# Presentation: 672020

## COMPARISONS OF NON-INFERIORITY METHODS FOR MORTALITY TRIALS*

- By Mark Rothmann
- Division of Biometrics I, CDER, FDA
- *  The views expressed in this article do not necessarily represent those of the U.S. Food and Drug Administration
- Presented November 4, 2002 at the 2002 PharMa meeting in Bethesda

## Outline

-  Notation, Terms and Set up
-  Methods and Comparisons
-   -  Two CI method
-   -  large sample normal test statistic
-   -  Delta method CI approach
-  Concluding Remarks

## Notation

- T - Experimental Treatment
- C - Active-Control
- P - Placebo or other reference therapy
- HR - Hazard Ratio

## Definitions of the Proportion of Effect Retained, 

## Hypotheses 

- We will assume that HR(P/C) > 1. When testing
- whether the treatment maintains 100***0*%** **of the effect of the
- active control, the hypotheses are:
-                              H0: *** * _<_ ***0*  vs.
-                              Ha: **  > ***0*

## Making Inferences

-  Inferences about HR(T/C) are made from a trial
- comparing the experimental treatment with the
- active control.
-  Historical trials (a meta-analysis or a single trial)
- and changes in study conditions are used to model
- (make inferences about) HR(P/C).  A normal model
- will be used for estimator of log HR(P/C).

## Modeling the active-control effect

- The current active-control effect may be modeled
- based on previous trials comparing the active-control
- with a placebo or other reference therapy and any
- necessary adjustments (for differences between the
- current trial and previous trials).

## Assumptions

-  Type I error rate calculations assume that the
- active-control effect for the current trial can be
- correctly modeled and that
-  the stopping rule (design) for the current trial is independent
- of the results for estimating the active-control effect.
- Submitted a paper recently dealing with case where the design
- of the current trial depends on the results for estimating
- the active-control effect.

## 95% CI Lower Limit Method 

- A two 95% confidence interval procedure:
- -   Define the “non-inferiority” cutoff as
-      (1- 0lower limit of 95% CI for log HR(P/C)).
-      If the 95% confidence interval for log HR(T/C) lies entirely
-      beneath this cutoff, “non-inferiority” is inferred.

## As a Two 95% CI Approach

-                           0

-    95% CI for log HR(T/C)         95% CI  for log HR
-                                                     representing 1000%
-                                                     of the control effect

**    **** **

## Type I error rate

- If the current active-control effect is correctly modeled, using
- the 95% CI lower limit for HR(P/C) to define the effect will
- give a type I error somewhere between roughly
- 0.0027 and 0.025.
- The type I error rate graph follows where

## Type I error rate vs. square root of the information ratio when using the 95% CI lower limit method

## Test Statistic

- Compare *Z****  with –1.96 leads to an approximate
- one-sided 0.025 level test, if the current active-control effect is correctly modeled.

## A Hasselblad & Kong* Procedure

- A “95%” confidence interval is calculated using a normal
- distribution with standard deviation (error) given by

- If this C.I. is entirely above ***0*, “non-inferiority” is concluded.
- * Hassalblad and Kong (2001) ***Drug Inform J*****, 35 **435-449

## Assessing Normality

- two normally distributed random variables.  When the
- denominator of this ratio acts like a non-zero constant
- with respect to the numerator then  will have an
- approximate normal distribution.

- Note that

- involves a ratio of

## Assessing Normality

- For **1 = logHR(T/C) and **2 = logHR(P/C)  with **1 and **2
- as the standard deviations of the corresponding estimators of
**1 and **2. Normality will approximately hold, for example,
- when
- a) |* ***2/**2| is large or
- b) **1 /**2  is small and |**2/**2|  is not small (> 2.5 – for example).

## Simulation Model

## Simulation Results - H & K

## Application: Xeloda vs 5-FU+LV Trial

- Indication: First-line Metastatic Colorectal Cancer
- Experimental Treatment (T): Xeloda
- Active Control (C): 5-FU+LV
- Reference (Placebo; P):5-FU

## 5-FU vs 5-FU+LV

- Random Effects Meta- analysis Model results based on
- ten trials

**          ****Survival Summary **
**HR(P/C)****	****log HR      SE(logHR)**
- 	1.264     0.234          0.075
- 95% C.I. for HR(P/C) is (1.091, 1.464)

## Survival Results for Study SO14695

**log HR(T/C)         SE****	****	****95% CI**
-   - 0.00359	       0.08675	         0.84-1.18

## Results

**95% Lower Limit Method:** The largest percent retained
- that is rejected is -91%.
**H&K:** 95% C.I. (28.9%, 174%)
**Z*:**      95% C.I. (9.2%, 196%)
**Z*:**   90.5% C.I. (28.9%, 175%)

## Concluding Remarks

-  It is important to determine an appropriate model for the
- active-control effect.
-   Once modeled, maintaining an appropriate type I error
- probability under reasonable  assumptions should be a goal.
- The type I error probability will depend on whether.