Choosing between Cox proportional hazards and logistic models for

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Choosing between Cox proportional hazards and logistic models for
Journal of Applied Statistics, Vol. 30, No. 1, 2003, 37–47
Choosing between Cox proportional hazards
and logistic models for interval-censored data
via bootstrap
JOSÉ EDUARDO CORRENTE1, LICIANA V. A. S. CHALITA2,
JEANETE ALVES MOREIRA3, 1Departamento de Ciências Exatas, ESALQ/
USP, CP 9, Piracicaba, Brasil; 2Departamento de Bioestatı́stica, IB/UNESP, CP 510,
Botucatu, Brasil and 3Departamento de Estatı́stica, Universidade Federal do Rio
Grande do Norte, Lagoa Nova, Natal, Brasil
 This work develops a new methodology in order to discriminate models for
interval-censored data based on bootstrap residual simulation by observing the deviance
difference from one model in relation to another, according to Hinde (1992). Generally, this
sort of data can generate a large number of tied observations and, in this case, survival time
can be regarded as discrete. Therefore, the Cox proportional hazards model for grouped data
(Prentice & Gloeckler, 1978) and the logistic model (Lawless, 1982) can be fitted by means
of generalized linear models. Whitehead (1989) considered censoring to be an indicative
variable with a binomial distribution and fitted the Cox proportional hazards model using
complementary log-log as a link function. In addition, a logistic model can be fitted using
logit as a link function. The proposed methodology arises as an alternative to the score tests
developed by Colosimo et al. (2000), where such models can be obtained for discrete binary
data as particular cases from the Aranda-Ordaz distribution asymmetric family. These tests
are thus developed with a basis on link functions to generate such a fit. The example that
motivates this study was the dataset from an experiment carried out on a flax cultivar
planted on four substrata susceptible to the pathogen Fusarium oxysoprum. The response
variable, which is the time until blighting, was observed in intervals during 52 days. The
results were compared with the model fit and the AIC values.
1 Introduction
The development of methods to analyse survival data is one of the areas in statistics
that have increased the most in the last few years. This is because, in many practical
Correspondence: J. E. Corrente, Departamento de Ciências Exatas, ESALQ/USP, CP 9, 13418-900
Piracicaba-SP, Brasil. Email: [email protected]
ISSN 0266-4763 print; 1360-0532 online/03/010037-11
DOI: 10.1080/0266476022000018493
© 2003 Taylor & Francis Ltd
38
J. E. Corrente et al.
situations, researchers are interested in the survival time until the occurrence of an
event, such as the failure time of a component or the time until the death of a subject,
etc. However, it is not always possible to observe the exact time when an event
occurs, but only its interval is identified. These sorts of data are known as intervalcensored or grouped survival data and they generally present tied observations.
In the case of a low number of ties, the analysis of these data can be made by
using the Cox proportional hazards model (Cox, 1972) by means of the exact
partial likelihood, or by approximations proposed by Breslow (1972, 1974), Peto
(1972), Efron (1977) and Farewell & Prentice (1980). When the number of tied
observations is large, time can be considered as discrete and a model can be fitted
to the probability of occurrence of an event since it did not occur in the previous
interval. Such fits can be made using the Cox proportional hazard model for
grouped data (Prentice & Gloeckler, 1978) or logistic model (Lawless, 1982).
These methods are more suitable when the time intervals are disjoint. In the case
of an overlapping interval, they cannot be directly applied (Finkelstein, 1986).
The fitting of models for discrete times in the presence of interval censoring
becomes easier when generalized linear models fit (Nelder & Wedderburn, 1972).
After models are fitted, there is an interest in knowing which of them better
explain the experimental data. For this, Colosimo et al. (2000) proposed two score
tests from a more general distribution: the Aranda-Ordaz asymmetric transformations family, which has the logit transformation (logistic model), and complementary log-log (Cox proportional hazards model), as particular cases.
Therefore, this work aims to propose a new methodology to choose between the
two discrete models for interval-censored data by fitting the two cited models; that,
is, using a bootstrap residual simulation and by the analysis of the deviance
differences, according to Hinde (1992). To that end, the dataset from an experiment
carried out on a flax cultivar planted on four substrata (Höper et al., 1995) and
susceptible to the pathogen Fusarium oxysoprum was used. The results were
compared with the model fit and the AIC values.
Section 2 presents the proportional hazard and the logistic models for intervalcensored data, including the score test proposed by Colosimo et al. (2000). The
bootstrap residuals techniques are shown in Section 4. The applications and results
of the dataset analysis are found in Section 5.
2 Models for interval-censored data
Let the lifetimes Tl , ló1, . . . , n, where n is the sample size, be grouped into k
intervals, Ii ó[aiñ1, ai ), ió1, . . . , k, where 0óa0 \a1 \ . . . \ak óê and assume
that every censoring takes place at the end of the interval. Let Di be the set of the
subjects that failed in the interval Ii ; Ri the set of subjects at risk at the beginning
of the interval Ii and *li an indicative variable for the subject’s lifetime l at the
interval Ii (*li ó0, if the lth subject is censored at Ii and *li ó1 otherwise). Then,
the likelihood function for life tables with explanatory variables can be written
considering the following steps.
(i) Failure contribution for the likelihood:
P(Tl é Ii D xl )óP(aiñ1 OTl \ai D xl )óS(aiñ1 D xl )ñS(ai D xl )
ó[(1ñp1(xl )) . . . (1ñpiñ1(xl ))]pi(xl )
(1)
Cox proportional hazards and logistic models
39
(ii) Censored contribution for the likelihood (supposing right censoring):
P(Tl PIi D xl )óS(ai D xl )ó[(1ñp1(xl )) . . . (1ñpi(xl ))]
(2)
(iii) Based on equations (1) and (2), the likelihood function for the life table
with explanatory variables is given by:
k
Ló< < (pi(xl ))*li (1ñpi(xl ))(1ñ*li )
(3)
ió1 léRi
where pi(xl ) is the probability for the l-th subject failing at ai , given it survived until
aiñ1 in the presence of the explanatory variables xl , for all values of l.
Equation (3) corresponds to the likelihood function for variables with Bernoulli
distribution, once *li is a binary variable with probability pi(xl ) (that is, either
subject fails or not in the interval Ii ). The probability pi(xl ) can be modelled using
Cox proportional hazards or the logistic model.
2.1 Cox proportional hazards model
The Cox proportional hazards function for the lifetime T and the explanatory
variables xl is given by:
h(t D xl )óh0(t) exp(@xl )
and the survival function is:
t
S(t D xl )óexp ñ
h(u D x)du ó[S0(t)]exp(@xl )
0
(4)
where S0(t) is the baseline survival function.
Modelling pi(xl ) according to equation (4), we have:
pi(xl )ó1ñ
S0(ai )
S0(aiñ1)
exp(@xl )
(5)
Replacing pi(xl ) obtained from equation (5) in (3), the likelihood function for
life table data with explanatory variables xl can be written by:
k
L(,)ó< < [1-iexp(@xl )]*li [iexp(@xl )]1ñ*li
(6)
ió1 léRi
where i óS0(ai )/S0(aiñ1).
The parameter estimation is made by maximizing the log-likelihood function
(6), which is given by:
k
ln L(, )ó ; ; [ñ(1ñ*li ) exp(ci ò@xl )ò*li ln(1ñexp(ñexp(ci ò@xl )))]
ió1 léRi
where ci óln(ñln(i )), as suggested by Prentice & Gloeckler (1978).
40
J. E. Corrente et al.
2.2 Logistic model
According to the logistic model, we have:
pi(xl )ó1ñ(1òci exp(@xl ))ñ1
(7)
where ci ópi(0)/(1ñpi(0)), ió1, . . . , k.
Then, the likelihood function can be written as:
k
L(, )ó< <
ió1 l é Ri
ci exp(@xl )
1òci exp(@xl )
*li
1
1òci exp(@xl )
1ñ*li
A reparametrization ai óln(ci ) can be made as suggested by Lawless (1982), in
order to obtain improved convergence and the normal approximation. Again, the
parameters are estimated by maximizing the log-likelihood of the reparametrized
function, given by:
k
log(L(, ))ó ; ; *li (ai ò@xl )ñlog(1òexp(ai ò@xl ))
ió1 l é Ri
2.3 Fitting models using generalized linear models
Generalized linear models provide a simple way to fit data with interval censoring
to discrete models. Whitehead (1989) fitted the Cox proportional hazards model
for interval-censored data making the response variable *li a censoring indicator,
which can be considered as a binary response with the complementary log-log link
function.
Then, applying the complementary log-log transformation in equation (5),
we have:
ln(ñln(1ñpi(xl )))ó@xl òln ñln
S0(ai )
S0(aiñ1)
or
ln(ñln(1ñpi(xl )))ó@xl òci ó@xl òc*i ógli
(8)
where c*i óci óln(ñln(S0(ai )/S0(aiñ1))) corresponds to the effect of the interval
time, that is, these parameters are incorporated in the model to fit the correspondent
terms to the k-levels of the factor associated with the interval time.
To fit a logistic model, the response variable is also the indicator of failure *li ,
and the link function is the logit.
The logistic transformation applied in equation (6) yields:
ln
pi(xl )
óln(ci )ò@xl óc*i ò@xl ógli
1ñpi(xl )
(9)
where c*i óln(ci ) corresponds to the effect of the time interval.
Becker & Melbye (1991) described how a loglinear model can be used to
calculate the non-parametric maximum likelihood estimate for the survival function
for interval-censored data.
Colosimo et al. (2000) have proposed a score test to discriminate between the
Cox proportional hazards and logistic models
41
two discrete models based on the family of asymmetric Aranda-Ordaz (1981)
transformations, written as:
Vj (pi(xl ))ó
ln
(1ñpi(xl ))ñj ñ1
j
ln(ñln(1ñpi(xl )))
for jÖ0
for jó0
Then the model to be fitted to pi(xl ) is:
pi(xl )ó
1ñ(1òj exp(gli ))ñ1/j
for j exp(gli )[ñ1
1
for j exp(gli)Oñ1
(10)
where the particular cases of link functions are: logit, gli óln[pi(xl )/(1ñpi(xl ))],
when jó1 and complementary log-log, gli óln[ñln(1ñpi(xl ))], when j0.
In order to test jó0 or jó1, the score test is preferable once one can no longer
estimate j as in the likelihood test.
The statistics of the score test will be obtained from the two interest hypotheses:
H0 :jó0 for the Cox model and H0 :jó1 for the logistic model.
The log-likelihood of the function (3) is:
k
l(h)ó ; ; (*li ln(pi(xl ))ò(1ñ*li )ln(1ñpi(xl )))
ió1 l é Ri
where pi(xl ) is fitted according to equation (10);
hó(j, , *)@ is a vector of dimension 1òpòk;
gli óc*i ò@xl ,
and
c*i ó
ci
for jó0
ln(ci )
for jó1
for ió1, . . . ,k.
Then, the score function is given by
Uj (h)ó
Ll(h)
Lj
Under the hypothesis H0 :jó0 (Cox model), we have:
Uj (h0 )óUj (0, , *)
(1ñ*li )exp(exp(gli ))ñ1
1 k
(exp(gli ))2
óñ ; ;
2 ió1 l é Ri
1ñexp(exp(gli))
and the statistic score test for this hypothesis is
Sró
U j2 (ĥ0)
Ijj (ĥ0 )ñ1
42
J. E. Corrente et al.
where ĥ0 ó(0, ˆ , ˆ *) are the MLE of h0 ó(0, , *) and Ijj (ĥ0 )ñ1 is the observed
information matrix.
Similarly, the statistic of the score test for the logistic model can be obtained
considering the hypothesis H0 :jó1, and it is given by:
Sr1 ó
U j2 (ĥ1 )
Ijj (ĥ1 )ñ1
where ĥ1 ó(1, ˆ , ˆ *) are the MLE of h1 ó(1, , *) and Ijj (ĥ1 )ñ1 is again the
observed information matrix.
In addition, it was shown that Sr and Sr1 have an asymptotic chi-square
distribution under the null hypothesis with one degree of freedom.
3 Bootstrapping
3.1 The bootstrap method
The bootstrap technique was introduced by Efron (1979) as a computationally
intensive method in order to estimate the standard error of parameter h. Basic and
practical applications of the bootstrap method can be found in Hall (1992), Efron
& Tibishirani (1993) and Davison & Hinkley (1997). An approach to the theoretical
properties of this method can be better seen in Shao & Tu (1995). The asymptotic
properties and some regular conditions were first discussed in Bickel & Freeman
(1981).
Among many applications of the bootstrap, there is a particular interest in
regression methods. The main point is related to the form of bootstrapping, since
if it is made consistent with the adopted model, the obtained asymptotic result
must be the same as that obtained by the classical methods. Then, for the regression
model yóXòe, where y is the response variable, X is the matrix of the
explanatory variables and is the parameter vector, and the sampling with
replacement from suitably standardized residuals provides a model-based simulated
dataset. The studentized residuals are pointed out by Davison & Hinkley (1997)
as the best ones.
The bootstrap method approach can also be made for generalized linear models.
In this way, the Cox and the logistic fit can be obtained by simulating residues. A
non-nested simulated model is used by Hinde (1992). This problem consists of a
bootstrap selection, in which two models Mk and Mq , with independent parameters
k and q, for instance, can be fitted and the Lk ñLq , which corresponds to the
likelihood ratio statistic, is used as a discrepancy measure between the two models.
Another discrepancy measure that can also be used is the deviance differences
between the two models, which can have different specifications for different errors
and link functions.
Therefore, in order to fit the Cox proportional hazards and logistic models for
interval-censored data using X as a matrix of explanatory variables, errors were
calculated in relation to the fitted value gli corresponding to the proportional
hazards model given in equation (8) and the proportional odds given in equation
(9), where
róñˆ ó(IñX([email protected][email protected])
Cox proportional hazards and logistic models
43
and the new values of are obtained by
*óyòr*
where r* is calculated by bootstrapping the errors r.
For this case, taking into account the presence of the explanatory variable and
possible leverage values and asymmetry in the residues, we take the standardized
Pearson residuals, given by
rPli ó
gli ñĝli
{cli îV (ĝli )(1ñhli )}1/2
ló1, . . . , n; ió1, . . . , k where the hli’s are the elements of the X([email protected][email protected] and
cliîV (ĝli ) is the variance function associated with the fitted model.
The sampling with replacement from suitably standardized Pearson residuals
was elaborated and, in order to measure the discrepancy between the two studied
models, the deviance differences under each model were used. An S-Plus program
was built for the bootstrap simulation. This is available from the author.
4 Applications
The soil ability to limit disease severity despite a high pathogen inoculum density
has been defined as soil disease suppressiveness. The soil suppressiveness to diseases
induced by soil-borne plant pathogens is generally assessed by bioassays performed
under standardized climatic conditions. The soil is infested with increasing doses
of pathogen, then seeds of the susceptible host plants are sown and the plants are
allowed to grow. The number of plants affected by the disease is regularly recorded
with time.
Höper et al. (1995) used this procedure to compare three soil–clay mixtures and
natural soil when infested by the pathogen Fusarium oxysporum. A sample of 286
susceptible flax cultivars was randomly selected for the three treatments (different
soil–clay mixtures) and natural soil. Each cultivar status (alive or dead) was
evaluated twice a week during the period of study (52 days). A great number of
ties occurred at each evaluation time justifying again the appropriateness of discrete
models.
In order to choose between the two cited models, the procedure to be followed,
considering Mc and Ml as the proportional hazards and the logistic models,
respectively, will be:
Step 1. Fit the models Mc and Ml with the original dataset and calculate the
deviance differences between the two models, given by:
dobs óDc ñDl
where Dc and Dl are the deviances referring to the two fitted models.
Throughout the estimated parameters of the initial fit of Mc, the errors
are obtained and a bootstrap simulation is made B times, obtaining B
bootstrap residues. With these B bootstrap residues, new bootstrap
observations are generated and, then, fit again to the models Mc and Ml ,
calculating the deviance differences given by:
dcb óDc ñDl
for bó1, . . . , B.
44
J. E. Corrente et al.
Step 2. Analogously, throughout the estimated parameters of the initial fit of Ml ,
the errors are obtained and a bootstrap simulation is made B times,
obtaining B bootstrap residues. With these B bootstrap residues, new
bootstrap observations are generated and, then, fit again to the models
Mc and Ml , calculating the deviance differences given by:
dlb óDc ñDl
for bó1, . . . , B.
The value of dobs is then compared with the simulated datasets {dcb} and {dlb} in
order to point out the evidence of one model in relation to the other. This evidence
was verified by the histograms shown in Fig. 1 and by calculating the significance
levels, given by:
significance leveló
1 B
; I(dcb [dobs ) for the proportional hazards model
B bó1
significance leveló
1 B
; I(dlb [dobs ) for the proportional odds model
B bó1
where I(.) is an indicative function, and the values of dobs are considered positive.
If the value of dobs is negative, just change the sign of the inequality in the indicative
function.
F. 1. Histogram of the deviance differences from 1000 bootstrap simulations considering standardized
Pearson residues under Cox and logistic models for the dataset of the flax cultivar.
Cox proportional hazards and logistic models
45
T 1. Analysis of deviances obtained when fitting the Cox and
logistic models for flax cultivar data
Deviances
Sources
d.f.
Interval
Substratum
Residual
8
3
23
Cox
Logistic
188.427**
17.378**
20.089
188.427**
18.253**
19.214
** Significant at 1%
T 2. Estimated values for the interval and substratum in the Cox and logistic
models
Cox
Parameters
c*1
c*2
c*3
c*4
c*5
c*6
c*7
c*8
c*9
a*2
a*3
a*4
Estimate
ñ3.1356
ñ1.8503
ñ1.6755
ñ1.1102
ñ0.8603
ñ0.1089
ñ0.4282
0.2280
0.2118
ñ0.5342
ñ0.1758
ñ0.6375
Logistic
S.E.
0.3455
0.3799
0.3784
0.3655
0.3663
0.3594
0.3931
0.3942
0.4831
0.1748
0.1735
0.1734
Estimate
ñ3.0650
ñ1.7335
ñ1.5489
ñ0.9108
ñ0.6244
0.2950
ñ0.1052
0.7773
0.7513
ñ0.6327
ñ0.2075
ñ0.7845
S.E.
0.3567
0.3902
0.3898
0.3787
0.3825
0.3851
0.4265
0.4577
0.6002
0.2092
0.2088
0.2080
Then, fitting the models Mc and Ml , the residual deviances were 20.089 and
19.214, respectively, as shown in Table 1. In this way, we can observe a significant
effect on the interval and substratum for the two models and the value of the
observed deviance differences is dobs ó0.87. The parameter estimates and their
respective standard errors are given in Table 2. In order to compare the fit, the
Akaike Information Criteria (AIC) was calculated for both models and the values
were 159.2 and 160.1 for the Cox and logistic models, respectively. These two
values are very similar, thus showing that both models can be considered reasonable,
which indicates that one is not better than the other.
In spite of the similarity of the deviances and the AIC for the two considered
models, it is also interesting to elect one of the models as the one that fits the
dataset better. Then, calculating the standardized Pearson residuals for the two
models, 1000 bootstrap replications of these values were made, calculating the new
bootstrap observations, and supposing the Cox model to be the true model. After
that, the same was done for the logistic model. The distribution of the deviance
differences is given in Fig. 1.
The significance level for each fit was:
significance level for Mc ó0,186
significance level for Ml ó0,313
46
J. E. Corrente et al.
The histograms and the significance levels clearly indicate that the observed
deviance differences are consistent with the logistic model rather than with the
proportional hazards, which agrees with the deviance of the model fit and the AIC
values.
By observing the interval estimates, the interval [49;52) can be said to have
presented the largest effect on the probability of a plant not blighting, that is, in
this interval (1ñpi(xl )) is smaller than the others in all substrata. As for soil type,
it was verified by an approximated Student-t test for the substratum parameters,
to the level of 5%, that the plants placed in natural soil have smaller resistance to
blighting caused by the studied pathogen than the plants in the other substrata.
Conclusions
Through the obtained results, it is concluded that the bootstrap method used,
based on the standardized Pearson residuals for data with interval censoring,
provided similar results considering the deviance of the model fit and the AIC,
when choosing between the Cox proportional hazards and the logistic models,
which suggests that the logistic model should be fitted to the blighting occurrence
data in the flax cultivar susceptible to the pathogen Fusarium oxysporum planted in
four substrata.
Furthermore, other types of residues can be used for the bootstrap method by
taking into account more specific data characteristics.
To that end, tests are being developed in order to compare the bootstrap method
and verify if the use of other residues also leads to similar results.
References
A-O, F. J. (1981) On two families of transformations to additivity for binary response data,
Biometrika, 68, pp. 357–364.
B, N. G. & M, M. (1991) Use of log-linear model to compute the empirical survival curve
from interval censored data, with application to data on tests for HIV positivy, Australian Journal of
Statistics, 33, pp. 125–133.
B. J. & F, D. A. (1981) Some asymptotic theory fo the bootstrap, The Annals of Statistics,
9, pp. 1196–1217.
B, N. (1972) Contribution to discussion of paper by D. R. Cox, Journal of the Royal Statistical
Society B, 34, pp. 216–217.
B, N. (1974) Covariance analysis of censored survival data, Biometrics, 30, pp. 89–99.
C, E. A., C, L. V. A. S. & D́, C. G. B. (2000) Tests of proportional hazards
and proportional odds models for grouped survival data, Biometrics, 56, pp. 1233–1240.
C, D. R. (1972) Regression models and life-tables (with discussion), Journal of the Royal Statistical
Society B, 34, pp. 187–220.
D, A. C. & H, D. V. (1997) Bootstrap Methods and their Application (Cambridge, Cambridge
University Press).
E, B. (1977) The efficiency of Cox’s likelihood function for censored data, Journal of the American
Statistical Association, 72, pp. 557–565.
E, B. (1979) Bootstrap methods: another look at jackknife, Annals of Statistics, 7, pp. 1–26.
E, B. & T, R. J. (1993) An Introduction to the Bootstrap (New York, Chapman & Hall).
F, V. T. & P, R. L. (1980) The approximation of partial likelihood with emphasis on
case-control studies, Biometrika, 67, pp. 273–279.
F, D. M. (1986) A proportional hazards model for interval-censored failure time data,
Biometrics, 42, pp. 845–854.
H, P. (1992) The Bootstrap and Edgeworth Expansion (New York, Springer).
H, J. (1992) Choosing between non-nested models: a simulation approach. Technical Report,
University of Exeter (Unpublished).
Cox proportional hazards and logistic models
47
H̈, H., S, C. and A, C. (1995) Importance of physical and chemical soil
properties in the suppressiveness of soils to plant diseases, European Journal of Soil Biology, 32,
pp. 41–58.
L, J. F. (1982) Statistical Models and Methods For Lifetime Data (New York, John Wiley & Sons).
N, J. A. & W, R. W. M. (1972) Generalized linear models, Journal of the Royal
Statistical Society, Series A, 135, pp. 370–384.
P, R. (1972) Contribution to discussion of paper by D. R. Cox, Journal of the Royal Statistical
Society, B, 34, pp. 205–207.
P, R. L. & G, L. A. (1978) Regression analysis of grouped survival data with
application to breast cancer data, Biometrics, 34, pp. 57–67.
S, J. & T, D. (1995) The Jackknife and Bootstrap (New York, Springer).
StatiSci (1993) S-Plus for Windows—User’s Manual (Seattle, StatiSci).
W, J. (1989) The analysis of collapse clinical trials, with application to a comparison of two
ulcer treatments, Statistics in Medicine, 8, pp. 1439–1454.

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