C. V. Gonzalez et al. / Modern Chemotherapy 2 (2013) 51-56 53

2

jjj iijiigiiiii

i1, D

ˆ

SWtYYYYYdY1

jg

d,

The quantities are linearly depen-

dent since j1

, K is zero. Therefore, the test

statistic is constructed by selecting any K − 1 of the j

1K

Zι,,Zι

j

Zι

Zs

(the first K1, say). The estimated variance-covariance

matrix of the resulting vector is given by the (K − 1) x (K

− 1) matrix formed by the appropriate jg . Finally,

the test statistic is given by the quadratic form:

ˆ

SS

t

1

1K1 1K1

Zι,,ZιZι,,Zι

X

If the null hypothesis H0 is true and the sample size is

large, X is approximately distributed as a chi-square with

K − 1 degrees of freedom. An α-level test of H0 thus re-

jects the null hypothesis when X is greater than the upper

α-quantile of this chi-square. In particular, when K = 2,

as is the case in our data set, X should be distributed as a

chi-square with 1 degree of freedom under H0.

A variety of weight functions have been proposed in

the literature (see [4-8], and [3] for a review). The most

common and widely used test has W(t) = 1 for all t. This

test is referred to as the Mantel-Haenszel or log-rank test,

and is available in any modern statistical software. It has

optimum power to detect alternatives where the hazard

rates in the K treatment groups are proportional to each

other.

Fleming and Harrington proposed (see [3]) a very

general class of tests that includes the Mantel-Haenszel

test as a special. Let Ŝ(t) be the Kaplan-Meier estimator

of the common survival function under H0, based on the

combined treatment groups. The weight function in the

Harrington-Fleming’s test is, at time ti:

q

p,qii 1i 1

ˆˆ

Wt St1St,p0,q0

p

(4)

Here, the survival function at the previous death time

is used as a weight for mathematical reasons (this en-

sures that these weights are known just prior to the time

at which the comparison is to be made). Letting p = q = 0

in (4) results in the Mantel-Haenszel test. Letting p = 1

and q = 0 results in a version of the Mann-Whitney-

Wilcoxon test. When p > 0 and q = 0, Wp,q give the most

weight to early departures between the hazard rates in the

K groups, whereas when p = 0 and q > 0, the corre-

sponding tests give most weight to departures which oc-

cur late in time. By an appropriate choice of p and q, one

can construct tests which have the greatest power against

alternatives where the K hazard rates differ over any de-

sired region.

We applied this methodology to our data sets. Flem-

ing-Harrington test (with p = 0.5 and q = 0.5) is more

sensitive to detect differences when the curves have a

delayed separation in time that is why sometimes the

results are significant. Mantel-Haenszel test is appropri-

ate when there is a proportional separation of curves.

3.3.2. Stratifi ed Test

As mentioned above, the log-rank tests test is useful

when each treatment group is homogeneous that is, when

the survival distribution is the same for every patient

within a group. A violation of this homogeneity usually

indicates that one needs to adjust the analysis for some

other (than the treatment group) covariate. For example,

previous studies suggest that an evaluation of CIMA

Vax®EGF efficacy should be stratified over age, since

homogeneity only holds within the two subpopulations

of patients under (respectively over) 60 years of age. One

possible approach to this issue is to base the decision on

a stratified version of one of the tests discussed above.

This approach is feasible when the covariate we adjust

for is categorical and its number of levels is not too large,

or when it is continuous but can be discretized into a

workable number of levels. In the sequel, we discuss

how such stratified tests are constructed, and how they

can be used to analyze our data.

Suppose that the covariate we need to adjust for is

discrete (or continuous and discretized), with M levels.

Then, we wish to test the hypothesis

0,strat 1s2sKs

H:htht ht,

for s1,M andtι

(5)

against the alternative that at least one of the hjs is dif-

ferent from the others for some s and some t . A strati-

fied test is constructed similarly as in (2) and (3) (for the

weighted version of the test), except that all quantities

are calculated by using only the data from the s-th stra-

tum, yielding Zjs() and s. The same weight functions as

in the previous section can be used for the stratified tests.

A global test of H0,strat in (5) is obtained by summing all

the within-stratum quantities, such as: Zj() = s = 1,···,M

Zjs() and ŝjg = s = 1,···,M ŝjgs. Finally, the stratified test sta-

tistic is defined as

t

1

1K1 1K1

ι,, ιZι,,ZιZZ

strat

X

where is the (K − 1) x (K − 1) matrix obtained from

the ŝjg's. If the null hypothesis H0,strat in (5) is true, and

the sample size is large, strat is approximately distrib-

uted as a chi-square with K − 1 degrees of freedom. An

α-level test of H0 thus rejects the null hypothesis when

strat is greater than the upper α-quantile of this

chi-square. In particular, when K = 2, as is the case in our

data set, strat should be distributed as a chi-square

with 1 degree of freedom under H0,strat.

X

X

X

4. RESULTS

We analyzed the data obtained from the phase II and

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