1. Clinical Trials with Two Endpoints

2. EXAMPLES HIV Vaccine Trial: Outcome 1: Incidence of HIV infection Outcome 2: Reaching Viral Set-point Cardiovascular: Stroke Heart Attack Cognitive: Evaluation of Memory Occurrence of Dementia Insomnia: Time Till Sleep Length of Continuous Sleep Diet: Pounds lost MmHg BP Lowered

3. Difficulty: Want an overall test that rejects the null only 5% of the time. Somehow, combine two endpoints. There are better ways than Bonferroni that can consider when Modest effects are seen in each outcome. Dealing with the difficulty: Combine Endpoints into a Composite Endpoint. Here, you condense results into a single outcome. May complication interpretation of statistic (Burden of Illness Scores, total time awake at night, lost 10 lbs or lowered BP ) Combine Tests. Perform a statistical test on each endpoint separately, and then consider results from both tests together. Combine P-values.

4. Two Outcomes: H 0 : Θ 1 =0, Θ 2 =0 H A : Θ 1 ≥0, Θ 2 ≥0 (with one strict inequality) How do you test this hypothesis? ONE-SIDED TESTS Why? Believe treatment can only be beneficial Only care if treatment can be beneficial One Outcome: H o : Θ=0 H A : Θ > 0 Reject when Z 1 > α 0.05

5. EXAMPLE of a STUDY with TWO OUTCOMES: DIET: Outcome 1: Weight Loss Outcome 2: Blood Pressure Lower W i = Starting Weight – Final Weight BP i =Starting SBP – Final SBP

6. Option 1: Simes Test Reject H 0 when max(Z 1,Z 2 ) > k x or min(Z 1,Z 2 ) > k n

7. Option 2: Fisher’s Test Reject H 0 when (Z 1 Z 2 ) > k

8. Option 3: Linear Combination of Z Statistics Reject H 0 when Z 1 + Z 2 > k

9. Options 4 - 6: Weighted Versions of Statistics Reject H 0 when w 1 Z 1 + w 2 Z 2 > k When w 1 is large, rejection is primarily based on Z 1

10. Ideal Test? Most Powerful? Only one alternative ( θ 1, θ 2 ) Reject H o when Z 1 θ 1 +Z 2 θ 2 >k H A : E(Z 1 ) = θ 1 E(Z 2 ) = θ 2

11. Ideal Test? Most Powerful? Only one alternative ( θ 1, θ 2 ) Reject H o when Z 1 θ 1 +Z 2 θ 2 >k H A : E(Z 1 ) = θ 1 E(Z 2 ) = θ 2

12. Ideal Test? Most Powerful? ( θ 1, θ 2 ) Reject H o when Z 1 θ 1 +Z 2 θ 2 >k (Z 1,Z 2 ) Z 1 θ 1 +Z 2 θ 2

13. Best test statistic under known alternative: Z 1 θ 1 + Z 2 θ 2 Unknown (θ 1,θ 2 )? Estimate the parameters from the data: When there are no restrictions on the Parameter space However, by believing our treatment can only be beneficial, we implicitly State θ 1, θ 2 ≥ 0 Z 2 > 0 otherwise Z 1 > 0 otherwise

14. Rejection Region for

15. Likelihood Ratio Test Switch notation, return to variation of original example 1) Assume subjects are paired. 2) Let X i1,X i2 be the difference in weight and blood pressure change for Pair i. TREATMENT WORK ≈ μ 1 ≡ E(X 1 ) > 0 and/or μ 2 ≡ E(X 2 ) > 0 Density: Just need to find the μ that maximizes the denominator. And then can Use 2*log(λ) as the likelihood ratio test. LR

16. Likelihood Ratio Test (Continued) x 1 > 0 otherwise x 2 > 0 otherwise MLE’s (assuming X 1 and X 2 are independent) log …(skipping the algebra) Always use LRT? --It is very difficult to maximize μ when X i1 and X i2 are not independent Use LRT for POC HIV Vaccine Trial? --The Z statistics for incidence of viral infection and reaching viral set-point are independent.

17. Properties of Likelihood Ratio Test Combining Most Powerful Rejection Regions:

18. Properties of Likelihood Ratio Test Combining Most Powerful Rejection Regions:

19. Properties of Likelihood Ratio Test Combining Most Powerful Rejection Regions:

20. R α (Θ) = The rejection region that gives the most powerful α-level test under the alternative hypothesis, Θ = (θ 1,θ 2 ). p α (Θ) = The power of the most powerful α-level test under the alternative hypothesis, Θ = (θ 1,θ 2 ),. Uniformly Most Powerful Invariant Test (UMPI) Notation: If we can write the rejection region as the union of R α (Θ) such that p α (Θ) is the same for equivalent Θ’s, we will say that test has composition invariance. When does composition invariance imply that the test is Uniformly Most Powerful Invariant? THE END Question