All-or-None procedure: An outline Nanayaw Gyadu-Ankama Shoubhik Mondal Steven Cheng

History of All or None procedure Min test Test by Cappizi and Zhang Min test based on Restricted Null Space Average Type I Error Approach Discussion References Summary

Why All or None Procedure All or none method was evolved in 1982 in context of Quality control by Berger(1982) to test quality of a product based on several parameters. He compared producer’s and consumer’s risk. Disease like migraine, Alzheimer’s disease and Arthritis are characterized by more than one endpoints. The primary objective of a clinical trial is met if the test drug shows a significant effect with respect to all the endpoints.

Two types of Multiplicity First case is to when treatment is effective if it improves at least one of the multiple endpoints The second case is to when treatment is effective when it improves on all the multiple endpoints. Multiple primary endpoints in second case is called co-primary endpoints where simultaneous improvement is required to declare a treatment effective

Example of co-primary endpoints

General Sense formulation of the multiple endpoint problem pertains to the requirement that the treatment be effective on all endpoints This problem is referred to as the reverse multiplicity problem represents the most stringent inferential goal for multiple endpoints. IU framework, the global hypothesis is defined as the union of hypotheses To reject null hypothesis, one needs to show that all individual hypotheses are false

Formulation of IU test

Min Test the goal of demonstrating the efficacy of the treatment on all endpoints requires an all-or-none or IU procedure of the union of individual hypotheses Reject all hypotheses if tmin= min 1≤i≤m ti ≥ tα(ν), where tα(ν) is the (1 − α)-quantile of the t- distribution with ν = n1 + n2 − 2 df This procedure is popularly known as the min test (Laska and Meisner) We take α = for one sided

Min test Contd. Advantage: this procedure does not use a multiplicity adjustment (each hypothesis Hi is tested at level α Disadvantage: Min test is not as powerful as it looks Maximum power occurs when no treatment effect at one-point and infinite treatment effects at other points. This leads to marginal t-test

Power Comparison of Min Test

Power of the min test is always less than the power of individual test When endpoints are highly correlated the power is almost same because all the endpoints be merged to one endpoints When endpoints are independent then the overall power is product of individual power As correlation between endpoints increases, power increases for min test when power for individual test is fixed

Sample Size Increment For Min Tesr there are three co-primary end-points and correlation among the test statistics is 0.2 the overall power for detecting size corresponding to an 80%power at the individual subhypothesis level is only 55% Increase with the number of co-prmary endpoint and the decrease in correlatiom

Test by Cappizi and Zhang Cappizi and Zhang (1996) suggested another alternative to the min test which requires that the treatment be shown effective at a more stringent significance level α1 on say m1 α1 on the remaining m2=m-m1 For m=2, m1=1, m2=1, they take α1=0.05, α2= 0.1 or 0.2 They did not consider the null space for no treatment effect on at least one endpoint This rule does not control the experimentwise type I error

Restrict Null Space Chuang-Stein et al. (2007) propose to take a restricted null space. Considering type I error between(0,0) (M,0) and (0,M). Adjusted significance level is minimal Still not much gain in terms of power

Restrict the null space

Adjusted significance level

Average Type I Error Approach Another approach to this formulation adopts a modified definition of the error rate to improve the power of the min test in clinical trials with several endpoints. Instead of looking at the maximum false positive rate over a restricted null space, Chuang et al. proposes to look at the “Average” false positive Rate over that space They find an upper bound of this Average False positive Rate.

Average Type I Error

Formula of Average Type I Error

Adjusted significance level

Sample size increment

Discussion Here assumption is that all points in restricted null space are equally likely. Have to use higher significance level to manage the lower overall significance level. The level of significance is a function of number of co- primary endpoints and the correlation among the endpoints. If the endpoints are highly correlated the level of significance will be very close to 2.5%, because high correlation essentially reduces multiple primary endpoints to a single endpoint.

Discussion The sample size needed to maintain a desirable power under the new approach is much smaller than IU test. Assumption of equally likely is not realistic always. This work is still not scientifically justifiable.

Refrences 1) Roger L. Berger(1982), Multiparametr Hypothesis Testing and Acceptance Sampling., Tech-nometrices. 2) Offen et al.(2007), Multiple Co-primary Endpoints: Medical and Statistical Solutions., Drug Information Journal. 3)Eugene M Laska and Morris J. Meisner(1989), Testing Whether an Identied Treatment Is Best., Biometrics. 4)Chuang et al.(2007), Challenge of multiple co-primary endpoints: A new approach.,Statistics in Medicine.