**Help Aids **Top

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

where is the hazard rate of patients in experimental group in
stratum A

is the hazard rate of patients in control group in stratum A

is the hazard rate of patients in
experimental group in stratum B

is the hazard rate of patients in control group in stratum B

is the weight of

is the weight of

**Procedure**:

- Enter

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

b) value of β, the probability of type II error, or (1-power) of the test

c) value of K, the proportion of sample in each stratum, the sum of K’s in all strata must be equal to 1

d) value of δ, the minimum hazards ratio, it is specified in alternate hypothesis

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

f) value of Q_{C}, the **proportion of patient in control group compared with
experimental group in each stratum**

g) value of Q_{E}, the **proportion of patient in experimental group compared
with control group in each stratum, where Q _{C} + Q_{E} = 1**

h) value of T_{0}, **accrual duration (month), the length of time to recruit patients for
study in each stratum**

**i****) value of **T-T_{0}, **follow-up duration (month, the length of study time
of all recruited patients to the end of study T **

- Click the button “Calculate” to obtain

a) value of N, the sample size of patients in the experimental arm in each stratum

b) value
of D_{C},
the n**umber of
deaths in control group in each stratum**

** c) value
of **D_{E}, the n**umber of deaths in experimental group in each
stratum**

**
d) value of **D_{T}, the
total number of deaths at the end of study in each stratum

e) value
of N_{T}, the total sample size for all strata

f) value
of N_{C}, the total sample size for control group for all strata

g) value
of N_{E}, the total sample size for experimental group for all strata

**Assumption:**

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

**Theory:**

In some
cases, the trial may be designed with the intent of conducting a pooled (e.g.,
Mantel-Haenszel) analysis over strata. For simplicity
we consider the case of two strata, say A and B, with sample sizesand(), where each stratum may have an associated recruitment and
follow-up period () and associated hazards under the alternative hypothesis (), For example, strata A and B may correspond to a prognostic
factor (e.g., severity of disease), or stratum A mat refer to a feasibility or pilot
phase and stratum B to the main trial.

For
each cases, the test statistic would employ pooled estimators of the
within-stratum differences in hazard rated: , where,are the estimated hazards from stratum A ofpatients, and likewise for,ofpatients. Further, let,be the sample fractions of the two treated groups in stratum
A (), and likewise, ,for stratum B.

Under
the assumption of uniform entry with no losses to follow-up,

** **,** **

** **where , ,

, ,

Under the null hypothesis, it then follows that

Under the alternative hypothesis,

where,

and ,

If we then express the strata samples sizes as fractions of the total, , , it is easily shown that

where

The
basic equation relating sample size and power is

where,

andare the strata sample fractions (e.g.,)

**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:**

Consider the case of two prognostic strata,
such as early-versus late-stage disease, with stratum-specific hazards and
relative strata size *K _{A}*=0.43
and

It is a two sided test with equal-sized
group (*Q _{CA}*=

**Reference**:

- Lachin and Foulkes (1986)
*Biometrics*42: 507-519