A multiple imputation method based on weighted quantile regression models for longitudinal censored biomarker data with missing values at early visits

Ankylosing spondylitis is a chronic inflammatory disease characterized by inflammatory spinal pain usually beginning in the second to fourth decades of life that can result in chronic pain, that can result in functional impairment and diminished quality of life, and, in some patients, complete spinal fusion. In PSOAS, participants meeting the modified New York (mNY) Classification Criteria for AS [26] were enrolled from one of the five study sites (Cedars-Sinai Medical Center in Los Angeles, California, the University of Texas Health Science Center at Houston (UTH), the NIH Clinical Center, the University of California at San Francisco (UCSF), and the Princess Alexandra Hospital in Brisbane, Australia (PAH)¹ and were followed for up to 13 years (through two cycles of NIH funding: 2002–2006 and 2007–2016). At each study visit, spaced 6 months apart, the patients underwent comprehensive clinical evaluation for disease activity and functional impairment. Self-reported outcomes were measured at 6-month intervals and radiographic data, including AP pelvis X-rays, AP and lateral lumbsacral spine films and lateral cervical spinal films were collected every 2 years in order to assess longitudinal radiographic damage which was defined by scoring the Bath Ankylosing Spondylitis Radiology Index (BASRI) [17] and the modified Stoke Ankylosing Spondylitis Spine Score (mSASSS) [3]. C-reactive protein (CRP) levels and erythrocyte sedimentation rate (ESR) as well as medication usage were also determined at each clinical visit.

One of the objectives of PSOAS was to evaluate factors associated with longitudinal radiographic severity and rate of progression in AS patients. Specifically, we focused on evaluating longitudinal association between CRP level and radiographic damage which is assessed by mSASSS values (range 0–72) at each X-ray visit. We considered analysis cohort of 295 patients who were confirmed AS by mNY criteria and had at least 4 years of radiologic follow up data (as of August 2016). However, we faced with two challenges in analyzing CRP data in relation to mSASSS. First, we found that 13.3% of CRP values were left-censored due to being below the limit of detection. Another issue is related to unobserved CRP data during early visits for some patients which was by study design. Specifically, of the 295 patients with at least 4 years of radiologic follow up, 37% have been followed since first cycle of funding, i.e., Study I (enrolled from 2002–2006)² and then consented to Study II (enrolled from 2007–2016) for their continued participation. The inclusion of these patients increases the study power by increasing the number of patients who have been followed for 10+ years. However, CRP levels for 111 patients among our study cohort were not collected for up to first two consecutive X-ray visits because initially blood sample collection (e.g., CRP and ESR) was not a part of the protocol for some patients in Study I. Different scenarios in which missing CRP data are generated are presented with detailed descriptions in Fig. 1. An important feature of CRP data was that it has a monotonic missing pattern; i.e., if a value was missing at visit t then the values for all previous visits (i.e., k,1≤k≤t−1) were also missing. This monotonic pattern was found in 92% of 111 patients. We also discovered that the number of patients who had CRP data missing at their first visit only (73.9%) was higher than the number of those who had missing data for both first and second visits (26.1%). There were only 5 patients who had CRP levels missing at all first three visits. Also we noted that patients who were enrolled earlier had a higher number of visits with unobserved CRP data.

Our approach for assessing the longitudinal association between CRP and radiographic damage (i.e., mSASSS) includes four steps: Step (1) modeling missing data processes, Step (2) applying the inverse weighting techniques to censored quantile regression (CQR) using the probabilities of missing early visits that are estimated from the modeled missing data process, Step (3) employing multiple imputation process for both censored and missing CRP data at early visits, based on quantile estimates from established weighted CQR in Step (2), and Step (4) conducting longitudinal regression analyses where imputed CRP data are treated as an independent variable and mSASSS as a response variable. Natural log-transformed CPR was used in our analyses to reduce its highly right-skewed distributions.

We conducted simulation studies to investigate the performance of our developed MI methods through different scenarios. We considered the following four different scenarios of longitudinal data structures, as well as different levels of censoring for generating biomarker data.

In order to generate a longitudinal outcome variable that mimics the distribution of mSASSS (=y _it ) in PSOAS, we used the following regression model

for the i-th patient and the t-th visit, where α₀=5, α₁=−4, α₂=−6, w _it represents the longitudinal structure variable time (t=1, ⋯,4) and \(z^{*}_{it}\) denotes complete biomarker data which have been generated based on the aforementioned four scenarios. An error term ε _it was generated from multivariable normal distribution based on exchangeable covariance structure with a correlation coefficient ρ of 0.3. We then produced missing and censored values for biomarker data z _it that mimic the missing data pattern of CRP levels in PSOAS. We used a logistic regression to model probability of observed biomarker data η _it , based on variables w _it , y _it and the first observed biomarker data after time t (i.e., \(z_{it'}, t<t'\leq 4\phantom {\dot {i}\!}\)). Based on probability η _it , we calculated the patient-level probability weights (=π _i ) under monotone missing data mechanism through a specific function of η _ij , as shown in Appendix B. For generating censored data, we chose the detection limit c, as the (100×r)-th percentile of the simulated biomarker data, where r is the censoring rate (i.e., r=0.1, 0.15, 0.2, 0.3). We simulated data for 75% of patients who had missing data for up to first 3 visits (i.e., missing at first visit only, first and second visits, or all first three visits), and 25% of patients who had complete measurements up to visit 4. For each scenario, three hundred simulation datasets with sample size of 250 were generated. Details regarding parameters used for data generation and covariance structures for each scenario are described in Appendix B.

For multiple imputation, we fitted weighted CQR (wCQR) models using both known probability weights π (MI-wCQR ₁) and \(\hat {\pi }\) that was estimated through the aforementioned logistic regression model. Since the observed biomarker data z_i,t+j in this logistic regression model had also censored values, we considered fitting two separate wCQR models, one based on imputed censored data by DL/2 (MI-wCQR ₂) and the other using uncensored data only (MI-wCQR ₃), in order to see the impact of these two approaches on parameter estimates. We also considered unweighted CQR method (MI-CQR) which accounts for censoring but ignores the missing data mechanism. Other imputation methods were further applied, that included Markov Chain Monte Carlo (MCMC)-based MI methods [11, 14] through on Bayesian frame work for a monotone missing data, which were implemented in the function ‘mice’ of the R package mice [25], imputing only missing values (MI-MCMC ₁), as well as imputing both censored and missing values (MI-MCMC ₂). MCMC-MI algorithm obtains the posterior distribution of parameters by sampling iteratively from conditional distributions based on Gibbs sampling method.

Using imputed biomarker data generated from different MI methods described above, we conducted longitudinal regression analysis using model 1, for each scenario. For assessing the performance of each estimator, we calculated bias and ratio of the mean squared error (MSE) of the omniscient estimator (OMNI), the gold standard, which is based on the complete data without censoring, to that of each estimator. Throughout we refer to this ratio of MSEs as relative efficiency (RE), which will be used for comparing the performance of the aforementioned methods. Moreover, we also conducted complete case analysis using only observed biomarker data where censored data were imputed by a single value of DL/2 (CC-DL/2).

Censored or missing CRP data at early visits in PSOAS were imputed based on six different approaches (CC-DL/2, MI-MCMC ₁, MI-MCMC ₂, MI-CQR, MI-wCQR ₂, MI-wCQR ₃) that are described in “Simulation studies” section. For each imputed dataset, we assessed the longitudinal association between natural log-transformed CRP levels and mSASSS while controlling for potential confounding factors, that included Bath Ankylosing Spondylitis Disease Activity Index (BASDAI), medication usages of Tumor Necrosis Factors inhibitors (TNFi), and Nonsteroidal Anti-Inflammatory drugs (NSAIDs) as well as demographic information such as sex, race, disease duration, co-morbidity, education and smoking status. Multivariable mixed effect Poisson regression models were conducted for each imputed dataset, to account for the correlations of repeated measures within a patient.

Let \(z^{*}_{it}\) be the biomarker measurement for the i-th subject at time t assuming all subjects are to be observed at the same time. Suppose we define the linear regression model \(z^{*}_{it}= \boldsymbol x_{it}^{T}\boldsymbol \beta +e_{it},\quad i=1,\cdots, n; \quad t=1,\cdots, m \label {latent}\), where x _it is a p×1 vector of covariates that can include the time of measurement, β is an unknown p×1 vector of regression parameters and the random errors e _it are correlated within the subject to reflect the serial correlations of repeated measurements within each individual. If the τ-th conditional quantile of e _it given \(\boldsymbol x_{it}^{T}\) is assumed to be zero, a quantile regression model related to the τ-th quantile of response variable, \(q_{\tau }\left (z^{*}_{it}\right)\), conditional on x _it has the form \(q_{\tau }\left (z^{*}_{it}\right)=\boldsymbol x_{it}^{T}\boldsymbol \beta _{\tau }, \quad 0<\tau <1\), where β _τ is a vector of quantile specific regression parameters corresponding to the coefficient β in the linear regression model above. When there exists a lower detection limit, say c, \(z^{*}_{it}\) is a latent variable and we cannot observe the biomarker measurement if it has a value below c and we only observe \(z_{it}=z^{*}_{it}\), if \(z^{*}_{it}>c\). This leads to the longitudinal censored quantile regression (CQR) model defined as \(z_{it}= max\left (c, \boldsymbol x_{it}^{T}\boldsymbol \beta +e_{it}\right)\). We can define the objective function for longitudinal censored data as

The loss function ρ _τ (u)=u{τ−I(u≤0)}, with I(·) being an indicator function, represents the contribution by residuals. The estimates resulting from (2) are equivalent to the solution of estimating equation

To apply the weighting techniques to the censored quantile regression model for handling missing data at early visits, let O _i be a random variable indicating the time point when the data collection was started for the i-th subject. O _i can take the values between 1 and m. If the subject has completed 1 ∼m follow-up visits then O _i =1, and if the subject had missing data from visit 1 to m-1 then O _i =m. We denote \(\boldsymbol z_{i}^{o}\) as the observed response history since the data collection was started, and X _i ={x_i1,⋯,x _im } ^T as a set of covariates that were observed from the complete study visits 1 ∼m. When the biomarker measurements are MAR, the conditional probability of missing early visit from the baseline to the o _i −1 occasion for the i-th subject is \(\pi _{io_{i}}=Pr\{O_{i}=o_{i}|\boldsymbol z_{i}^{o}\), X _i ,γ} (o _i =1,⋯,m), where \(\pi _{io_{i}}>0\) and γ is a parameter vector of the regression model. Now the weighted estimating equations for censored quantile regression model can be defined as

The basic idea of weighted estimating equations is to weight each subject’s contribution by the inverse probability of missing early visits to a given occasion. After we define \(\boldsymbol x_{it}^{w}=\frac {1}{\pi _{io_{i}}}\boldsymbol x_{it}, z_{it}^{w}=\frac {1}{\pi _{io_{i}}}z_{it}\) and \( {c_{i}}^{w}=\pi _{io_{i}}^{-1} {c}\), Eq. (4) can be written in the same form as the unweighted estimating Eq. (3) as follows:

Thus, the corresponding objective function is in the form of

Now the traditional censored quantile regression estimation algorithm ([4, 20]) can be straightly applied to minimize this objective function. Details related to estimation procedures, inference, and asymptotic properties of parameter estimators were discussed in Lee and Kong [10].

If the missing early visit data arise from the MAR mechanism, estimation of probability of missing early visits is straightforward. To illustrate the missing data process, denote R _it as the missing status of response variable z _it , i.e., R _it =1 if z _it is observed and 0 otherwise. Then R _ij =1 implies that \(\phantom {\dot {i}\!}R_{ij'}=1\) for all j^′>j given the monotone missing pattern. To indicate when the data collection is started, we define a random variable O _i as \(O_{i}=1+(m-\sum _{t=1}^{m} R_{it})\). The probability of missing early visit \(\pi _{io_{i}}\) from baseline to occasion o _i −1 can be given by

where \(z_{io_{i}'}^{o}\) is the first observed z value after time o _i (i.e., o _i <oi′≤m). When we define the probability of being observed at time t for the i-th subject as \(\eta _{it}=Pr\left (R_{it}=1| R_{i,t+1}= \cdots = R_{im}=1, z_{it'}^{o}, \boldsymbol X_{i}, \boldsymbol \gamma \right)\), t<t^′≤m, we can carry out the probability of missing early visits in terms of η _it as follows:

where γ is the parameter vector of the regression model for η _it . Then appropriate regression models such as logistic regression model can be used to estimate η _it , and then we can calculate \(\pi _{io_{i}}\) based on the equation above.

Multiple imputation techniques [21] have been widely used for the general handling of missing data. However, the censoring and monotone missing mechanisms should be incorporated in the imputation models to deal with the complexity of missingness. Based on the weighted censored regression of quantiles that was introduced in the previous section, we propose the multiple imputation procedure to fill in the data that are left-censored or missing at early visits. The conditional censoring probability of z _it , ω(X _it )=Pr(z _it <c|X _it ) can be estimated using a logistic regression model \(logit[\omega (\boldsymbol X_{it}) ]=\boldsymbol X_{it}^{T}\boldsymbol \delta \), where δ is unknown parameter vector and then v is sampled from uniform distribution UNIF (0,ω(X _it )) in order to impute the censored value z _it by its conditional quantile of \({z}_{it}^{*}=\boldsymbol X_{it}^{T} \hat {\boldsymbol \beta }_{v}\) which is estimated through fitting weighted censored quantile regression model where z _it is treated as the dependent variable as described in the previous section. We also draw u from UNIF(0,1) to fill in the missing value by \({z}_{it}^{*}=\boldsymbol X_{it}^{T} \hat {\boldsymbol {\beta }}_{u}\), that is the u-th conditional quantile of z _it given X _it . Once the imputed datasets are generated, any analysis designed for the complete dataset can be applied to each of M imputed datasets. To obtain the parameter estimators of interest in the regression model \(y_{it}= \alpha _{0}+\alpha _{1} z^{*}_{it}+\alpha _{2} w_{it}+\epsilon _{it}\), we define the combined MI estimator as \(\hat {{\boldsymbol {\alpha }}}_{MI}=M^{-1}\sum _{k=1}^{M} \hat {{\boldsymbol {\alpha }}}_{k}\). Wang and Feng [28] discussed the asymptotic properties of their proposed multiple imputation procedure based on the conditional quantile function and suggested bootstrapping because the asymptotic variance of MI estimators takes complex forms and it is difficult to estimate directly. We adopted a bootstrap method by resampling the paired observations with replacement based on 500 bootstrap samples to obtain the standard errors for estimated parameters and p-values that were calculated by using the normality of estimated parameter \(\hat {{\boldsymbol {\alpha }}}_{MI}\).