**Help Aids **Top

**Application: **The
determination of the number of patients needed in a prospective comparison of
survival curves, when the control group patients have already been followed for
some period, following hypotheses are usually used:

**Procedure**:

- Enter

a) value of α, the probability of type I error

b) value of δ, the minimum hazards ratio, it is specified in alternate hypothesis (i.e. )

c) value of M_{S}, the control median survival (month), it is estimated
from existing control data

d) value of r, the accrual rate (per
month), it is estimated for future accrual

e) value of n_{C},
the number
of deaths observed in the control, it is obtained from existing control data

f) value of y_{C}, the number of patients
still at risk in the control, it is obtained from existing control data

g) value of τ, the
length of the planned continuation period (month)

h) value
of T, the length of the actual period for the new study (month).

- Click the button “Calculate” to obtain the power of the test and the sample size of patients in the experimental arm.
- Adjust the variable T (accrual time) until desire power is obtained (e.g. 80%).

**Assumption:**

1.
Time
to survival is exponential distributed with hazard rate λ.

2.
Prospective
studies are used.

3.
There
is no withdrawal or losses to follow-up.**
**

**Formula:**

Null
hypothesis for two exponential distributions has a rejection region. (1),

where ,

number of failures observed in group
*i*,

total patient years of observation
time in group *i*,

is the upperpercentile of the standard normal distribution.

For, whereis the true failure rate in group *i*, (1) is the critical region of
sizefor the one-sided alternative.

The
power of the test for a specific value ofis approximately (2),

where is the cumulative standard normal distribution function,

(3),

(4),

(5),

* n _{C}*

* S _{i} *is the total additional observation time
to be observed in group

*y _{C}* is the number still at risk in the
control group,

*r* is the accrual rate,

T
is the length of the accrual period for the new study,

is the length of the planned
continuation period.

Note: (If withdrawals are expected to occur at
rate, thenandin equation (3) and (4) should be replaced byand, respectively.) To calculate the power of a particular study,
one replacesbyand placesby, whereis the total observation time already observed in the control
group. If additional follow-up of remaining controls were not possible, one
would setto zero in (2).

**Notations:**

**α****:** The
probability of type I error (significance level) is the *probability of rejecting the true null
hypothesis. *

**β****:** The
probability of type II error (1 – power of the test) is the *probability of
not rejecting the false null hypothesis.*

**Example:**

Suppose one wishes
to design a trial to compare new treatment E with control treatment C, on the
basis of survival duration. Historical data for C from a recently compared
trial are available: a total 100 patients, of whom 50 have already died (*n _{C}*=50,

**Reference**: Dixon & Simon (1988) *J Clin Epidemiol* 41:1209-1213