6a.3 - Example: Discarding Ineffective Treatment

An approach for discarding an ineffective treatment in an SE study, based on the exact binomial method, is as follows. Suppose that the lowest success rate acceptable to an investigator for the treatment is 0.20. Suppose that the investigator decides to administer the treatment consecutively to a series of patients. When can the investigator terminate the SE trial if he continues to find no treatment successes?

Example

Determine when the exact confidence interval for p no longer contains a certain value Section

SAS Example: Modifications to the exact confidence interval program used earlier can be made to determine when the exact confidence interval for p no longer contains a certain value.

***********************************************************************
* This is a program that illustrates the use of PROC FREQ in SAS for  *
* determining an exact confidence interval for a binomial proportion. *
***********************************************************************;

proc format;
value succfmt 1='yes' 2='no';
run;

data Example_1;
input success count;
format success succfmt.;
cards;
1 03
2 16
;
run;

proc freq data=Example_1;
tables success/binomial alpha=0.05;
weight count/zeros;
title "Exact and Asymptotic 95% Confidence Intervals for a Binomial Proportion";
run;

SAS PROF FREQ (trial-and-error) indicates that the exact one-sided 95% upper confidence limit for p, when 0 out of 14 successes are observed, is 0.19. Thus, if the treatment fails in each of the first 14 patients, then the study is terminated.

Try it!

What is the upper 95% one-sided confidence limit for p when you have seen no successes in 5 trials?

Did you get 45% with the exact limits?

Notice also how clearly wrong the asymptotic limit is in this situation.

`Exact and Asymptotic 95% Confidence Intervals for Sensitivity`
`The FREQ Procedure`
`Positive`	`Frequency`	`Percent`	`Cumulative Frequency`	`Cumulative Percent`
`yes`	`0`	`0.00`	`0`	`0.00`
`no`	`5`	`100.00`	`5`	`100.00`


`Binomial Proportion for success = yes`
`Proportion`	`0.0000`
`ASE`	`0.0000`
`90% Lower Conf Limit`	`0.0000`
`90% Upper Conf Limit`	`0.0000`
`Exact Conf Limits`
`90% Lower Conf Limit`	`0.0000`
`90% Upper Conf Limit`	`0.4507`


`Test of H0: Proportion = 0.5`
`ASE Under H0`	`0.2236`
`Z`	`-2.2361`
`One-Sided Pr < Z`	`0.0127`
`Two-Sided Pr > \|Z\|`	`0.0253`

Sample Size = 5

Try it!

Here is another one to try... How many straight failures would it take to rule out a 30% success rate?

The answer is 9 ...

`Exact and Asymptotic 95% Confidence Intervals for Sensitivity`
`The FREQ Procedure`
`Positive`	`Frequency`	`Percent`	`Cumulative Frequency`	`Cumulative Percent`
`yes`	`0`	`0.00`	`0`	`0.00`
`no`	`9`	`100.00`	`9`	`100.00`


`Binomial Proportion for success = yes`
`Proportion`	`0.0000`
`ASE`	`0.0000`
`90% Lower Conf Limit`	`0.0000`
`90% Upper Conf Limit`	`0.0000`
`Exact Conf Limits`
`90% Lower Conf Limit`	`0.0000`
`90% Upper Conf Limit`	`0.2831`


`Test of H0: Proportion = 0.5`
`ASE Under H0`	`0.1667`
`Z`	`-3.0000`
`One-Sided Pr < Z`	`0.0013`
`Two-Sided Pr > \|Z\|`	`0.0027`

Sample Size = 9