The recent statistical literature attests to a considerable

current interest in specialised statistical methods for

the analysis of time to occurrence or failure time data in

medical research.

Frequently the primary objective of a failure time Study

concerns with the association between certain co-variates

Z = (Z1, ..., Zp) and the time T > 0 to the occurrence of a

certain event. For example a clinical study may be designed to

compare several treatment programmes in respect to the time T to

recurrence of a disease or response to treatment of a disease.

The regression vector Z would include indicator components for

treatment as well as other prognostic factors

Prabhakar, R

Venkatesan, P

Viswanathan, K

NIRT Institutional Repository

Technical Report - 7/87AN APPLICATION OF COX REGRESSION MODEL TOTHE ANALYSIS OF GROUPED PULMONARYTUBERCULOSIS SURVIVAL DATAP. VENKATESAN*K. VISWANATHAN+R. PRABHAKAR**Tuberculosis Research Centre,I.C.M.R., MADRAS, INDIA.+ Department of Statistics,University of Madras;MADRAS, INDIA.December 1987.AN APPLICATION OF COX REGRESSION MODEL TO THEANALYSIS OF GROUPEDPULMONARY TUBERCULOSIS SURVIVAL DATAP. VENKATESANDepartment of StatisticsTuberculosis Research CentreI.C.M.R., MADRASandK. VISWANATHAN R. PRABHAKARDepartment of Statistics Tuberculosis Research Centre, (ICMR)University of Madras MADRASMADRAS.1. INTRODUCTION:The recent statistical literature attests to a consider-able current interest in specialised statistical methods forthe analysis of time to occurrence or failure time data inmedical research.Frequently the primary objective of a failure time Studyconcerns with the association between certain co-variatesZ = (Z 1, ..., Zp) and the time T > 0 to the occurrence of acertain event. For example a clinical study may be designed tocompare several treatment programmes in respect to the time T torecurrence of a disease or response to treatment of a disease.The regression vector Z would include indicator components fortreatment as well as other prognostic factors.The distribution of failure time T can be represented inthe usual manner in terms of density or distribution functionsas well as in more specialised ways such as the hazard function.- 2 -If the underlying process is continuous, hazard function attime t among individuals with covariates Z is defined ash (t, Z) = lim¶ tP[t < T < t + ¶ t/T> t, Z]/ ¶ t (1.1)-> 0The hazard function gives the risk of failure at any time giventhat the individual has not failed prior to t. Since thenotion of failure rate is basic and conceptually simple itprovides a starting point for modelling the associationbetween Z and t.2. COX REGRESSION MODEL:One such model introduced by Cox (1972) provides thebasis for most of the discussion of this paper. This modelpresumes that covariatos affect the hazard function in a multi-plicative manner according toh(t ; Z) = ho(t) cZ b(2.1)where h o(t) is an unspecified function of time, b is a columnvector of the regression coefficients and the factor ebdes-cribes the risk of failure for an individual with regressionvariables Z relative to that of starting value Z = 0. Sincethe ratio of hazard functions corresponding to any two Z valuesis a constant it is often referred to as Proportional Hazard(PH) Model.- 3 -Cox model is a particular case of the general model thathas hazard functionh(t; Z) = ho g ( Z , b ) (2.3)The Cox model is also equivalent to the assumption of Lehmannalternatives1 - F(t; Z) = [l-F(t,O)]exp Z  b(2.4)The PH model provides a unified approach to inferences froma variety of typos of failure time data.3. ESTIMATION OF b:The approach taken here is a distribution free one whereno specific form is assumed for h o(t). The main idea forthis originated with Cox (1972) who proposed a method forestimating b in the absence of the knowledge of h o(t) and alsoestimating for h o(t).Consider a random sample of n individuals with k distinctobserved lifetimes and (n-k) censored times, let the k observedtimes bet(l) < t(2) < ... < t(k)and R i  = R(t(i)) risk set at time t(i)The likelihood function for estimating b in the absence ofknowledge of h o(t) is(3.1)- 4 -where Z(i) is the regressor vector associated with the indi-viduals observed to fail at t(i)(Cox 1975). The motivationis the probability an individual being fail ish(t, Zi) = h(t, z l  )= ez b/  S eZl bl Î R(t)(3.2)The likelihood is formed by the product of all such factorsover the k uncensored lifetimes. The estimate b ^ of b isobtained by solving the set of equationsThe another method used for estimating b isk Si bL (b) = p [ei=l/ ( S eZl  b) di ]l Î Ri(3.4)where di is the number of lifetimes equal to t(i) and S iis the sum of the regression vector Z for these di individuals.4. STRATIFICATION:When a factor does not affect the hazard multipli-catively, stratification may be useful in model building.Hazard function for an individual in stratum f with regressorvariable Z ish(t/Z) = hoj (t) eZ b- 5 -That is individuals in the same stratum have the same proportionalhazards function but it is not necessarily the case for indi-viduals in different strata. It is also assumed that therelative effect of the regressor variables is the same ineach stratum. This condition sometimes needs to be relaxedwith b varying from stratum to stratum. A partial likelihoodfunction L j  (b) is obtained for each stratum and then theoverall partial likelihood function for b isL (b) = L i  (b) ... L s (b) (4.1)After estimating b by maximising L (b) the survivor functionscan be estimated bysoj (t) = exp [- ò t hoj (u) du] j = 1,2, . . . , s (4.2)05. PARAMETRIC MODELS:If h(t, Z) is specified upto a finite number of para-meter q, ordinary parametric likelihood methods can be applied.However, the presence of censoring except in a few extremelyspecial cases precludes the possibility of tractable exactdistribution theory. One parametric model that has foundextensive application is the Weibull Model is given byho(t) = a p (a t)p-l (5.1)This model is the intersection of PH and AFT models. Weibullis characterised by that model in which the regressor variables- 6 -act multiplicatively both upon hazard rate and upon failuretime. The asymptotic, theory is readily applied to the Weibullmodel with q = (a, p, b). The Weibull model can be written aslog linear form asy + Z b = d + s vwhere y = log t, d = - log a, s = p-1, b ' = p-1  b.The error variable v has an extreme value density(5.2)f(v) = exp(v - e v). (5.3)Other parametric cases include lognormal, log logistic andgeneralised gamma regression model corresponding to errorquantities v that are normal, logistic and log gamma res-pectively. On fully parametric set up the precise conditionsof censoring mechanism, regressor variables and the form ofh(t, Z) that would be sufficient to ensure modelling is notyet known fully.In studies involving human subjects no theoretical formfor survival distributions can be assumed and in such situat-ions techniques based on PH model appear to be superior toother models. The relative efficiency of Cox regression modelcompared with likelihood methods that assume full parametricform for the hazard function is generally good. (Breslow 1974,Kalfleisch 1974, Efron 1977, Oakes 1977).- 7 -6. A MODEL BASED ON GROUPING DATA FROM COX MODEL:In many instances the life time T of individuals withregressor variable Z might be assumed to come from a continu-ous proportional hazard model. If the life times are groupedinto k classes, then a group regression model is obtained forwhich survival probability for the i-th interval [a i , a i+1)is given byPi (Z) = S(a i /Z) = S o(a i )exp(Z b) = Pi (O)exp(Z b)(6.1)i = 1, 2, ..., k with P O(O) =0.pi (O)where p i (O)=     i = 1, 2, . . . , k.pi-1 (O)Use of this model produces a likelihoodp k [ pi=l           l Î Di(lp i (Z l ))l Îp Ri  -D i  p i (Z l )] (6.2)which can be used to estimate  b and p 1, p 2, ..., pk. Thisapproach was first introduced by Kalbfleisch and Prentice(1973). The m.l.e. for (6.2) is discussed by Prentice andGloeckler (1978). The log likelihood can be written aslog L (b, p) = p k [ S log1- p iexp(Z1 b)i=l       l Î Di exp (Z 1 b)+ S log (p iexp(Z l  b)  p il Î Ri(6.3)Prentice and Gloeckler suggest using the parametersg i  = log (- log  p i ) i = 1, 2, ..., k. ( 6 . 4 )- 8 -The g i  are unrestricted and they found that convergence in aNewton-Raphson iteration procedure was improved when g = (g i  ... g k)was used in the place of p.7. APPLICATION TO PULMONARY TUBERCULOSIS DATA:The studies at the Tuberculosis Research Centre, Madrashave shown a negative relationship between time to response andthe covariates Extent of Disease, Extent of Cavity and sputumculture positivity on admission (TRC 1966). Differentialtreatment methods between lightly and heavily diseased patientscould distort the magnitude of relationship between two treat-ment groups. The estimation of joint relationship of Time toresponse with each of the covariates Extent of Disease, Cavityand culture Positivity not only gives the potential for makingdetailed and accurate assessment but also may provide valuableinsight into the response mechanism. Of particular interest interms of the response mechanism is the ability of Relative Risk(RR) models that are multiplicative or additive to describedata. In general terms a multiplicative RR model would corres-pond to a process where in an intial exposure induces a fract-ional increase in the age specific failure rate which then servesas a new base line rate that the RR function for the secondexposure variable multiply.- 9 -8. METHODS AND MATERIALS:Tuberculosis Research Centre (ICMR) Madras is conductingstudies since 1956 in pulmonary and extrapulmonary tuberculosispatients to assess the therapautic effects when drugs are admin-istered to the patients. Various tests have been carried out toassess the eligibility of patients to the study before admis-sion and periodically to assess the response to treatmentsafter admission. The assessment of response to treatment isbased on clinical, radiological, bacteriological, biochemical,immunological and other investigations. The patients arefollowed up to a maximum period of 5 years.Our study population consists of 261 patients takenfrom a study conducted at the Tuberculosis Research Centre,Madras (TRC 1983). The patients belong to 2 treatment groupsof a controlled clinical study of 3 short course regimens.Out of the two group one contained rifampicin (R/7 series) andthe other no rifampicin (Z/7 series).R/7 Series: Daily chemotherapy for 2 months with rifampicin (R)12 mg/kg body weight plus streptomycin sulphate (S)0.75 g plus isoniazid (H) 400 mg (incorporatingPyridoxine 6 mg) plus pyrazinamide (Z) 40 mg/kgfollowed by twice weekly chemotherapy for 5 monthswith streptomycin 0.75 g plus isoniazid 15 mg/kgplus pyrazinamide 70 mg/kg the total durationbeing 7 months.- 1 0 -Z/7 Series: The same as the R/7 series but without rifampicinthe total duration being 7 months.Of the 261 patients (132 R/7, 129 Z/7) 67% were males,24% of the patients were less than 25 years of age, 63% werebetween 25 and 44 and 13% were greater than 45 years. Thepatients in the two groups were almost similar with respect toage, sex and weight. Response to treatment times were availablein monthly intervals. Over 30 disease characteristics woreavailable. A regression vector of 16 components is used toillustrate the above procedures. The components used are asfollows:1. Age - 3 components (1 for < 25 years, 2 for 25-45, 3 for > 45)2. Sex - 2 components (1 male, 2 female)3. Extent of Disease (EOD)- 3                    (BODI, EOD2, EOD3)components4. Extent of Cavity (BOC) - 3 components (EOC1, EOC2, EOC3)5. Sputum Culture (CUL) - 3 components (CULl, CUL2, CUL3).An additional indicator variable distinguishing twotreatment groups taking value 0 for Z/7 series and 1 for R/7series is considered. The final component is another indicatorvariable for patients response time is censored (0) or un-censored (1).The radiographic and bacteriological findings onadmission to treatment were given in Table-l.- 11 -TABLE-1---------------------------------------------Percentage of PatientsCharacteristic Levels ------------------R/7 Z/7---------------------------------------------EOD EOD1 29 27EOD2 39 39EOD3 32 34EOC EOC1 43 39EOC2 35 30EOC3 22 31CUL CULl 42 44CUL2 43 38CUL3 15 18The above variables are related to the hazard functionto response time to treatment for tuberculosis using the PHmodel. A partial likelihood (Cox 1975) and Newton Raphsoniteration is applied to the data which gives a maximumpartial likelihood estimates for the corresponding regressioncoefficients as given in Table-2. Many refinement are necessarybefore claiming any association.From Table-2 we see that the estimated regressioncoefficient of R/7 series b = - 0.344 with an estimatedstandard error of b from the observed information matrix is0.117, giving a T-value of 2.94 which is significant at 0.02level. This suggests that the response time is reduced by an- 12 -estimated multiplicated factor exp (b) = 0.71 in comparisonto Z/7 series. The other estimates of b for the covariatesEOD, EOC and CUL are not significant. But we observe fromthe table that the b 's increases as we move from the lowerlevels to the higher levels of all the characteristics.For example the duration of response is prolonged by afactor exp (.083) = 1.09 for a patient with slight orlimited disease (EOD1). The factors for moderate (EOD2)and extensive (EOD3) disease are 1.29 and 1.39 respectively.For a patient with 2 or more covariates the duration willbe prolonged by exp (sum of the b 's). For example theduration of response to treatment for a patient havingmoderate disease (EOD2), extent of cavity (EOC2) and cultureresult 3 + (CUL3) will be prolonged by a factorexp (.173 + .238 + .112) = 1.67. If this patient istreated with R/7 series the duration will be reduced bya factor 0.71 in comparison to a patient in Z/7 series.TABLE-2M.L.E. FIT OF COX MODELS TO DATA ON 261 PULMONARY TUBER-CULOSIS PATIENTS.-----------------------------------------------------Covariate b SE (b) Sig.---------------------------------------------------------EOD1 0.083 0.101 NSEOD2 0.173 0.210 NSEOD3 0.255 0.314 NSEOC1 0.073 0.100 NSEOC2 0.238 0.348 NSEOC3 0.255 0.368 NSCUL1 0.026 0.151 NSCUL2 0.068 0.250 NSCUL3 0.112 0.254 NSR/7 0.344 0.117 0.01--------------------------------------------------NS = Non-Significant- 13 -9. DISCUSSION:An important point particularly with strong regres-sion effects concerns the appropriateness of PH assumption.One possibility of t esting proportionality is to stratifythe data and fit separate PH models for each stratum specificplots of log (-log p(t,Z) at specified Z values should thenthe separated by approximately a constant difference overtime if the proportionality assumption is appropriate.More formal procedures arc available by introducing timedependent covariates into the model and testing for b = 0.The starting point is usually to obtain estimators for thevalue of b which will indicate whether individual covariatesinfluence prognosis, whether the effect is favourable orunfavourable and the magnitude of the effect. The standar-dised component of b can be aged to determine whether thecorresponding covariate has a statistically significanteffect. A significant b i indicates that a highly significantprognostic factor is Z i . The value of b i  are used toestimate the effect of a particular covariate.A number of parametric models have been proposed manyof which arc unified in Farewell and Prentice (1977). LeastSquare Models arc proposed by Miller (1976). The later modelsare principly linear in logarithm of failure times. Log linearmodels are equally good.- 14 -10. REFERENCES :1. BRESLOW, N.E. (1974): Biometrics 30, 89-100.2. COX, D.R. (1972) : J.R.S.S., B-34, 187-200.3. COX, D.R. (1975) : Biometrika 62, 269-276.4. EFRON, B. (1977) : JASA, 72, 555-565.5. FAREWELL, V.T. and PRENTICE, R.L. (1977): Technometrics19, 69-75.6. KALBFLEISCH, J.D. (1974): Biometrika 61, 31-38.7. KALBFLEISCH, J.D. and PRENTICE, R.L. (1973): Biometrika60, 267-278.8. MILLER, R.G. (1976): Biometrika, 63, 449-464.9. OAKES, S.D. (1977): Biometrika 64, 441-448.10. PRENTICE, R.L. and GLOECKLER, L.A. (1978): Biometrics,34, 57-67.11. TUBERCULOSIS CHEMOTHERAPY CENTRE, MADRAS (1966): Bull.Wld. Hth. Org. 34, 483-515.12. TUBERCULOSIS RESEARCH CENTRE, MADRAS (1983): TUBERCLE, 64,73-91.REPORT DOCUMENTATION PAGE:--------------- ------ -------------- -------------------------- ---------1. Report No. 7/87  2. Library Catalogue No.------------------------------------------------------------- 3. Title (and Sub-title):  4. Classification/Section: i.'An Application of Cox Reg-  Inferenceression Model to the Ana- ii. Survival Analysislysis of Grouped Pulmonary Tuberculosis Survival Data' 5. Authors:  6. Contract/Sponsoring Agency  P. VENKATESANK. VISWANATHAN andR.PRABHAKARUniversity of Madras       ------------- ------------------------ ----------------------------- 7.----- Performing Organisation  8. Name and Address:Type of Report:Department of Statistics       Research PaperUniversity of MadrasMadras 600005----------------------------------- -------------------------- ---------- 9. Report Date: 10. Number of Pages:December 1987 14 11. Key-words:Covariates - Hazard function - PH Model - AFT Model -Partial Likelihood - Survival Analysis - Clinical Trial   - Prognostic factor - Pulmonary Tuberculosis - NewtonRaphson Iteration Method. 12. Short Abstract:The Technical Report is based on the paper presentedat the 5th Conference of Indian Society for MedicalStatistics hold at Srinagar in September 1987. TheCox model and its grouped version are applied to thetuberculosis data to explain the treatment differences  by disease status and other demographic characteristics   at diagnosis.Place: MADRAS 600005Date : DECEMBER 1987.(K.N. PONNUSWAMY)Professor and Head,Dept. of Statistics,University of MadrasMADRAS 600005.

An application of Cox regression model to the analysis of grouped pulmonary tuberculosis survival data

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An application of Cox regression model to the analysis of grouped pulmonary tuberculosis survival data

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