred (conditioning on A) Pr(B given that A is true) = Pr(B | A)

33. Conditional probabilities Categorical data analysis odds ratio = ratio of odds of two conditional probabilities Conditional probabilities in survival analysis of the form : Pr(live till time t 1 +t 2 | survive up till time t 1 )

34. Basic probability Example: automatic blood-pressure machine 84% hypertensive and 23% normotensives are classified as hypertensive Given 20% of adult population is hypertensive We now know: Pr(machine says hypertensive | truly hypertensive) What is Pr(truly hypertensive| machine says hypertensive)?

35. Basic probability Machine diagnosed as hypertensive (D) Hypertension (H)YesNo Yes No

36. Basic probability Positive predictive value — Probability that a randomly selected subject from the population actually has the disease given that the screening test is positive Negative predictive value — Probability that a randomly selected subject from the population is actually disease free given that the screening test is negative

37. Basic probability Sensitivity — Probability that the procedure is positive given that the person has the disease Specificity — Probability that the procedure is negative given that the person does not have the disease Review examples 3.26, 3.27, and 3.28 in Rosner

38. Measures of Occurrence –Measure using proportions (e.g., prevalence, odds) –Rates (e.g., incidence, cumulative incidence) Measure of Association –Based on odds (e.g., odds ratio) –Based on probabilities (e.g., risk ratio)

39. Absolute Measures of Disease Occurrence Point prevalence = proportion of cases at a given point in time –cross-sectional measure Incidence = number of new cases within a specified time interval –prospective measure

40. Absolute Measures of Disease Occurrence Example: Consider four individuals diagnosed with lung cancer Proportion of death = 2/4 = 0.5 Rate of death = 2/(3+5+2+1) = 0.18 deaths per person year PersonYears of Follow-upStatus 13Dead 25Alive 32 41Dead

41. Absolute Measures of Disease Occurrence Two kinds of quantities used in measurement: –Proportion: the numerator of a proportion as a subset of the denominator, e.g., prevalence –Rate: # events which occur during a time interval divided by the total amount of time, e.g., incidence rate

42. Absolute Measures of Disease Occurrence Remarks: 1)Diseases of long duration tend to have a higher prevalence 2)Incidence tends to be more informative than prevalence for causal understanding of the disease etiology 3)Incidence is more difficult to measure & more expensive

43. Absolute Measures of Disease Occurrence 4) Prevalence & incidence can be influenced by the evolution of screening procedures and diagnostic tests 5) Both incidence and prevalence rates may be age dependent

44. Absolute Measures of Disease Occurrence Odds = ratio of P(event occurs) to the P(event does not occur). Example: The probability of a disease is 0.20. Thus, the odds are 0.20/(1-0.20) = 0.20/0.80 =0.25 = 1:4 That is, for every one person with an event, there are 4 people without the event.

45. Absolute Measures of Disease Occurrence Risk of disease in time interval [t 0, t 1 ) P(t) = Pr(developing disease in interval of length t = t 1 - t 0 given disease free at the start of the interval) Average Prevalence = Incidence x Duration duration = average duration of disease after onset

46. Measures of Disease Association So far we have discussed –Prevalence –Incidence rate –Cumulative incidence rate –Risk of disease within an interval t All absolute measures Next, relative measures and associations –Exposed (E) versus Unexposed ( )

47. Measures of Disease Association Population versus sample –Probabilities (population) are denoted by symbols such as = P(disease within the exposed population) –Sample estimates are denoted by

48. Measures of Disease Association Exposed E Not ExposedTotal Disease D abn1n1 No Disease cdn0n0 Total m1m1 m0m0 n

49. Conditional distribution Exposed E Not ExposedMargin Disease D No Disease Margin1

50. Conditional distribution Exposed E Not ExposedMargin Disease D No Disease Margin1

51. Measures of Association Odds ratio: Odds of disease among exposed divided by odds of disease among unexposed

52. Measures of Association OR > 1 implies a positive association between disease and exposure OR < 1 implies a negative association between disease and exposure OR for disease = OR for exposure

53. Measures of Association Risk ratio = ratio between P(disease for exposed) and P(disease for unexposed), both P(.) measured within the same duration of time

54. Measures of Association? Risk Difference (Excess Risk): RD = 1 - 0 RD not scale free e.g., What is the meaning of these two equal differences RR = 0.009. RD = 0.010-0.001 vs. RD = 0.210-0.201 Attributable Risk for Exposed Persons: AR = ( 1 - 0 ) / 1 = 1 – 1 / RR

55. Measurements of risk and relative risk in different sampling designs Cross-sectional Cohort Case-control

56. Measures of Disease Association Exposed E Not ExposedTotal Disease D abn1n1 No Disease cdn0n0 Total m1m1 m0m0 n

57. Cross-Sectional Sampling Randomly sample n subjects from population at time t and determine disease and exposure status. Important: n is fixed for this design. 1)a/m 1 estimates prevalence of disease at t among exposed 2)b/m 0 estimates prevalence of disease at t among unexposed 3)ad/bc estimates the OR for disease and exposure

58. Odds Ratio p 1 = a/m 1 = disease risk among exposed p 0 = b/m 0 = disease risk among unexposed If p 1 and p 0 are small (rare disease) and the time interval is relatively short, it can be shown that OR ≈ RR

59. Cross-sectional Sampling Cross-sectional design not prospective Can only test for association between exposure and prevalence and not incidence Cannot test hypotheses about causality

60. Cohort Sampling Sample n disease-free individuals from the population at time t 0 and follow them until time t 1. Measure exposure history for each subject and observe which subjects develop disease in interval [t 0, t 1 ) Important: m 1, m 0, and n are fixed

61. Cohort study: Estimates of risk 1)p 1 = a/m 1 estimates risk of developing disease in interval among exposed 2)p 0 = b/m 0 estimates risk of developing disease in interval among unexposed 3)RR ≈ p 1 / p 0 4)OR = ad / bc 5)IR (incidence rate): i ≈ p i / t for i = 0, 1 (and small t) 6)RD (risk difference): RD ≈ 1 – 0 ≈ (p 1 – p 0 ) / t

62. Case-Control Sampling Sample n 1 cases and n 0 disease free controls from target population during interval [t 0, t 1 ) Important: n 1, n 0, and n are fixed

63. 1)a/m 1 and b/m 0 do not estimate population disease risks 2)a/n 1 estimates Pr(prior exposure | disease incidence in [t 0, t 1 ) 3)c/n 0 estimates Pr(prior exposure | no disease incidence in [t 0, t 1 ) 4)OR = ad / bc 5)RR ≈ OR for rare disease or short time intervals 6)IR (incidence rate) or disease risks cannot be estimated; RD (risk difference) cannot be estimated

64. Hypothetical example Frequency of disease and exposure in a target population p 1 = ? p 0 = ? RR = p 1 / p 0 = ? OR = ? Exposure Not Exposure Total Disease83240 No Disease92868960 Total1009001000

65. Hypothetical example Frequency of disease and exposure in a target population p 1 = 8 / 100 = 0.08; p 0 = 32 / 900 = 0.036 RR = p 1 / p 0 = 0.08 / 0.036 = 2.25 OR = (8 x 868) / (92 x 32) = 2.36 Exposure Not Exposure Total Disease83240 No Disease92868960 Total1009001000

66. Cohort Study 50% of exposed individuals sampled 25% of unexposed individuals sampled p 1 = 4 / 50 = 0.08; p 0 = 8 / 225 = 0.036 RR = p 1 / p 0 = 0.08 / 0.036 = 2.25 OR = (4 x 217) / (46 x 8) = 2.36 ExposureNot ExposureTotal Disease4812 No Disease46217263 Total50225275

67. Case-Control Study 100% of diseased individuals sampled 25% of disease-free individuals sampled p 1 = 8 / 31 = 0.26 ≠ 0.08; p 0 = 32 / 249 = 0.13 ≠ 0.036 RR = p 1 / p 0 = (8/31) / (32/249) = 2.01 ≠ 2.25 OR = (8 x 217) / (23 x 32) = 2.36 Exposure Not Exposure Total Disease83240 No Disease23217240 Total31249280

68. Odds ratio The odds ratio is equally valid for retrospective, prospective, or cross-sectional sampling designs That is, regardless of the design it estimates the same population parameter

69. Take home messages –Occurrence of disease measured by prevalence, or proportion –Incidence measured by incidence rates, or proportion per unit time –Risk is probability of developing disease over a specified period of time

70. Take home messages –Association of disease with exposure measured by odds ratios and risk ratios –Odds ratios are valid for cross-sectional, cohort, and case-control designs, risk ratios are not

71. HW #1 Due May 29 Can talk to others but turn in own work