HSRP 734: Advanced Statistical Methods May 22, 2008

1. HSRP 734: Advanced Statistical MethodsMay 22, 2008

2. Course Website • Course site in Public Health Sciences (PHS) website: http://www.phs.wfubmc.edu/public/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 dichotomous outcome is the log(odds) of the outcome.

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

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 (H0: m = 0)? • P-value = probability of obtaining a result as or more “extreme” than observed if H0 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 • Set up a null and alternative hypothesis • Calculate test statistic • Calculate the p-value for the test statistic • Based on p-value make a decision to reject or fail to reject the null hypothesis • Make your conclusion

16. Hypothesis Testing

17. Hypothesis Testing • Type I error (a) = the probability of rejecting the null hypothesis given that H0 is true (the significance level of a test). • Type II error (b): the probability of not rejecting the null hypothesis given that H0 is false (not rejecting when you should have). • Power = 1 - b

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 734Lecture 1: Measures of Disease Occurrence and Association

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

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

31. Basic Probability • Conditional probability • Restrict yourself to a “subspace” of the sample space

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 t1+t2 | survive up till time t1)

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

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

37. Basic probability • Sensitivity— Probability that the procedure is positivegiven that the person has the disease • Specificity— Probability that the procedure is negativegiven 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

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: • Diseases of long duration tend to have a higher prevalence • Incidence tends to be more informative than prevalence for causal understanding of the disease etiology • 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 [t0, t1) P(t) = Pr(developing disease in interval of length t = t1 - t0 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

49. Conditional distribution

50. Conditional distribution