A TWO-STAGE LINEAR MIXED-EFFECTS/COX MODEL FOR LONGITUDINAL DATA WITH MEASUREMENT ERROR AND SURVIVAL

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1 A TWO-STAGE LINEAR MIXED-EFFECTS/COX MODEL FOR LONGITUDINAL DATA WITH MEASUREMENT ERROR AND SURVIVAL Christopher H. Morrell, Loyola College in Maryland, and Larry J. Brant, NIA Christopher H. Morrell, Mathematical Sciences Department, Loyola College in Maryland, 4501 North Charles Street, Baltimore, MD Introduction Recently a number of papers have considered both longitudinal changes in a variable and the associated effect on the length of time to the occurrence of an event (Altman and De Stavola, 1994; Bycott and Taylor, 1998; Dafni and Tsiatis, 1998; De Gruttola and Tu, 1994; Hogan and Laird, 1997; Tsiatis et al., 1995; Wulfsohn and Tsiatis, 1997). Some of these and other papers also address the issue of the effect of measurement errors in the covariate on the estimate of the parameters in a Cox model (Hu et al., 1998; Hughes, 1993; Pepe et al. 1989; Prentice, 1982). It is well known that measurement error in the covariates leads to underestimates of the parameters in the Cox model (Prentice, 1982). This underestimation is sometimes referred to as regression dilution bias (Hughes, 1993). A number of these papers present models that jointly account for the repeated measurements over time and their association with the outcome event. To fit the joint models, the EM algorithm may be used to estimate the parameters from a suitably defined likelihood, or Markov-Chain Monte Carlo methods may be applied. These methods require extensive time to develop the software routines to fit these models. Furthermore, any model reformulation would require additional time to modify the procedure. In this paper, we explore a two-stage procedure similar to those proposed by Bycott and Taylor (1998) and Dafni and Tsiatis (1998) and evaluate its properties using a Monte Carlo study. The procedure first models the repeated measurements over time using a linear mixed-effects (LME) model (Laird and Ware, 1982) and then uses individual predictions from this LME model as either fixed- or timedependent covariates in a Cox model. In the first stage the pattern of change for each individual is estimated by a LME model. This model will smooth out irregularities in the observed data and also account for measurement error or other noise. As an example, we investigate the effects of systolic blood pressure on the time to development of coronary heart disease (CHD) using data from the Baltimore Longitudinal Study of Aging (BLSA). We study the effect of various numbers of repeated measurement times and the number of observations at each measurement time on the Cox model parameter estimate. We consider data for individuals with only one measurement time, individuals with up to three measurement times (or visits), and individuals with up to five visits. In addition, we consider single observations at each visit versus up to four repeated blood pressure measurements per visit. We model these various longitudinal data sets with a LME model using a backward elimination procedure as described in Morrell et al. (1997). Using the estimated random effects from the LME models, empirical Bayes estimates of the beginning systolic blood pressure for each individual are obtained for use in a fixed covariates Cox model relating systolic blood pressure to time to the occurrence of CHD. In addition, the LME models are used to estimate the blood pressure for each individual at each event time in a timedependent or updated covariates Cox model (Altman and De Stavola, 1994). This method may be applied using commercially available software that can fit both the LME model as well as both fixed and timedependent covariate Cox models (for example, proc mixed and proc phreg, (SAS, 1996)). This procedure is similar to the one employed by Tsiatis et al. (1995) in their study of CD4 counts in AIDS patients. However, we also investigate the effects on the Cox model parameters of varying numbers of visits and varying numbers of measurements per visit. This twostage procedure has the advantage of readily allowing for the inclusion of other covariates or other quantities derived from the LME model, for example, rate of change in the covariate, which may influence the time to event. However, caution must be exercised in using the time-dependent Cox model if the survival/censoring time for individuals are far beyond the last follow-up time used in the LME model for that individual since extrapolation of the predicted path well beyond the observed data may be risky. Finally, the properties of the Cox model parameters obtained using these methods are evaluated using a Monte Carlo simulation study with various levels of censoring. 2. Statistical Methods 2.1. Linear Mixed-Effects Model Using the notation in Laird and Ware (1982), the model for the i th cluster or individual is y = Xi + Zi bi + ei, i =1,..., m Reprinted from the 2000 Proceedings of the Biometrics Section of the American Statistical Association i 198

2 where y i is the n i 1 vector of observations for cluster i, X i is the design matrix of independent variables for cluster i, β is the p 1 vector of parameters for the fixed effects in the LME model, Z i is the design matrix of predictor variables for the random effects, b i is the q 1 vector of random effects for cluster i, e i is the error vector, and m represents the number of different clusters. It is assumed that b i ~ N (0, D) where D is a q q variance-covariance matrix of the b i, e i ~ N (0, σ 2 I), and the b i and e i are independent. The fixed effects, β, give the population average intercept and slopes. They model the systematic variation in the data that can be linked to explanatory variables that differ among the clusters and other explanatory variables that vary within the clusters. In contrast, the random effects, b i, account for the heterogeneity among the clusters by allowing their intercepts and partial slopes for each cluster to differ from the overall average. Finally, the random errors, e i, account for the unexplained variation in the data Cox Model The proportional hazards model, as formulated by Cox (1972), studies the relationship of a number of covariates to the time to survival in the presence of censoring. In the Cox model the hazard rate or intensity of failure for the survival time of a subject with covariate vector X C is λ(t X C ) = λ 0 (t)exp(θ X C ), t 0, where θ is a vector of unknown regression coefficients and λ 0 (t) is an arbitrary baseline hazard function. The problem is to estimate the regression parameter, θ, and baseline hazard function, λ 0 (t), in the presence of right censored survival data. This is accomplished by maximizing the partial likelihood (Kalbfleisch and Prentice, 1980). However, if the covariate, X C, is measured with error, the estimate of θ will be biased downward (Prentice, 1982). In addition, if the covariate vector, X C, is repeatedly measured over time, then the timedependent or updated covariates Cox model (Altman and De Stavola,1994) may be used to assess the association of the covariates with survival. To achieve this it is necessary to specify the value of the covariates at each survival time Determining the Covariate Values Follow-up studies that examine the relationship between a set of covariates and the time to the occurrence of an outcome rely on measurements that may often be only a single measurement of each covariate. If these single values are measured with error or have short-term biological variability, the parameter estimate of the association to the outcome event will be biased downward. If multiple measurements are available at each time point, these multiple values may be used to overcome some of the problems associated with measurement error, for example, by using the average value at each time point. A second issue that needs to be addressed is how to efficiently use the repeated values over time to estimate the association of the covariates with the time to the event. Altman and De Stavola (1994) mention the possibility of estimating the time path for the covariate and using these time paths in the Cox model as time-dependent or updated covariates. This paper addresses both issues of measurement errors and updated covariates by using a two-stage procedure. First, the linear mixed-effects model is used to describe the repeated measurements or patterns of change in the data for each subject. This will reduce the problems associated with measurement errors and permit the time path of the covariates for each subject to be estimated. Second, empirical Bayes shrinkage estimates from the mixed-effects model are used either to estimate the baseline value of the covariates, which may be used as fixed covariates in the Cox model, or to estimate the value of the covariate at each event time in an updated covariates analysis. This updated covariates analysis is simple to implement in SAS. The time-dependent covariate is specified in the model statement and then a formula obtained from the final LME model is given to evaluate the covariate. This formula will include, as variables, the estimated random effects so that individual-specific values of the covariate may be calculated at each time point. 3. Data Analysis For this paper, systolic blood pressure measurements that have been collected on 932 male participants of the Baltimore Longitudinal Study of Aging (BLSA) are used for examining the risk of systolic blood pressure on time to an event of coronary heart disease (CHD). CHD is defined as the occurrence of a coronary death, or the diagnosis of a myocardial infarction by history or pathologic Q- waves, or angina pectoris. Out of the 932 participants 185 experienced a coronary event. Table 1 displays some descriptive statistics for the CHD cases and censored events. The participants with CHD tend to be older and have shorter survival times. We used a maximum of 5 visits or repeated examinations per participant chosen to be as uniformly spaced over the interval of study as possible. 199

3 Table 1. Descriptive statistics (mean standard deviatio n) of BLSA sample data. Systolic Blood Pressure Censored (n = 747) CHD ( n = 185) Age at baseline Survival time There was an average of 3.1 visits per participant. In addition, up to 4 repeated blood pressure readings were recorded at each visit during the 22 days the participants spend at the BLSA. There was an average of 3.0 blood pressure readings per visit. To investigate the association of systolic blood pressure on the time to CHD, the data is analyzed in various ways: 1. The first blood pressure from the first visit is used in a fixed-covariate Cox model. 2. The mean of the repeated values at the first visit is used in a fixed-covariate Cox model. 3. Repeated values at first visit are modeled using a LME model. The beginning blood pressure is estimated from the LME model for use in a fixedcovariate Cox model. 4. The pattern of change based on a single value per visit with a maximum of 3 visits per individual is obtained using a LME model, then a) estimate the beginning blood pressure from the LME model for use in a fixed-covariate Cox model and b) estimate the blood pressure at each event time from the LME model for use in an updated covariate Cox model. 5. Model the pattern of change based on multiple values per visit with a maximum of 3 visits per individual using a LME model then estimate the Cox parameters as in 4 above. 6. Model the pattern of change based on a single value per visit with a maximum of 5 visits per individual using a LME model then estimate the Cox parameters as in 4 above. 7. Model the pattern of change based on multiple values per visit with a maximum of 5 visits per individual using a LME model then estimate the Cox parameters as in 4 above. These possibilities allow us to address how the association of systolic blood pressure on time to CHD depends on the number of repeated visits and the use of single or multiple measurements per visit. The LME model used will depend on the number of visits available in each case. For repeated measurements at first visit, the LME model will only contain fixed and random intercept terms. If y ij is the jth blood pressure for the ith participant, y ij = β 0 + b i 0 + e ij. If we use a maximum of 3 visits, there is only an average of 2.3 visits per participant. In this case we entertain models that contain up to linear terms in follow-up time with both fixed and random effects for time as well as terms in age at first visit (fage), fage 2, fage 3, and interactions of these terms with follow-up time in the LME model. Using a backward elimination procedure (Morrell, Pearson, and Brant, 1997) the final model for the kth observation at visit j for participant i is 2 y ij k = β 0 + b i 0 + (β 1 + b i 1 ) time ij + β 2 fage i + β 3 fage i + β 4 fage 3 i + β 5 time ij fage i + e ij k for both single and multiple observations per visit. With a maximum of 5 visits, follow-up time 2 terms are included as both fixed and random effects as well as interactions with fage. The final model is y ij k = β 0 + b i 0 + (β 1 + b i 1 ) time ij + (β 2 + b i 2 ) time 2 ij + β 3 fage i + β 4 fage 2 i + β 5 fage 3 i + β 6 time ij fage i + e ij k for both single and multiple observations per visit. Figure 1 shows the expected longitudinal trends for two of these data sets. The mean of the baseline systolic blood pressures obtained for the CHD cases and censored observations via the various methods were calculated. The values predicted from the LME models all have smaller standard deviations than for a single observed value or the mean of the observed values. This indicates that some of the noise in the original data has been eliminated by obtaining predictions from the LME model. In addition, using predicted values from LME models based on single values always lead to smaller standard deviations of the predicted values illustrating that these estimates are subject to more shrinkage than when prediction is based on multiple observations per visit. For single observations per visit, the means of the predicted values from the LME model for the CHD cases are smaller than for multiple values per visit whereas the means for the censored cases are similar for the single and multiple values per visit cases. The Cox models are fit with the systolic blood pressure represented as described above. The parameter estimates and the standard errors from the Cox model are given in Table 2. The table shows that using the mean of observation at first visit provides a larger parameter estimate than when only a single value measured at baseline is used. Similarly, estimating the baseline value at first visit from the LME model using multiple measurements at first visit additionally increases the estimate. Interestingly, for this data, the estimates based on repeated visits with single values per visit provide larger estimates than when multiple measurements are used at each visit. 200

4 This may be due to the estimates in the single value cases having more shrinkage towards the mean than in the multiple observations case or due to the fact that, or average, the covariates were smaller for the CHD cases for the single versus multiple observations per visit condition. In addition, increasing from a maximum of 3 visits to a maximum of 5 visits gives a modest increase for single values but almost no change for multiple measurements. In all cases the parameter estimates from the updated covariates Cox model are smaller than their counterparts in the fixed covariates Cox model although a similar pattern holds between 3 and 5 visits and single versus multiple measurements. In this analysis the true value of risk relating systolic blood pressure to time to CHD is unknown. Consequently, it is not clear which of these approaches provides for the best results. Therefore, a simulation study is used to assess the procedure. 4. Simulation Study To assess the bias and precision of these procedures a Monte Carlo study is performed using FORTRAN routines to fit the LME model and estimate the Cox parameters. For simplicity, in the simulation study we assume that each participant has 5 visits and 4 observations at each visit. To generate the data for the Monte Carlo simulation study we adapt the method Table 2. Parameter estimates and standard errors from the Cox model for the various methods of obtaining the SBP covariate values. Systolic Blood Pressure Baseline Cox Model Observed single value ( ) Mean of values at first visit Predicted from LME model: Multiple values at first visit Single value with at most 3 visits Multiple values with at most 3 visits Single values with at most 5 visits Multiple values with at most 5 visits ( ) ( ) ( ) ( ) ( ) ( ) Updated Covariates Cox Model ( ) ( ) ( ) ( ) developed by Dafni (1993) for generating the time to event and the censoring/event variable. The LME model is fit using a FORTRAN routine developed by Mary Lindstrom (Lindstrom and Bates, 1988). The parameters of the Cox model are estimated using the routines developed from Kalbfleisch and Prentice (1980). For a sample data set, the simulation algorithm is checked against the results produced by SAS to ensure that identical results are obtained. In the simulation study we only use linear growth-curve models. The simulation uses the same subsets of data as used in the analysis of the SBP/CHD data. For the case of three repeated measurements, the first, third, and fifth observations are used in the analysis. In addition, the Cox model parameters are estimated using the Atrue@ baseline value of the covariate (before error is added) as a fixed covariate as well as the Atrue@ path of the covariate in a time-dependent analysis. Some of the parameters of the LME and Cox models are chosen to approximately match the BLSA data. In particular, the longitudinal rate of change in the covariate is chosen to be 1, the covariance matrix of the random effects is , where the entry below the main diagonal gives the correlation, the error standard deviation is 8, the Cox parameter is 0.1, and three values of the censoring mean are chosen to obtain mild, moderate, and severe censoring. For the purposes of space, only the censoring pattern closest to the actual data is presented here. Table 3 contains the results of the proportional hazards model regressions. In the simulations, on average 27.1%, 51.0%, and 73.0% of the cases are events. The first row of the table shows that when the true values of the covariate are used in the Cox model the parameter estimates are unbiased with coverage proportions close to the nominal level. As expected, a single value provides estimates that are severely biased towards zero. While the use of the mean of the values at first visit somewhat improves this situation, in these two cases for all the 500 replications the estimates were less than 0.1 for each of the simulations. In all cases the standard errors produced from the Cox model are very similar to the standard deviation of the Monte Carlo estimates. This indicates that the standard errors provided by the Cox model appropriately assess the uncertainty in the estimate. When the LME model is used to predict the covariates in the Cox model, some general observations may be made. As the number of visits 201

5 increases the parameter estimates become less biased, using multiple observations as opposed to single observations improves the estimates, and the updated covariates estimates provide improvements over the baseline covariates. This latter observation is not surprising since the hazard function was assumed to be a function of the linear trend for each individual. These results agree with Dafni (1993) who showed that the two-stage model gave results that were still slightly biased towards zero, even though they eliminated much of the bias of the uncorrected values. Interestingly, as the proportion of events increases, the Cox model parameter estimates become more biased and the standard errors of the estimates decrease. This leads to lower coverage proportions. Based on the CHD data example, the parameter estimates from the Cox model were larger for the single value at 3 or 5 visits than for the multiple values scenario. In contrast, the simulations show that on average the parameters based on single values will be more biased towards zero than those based on multiple values. 5. Conclusions We have demonstrated the use of a two-stage model for using longitudinal data that may be measured with error or possess short-term biological variability to assess the risk of time to developing an event. Our procedure may be applied with commercially available software. For example, first proc mixed in SAS (SAS Institute Inc., 1996) may be used to fit the LME model and obtain the estimated fixed and random effects from the covariate data and then proc phreg may be used to fit both the fixed-covariate and time-dependent or updated covariate Cox model. Wulfsohn and Tsiatis (1997) developed a unified model to simultaneously estimate the covariate and failure time processes. Based on their example, their model provides less biased estimates than the twostage process. However, software to fit their model is not generally available. In addition, the two-stage process readily allows for the incorporation of additional fixed and time-varying covariates and for the inclusion of functions of the time-varying covariates. For example, the rate of change in the covariate may be evaluated as the derivative of the function that defines the pattern of change over time and this derivative may be used either as a covariate at baseline to evaluate how the initial rate of change impacts time to event or as a time-varying covariate. However, caution must be exercised in using the time-dependent Cox model if survival/censoring time for individuals are far beyond the last follow-up time used in the LME model for that individual since extrapolation of the predicted path well beyond the observed data may be risky. Acknowledgements We also thank Dr. Urania Dafni for her assistance on the method of generating the data for the Monte Carlo simulations. References D.G. Altman and B.L. De Stavola, APractical problems in fitting a proportional hazards model to data with updated measurements of the covariates,@ Stat. Med., vol. 13, pp , P. Bycott and J. Taylor, AA comparison of smoothing techniques for cd4 data measured with error in a timedependent Cox proportional hazards model,@ Stat. Med., vol. 17, pp , D.R. Cox, ARegression models and life tables@ (with discussion), J. Roy. Statist. Soc. B, vol. 43, pp , U.G. Dafni. Evaluating Surrogate markers of clinical outcome when measured with error. Ph.D. dissertation, Biostatistics Library, Harvard School of Public Health, U.G. Dafni and A.A. Tsiatis, AEvaluating surrogate markers of clinical outcome when measured with error,@ Biometrics, vol. 54, pp , V. De Gruttola and X.M. Tu, AModeling progression of CD-4-lymphocyte count and its relationship to survival time,@ Biometrics, vol. 50, pp , J.W. Hogan and N.M. Laird, AModel-based approaches ro analysing incomplete longitudinal and failure time data,@ Stat. Med., vol. 16, pp , P. Hu, A.A. Tsiatis, and M. Davidian, AEstimating the parameters in the Cox model when covariate variables are measured with error,@ Biometrics, vol. 54, pp , M.D. Hughes, ARegression dilution in the proportional hazards model,@ Biometrics, vol. 49, pp , J.D. Kalbfleisch and R.L. Prentice, The Statistical Analysis of Failure Time Data, John Wiley and Sons, New York, N.M. Laird and J.H. Ware, ARandom-effects models for longitudinal data,@ Biometrics, vol. 38, pp , M.J. Lindstrom and D.M. Bates, ANewton-Raphson and EM algorithms for linear mixed-effects models for repeated-measures data,@ JASA, vol. 83, pp ,

6 C.H. Morrell, J.D. Pearson, and L.J. Brant, ALinear transformations of linear mixed-effects The American Statistician, vol. 51, pp , M.S. Pepe, S.G. Self, and R.L. Prentice, AFurther Results on covariate measurement errors in cohort studies with time to response Stat. Med., vol. 8, pp , R.L. Prentice, ACovariate measurement errors and parameter estimation in a failure time regression model,@ Biometrika, vol. 69, pp , SAS Institute Inc. SAS/STAT Software: Changes and Enhancements through Release 6.11, Cary, NC: SAS Institute Inc., A.A. Tsiatis, V. De Gruttola, and M.S. Wulfsohn, AModeling the relationship of survival to longitudinal data measures with error. Applications to survival and CD4 counts in patients with AIDS,@ JASA, vol. 90, pp , M.S. Wulfsohn and A.A. Tsiatis, AA joint model for survival and longitudinal data measured with error,@ Biometrics, vol. 53, pp , Table 3. Mean (standard deviation) and standard errors of parameter estimates, coverage proportions for 95% confidence intervals for the Cox model parameter, θ H, and MSE ( 10-5 ) from the 500 Monte Carlo replications of the LME-Cox procedure for the various methods of obtaining the covariate values. Baseline Covariate Updated Covariates Covariate Parameter SE MSE Coverage Parameter SE MSE Coverage Without error ( ) ( ) ( ) ( ) Observed single value ( ) ( ) Mean of values at first visit ( ) ( ) Predicted from LME model: Multiple values at first visit ( ) ( ) Single value at visits ( ) ( ) ( ) ( ) Multiple values at 3 visits ( ) ( ) ( ) ( ) Single values at visits ( ) ( ) ( ) ( ) Multiple values at 5 visits ( ) ( ) ( ) ( ) H True value of θ = 0.1. On average, 27.1% of the cases are events. Single Observations With at Most 3 Visits Multiple Values at All Visits SBP 120 SBP Age Figure 1. Expected longitudinal trends based on the linear mixed-effects model for two of the data sets. Age 203