Equivalence and Noninferiority Tests for Quality, Manufacturing and Test Engineers

88 Equivalence and Noninferiority Tests

Computational considerations:

Computationally, Test 7.1 is a combination of Test 2.5 (regression slope) and

Test 2.2 (comparison of two means, xed Δ). In the example used here, the

two systems are considered equivalent at time T = 3 if the reliability of the

evaluation system, Re(T), is at least 99 percent of Rc(T) (i.e., δ = 0.01).

JMP Data Table and formulas (Figure 7.2 and Figure 7.3). The columns

reandrc in Figure7.2 represent the reliability functions for the time-to-event

variables, Te and Tc. You can use the “Fit Model” function to obtain coef-

cient estimates. Make sure to check the “No Intercept” option box.

Right-click on the Parameter Estimates Chapter of the Fit Model output,

and select the “Make into Data Table” option. Create new columns to com-

pute the standard error squared, and a column that indicates the row num-

ber in each parameter estimate table (called “order” in this example). Join the

tables for Te and Tc parameter estimates using the Tables -> Join function,

with “order” as the matching column. Then compute Dk, the difference in

the parameter estimates, for the linear (order = 1) and quadratic (order = 2)

terms of the two models.

• The models and the data (Figures7.4 and 7.5):

FIGURE 7.2

Test 7.1, JMP screen 1.

89Reliability

FIGURE 7.3

Test 7.1, JMP screen 2.

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

R(t)

7

6.565.554.543.532.721.5

Tc

YRc Rc(t) - model

FIGURE 7.4

R

c

(t) (JMP overlay plot).

90 Equivalence and Noninferiority Tests

Test 7.1, JMP screen 3.

1

1.5 2 2.5 3 3.5 4

Te

4.5 5 5.5 6 6.5

1.7

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

R(t)

Re(t) - modelReY

FIGURE 7.5

R

e

(t) (JMP overlay plot).

91Reliability

Test 7.1, JMP screen 4.

Create “dummy variable” columns (“indlinear” and “indquad”) to allow the

standard errors for the linear and quadratic coefcient estimates to be com-

puted in the same row of an output table. That is, for example, indlinear=1

for row 1 and 0 for row 2 of the estimate table. D1 is indlinear*Dk; D2 is

indquad*Dk.

92 Equivalence and Noninferiority Tests

Test 7.1, JMP screen 5.

The test statistic requires as input a particular time to event (T) and δ.

Since in this example the test statistic is less than 0, the null hypothesis is

not rejected, and the conclusion is that the two systems are not equivalent at

time T = 3.

Test 7.2 Nonparametric Test for Comparison of Reliability

Parameters:

T = time to failure

h

c

(t) = hazard rate function, comparator system (no form specied)

h

e

(t) = hazard rate function, evaluation system (no form specied)

Ht

hd

() ()

cc

t

0

∫

=τ

τ = cumulative hazard function, comparator system

Ht

hd

() ()

ee

t

0

∫

=τ

τ = cumulative hazard function, evaluation system