62 Equivalence and Noninferiority Tests
Test 4.2 Two Exponential Rate Parameters (One-Sided)
Parameters:
λ
c
= exponential rate parameter (number of events per unit time),
comparator system
λ
e
= exponential rate parameter (number of events per unit time), evaluation
system
n = number of times to event, comparator system (no censoring)
m = number of times to event, evaluation system (no censoring)
δ
0
= proportionality constant, δ
0
> 1
T
c
= time-to-event variable, comparator system
T
e
= time-to-event variable, evaluation system
E[T
c
] = expected value of T
c
= 1/λ
c
E[T
e
] = expected value of T
e
= 1/λ
e
Hypotheses:
H
0
: λ
e
> δ
0
λ
c
H
1
: λ
e
≤ δ
0
λ
c
Or, alternatively:
<
δ
HE
TE
T:[]
1
[]
ec
0
0
≥
δ
HE
TE
T:[]
1
[]
ec
1
0
Data:
T mean time to event, comparator system
c
=
T mean time to event, evaluation system
e
=
Critical value(s):
Reject the null hypothesis, H
0
, if
TTtSE
1
0
ec
0
−
δ
+≥
β
63Exponential Rate Parameters
where
SE
T
m
T
n
ec
22
0
2
=+
δ
and t
β
= 100(1 − β) percentile of a (central) t-distribution with m + n – 2 degrees
of freedom.
Discussion:
This test is a special case of Test 2.3, with the primary difference being
the estimate of the standard error, SE. Under the assumption that T has an
exponential distribution,
T
n
T
n
ˆ
[]
1
ˆ
2
2
2
σ=
λ
=
is the minimum variance estimate for the variance of
T
(Mann, Schafer, and
Singpurwalla, 1974).
Suppose H
0
: λ
e
= δ
a
λ
c
, δ
a
> δ
0
. Then the test statistic
TT
SE
1
ec
0
−
δ
has a noncentral t-distribution with noncentrality parameter:
nc
n
11
1
1
a 0
0
2
=
δ
−
δ
+
δ
.
The correspondence between Test 4.2 and Test 2.3 is
p
1
1
0
0
δ
=−
The exponential distribution assumption simplies the noncentrality,
since in this case
ET
1
()
=µ=σ=
λ
.
64 Equivalence and Noninferiority Tests
Example:
Suppose:
T
c
= 100
T
e
= 80
n = m = 100
δ = 1.001
so that
SE
T
m
T
n
(80)
100
1
1.001
(100)
100
12.798
ec
22
2
2
2
2
=+
δ
=+ ≈ .
Then the test statistic is
TTtSE
1
80
1
1.0001
(100) 1.653(12.798) 1.250
0
ec
−
δ
+≈−+
≈≥
β
.
Therefore, the null hypothesis, H
0
, is rejected.
Condence interval formulation:
TTtSE
1
ec
−
δ
+
β
is an approximate 100(1 − β) percent upper condence limit on
ET ET[]
1
[]
ec
−
δ
.
Computational considerations:
See the code for Test 2.3.

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