EPP 246 Clinical Biostatistics Lecture 4, Jan 26, 2007

1. EPP 246 Clinical Biostatistics Lecture 4, Jan 26, 2007 Lihong Qi lhqi@ucdavis.edu Division of Biostatistics Department of Public Health Sciences Rowe Program in Human Genetics UCD School of Medicine

2. Review of last class Clear state the study objective/hypotheses Valid study design with proper statistical tests Based on the test for the hypotheses Based on the primary end points Clinically meaningful difference of primary end points

3. Study objectives Safety equivalence non-inferiority superiority ------------------------------------------------------------------------------- Equivalence E/E E/N E/S Efficacy non-infer N/E N/N N/S Superiority S/E S/N S/S -------------------------------------------------------------------------------

4. Types of Hypotheses Test for equality H0: u1 = u0 vs H1: u1 != u0, 1: Rx, 0: control Test for non-inferiority H0: u0 � u1 >= delta vs u0� u1 < delta u0: standard, u1: Rx delta = difference of clinical importance Test for superiority H0: u1 � u0 <= delta vs H1: u1 � u0 > delta Test for equivalence H0: |u1 � u0| >= delta vs H1: |u1 � u0| < delta

5. Key values for sample size calculation Significance level, common range 0.01 � 0.05 Power, common values 0.80 and 0.90 �minimum detectable difference" between the treatment groups under consideration Larger difference => fewer patients standard deviation of the clinical parameter under consideration, or at least a possible range for it (for continuous outcome) derive either from similar studies in literature or from an internal pilot study.

6. Types of primary outcome Continuous Eg. Blood pressure Dichotomous Eg. Success vs failure Non-binary categorical variable Eg. blood pressure below, within, or above a normal range Time to event Eg. time to breast cancer

7. Example for binary variable A drug company is planning to conduct a trial to compare the efficacy, safety, and tolerability of two antimicrobial drugs in the treatment of patients with skin and skin structure infections Response: cure and failure (not cure) after 4 to 8 days Consider three situations: Equality of mean cure rates Non-inferiority or superiority of the test drug compared to the active control drug Therapeutic equivalence

8. Binary variable Assumptions for the example Significance level: 0.05 Power: 0.80 Equal size arms P0 = 0.65 P1 = 0.85

9. Sample size for testing non-inferiority Define no clinical importance if difference < -10%

10. Sample size for testing superiority Define no clinical importance if difference < 5% (superiority margin)

11. Sample size for testing equivalence p1 = 0.75, p2 = 0.80 Equivalence limit is 0.20

12. Non-binary categorical variable One categorical variable with more than two levels Multivariate dimension of a categorical variable Contingency table Hypotheses testing for Goodness-of-fit Independence/association

13. Test for goodness-of-fit Compare the distribution of the primary outcome with reference distribution using Pearson�s test ---------------------------------------- X x1 x2 � xr n1 n2 � nr total = n ----------------------------------------

14. Test for goodness-of-fit (cont) H0: for all k vs for some k Pearson�s chi-square statistic ~ under alternative, non-central chi-square random variable with r-1 degree of freedom and the non-centrality parameter a: significance level b: 1 - power

15. Test for goodness-of-fit: sample size

16. An example --- pilot study of hypertension A pilot study to evaluate clinical efficacy of a test compound on subjects with hypertension. Objective: compare the distribution of the proportions of subjects whose BP are below, within and above normal range after treatment with that from historical control. Suppose proportions of patients after treatment below: 20%, within: 60%, above: 20% r = 3, (p1,p2,p3) = (0.2,0.6,0.2) Historical data (p10,p20,p30)=(0.25,0.45,0.3) a = 0.05, b=1-0.8=0.2

17. pilot study of hypertension (cont)

18. Test for independence � single stratum rxc two-way contingency table Two variables X (Eg. treatment group) and Y (Eg. outcome categories) y1 y2 �� yc row total ------------------------------------------------------- x1 n11 n12 n1c n1. x2 n21 n22 n2c n2. ..�. xr nr1 nr2 nrc nr. ------------------------------------------------------ Col total n.1 n.2 n.c n (overall total)

19. Test for independence � single stratum (cont) Pearson�s test and Likelihood ratio test under H1, test statistic ~ non-central chi-square random variable with (r-1)(c-1) df and non-centrality parameter

20. An example: categorized hypotension Pilot study to compare two treatment (treatment and control) for the categorized hypotension. hypotension ----------------------------------------------------------------------- below normal above ------------------------------------------------------------------------ treatment 2 7 1 10 control 2 5 3 10 ------------------------------------------------------------------------- 4 12 4 20

21. Trial for categorized hypotension Plan a larger trial to confirm the difference Significance a = 0.05, b = 1-0.8 = 0.2 Degree of freedom of chi-square variable X = treatment: 2-1 Y = categorized hypotension: 3-1

22. Time �to-event data Time-to-event: time to the occurrence of an event Censoring: exact value unknown Assume right censoring � event time >= observed censoring time Survival function X: time to event, continuous Probability the event will occur after time x Hazard function/hazard rate the probability that an event will occur within a small time interval, given the subject has survived up to the beginning of the interval; risk of having an event at time x.

23. Cox�s proportional hazards model Baseline hazard Most commonly used for survival (time-to-event) data uses all available information, including patients who fail to complete the trial (censored) models the hazard ratio (relative risk) of death, or other event of interest, for individuals, given their prognostic variables.

24. Cox�s proportional hazards model (cont) The hazard ratio is an estimate of the ratio of the hazard rate in the treated versus the control group. Assume constant hazard ratio over time Assume censoring time and failure time are independent

25. Sample size using Cox�s model, test for equality z=1: treatment group, z=0:control vs proportion of patients in treatment, control group probability of observing an event a th upper percentile of the standard normal distribution

26. Cox�s model, test for non-inferiority/superiority Vs Superiority or non-inferiority margin Rejection of H0 indicates superiority over the reference level Rejection of H0 indicates non-inferiority against the reference level

27. Cox�s model, test for equivalence vs

28. An example, infection of burn wound Goal of burn wound management is to prevent or delay the infection. Outcome: time to infection A trial to compare a new therapy with a routine bath care in terms of the time to infection Hazard ratio of 2 for routine bathing care/test therapy is considered of clinical importance. 80% of patients� infection may be observed (=d) n=n1=n2 (p1=p2=0.5), equal size treatment groups Significance level: a=0.05 Power = 0.8, b=1-0.8=0.2

29. infection of burn wound Test of equality Test of superiority, Test of equivalence,

30. Sample size for survival data Key values Hazard ratio = hazard rate in treatment : hazard rate in control Probability of observing an event Sample size ratio between treatment and control Significance level, power Some methods need Length of accrual period Follow-up time Follow-up loss rate Treatment crossover rate

31. The rule of 50 http://www.childrensmercy.org/stats/size/quick.asp The rule of 50 applies when your outcome measure is a discrete event such as morbidity or mortality. The rule works well if that event is relatively rare. If you want enough subjects to be able to detect a halving of risk from your control group, be sure to collect enough data so that you will have at least 50 events in your control group. Then sample the same number of subjects in your treatment group.

32. The rule of 50 (cont) For example, patients using a control medication will have a risk over five years for a heart attack of roughly 8%. You want to try a new drug to see if it can reduce the risk to 4%. You would need a large enough sample in each group to ensure that at least 50 patients in the control group will have a heart attack. A control group of 625 subjects would suffice (8% of 625 is 50). With the same number of treated subjects, you would have a total of 1,250 patients in your study. This is just an approximation. The sample size that provides 80% power for detecting a halving of risk is actually 553 per group, not 625.

33. Some useful websites Sample size calculation of survival data http://cct.jhsph.edu/javamarc/index.htm Simple Interactive Statistical Analysis http://home.clara.net/sisa/ Statistical Rules of Thumb http://www.vanbelle.org/