Multi-Trajectory Modeling to Predict Acute Kidney Injury in Chronic Kidney Disease Patients

Abstract

Risk-stratifying chronic disease patients in real time has the potential to facilitate targeted interventions and improve disease management and outcomes. We apply group-based multi-trajectory modeling to risk stratify patients with chronic kidney disease (CKD) and its major complications into distinct trajectories of disease development and predict acute kidney injury (AKI), a serious, under-diagnosed outcome of CKD that is both preventable and treatable with early detection. Utilizing Electronic Health Record data of 1,947 patients, we identify eight risk groups with distinct trajectories and profiles. We observe that a higher estimated probability of AKI generally coincides with a higher risk group. Overall, at least 75% of patients stabilize into their final groups within less than two years from diagnosis of CKD Stage 3. Model calibration confirms that the estimated outcome probabilities are highly correlated with AKI incidence, providing group-specific and individual level predictions to improve clinical management of AKI in CKD patients.

Introduction

Illness trajectories have long been considered an important component of palliative care, such as in the management of kidney failure¹. More recently, there has been increasing focus on trajectories of disease progression for common chronic conditions^2-6. Tracking the progression of a chronic condition from its early stages, including the multiple associated comorbidities and complications that develop contemporaneously, and stratifying patients into distinct risk groups, has the potential to facilitate targeted interventions^2,3,6. Furthermore, making predictions in real time for individual patients about critical disease outcomes based on these trajectories and estimating confidence intervals around the predictions may provide clinicians with the capability to anticipate the future trajectory of disease progression, thereby improving timely and appropriate clinical decision-making^{7, 8}. In this paper, we demonstrate the application of group-based multi-trajectory modeling to risk stratify patients with chronic kidney disease (CKD) and its significant complications into distinct trajectories of disease development and predict acute kidney injury (AKI), a serious, under-diagnosed outcome of CKD that is both preventable and treatable with early detection^9-12.

CKD is a costly, complex, and high mortality health condition affecting 26 million adults in the US, with another 73 million at increased risk of the disease⁹. It leads to a progressive deterioration of the kidney function over five stages. Prevalence is estimated to be 8-16% worldwide¹³. CKD patients make up only 1.5% of Medicare population but cost $30 billion annually, almost 10% of Medicare costs, due to the high incidence of co-morbidities and complications¹⁴, particularly related to AKI, caused by a sudden decline in kidney function, and cardiovascular disease^10-12. Even small acute changes in kidney function can result in high morbidity and mortality, an increasing incidence of AKI with CKD progression and associated hospitalizations¹⁰. These adverse outcomes are potentially preventable or can be mitigated through early identification and treatment of individuals at risk as they progress through the five stages^10-12. Early recognition and management of AKI is thus critical to delaying CKD progression and development of complications. Using clinical and demographic characteristics of CKD patients, longitudinal data on laboratory markers of CKD as well as its complications, and AKI diagnosis as the outcome measure, we identify distinctive groups of individual trajectories within the patient population and provide trajectory-group-specific estimates and individual-level predictions of the probability of AKI.

Data

Our data set comes from a leading nephrology practice in southwestern Pennsylvania. We extract patient data from the Electronic Health Record for the years 2009 to 2013, with access to all laboratory measurements for this period. Specifically, we include patients diagnosed with CKD Stage III between January 1, 2009 and November 19, 2012, since patients are normally referred to a nephrologist when diagnosed to be in this stage. Patients who received a kidney transplant on or after January 1, 2009 were removed from the analysis since their level of kidney function differs sharply from those who did not receive a transplant (keeping them would likely distort the identified disease progression). The population is split almost evenly among female and male patients. Most patients are retirees, with median age of 70 years. 94% of the patients are white. About half the patients have a diagnosis of CKD Stage III, while the remaining patients progress to more advanced stages of CKD over the study period.

In prior research, we used the current data set for trajectory modeling of CKD and related complications without consideration of an outcome variable^2,3. We considered some of the main complications of CKD, including Anemia¹⁵, Secondary Hyperparathyroidism¹⁶, Hyperphosphatemia¹⁷, and Metabolic Acidosis¹⁸. Our data set allows us to track these conditions via corresponding laboratory measurements. For CKD, we use the estimated glomerular filtration rate (eGFR), derived using serum creatinine laboratory measurements, and calculated by the CKD-EPI equation according to best practice¹⁹. The respective laboratory measurements of Hemoglobin (HGB), parathyroid hormone (PTH), phosphate (PO4), and carbon dioxide in bicarbonate form (HCO3) are used as markers for the considered complications.

In the cohort, 1,367 patients develop Anemia, while 1,476 of them are diagnosed with Secondary Hyperparathyroidism. 438 patients have Acidosis, but only a 100 suffer from Hyperphosphatemia. After data cleaning and processing, 1,944 patients diagnosed with CKD Stage III on or after January 1, 2009 form the cohort for our analysis.

As epidemiologic evidence has mounted, it is now recognized that acute kidney injury (AKI) and chronic kidney disease (CKD) are not distinct phenomena, but closely interwoven¹. As pointed out by Chawla et al., it has been discovered that the risk factor most associated with AKI is a pre-existing CKD diagnosis, which increases risk by as much as ten times¹¹. We thus augment our previous analysis by including AKI as an outcome variable in our trajectory model.

Over the study period, 39.65% of patients suffer from at least one instance of acute kidney injury (AKI), extracted from the EHR using ICD9 codes 585.5, 6, 8 and 9, and ICD10 codes N17.0, 1, 2, 8, and 9. The median number of diagnoses per patient is equal to four. However, most of these diagnoses are made in rapid succession after a patient has been hospitalized, such that we only keep track of whether a patient has at least one recorded instance of AKI over the entire study period. Nine percent of the patients had a prior history of AKI before the year 2009. Approximately 38% of these patients encounter another episode of AKI in the study period, which is not significantly different from the overall AKI rate.

Methods

Single Trajectory Model

Let Y_ij define the multivariate response variable of individual i for the j-th marker, for example eGFR. Then, $Y_{ij}={(Y_{ij1},\dots ,Y_{ijT})}^{\text{T}}$ is a vector of length T, which holds the quarterly lab results for the marker in question; T denotes the total number of time periods. For a single response, the group-based trajectory model posited by Nagin²⁰ assumes the following density for a sequence of longitudinal measurements $y_{ij}={(y_{ij1}\dots ,y_{ijT})}^{\text{T}}$:

$$f(y_{ij})={\displaystyle \sum _{k=1}^K{p}_k{f}_{Y_{ij}}(y_{ij}|C=k),}$$ (1.1)

where K denotes the total number of groups, p_k denotes the probability of belonging to group k, and $f_{Y_{ij}}(y_{ij}|C=k)$ is the conditional density of the observed data vector given class k. The probabilities p_k of this mixture model are not estimated directly, but related via the SoftMax function to a k-dimensional vector θ of class coefficients, and time-stable covariates x with associated weight vectors w_k. Through the time-stable covariates, the model permits the group memberships to vary by individual:

$$p_{ik}=\frac{e^{{\theta }_k+x_i{}^{\text{T}}{w}_k}}{{\displaystyle \sum _{l=1}^K{e}^{{\theta }_i+x_i{}^{\text{T}}{w}_i}}}.$$ (1.2)

The response vector for each outcome is modelled as a multivariate normal random variable

$$Y_j|C=k~N\left({\mu }_{jk},{\sigma }_j^{2}I\right),$$ (1.3)

where the elements of the mean vector are related to the period t (= time in quarters since diagnosis of CKD Stage III) of the individual patient as follows:

$${\mu }_{jkt}={\beta }_{jk0}+{\beta }_{jk1}t+{\beta }_{jk2}{t}^2+{\beta }_{jk3}{t}^3.$$ (1.4)

As can be seen, our group-based trajectory model assumes that the trajectories in each group have a simple polynomial form. From our experience, a polynomial order above three is rarely necessary, which is why we have constrained the model to have cubic terms at most.

Multi-Trajectory Model

Multi-trajectory modeling is an extension of the single trajectory model that jointly models the trajectories of multiple outcomes²¹. In this model extension, the density for J outcomes becomes

$$f(y_i)=f(y_{i,j=1,\dots ,J})={\displaystyle \sum _{k=1}^K{p}_{ik}}{\displaystyle \prod _{j=1}^J{f}_{Y_{ij}}(y_{ij}|C=k).}$$ (1.5)

Model fitting and inference is carried out via the traj procedure from the Stata package of the same name²², which implements a Newton-Raphson optimization algorithm for maximum likelihood estimation. Following the suggestion given by Jones et al., the Bayesian information criterion (BIC) is used to perform model selection and to determine the number of groups K²².

Using the Multi-Trajectory Model to Predict the Probability of Acute Kidney Injury

We apply a recent extension of group-based trajectory modeling⁸, which estimates the joint distribution of the trajectories and an outcome of interest. The extended model provides trajectory-group specific estimates of the probability of the outcome, which is acute kidney injury (AKI). We denote these estimates by ${\widehat{\alpha }}_k$. Combining the ${\widehat{\alpha }}_k$ estimates with the posterior probability of group membership (PPGM), we calculate individual level predictions for the probability of AKI at each time point as follows:

$${\widehat{\alpha }}_{it}={\displaystyle {\sum }_{k=1}^K{\text{PPGM}}_{it}^k{\widehat{\alpha }}_k}$$ (1.6)

The posterior probability of group membership in group l can be computed using Bayes' rule as

$${\text{PPMG}}_{it}^l=\mathrm{Pr}\left(C=l|{\left\{Y_{ij}^t=y_{ij}^t\right\}}_{j=1,\dots ,J}\right)=\frac{p_{il}{\displaystyle \prod _{j=1}^J{f}_{Y_{ij}}\left(y_{ij}^t|C=l\right)}}{{\displaystyle \sum _{k=1}^K{p}_{ik}}{\displaystyle \prod _{j=1}^J{f}_{Y_{ij}}\left(y_{ij}^t|C=k\right)}},$$ (1.7)

where the number of biomarkers J is equal to five for the purposes of this paper. For the conditional densities, it follows from Equation 1.3 that

$$f_{Y_{ij}}\left(y_{ij}|C=k\right)={\displaystyle \prod _{t=1}^T\varphi \left(\frac{y_{ijt}-{\mu }_{jkt}}{{\sigma }_j}\right)}.$$ (1.8)

in which ϕ is the density function of the standard normal distribution. For the data used for model fitting, T=18 (measured in quarters). Notice that we can calculate the PPGMs without access to eighteen quarters of data per patient by conditioning only on the measurements y_jjt that have been observed so far after the initial CKD diagnosis. This way, we can calculate posterior probabilities that take all currently available information into account and may serve as a prognostic tool for clinicians to detect high-risk patients early on and not when it may already be too late.

Results

We fit a multi-trajectory model with eight groups. For comparison purposes, we chose the same number of groups as in our previous analysis in which we used the Bayesian information criterion (BIC) as a model selection criterion to pick the number of groups as well as the order of the polynomials. The updated model incorporates a binary indicator for AKI occurrence as an outcome variable. The eight-group model with all five biomarkers as well as the AKI outcome variable, but without any other covariates, forms our baseline model. The estimated trajectories for the biomarkers of CKD and related complications are displayed in Figure 1. The results do not differ significantly from the ones obtained previously³. The groups are roughly the same in size, with the most extreme groups, one and eight, having slightly lower relative frequencies than the others. Table 1 reports the AKI probability estimates associated with each trajectory. The results show clear differences in ${\widehat{\alpha }}_k$ across trajectory group, from a low of .12 for group 8 to a high of .78 for group 1.

Figure 1.

Fitted trajectories of the baseline eight-group multi-trajectory model for the five considered biomarkers, ordered increasingly by the estimated probability of AKI (${\widehat{\alpha }}_k$). Group size proportions are displayed next to the group label inside the parentheses.

Table 1.

Table of the estimated group-level AKI probabilities ${\widehat{\alpha }}_k$ alongside 95% Wilsonian confidence intervals.

	Group 1	Group 2	Group 3	Group 4	Group 5	Group 6	Group 7	Group 8
Average	0.717	0.531	0.513	0.449	0.353	0.299	0.227	0.168
Lower	0.647	0.465	0.453	0.393	0.292	0.245	0.184	0.124
Upper	0.777	0.596	0.571	0.507	0.419	0.36	0.276	0.224

While all patients in the cohort have a diagnosis of CKD Stage III and thus suffer from kidney damage, we see that some of the identified groups are characterized by trajectories that show almost no change in eGFR values (groups 5-8). On the other hand, groups one to four show a clear deterioration in kidney function after the initial diagnosis. We ordered the groups decreasingly according to the estimated group-level probability of AKI (${\widehat{\alpha }}_k$). As can be seen, a higher estimated probability of AKI generally coincides with a worse eGFR – apart from group seven, eGFR trajectories are monotonically getting better from left to right in Figure 1. Patients assigned to group seven tend to have very low phosphate, indicating that they may have developed severe hyperphosphatemia. However, low phosphate is also associated with AKI, which may explain the larger AKI rate of group seven compared to group six.

Figure 2 shows for each group how the individual-level AKI probabilities develop over time via the quantiles of the estimates. We can see that as time passes after the initial CKD diagnosis, the individual estimates in any group converge towards the group-level estimates ${\widehat{\alpha }}_k$, indicating that individuals are assigned to a single group with a higher and higher probability. The probability estimates vary considerably across groups, confirming the sentiment that AKI and CKD should not be looked at in isolation. In the next section, we formally investigate the performance of the estimated model.

Figure 2.

For each of the eight groups, the median ${\widehat{\alpha }}_{it}$ of all individuals in the respective group is displayed as the green line, with a confidence band of the group-specific AKI probabilities ${\widehat{\alpha }}_k$ overlaid in gray (the point estimates and lower- and upper bounds for the final period are displayed in Table 1). Also displayed are lines for the 10^th and 90^th percentile of ${\widehat{\alpha }}_{it}$ (in blue and red, respectively).

Model Evaluation

We used a five-fold cross validation scheme to obtain unbiased estimates of model performance. Randomly splitting the data set into five equally sized folds, we estimated the baseline model in each case, using all data aside from the fold in question for model training. The estimated model coefficients were then used to calculate the PPGMs of every individual in the held-out fold and subsequently their outcome probabilities ${\widehat{\alpha }}_{it}$ at the various time-periods. In this scheme, the data of every individual is used once to calculate individual-level predictions, without his or her data having been used to train the model from which the predictions were obtained.

Besides the baseline model, which fits trajectories for all five biomarkers jointly with the outcome, we have also included time-stable covariates, namely demographic variables and indicators for the existence of diabetes and hypertension. Since a patient may have AKI more than once over their lifetime, we have additionally estimated a model with an additional risk factor: a binary indicator for an occurrence of AKI prior to the study period. This coefficient turned out to be non-significant for all groups, which is why we report in this section the results of the baseline model.

We have performed two sanity checks to ensure that the model is well calibrated. First, we confirm that the estimated outcome probabilities are highly correlated with the incidence of AKI by regressing AKI occurrence for each patient on the outcome probabilities ${\widehat{\alpha }}_{it}$ for varying times. The results are reported in Table 2. If the probabilities are well calibrated, the resulting regression should have an intercept value close to 0 and a slope close to 1, thus forming a 45-degree line through the first quadrant. This is indeed what we find for T=6, 12, and 18. All intercept estimates are near 0 and the slope estimates are very close to 1, particularly for T=12 and 18.

Table 2.

Displayed are coefficients for regressions of AKI occurrence on estimated outcome probabilities at time 6, 12, and 18.

Outcome
	(1)	(2)	(3)
T=6	0.912***
	(0.070)
T=12		0.963***
		(0.069)
T=18			0.995***
			(0.068)
Constant	0.037	0.016	0.002
	(0.030)	(0.029)	(0.029)
Observations	1,947	1,947	1,947
Residual Std. Error (df = 1945)	0.469	0.467	0.465

^***Notes: Significant at the 1 percent level.

Second, we show that, at various fixed points in time, the average of the predicted outcome probabilities, ${\widehat{\alpha }}_{it}$, closely corresponds to the actual incidence rates of individuals with ${\widehat{\alpha }}_{it}$ inside one of several initially chosen bins of ${\widehat{\alpha }}_{it}$. For this check, we binned the predicted outcome probabilities into intervals with a width of 0.1, which resulted in six ${\widehat{\alpha }}_{it}$ bins overall. The results of this test are reported in Table 3.

Table 3.

Displayed are the observed rates of AKI in bins of the estimated outcome probabilities ${\widehat{\alpha }}_{it}$ in the y-column. The alpha column shows the average of the ${\widehat{\alpha }}_{it}$ coefficients in the respective bin at the given time, with N being the number of patients assigned to the bin.

The endpoints of the six bins are displayed in the LL and UL columns of the table. Inspection of the table shows a very close correspondence between the binned averages of ${\widehat{\alpha }}_{it}$ and incidence of AKI. The predicted probabilities line up well with the actual incidence rates in each bin, even when only the first six quarters are considered. At a significance level of 5%, we do not observe a significant difference.

Table 4 is intended to examine whether the outcome probabilities ${\widehat{\alpha }}_{it}$ have prognostic value. Because an instance of AKI can occur anytime over the course of the 18-quarter observation window, the tabulations of the AKI incidents are limited to those occurring after a specified value of t. The first panel is for t=0 and thus includes all instances of AKI. Here again we see that the AKI probability estimates are well calibrated. The next two panels are for t=6 and t=12. Observed AKI rates (y) in each bin are now lower than the average of the estimated probabilities in that bin (alpha) because incidents of AKI prior to t are excluded from the calculation of y. Note, however, that a strong monotonic relationship between y and alpha still exists, which implies that larger estimates of ${\widehat{\alpha }}_{it}$ imply higher future risk of AKI.

Table 4.

Compared to Table 3, the observed rates of AKI in this table are calculated by only taking those patients into account who have an instance of AKI after the considered time epoch.

We further investigate the observation that the predicted probabilities line up with the actual incidence rates. Figure 3 shows the time taken until trajectory group assignment stabilizes for a patient, where we define PPGMs as stabilized once the highest probability class starts to match that of the final probabilities at the study horizon of 18 quarters. As we can see from the graph, patients are assigned to one of the risk groups early on, with almost 50% of patients being assigned to their final group already by the third quarter and 76.2% by the seventh quarter (less than two years), as indicated by the overlaid dashed line. Looking at the individual bars, we see that most patients stabilize in periods one, two, and three, though groups differ quite a bit in how soon patients are finally assigned to them, with group five being exceptional insofar as many patients are assigned to it already in the first period.

Figure 3.

The overlaid line shows the cumulative percentage of patients for whom the PPGM stabilizes at the given time, which is displayed on the x-axis. Patient’s trajectories are said to be stabilized when their highest probability class starts to match that of the final probabilities after 18 quarters, which is the study horizon.

Table 5 shows the group profiles of the eight detected risk groups in terms of their demographic variables and average biomarker values (calculated over all time periods). As can be seen, the groups differ considerably with regards to the percentage of African-Americans and whether their patients have diabetes, with the average biomarker values reflecting the relationships displayed in Figure 1.

Table 5.

Displayed are demographic variables and biomarker averages (over all periods) of the detected risk groups.

We see the usefulness of our model not primarily in purely predictive purposes, but rather as an analytical tool that may aid the work of clinicians in screening at-risk patients. Hence, we eschew some of the more traditional evaluation metrics for predictive models. However, while a black-box model with careful feature engineering and a broader set of predictors might return better predictions, it would not have the easy interpretability of the trajectory model, which manages to condense a variety of information into a single group variable. And in fact, predictions obtained by the model are reasonable when you consider that only a single variable (the group indicator) is used to predict whether a patient experiences AKI. When looking at the Receiver Operating Characteristic (ROC), the area under the curve (AUROC) stands at approximately 0.7 on the held-out data. As it stands, we believe the model to be good enough to discriminate between patients and thus to serve as a useful tool for medical practitioners. Using it, they may obtain a quick overview of a patient’s risk of AKI and how that risk is connected to the various biomarker values. AUROC values are slightly better than those of a logistic regression fitted using the lab values available at any given time, as displayed in Table 6. Additional markers, particularly that of proteinuria, will likely improve the performance of our model.

Table 6.

Areas under the curve of the Receiver Operating Characteristic (ROC) for both our multi-trajectory model and for comparison purposes a logistic regression at various time points. The logistic regression was estimated using the marker values available at the given point in time.

Model \ Time	t=6	t=12	t=18
Logistic Regression	0.677	0.67	0.672
Multi-Trajectory Model	0.672	0.686	0.692

Figure 4 highlights the potential utility of our approach for medical practitioners based on two example patients that were chosen for illustrative purposes. For both patients, the figure displays both the development of the PPGMs as well as their AKI predictions. While patient 186 starts out being assigned to group four, after a few periods he is placed with high probability in high-risk group one, which coincides with a high prediction for AKI. By contrast, patient 24 emerges as a low risk patient due to the progression of his biomarker values. In this example, neither of the patients ends up experiencing AKI over the course of the study period. It would be up to a physician to conclude whether patient 186 would warrant more scrutiny going forward given his high-risk status as determined by the model.

Figure 4.

For two patients, posterior probabilities of group membership (PPGM) are displayed on the left-hand side, alongside 95% confidence intervals. On the right-hand side, the estimated individual-level AKI probabilities are displayed, again with error bars representing a 95% confidence interval.

Conclusion and Discussion

Building upon our prior work on modeling disease progression of Chronic Kidney Disease (CKD) and related complications via the use of group-based multi-trajectory modeling, we extend the previous analysis by incorporating the occurrence of Acute Kidney Injury (AKI) as an outcome variable into the trajectory model. Current research suggests that CKD and AKI are not as distinct as once assumed, but rather interrelated, with CKD now not only being regarded as one of the main risk factors for AKI, but also a potential long-term consequence of an episode of AKI. Since AKI is an under-diagnosed event that is both preventable and treatable with early detection, there is a clear need for prognostic models that may equip clinicians with tools to anticipate disease progression and to inform timely and appropriate clinical decision-making. Using the estimated glomerular filtration rate (eGFR) as a biomarker for CKD and Hemoglobin (HGB), parathyroid hormone (PTH), phosphate (PO4), and carbon dioxide in bicarbonate form (HCO3) as markers for some of the main complications of CKD (Anemia, Secondary Hyperparathyroidism, Hyperphosphatemia, and Metabolic Acidosis), we have estimated an eight-group CKD progression model over a time horizon of 18 quarters, which jointly estimates the trajectories of these markers as well as the relationship between the individual groups and our outcome of interest, AKI. The eight groups identified by the group-based trajectory model (GBTM) show distinct trajectories for all biomarkers and differ sharply in terms of their estimated AKI probabilities, with the high-risk group having an upper estimate of 0.78, whereas the group-level estimates for the lowest-risk group start at 0.12. Confirming the strong relationship between AKI and CKD, we see that a higher group-level AKI probability goes hand in hand with a worse eGFR value. Several calibration checks have confirmed that the estimated model is well calibrated and has prognostic value.

The usefulness of GBTMs for clinicians is enhanced by the fact that group membership can be predicted when data is available only for a subset of all time periods. Predictions can be updated once additional data becomes available. While singular events such as an episode of AKI might cause the estimated group probabilities to change significantly, it is of interest to investigate the overall likelihood that patients change group assignments as new data becomes available. We have demonstrated that groups tend to stabilize early, with 46% of patients being assigned to the maximum a posteriori group after only three quarters of data, and more than 75% within two years. Since it is not too common for patients to switch groups over time, such a model could be utilized by clinicians without the fear of having to work with information that is likely to become obsolete soon.

By showing how group assignments and estimated AKI probabilities of two patients develop over time, we demonstrate how individual-level patient predictions together with the group-level information in the form of AKI probabilities and trajectory estimates may aid clinicians in early-detection and disease management. To move from this illustration to deployment of our methods in a clinical care setting, several existing limitations will have to be overcome. Our model may be sub-optimal due to a lack of mortality data and missing marker values for proteinuria, which is a critical risk factor for AKI besides CKD. Given that patients in our data set are quite homogeneous in terms of demographic variables such as race and age, it remains an open question how well results will generalize to a broader patient population.

Since the biomarkers are only infrequently observed when patients go to appointments with their doctors, diagnostic delays are inevitable. Further research should be undertaken to investigate what factors cause AKI or a deterioration in one or more of the biomarkers whenever they occur. Should prediction of the exact times at which AKI occurs be a goal, different model formulations would have to be explored. Furthermore, we did not take account of the fact that AKI is not a singular event, and that some patients might suffer from several episodes of AKI over time. A finer-grained analysis that takes the times and frequency of AKI episodes into account may provide further insights that our model does not currently address.

One might also question the model assumption that conditional on group membership, the observations of a trajectory at different times are uncorrelated with each other. However, this assumption is not as restrictive as it might sound: the biomarker trajectories are modelled to be conditionally independent only at the group and not the population level. Specifically, the model assumes that conditional on the latent group membership, the Gaussian noise added to the trend line is drawn from the same distribution at all time periods. These limitations aside, we believe that group-based trajectory modeling constitutes a simple yet powerful tool for risk stratification that provides easy-to-visualize-and-interpret developmental trajectories to clinicians. The ability to jointly estimate the trajectories together with an outcome of interest allows one to obtain easily interpretable risk profiles of patients. Equipped with this information, clinicians may be better able to anticipate the future trajectory of disease progression. This in turn could lead to an improvement in timely and appropriate clinical decision-making.