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Applied Stochastic Models in Business and Industry Updating prognostic indices via regression models
Updating prognostic indices via regression models
John O'quigley, Thierry Moreauএই বইটি আপনার কতটা পছন্দ?
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3
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1987
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english
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10.1002/asm.3150030404
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APPLIED STOCHASTIC MODELS AND DATA ANALYSIS, VOL. 3, 227236 (1987) Biology, biometrics, biomedicine and health care UPDATING PROGNOSTIC INDICES VIA REGRESSION MODELS JOHN O’QUIGLEY’ Fred Hutchinson Cancer Research Center, Division of Public Health Sciences, 1124 Columbia Street, Seattle, Washington 98104, U.S.A. and Department of Biostaristics, University of Washington, U . S . A . AND THIERRY MOREAU Unite 169 INSERM, 16 av. P. V . Couturier, 94807 Villejuif, France SUMMARY In medical applications of survival analysis we are often interested in determining prognosis, and the adoption of statistical models in the establishment of a prognostic index has become common practice. The aim of this presentation is to examine the problem of reevaluation or the modification of any such index in the light of information subsequent to the initial assessment. Examples are given to highlight the approach. KEY WORDS Survival models Regression Survival prediction Stochastic processes Prognostic indices 1. INTRODUCTION The use of regression models in the establishment of a prognostic index is to inject a degree of formalism into the problem while making maximum use of incomplete data sets (i.e. those containing censored observations). Thus the weighting of any prognostic variable is decided mechanically in the framework of a model which stipulates how survival probabilities depend on explanatory variables x, the weights depending also upon the particular explanatory variables being considered at a given time. Accounting for relationships and interrelationships between the explanatory variables themselves and survival is considered to be the main advantage of such a multivariate approach. However, if we reexamine the usual underlying statistical formulation of the problem, we see the emphasis put on the idea of conditional probability. Thus analysis frequently begins by looking at the hazard function h(t,x) for the random variable T, defined loosely by h(t,x),At=pr(t < T < t + A t l T > t; X = x ) the appro; ximation becoming more exact as At approaches zero. Subsequently prognostic indices will be evaluated after consideration of S ( t , x), the survivorship function, defined by S(t, x) = pr(T > t I X = x ) The two functions are of course mathematically interchangeable, in all relatistic cases, in as much as given one we can obtain the other. However, the essentially ‘smoothing’ role of the * On leave from Unite 292 INSERM, France 87550024/87/040227 10$05.OO 0 1987 by John Wiley & Sons, Ltd. Received 7 November 1986 Revised 24 March 1987 228 J. O’QUIGLEY AND T. MOREAU second over the first can hide interesting phenomena in practice. For a number of situations, as time goes by, the prognostic outlook may change considerably. Overall survival masking much of interest, a study of conditional survival can be revealing. An interesting example’ in which the chance of conception in natural reproduction varies in an unanticipated way with elapsed time highlights this important point. It might be considered helpful to regard such phenomena as the simple intervention of the time element into the equation, a less simple intervention occurring when the prognostic weights given to certain components of x change. This is particularly likely to be the case in longterm studies where the simple fact of having survived a certain time, given, for instance, an initially unfavourable prognosis, suggests reevaluation. Section 2 examines the problem of updating using the proportional hazard model, modified in a simple way to accommodate timevarying prognostic weights. A third and closely related situation concerns prognostic variables which themselves are timedependent. Standard approaches to this problem which rely on stochastic models for the covariates are discussed in Section 3. Sections 4 and 5 examine simple models which can relax the Markov assumption if necessary and lead to simple expressions for survival probabilities conditional upon a hypothesized evolution of the covariates. 2. STANDARD AND PIECEWISE PROPORTIONAL HAZARDS MODELS The most popular model is that suggested by Cox4 which specifies the hazard for the ith subject as h ( t ,xi) = ho(l)exp(BTxi) (1) where x; is a pvector of explanatory variables ( x i l , ...,xip) and B a pvector of fixed but unknown parameters (PI, ...,pp). The function ho(t) is of an unspecified form and corresponds to the hazard for a, possibly hypothetical, subject for whom x = 0. Assuming this function to be constant or a power function of time leads to the well known exponential and Weibull survival models. Note that the model in (1) implies that differences in prognosis as measured by @ do not change with time. The prognoses themselves are allowed to change and to examine this we should look at the conditional survivorship function defined as A ( t , u ) = pr(T> t + u I T > u ) = S ( t + u ) / S ( u ) In the case of model (1) the logarithm of the survivorship function can be estimated by log S(t,,x = €) = exp(fiTE)log S(tr, x = 0) where fi (2) is the maximum partial likelihood estimate of 0 and with do(t)= J t n C 0 i=l [ Y ; ( ~ ) e x p ( f i ~ ~ i )dN(s) ] where the integral is of the LebesgueStieltjes type. The predictable process Y;(t) takes the value one if the ith subject is at risk of failure just before time t and is zero otherwise.6 The UPDATING PROGNOSTIC INDICES VIA REGRESSION MODELS 229 function N ( t ) is the sum of n univariate counting processes for which Ni(t) = ' h(s, x i ) ds + M ( t ) 0 where M i @ )is a local squareintegrable martingale.' Note that this formulation accounts in a mechanical way for tied failure times in the same way as Breslow.' The distribution theory, however, does not permit ties (see, for instance, Andersen and Gill9), so that while, in the presence of light tying, we can feel reasonably comfortable with the approximation given by the asymptotic theory, more severe tying suggests employing a descrete model. Martingale theory has been applied to a discrete model, lo although for our purposes we will assume the continuous model to be a satisfactory approximation. It follows that log A(tl  t,, t,; x = t )= exp(fiTt)IogA(tr r,, tu; x = 0 ) (3) where Act,  t,, tu; x = 0 ) = exp[Ao(t,)  Ao(tr)l Confidence intervals for A ( . ) can be obtained in a straightforward way by a simple * ). These results indicate that modification of the results of Tsiatis" or Bailey" for E [ A ( )] will converge to its population counterpart. However, it may be preferable to work with log a( * ). Following Andersen and Gill9 we see that martingale theory can be directly applied to prove the consistency of log A( . ). Furthermore, the simulation results of LinkI3 suggest that log A ( * ) is preferable to either a( * ) or logit a( * ). These latter quantities involve more linear approximation and this would imply that log [  log A( ) ] , a formulation linear in the parameter estimates, would outperform the other competitors. The variance results for log [  log )] l4 require very minor modification to be applicable to log [  log A( * )] . However, further simulation by Link l5 suggests that these latter variance results may be overconservative at small values of A( * ). More work is clearly needed in order to be able to obtain firmer guidelines, although in most practical applications we would expect a fairly close correspondence between the different approaches. An additional refinement could be introduced by smoothing the hazard function. 16,'' Model (1) can be modified simply to accommodate /3s which are not constant through time. Divide the time axis into r intervals bo, bl, ... ,b,, where bo = 0 and b, = 00, and rewrite the model as s( s(  h(t, xi) = ho(t)exp (B + yh)'Xil for bh1 < t C bh (4) h = 1, ...,r Without loss of generality y l = 0. This model has been considered by Gore et af. in the context of nonproportional hazard effects in breast cancer survival and by Moreau et al. l8 in the context of establishing an alternative hypothesis in testing the proportional hazards assumption. The simple relationship between the loglinear transformation for the hazard function and the corresponding power transformation for the survivorship function no longer holds. Nonetheless, with the aid of a little extra notation, we can find expressions only marginally more difficult to evaluate in practice. In order t o keep the presentation uncluttered we write /3d to denote 0 + yd. Let thj represent the j t h distinct failure time in the hth interval. Then pr(T > f h / I x = [) = S(fh/; x = t ) 230 J . O'QUIGLEY AND T. MOREAU and is estimated by where where 3. STOCHASTIC MODELS The previous section has shown how the simple flow of time can intervene to modify the prognostic outlook. This and the remaining sections consider the handling of explanatory variables which themselves change with time. The most straightforward approach is to divide the explanatory variables into discrete classes creating a total of say k possible disease states. We can imagine the disease process taking the patient through these states in either direction with a state representing death or failure. Transition rates are taken to be exponential for tractability of calculation and the state k + 1 represents death or failure. To make progress it is necessary to make the Markov assumption whereby the probability of transition to the death state depends only on the current state and not on the time spent in that state or on states previously occupied. Some general discussion is given by Chiang l9 and detailed calculations together with an example using cancer markers are given by Kay.20 A similar where time was taken to be discrete so that the probabilities of remaining in a given state need to be estimated, allowed the possibility of passing from any one state to any other without necessarily going via the intermediary stages. A state corresponding to missing information was included and interestingly enough this turned out to be the most dangerous place to be, with an estimated 40% chance of being in the death state at the next turn. The random variable time to death may be decomposed to give, for instance, time to disease progression plus time from disease progression until death. A study of these secondary random variables, socalled auxiliary variables, can lead to a more precise estimate of survival probabilities. 22323 The models used in this work are similar to those above although expressed in the terminology of competing risks. UPDATING PROGNOSTIC INDICES VIA REGRESSION MODELS 23 1 4. PARTIALLY STOCHASTIC MODELS It may sometimes be helpful to suppress the stochastic model for the covariates. This complicates the prediction problem (see Section 5 ) but facilitates the introduction of widerranging models not relying on the Markov property. For instance, prognosis may depend not only on the current but on the preceding value of a covariate. Consider the modified exponential model whereby for the ith subject the hazard function is constant and equal to Aji in the j t h interval. This socalled piecewise exponential model has already proved to be of value in significance testing. For a given interval (or state) the prediction problem requires integrals over the distribution of states as in the previous section. However, the idea of a conditional prognostic index discussed in the next section can provide a simple alternative approach.. Consider now the case where not only the current measure X ; ( j ) (taken to be binary for simplicity) but the previous measure X ; ( j 1) has an influence on A,;. A simple twoparameter model would be (YO + log A l i = PZXi(1) and ( ~ o + l o gA j i = P 2 X i ( j ) + P l [ X i ( j  l )  X i ( j ) l j > 1 (5) Obviously for the model to provide a reasonable description of the data, interval lengths would need to be of similar size. Figure 1 illustrates possible evolutions of the hazard level as a function of time. During the first time interval the subject will be at one of two levels depending on the current state of the covariate. Thereafter there will be only four possible levels determined by the current and the previous measure. Thus the prognosis will get better or worse by steps depending on the evolution of the subject. The symmetry of Figure 1 can be relaxed upon defining olo+log A j i = P 2 X i ( j ) + P l [ X ; ( j  1) X i ( j ) l + P i z X i ( j ) [ X ; ( j  1) X , ( j ) l Working with differences rather than a simple linear combination gives a more meaningful interpretation to the regression coefficients. The likelihood function is of course unchanged. In the majority of practical cases it may be anticipated that a oneparameter model with timedependent covariates or the twoparamater model, incorporating the effect of the previous as well as the current measure, would suffice. For the sake of generality, the likelihood is given in terms of p measures, meaning that as well as the current measure the p  1 previous measures ............................ 1........................ i........................ 1............................ 0 1 2 42 3 time B Figure 1: Possible levels for log X (ignoring constant) in first three time intervals 232 J . O'QUIGLEY AND T. MOREAU have an influence on the instantaneous failure rate. Thus a0 + log A;; = 5 $;(Pl,j) /=c, where and $;(PI, j ) = P I [ Xi(j+ I  P )  (1  S/p)Xi(j+ I  p + 111 a 6 carrying two subscripts being the Kronecker 6, which takes the value one when the subscripts have the same value and zero otherwise. The loglikelihood for the ith subject is log Li = 6; f: h $ ; ( ~ / ,j )~ i a o  eao c H, exp( 9 gi(o/,j)) = ; /=ch 1 /=c, where 6; = 1 if the subject fails in the hth interval and is zero should the subject be lost to followup sometime during this interval. Note that h is a function of i, the subscript being omitted for the sake of neatness. Hj is the length of time spent in the jth interval. Using the notation D;(x) and D?(x,y ) to represent first and second derivatives of the loglikelihood for the ith subject with respect to x , x and y, respectively, we see that D Z ( Pa~o ,) = D i ( P m )  6; 2 a $ i ( ~ /j ), / a P m /=c* For a total of n subjects, using an obvious notation, we have i= 1 ,=I i= 1 The model structure is of the form such that standard maximumlikelihood theory can be applied. Thus, if 0' = (ao,A , ...,p p ) u(e)= a log L/ae I = E [ v(e)uye)l then tests of the null hypothesis HO:8 = 00 can be carried out by referring the score statistic U' (00) I  ' ( ~ o U(0o), ) the maximumlkelihood statistic (8  0 0 ) I(Oo)(8 d o ) or the likelihood ratio statistic  2 log[ L(00)/L(8)]to tables of x 2 distribution with p + 1 degrees of freedom. Standard procedure can also then be used if we wish to test subsets of the null hypothesis or if we wish to estimate some parameters while treating those remaining as nuisance parameters (Rao,24 Chapter 6). UPDATING PROGNOSTIC INDICES VIA REGRESSION MODELS 23 3 Notice that in dividing the time interval in an appropriate way and defining the current measure to be the most recent, then the absence of updated information will automatically nullify the second covariate (and possibly others in the more general model) so that we return to the simple constanteffect model. Second differences as well as differences could be used as well as functions like [ X ; ( j 1)  X ; ( j ) ] which would be measuring instabilities rather than departures in a given direction. It is difficult to be more specific here, since the kind of analysis is going to depend on the example at hand. A balance must be set between the number of intervals we would like to define and the regularity of covariate measurements. Too few intervals will lead to abandoning the parametric power of the method. With many intervals it will probably be necessary to invoke some rule so that a measure holds good for more than one interval. ’ 5 . CONDITIONAL PROGNOSTIC INDICES Rather than integrate over the distribution of future values of the covariate to obtain a probability, a simple alternative is to argue conditionally. Thus we hypothesize different possible evolutions for the covariates and calculate a probability or possibly several probabilities on the basis of what might happen. This would amount, for example, to making a statement of the type ‘if he continues in a stable condition his life expectancy is five years, whereas if he deteriorates within the next three months and shows no further response to treatment then his life expectancy is only one year’. For the model outlined in the previous section we would simply have for the ith subject h A(u2  u l , u1) = j= 1 exp(  h,;Mj) (6) where Mj is the time spent in the j t h interval if a part or all of this interval lies within u1 and u2, otherwise Mj = 0. For different situations, for example, the current measure remaining where it is or changing and subsequently remaining unchanged, a calculation of probabilities is straightforward. Intermediary cases could also be evaluated if considered meaningful. The stepwise exponential model leading to (6) ought to be sufficiently flexible in most practical applications. Otherwise the semiparametric proportional hazards model with timedependent covariates could be used. Expressions similar to those given in Section 2 could then be derived where the covariates rather than the coefficients depend on the time interval. An alternative simpler approach” redefines the intervals so that the end points of these coincide with the failure times in the pilot study. Using an argument paralleling that of Breslow,8 we see that ( c I A (tr  t u , t u ; x = t )= exp J=U mj exp(@Ttj)/ c exp( PT~;,)) i C R, where [ j is the vector of hypothesized future covariate values and the notation x;, is used to denote the value of the covariate vector during the jth interval for the ith subject in the pilot study. R j represents the risk set at t j , at which time there are mj observed failures. Following Kalbfleisch and Prentice’ (p. 125), it is clear that any such conditional prognostic index cannot be given a probabilistic interpretation in terms of long run frequencies. This means model assumptions cannot be checked by comparing probabilities calculated via models with nonmodel (e.g. KaplanMeier) estimates. Nonetheless, formal methods for testing the model in the presence of timedependent covariates, 18926together with the assumption that the 234 J. O’QUIGLEY AND T. MOREAU prognostic behaviour of new patients does not differ dramatically from that of the reference set, will lead to meaningful indices providing overextrapolation is avoided. 6. DISCUSSION AND EXAMPLES The proportional hazards model of Section 2 has been used to evaluate prognosis in gastric cancer as a function of disease severity. 27 Groups 1 and 2 correspond to well defined, although not classically used, stages. The maximum partial likelihood estimate of /3 was 0.84 (SD = 0.1 l), corresponding to a strong prognostic effect. Considering median remaining lifetime and using an obvious notation we had that Al(l0,O) = a2(41,0) = 0.5 so that the prognostic effect at the outset was strongly reflected in the median survival time. However, maintaining the proportional hazards constraint, we found Al(46,30) = a2(66,30)=0.5 Thus a reassessment at 30 weeks gives a differential prognostic outlook very much weaker than that obtained at the outsetand this despite the proportional hazards constraint! The original analysis was undertaken using this constraint, justified by a residual plot,28 a method now known to be lacking in power. 29 A formal test of the proportional hazards assumption using recently developed algorithms 30 was strongly significant. Using the piecewise model of Section 2 with three intervals, (0,30), (30,60) and (60,00), we found that al(8,O)=A2(49,0)=0*5 (51,30) = A2(54,30) = 0.5 These results indicate that the proportional hazards constraint led to an underestimation of the strength of prognostic effects at the origin and to an overestimation of these effects when reassessed at 30 weeks. Indeed the piecewise model suggests that by this time differential effects have all but disappeared. A reexamination of data gathered for a controlled clinical trial of aspirin and dipyridamole in the secondary prevention of atherothrombotic cerebral ischemia31 was undertaken using the model of Section 4. Arterial blood pressure is considered the most important risk factor in stroke and in this study was measured regularly every four months (plus or minus two weeks, so that some adjustment was necessary). Two groups, identified at any instant by the timedependent covariable X ( X = 1 if distolic pressure exceeds 95 mm Hg and X = 0 otherwise), were established and a simple model using only the most recent measure applied. Ignoring the constant, the maximum likelihood estimate for /3 was 0.65 (SD = 0*20), indicating a highly significant effect. The simple model was used in preference to the twoparameter model (equation (5)) which could not be justified in view of an estimation for 01 not differing significantly from zero ( p 2 = 0.70, SD(&) = 0.30, 81 = 0.46, SD(p1) = 0.29). Following Section 5, the prediction problem requires the specification of future values and, for instance, if at four months X = 1 and so remains then a (32,4) = 0.81, whereas for X = 0 and so remaining A(32,4) = 0.90, meaning that a constantly elevated blood pressure approximately doubles the relapse rate at three years. Virtually identical results were obtained upon reanalysing the data using the timedependent proportional hazards model as described in Section 5. At first sight it may seem unnatural to be obliged to supply future values in order to obtain UPDATING PROGNOSTIC INDICES VIA REGRESSION MODELS 235 a probability. Direct comparison of modelbased probabilities and relative frequencies is thus impossible. Consequently we are obliged to believe the model gives a reasonable description of the data if we are to attach meaning to our estimated conditional probabilities. Nonetheless, such belief does not rely on blind faith and the goodnessoffit techniques mentioned would in practice play an important role. Furthermore, such a conditional index would seem to formalize the clinician’s natural assessment in which he would consider the effect of a number of possible evolutions for a given patient. The model of Section 4 only considered a single timedependent covariate. Multivariate extensions are straightforward and it can be seen that for covariables remaining constant with time the standard exponential model is recovered. REFERENCES 1. R. Kay and M. Schumacher, ‘Unbiased assessment of treatment effects on disease recurrence and survival in clinical trials,’ Stat. Med., 1, 4158 (1983). 2. D. Schwartz, ‘Importance de la duree d’infecondite dans I’appreciation de la fertilite d’un couple’, Population, 2, 237250 (1981). 3. S. Gore, S. Pocock and G. Kerr, ‘Regression models and nonproportional hazards in the analysis of breast cancer survival’, Appl. 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Lellouch, ‘On Schoenfeld’s test of fit for the proportional hazards model’, Biometrika, 73, 513515 (1986). 236 J . O’QUIGLEY AND T. MOREAU 27. S. A. Rashid, J . O’Quigley, A. T. R. Axon and E. H . Cooper, ‘Plasma protein profiles and prognosis in gastric cancer’, Brit. J . Cancer, 45, (3), 390394 (1982) 28. R. Kay, ‘Proportional hazard regression models and the analysis of censored survival data’, Appl. Stat., 26. 227237 (1977). 29. J . Crowley and B. E. Storer, ‘Comment on paper by Aitkin, M., Laird, N., Francis, B.’, J. Am. Stat. Assoc., 78, 277281 (1983). 30. J . O’Quigley and T. Moreau, ‘Cox’s regression model: computing a goodness of tit statistic’, Comput. Methods Prog. Biomed., 22, 253256 (1986). 31. M. G. Bousser, E. Eschwege, M. Haguenau, J . M. Lefauconnier, N., Thibult, D. Touboul and P . J. Touboul, “‘A.I.C.L.A.” controlled trial of aspirin and dipyridamole in the secondary prevention of atherothrombotic cerebral ischemia’, Stroke, 14, 514 (1983).