Equivalence and Noninferiority Tests for Quality, Manufacturing and Test Engineers

125

11

Computer Code—Power Curves

This chapter describes computer codes and procedures for generating the

power curves for various tests. Each test (except Test 6.1, for which only R

code is presented) will have the following parts:

Input parameters

SAS code

JMP Data Table and formulas

R

Test 1.1 Single Binomial Proportion

(One-Sided—Probability of “Success”)

Input parameters:

P0 = minimum desired probability of success

P1 = alternate values for probability of success (P1 ≤ P0)

N = sample size

Xc = critical value for number of successes (yields ~ 1 − beta probability of

rejecting the null hypothesis if probability of success = P

0

)

SAS code:

libname stuff 'H:\Personal Data\Equivalence & Noninferiority\

Programs & Output';

data calc;

set stuff.D20120730_One_Proportion_Power;

power = 1 - probbnml(P1,N,Xc-1);/* Pr{X > = Xc} */

run;

proc print data = calc;/*dataset calc has columns N, Xc, P0,

P1, power */

run;

126 Equivalence and Noninferiority Tests

Output:

The SAS System 09:04 Monday, July 30, 2012 1

Obs N Xc P0 P1 power

1 100 92 0.95 0.95 0.93691

2 100 92 0.95 0.94 0.85371

3 100 92 0.95 0.93 0.73397

4 100 92 0.95 0.92 0.59263

5 100 92 0.95 0.91 0.44940

6 100 92 0.95 0.90 0.32087

7 100 92 0.95 0.89 0.21651

8 100 92 0.95 0.88 0.13859

9 100 92 0.95 0.87 0.08448

10 100 92 0.95 0.86 0.04921

11 100 92 0.95 0.85 0.02748

12 100 92 0.95 0.84 0.01474

13 100 92 0.95 0.83 0.00762

14 100 92 0.95 0.82 0.00380

15 100 92 0.95 0.81 0.00183

16 100 92 0.95 0.80 0.00086

JMP Data Table and formulas:

Power for Test 1.1, JMP screen.

127Computer Code—Power Curves

Note that the column labeled “Power” is computed as described in the

formula view.

R:

> prob <- c(0.95,.94,.93,.92,.91,.90,.89,.88,.87,.86,.85,.84,

.83,.82,.81,.80)

> power <- 1 - pbinom(92,100,prob)

> power

[1] 0.8720395214 0.7483494398 0.5987792361 0.4471097512

0.3127949596

[6] 0.2060508618 0.1284536668 0.0761360984 0.0430807590

0.0233537817

[11] 0.0121651919 0.0061048616 0.0029577572 0.0013860090

0.0006291400

[16] 0.0002769869

> plot(prob,power)

0.80 0.85 0.90 0.95

0.0

0.2

0.4

0.6

0.8

Prob

Power

Test 1.1, R: power curve.

Test 2.1 Single Mean (One-Sided)

Input parameters:

muA = alternative values of the mean

muL = minimum desirable value for the mean

128 Equivalence and Noninferiority Tests

sigma = prior estimate of the standard deviation

n = sample size

SAS code:

libname stuff 'H:\Personal Data\Equivalence & Noninferiority\

Programs & Output';

data calc;

set stuff.d20120801_one_mean;

delta = (muA - muL)/sigma;

nc = sqrt(n) * delta;

tcrit = tinv(0.05,n-1);

power = 1 - probt(tcrit,n-1,nc);/* Pr{T' > = tcrit} */

run;

proc print data = calc;/*dataset calc has columns n muL muA

sigma delta nc power */

run;

The SAS System 09:11 Wednesday, August 1, 2012 1

Obs n muL muA sigma delta nc tcrit power

1 20 100 96.00 3.5 -1.14286 -5.11101 -1.72913 0.00053

2 20 100 96.25 3.5 -1.07143 -4.79157 -1.72913 0.00150

3 20 100 96.50 3.5 -1.00000 -4.47214 -1.72913 0.00390

4 20 100 96.75 3.5 -0.92857 -4.15270 -1.72913 0.00929

5 20 100 97.00 3.5 -0.85714 -3.83326 -1.72913 0.02033

6 20 100 97.25 3.5 -0.78571 -3.51382 -1.72913 0.04094

7 20 100 97.50 3.5 -0.71429 -3.19438 -1.72913 0.07597

8 20 100 97.75 3.5 -0.64286 -2.87494 -1.72913 0.13021

9 20 100 98.00 3.5 -0.57143 -2.55551 -1.72913 0.20668

10 20 100 98.25 3.5 -0.50000 -2.23607 -1.72913 0.30485

11 20 100 98.50 3.5 -0.42857 -1.91663 -1.72913 0.41961

12 20 100 98.75 3.5 -0.35714 -1.59719 -1.72913 0.54173

13 20 100 99.00 3.5 -0.28571 -1.27775 -1.72913 0.66005

14 20 100 99.25 3.5 -0.21429 -0.95831 -1.72913 0.76440

15 20 100 99.50 3.5 -0.14286 -0.63888 -1.72913 0.84815

16 20 100 99.75 3.5 -0.07143 -0.31944 -1.72913 0.90933

17 20 100 100.00 3.5 0.00000 0.00000 -1.72913 0.95000

18 20 100 100.25 3.5 0.07143 0.31944 -1.72913 0.97460

19 20 100 100.50 3.5 0.14286 0.63888 -1.72913 0.98814

20 20 100 100.75 3.5 0.21429 0.95831 -1.72913 0.99492

21 20 100 101.00 3.5 0.28571 1.27775 -1.72913 0.99800

129Computer Code—Power Curves

JMP Data Table and formulas:

Power for Test 2.1, JMP screen.

R:

> muA <- c(96,96.25,96.5,96.75,97,97.25,97.5,97.75,98,98.25,

98.5,98.75,99,99.25,99.5,99.75,100,100.25,100.5,100.75,101)

> muL <- 100

> n <- 20

> sigma <- 3.5

> delta <- (muA - muL)/sigma

> nc = sqrt(n)*delta

> tcrit <- qt(0.05,n-1)

> power <- 1 - pt(tcrit,n-1,nc)

> plot(delta,power)