Introduction to Biostatistics for Clinical and Translational Researchers

1. Introduction to Biostatistics for Clinical and Translational Researchers KUMC Departments of Biostatistics & Internal Medicine University of Kansas Cancer Center

2. Course Information • Jo A. Wick, PhD • Office Location: 5028 Robinson • Email: jwick@kumc.edu • Lectures are recorded and posted at http://biostatistics.kumc.edu under ‘Events and Lectures’

3. Inferences: Hypothesis Testing

4. Last Week Continuous outcome, compared between groups

5. Today • Yes/No or categorical outcome compared between groups? Chi-square tests • Time-to-event compared between groups? Survival Analysis • Association between two continuous outcomes? Correlation • What if we want to ‘adjust’ any of these for additional factors? Regression Methods

6. Chi-Square Tests

7. Inferences on Proportions • When do we do when we have nominal (categorical) data on more than one factor? • Gender and hair color • Menopausal status and disease stage at diagnosis • ‘Handedness’ and gender • Tumor response and treatment • Presence/absence of disease and exposure • These types of tests are looking at whether two categorical variables are independent of one another (versus associated)—thus, tests of this type are often referred to as chi-square tests of independence.

8. Inferences on Proportions • Remember, this is essentially looking at the association between two outcomes, where both are categorical (nominal or ordinal). • Assumptions? • ROT: No expected frequency should be less than 5 (i.e., nπ < 5) • If not met, Fisher’s Exact test is appropriate.

9. Inferences on Proportions • Example: Hair color and Gender • Gender: x1 = {M, F} • Hair Color: x1 = {Black, Brown, Blonde, Red} What the data should look like in the actual dataset:

10. Hair Color and Gender • The researcher hypothesizes that hair color is not independent of sex. • H0: Hair color is independent of gender (i.e., the phenotypic ratio is the same within each gender). • H1: Hair color is not independent of gender (i.e., the phenotypic ratio is different between genders).

11. Hair Color and Gender • Chi-square statistics compute deviations between what is expected (under H0) and what is actually observed in the data: • DF = (r – 1)(c – 1) where r is number of rows and c is number of columns

12. Hair Color and Gender • Does it appear that this type of sample could have come from a population where the different hair colors occur with the same frequency within each gender? • OR does it appear that the distribution of hair color is different between men and women?

13. Hair Color and Gender • Conclusion: Reject H0: Gender and Hair Color are independent. It appears that the researcher’s hypothesis that the population phenotypic ratio is different between genders is correct (p = 0.029).

14. Inferences on Proportions • Special case: when you have a 2X2 contingency table, you are actually testing a hypothesis concerning two population proportions: H0: π1 = π2 (i.e., the proportion of males who are blonde is the same as the proportion of females who are blonde).

15. Inferences on Proportions • When you have a single proportion and have a small sample, substitute the Binomial test which provides exact results. • The nonparametric Fisher Exact test can be always be used in place of the chi-square test when you have contingency table-like data (i.e., two categorical factors whose association is of interest)—it should be substituted for the chi-square test of independence when ‘cell’ sizes are small.

16. Survival Analysis

17. Inferences on Time-to-Event • Survival Analysis is the class of statistical methods for studying the occurrence (categorical) and timing (continuous) of events. • The event could be • development of a disease • response to treatment • relapse • death • Survival analysis methods are most often applied to the study of deaths.

18. Inferences on Time-to-Event • Survival Time: the time from a well-defined point in time (time origin) to the occurrence of a given event. • Survival data includes: • a time • an event ‘status’ • any other relevant subject characteristics

19. Inferences on Time-to-Event • In most clinical studies the length of study period is fixed and the patients enter the study at different times. • Lost-to-follow-up patients’ survival times are measured from the study entry until last contact (censored observations). • Patients still alive at the termination date will have survival times equal to the time from the study entry until study termination (censored observations). • When there are no censored survival times, the set is said to be complete.

20. Functions of Survival Time • Let T = the length of time until a subject experiences the event. • The distribution of T can be described by several functions: • Survival Function: the probability that an individual survives longer than some time, t: S(t) = P(an individual survives longer than t) = P(T > t)

21. Functions of Survival Time • If there are no censored observations, the survival function is estimated as the proportion of patients surviving longer than time t:

22. Functions of Survival Time • Density Function: The survival time T has a probability density function defined as the limit of the probability that an individual experiences the event in the short interval (t, t + t) per unit width t:

23. Functions of Survival Time • Hazard Function: The hazard function h(t) of survival time T gives the conditional failure rate. It is defined as the probability of failure during a very small time interval, assuming the individual has survived to the beginning of the interval:

24. Functions of Survival Time • The hazard is also known as the instantaneous failure rate, force of mortality, conditional mortality rate, or age-specific failure rate. • The hazard at any time t corresponds to the risk of event occurrence at time t: • For example, a patient’s hazard for contracting influenza is 0.015 with time measured in months. • What does this mean? This patient would expect to contract influenza 0.015 times over the course of a month assuming the hazard stays constant.

25. Functions of Survival Time • If there are no censored observations, the hazard function is estimated as the proportion of patients dying in an interval per unit time, given that they have survived to the beginning of the interval:

26. Estimation of S(t) • Product-Limit Estimates (Kaplan-Meier): most widely used in biological and medical applications • Life Table Analysis (actuarial method): appropriate for large number of observations or if there are many unique event times

27. Methods for Comparing S(t) • If your question looks like: “Is the time-to-event different in group A than in group B (or C . . . )?” then you have several options, including: • Log-rank Test: weights effects over the entire observation equally—best when difference is constant over time • Weighted log-rank tests: • Wilcoxon Test: gives higher weights to earlier effects—better for detecting short-term differences in survival • Tarome-Ware: a compromise between log-rank and Wilcoxon • Peto-Prentice: gives higher weights to earlier events • Fleming-Harrington: flexible weighting method

28. Early? Late? Proportional? Difference is early and maintained Early difference that fades Difference appears late

29. Inferences for Time-to-Event • Example: survival in squamous cell carcinoma • A pilot study was conducted to compare Accelerated Fractionation Radiation Therapy versus Standard Fractionation Radiation Therapy for patients with advanced unresectable squamous cell carcinoma of the head and neck. • The researchers are interested in exploring any differences in survival between the patients treated with Accelerated FRT and the patients treated with Standard FRT.

30. Squamous Cell Carcinoma

31. Inferences for Time-to-Event • H0: S1(t) = S2(t) for all t • H1: S1(t) ≠ S2(t) for at least one t

32. Squamous Cell Carcinoma Median Survival Time: AFRT: 18.38 months (2 censored) SFRT: 13.19 months (5 censored)

33. Squamous Cell Carcinoma Log-Rank test p-value= 0.5421 Median Survival Time: AFRT: 18.38 months (2 censored) SFRT: 13.19 months (5 censored)

34. Squamous Cell Carcinoma

35. Squamous Cell Carcinoma • Staging of disease is also prognostic for survival. • Shouldn’t we consider the analysis of the survival of these patients by stage as well as by treatment?

36. Squamous Cell Carcinoma Median Survival Time: AFRT Stage 3: 77.98 mo. AFRT Stage 4: 16.21 mo. SFRT Stage 3: 19.34 mo. SFRT Stage 4: 8.82 mo. Log-Rank test p-value = 0.0792

37. Inferences on Time-to-Event • Concerns a response that is both categorical (event?) and continuous (time) • There are several nonparametric methods that can be used—choice should be based on whether you anticipate a short-term or long-term benefit. • Log-rank test is optimal when the survival curves are approximately parallel. • Weight functions should be chosen based on clinical knowledge and should be pre-specified.

38. Publication Bias From: Publication bias: evidence of delayed publication in a cohort study of clinical research projects BMJ 1997;315:640-645 (13 September)

39. Table 4 Risk factors for time to publication using univariate Cox regression analysis Characteristic # not published # published Hazard ratio (95% CI)  Null 29 23 1.00 Non-significant trend 16 4 0.39 (0.13 to 1.12) Significant 47 99 2.32 (1.47 to 3.66) Interpretation: Significant results have a 2-fold higher incidence of publication compared to null results. Publication Bias From: Publication bias: evidence of delayed publication in a cohort study of clinical research projects BMJ 1997;315:640-645 (13 September)

40. Correlation

41. Linear Correlation • Linear regression assumes the linear dependence of one variable y (dependent) on a second variable x (independent). • Linear correlation also considers the linear relationship between two continuous outcomes but neither is assumed to be functionally dependent upon the other. • Interest is primarily in the strength of association, not in describing the actual relationship.

42. Scatterplot

43. Correlation • Pearson’s Correlation Coefficient is used to quantify the strength. • Note: If sample size is small or data is non-normal, use non-parametric Spearman’s coefficent.

44. Correlation

45. Inferences on Correlation • H0: ρ= 0 (no linear association) versus • H1: ρ> 0 (strong positive linear relationship) • or H1: ρ< 0 (strong negative linear relationship) • or H1: ρ≠ 0 (strong linear relationship) • Test statistic: t (df = 2)

46. Correlation

47. Correlation * Excluding France

48. Regression Methods

49. What about adjustments? • There may be other predictors or explanatory variables that you believe are related to the response other than the actual factor (treatment) of interest. • Regression methods will allow you to incorporate these factors into the test of a treatment effect: • Logistic regression: when y is categorical and nominal binary • Multinomial logistic regression: when y is categorical with more than 2 nominal categories • Ordinal logistic regression: when y is categorical and ordinal

50. What about adjustments? • Regression methods will allow you to incorporate these factors into the test of a treatment effect: • Linear regression: when y is continuous and the factors are a combination of categorical and continuous (or just continuous) • Two- and three-way ANOVA: when y is continuous and the factors are all categorical