Dynamic landmark prediction for mixture data

Summary

In kin-cohort studies, clinicians want to provide their patients with the most current cumulative risk of death arising from a rare deleterious mutation. Estimating the cumulative risk is difficult when the genetic mutation status is unknown and only estimated probabilities of a patient having the mutation are available. We estimate the cumulative risk for this scenario using a novel nonparametric estimator that incorporates covariate information and dynamic landmark prediction. Our estimator has improved prediction accuracy over existing estimators that ignore covariate information. It is built within a dynamic landmark prediction framework whereby we can obtain personalized dynamic predictions over time. Compared to current standards, a simple transformation of our estimator provides more efficient estimates of marginal distribution functions in settings where patient-specific predictions are not the main goal. We show our estimator is unbiased and has more predictive accuracy compared to methods that ignore covariate information and landmarking. Applying our method to a Huntington disease study of mortality, we develop dynamic survival prediction curves incorporating gender and familial genetic information.

1. Introduction

Prediction models are imperative for deciding treatments, communicating prognosis to patients, and designing clinical trials to assess disease-modifying therapies. A challenge to developing prediction models is when the models are based on so-called mixture data, whereby data are collected from multiple populations of interest, but we do not know from which population each observation came. Instead, we only know the probability an observation belongs to each population.

Such mixture data are commonly observed in kin-cohort studies (Wacholder and others, 1998) of mortality in Huntington disease (HD). HD is a rare, genetic disorder caused by a genetic mutation: at least 36 repeats of the trinucleotide cytosine–adenine–guanine (CAG) in the huntingtin gene (Rubinsztein and others, 1996). In kin-cohort studies of HD, genotype information is first obtained from (usually) diseased individuals known as probands. To increase the sample of individuals with the genetic mutation, disease history and mortality information are collected on the probands’ relatives using systematic interviews (Marder and others, 2003). Because of resource constraints, the relatives are not genotyped so we do not know if the relative is truly a carrier or non-carrier of the genetic mutation. But we may compute the probability a relative is a carrier or non-carrier because the mutation is genetically inheritable (Khoury and others, 1993) (see Section S.1. of the Supplementary material available at Biostatistics online for details). Data from the relatives are therefore an example of mixture data: the populations of interest are carriers and non-carriers of the genetic mutation, and each relative’s identity of being a carrier or non-carrier is unknown. Instead, we only know the probability of each relative being a carrier or non-carrier.

Early prediction models for mixture data involved parametric (Wu and others, 2002) and semiparametric (Zeng and Lin, 2007) assumptions of the populations from which data were collected. These models were susceptible to bias when the proposed (semi)parametric models were misspecified.

While nonparametric estimators helped bypass this misspecification, nonparametric maximum likelihood estimators (Wacholder and others, 1998; Chatterjee and Wacholder, 2001; Fine and others, 2004) for mixture data have been shown to be inefficient or not consistent (Wang and others, 2012). Wang and others (2012) and Ma and Wang (2014a) remedied these limitations by developing a class of nonparametric estimators that were consistent and efficient using Hilbert spaces and semiparametric projection (Tsiatis, 2006).

Unfortunately, these nonparametric estimators ignored any auxiliary covariate information that could improve prediction accuracy. Approaches to ameliorate this issue have been proposed but to a limited extent. Ma and Wang (2014b) introduced auxiliary information to a mixture data model where the distribution for each population was modeled semiparametrically. The distributional form, however, was chosen to represent HD after modeling the disease onset process for individuals with the gene mutation (Langbehn and others, 2010). This distributional form is not guaranteed to also represent the survival process for HD. We would need to evaluate the mortality process to accurately choose the distributional form. Wang and others (2017) proposed transformation models to incorporate auxiliary information for each population of the mixture data. The transformations may be completely unspecified, but one must accurately pre-specify the distributions for each mixture population to maximize prediction accuracy. Doing so is difficult without sufficient scientific knowledge. See Garcia and others (2016) for more comparison of all aforementioned methods.

In addition to incorporating auxiliary covariate information to improve prediction accuracy, it is also possible to improve prediction accuracy for individual patients by dynamically updating predictions over time. Dynamic landmark prediction (van Houwelingen and Putter, 2011; Parast and others, 2014) is one framework to achieve this goal whereby predictions for individuals are updated over time, conditioning on the event (e.g., death) not having yet occurred up to a landmark time (i.e., a reference point that takes into account all of the predictor information at that moment). Dynamic landmark prediction is adaptive and more accurate than traditional prediction models which often ignore new information available after baseline, and has not yet been applied to a mixture data setting.

We develop a nonparametric prediction model for mixture data that flexibly incorporates auxiliary covariate information and dynamic landmark prediction. Our contributions are 3-fold. First, we use a fully nonparametric approach to relate censored outcomes from mixture data to auxiliary covariate information. The approach is applicable to any time-to-event outcome and does not require pre-specifying distributional assumptions for the mixture populations. Second, we incorporate dynamic landmark to obtain personalized dynamic predictions over time that uses all updated covariate information. Third, we show our estimator gives more efficient marginal distribution estimates in settings where patient-specific predictions are not the primary goal.

The remainder of the article is as follows. Section 2 describes our proposed estimator, its asymptotic properties, and measures to evaluate prediction accuracy for mixture data. Empirical results in Section 3 show the improved finite sample performance over existing nonparametric methods both in terms of consistency and predictive accuracy. In Section 4, we apply our method to the kin-cohort mortality study of HD, examining the use of gender and CAG repeat length (of the proband) to estimate mortality in the relatives. We then create personalized dynamic risk trajectories over time for relatives whose genotype information is unknown.

2. Dynamic landmark prediction model for mixture data with covariates

2.1. Notation and model framework

Mixture data involve potentially censored outcomes observed from one of populations, but population identifiers are unknown. The observed data are , , where and with denoting a failure time and denoting a continuous random censoring time. We assume and are conditionally independent given covariates and , which denote discrete and continuous covariates, respectively. We assume and are one-dimensional, but we discuss extensions to the multivariate case in Section 5.

The failure times originate from one of populations, but we only observe the probability vector , where is the probability originates from population , and . We make the following assumptions about : (i) is known or exactly estimable from external data. When is measured with error, we could use measurement error techniques (Carroll and others, 2006) to assess and correct the misspecification, but this procedure is beyond the current scope of the paper. (ii) has a discrete distribution with probability mass function and finite support ; (iii) For each possible stratum and each neighborhood around , for all . This ensures there is a positive probability that an individual belongs to a mixture proportion group.

For the kin-cohort study of HD mortality (Section 4), there are populations referring to carriers and non-carriers of the genetic mutation. Data are collected on relatives, and for relative , denotes the age of death, and contains the probabilities a relative is a carrier and non-carrier, respectively. In the HD study, is one of vector values: . In terms of covariates, denotes the relative’s gender and denotes the CAG repeat length of the relative’s proband.

Our key objective is to develop a nonparametric prediction model for mixture data that incorporates the covariate information and dynamic landmark prediction. We will achieve this by estimating where

Here, denotes the population to which a subject truly belongs and can take values from . We use only for definition purposes, but is not observed in practice. The term is the cumulative distribution function for population with characteristics given that the failure time has not yet occurred up to time .

The time represents the landmark time which we will use to provide more informative estimates of risk. When , is simply a prediction calculated at baseline. When , provides updated predictions of survival. For example, in the HD study, where time is on the age scale, individuals in genetic counseling may be interested in learning their estimated survival probability to age 60 given their current age is 50 (i.e., the landmark time ) and their covariate information. At a follow-up visit 5 years later, the patient may now be interested to know the updated prediction of surviving to age 60 given they have now survived to age 55 (i.e., the landmark time ). Previous work outside the context of mixture data has shown that using updated knowledge of survival improves prediction accuracy (Parast and others, 2014). When covariates may also be updated before or at , these predictions could incorporate these updated values potentially further improving prediction accuracy. In this case, we would write and to be explicit about the dependence on . Since our numerical studies do not include updated covariates, we use the simplified notation of throughout.

2.2. Modified nonparametric kernel Nelson–Aalen estimator

2.2.1. Structure of the main estimator

To estimate , we take advantage of the finite support of . The finite support implies that each individual uniquely belongs to one of different subgroups based on his or her value. We will develop estimators for the subgroups defined by individuals whose , , and relate these estimators to .

For , let which is a valid survival function based on data for which . Let be the -dimensional vector containing the survival functions for all subgroups. The relationship between and is where is the matrix of mixture probabilities, and is an -dimensional vector of ones. From this relationship, we recover the estimator of interest as . But this solution is the least squares estimator with covariates and response . As with least squares estimation, we can improve its efficiency using a weighted least squares solution with weight matrix, :

(2.1)

Without covariates and landmarking, the estimator in (2.1) reduces to the estimator of Ma and Wang (2014a). In their work, is replaced by whose th component is the Kaplan–Meier estimator based on data for which . Our estimator thus generalizes the work of Ma and Wang (2014a) to include covariates and landmark times.

Forming the estimator in (2.1) requires estimating the survival vector and selecting the weight matrix to minimize variability. We discuss both choices next.

2.2.2. Components of the main estimator

To estimate , we use a modified nonparametric kernel Nelson–Aalen estimator (Du and Akritas, 2002). For , we set where

(2.2)

Here, denotes the set of individuals , denotes the counting process of event times, denotes the at-risk process, where is a known smooth symmetric kernel density function with bounded support and bandwidth with where . In all numerical examples, is the Gaussian kernel and we set , where is obtained from the bandwidth selection procedure of Scott (1992); the calculated can be shown to be and thus the resulting . If our only interest was estimating , it would be sufficient to require that ; however, because we use to estimate the marginal distribution (see Section 2.4) and to calculate accuracy estimates (see Section 2.3), we require under-smoothing such that (Cai and Zheng, 2011 ).

The form of the cumulative hazard, , accounts for different aspects of our mixture data. First, it uses the mixture probabilities by restricting estimation to those individuals for which . Second, it accounts for being discrete via the indicator function, and the continuity of using the kernel . Last, it accounts for landmarking via the restriction to , which modifies the estimation to those individuals known to survive past landmark time .

We now discuss how we chose in (2.1) to minimize the variability of . A few remarks help justify our choice. Each individual uniquely belongs to one of different subgroups based on his value. This means is constructed based on independent subgroups of data. The independence implies that is a diagonal matrix. Choosing the diagonal elements of as would asymptotically minimize the covariance of , but could be numerically unstable when the th subgroup is too small. Ma and Wang (2014a) showed this instability when estimating survival curves for mixture data without covariates or landmarking. In our setting, the instability could increase after incorporating landmarking and covariates because of a further reduction in the subgroup sample sizes.

To avoid this instability, Ma and Wang (2014a) found that there were minimal differences in estimation efficiency when each observation had equal weights or was weighted by its inverse variance. For this reason, they proposed to assign equal weights to each observation in the th subgroup. This corresponds to setting where is the number of individuals with . In our setting, we also found that using was numerically stable and led to minimally worse efficiency than when using . We thus also set .

2.2.3. Summary of the main estimator and asymptotic properties

Under our choice for , our estimator in (2.1) can be equivalently written as

where and is as in (2.2).

Under mild regularity conditions stated in Section S.2. of the Supplementary material available at Biostatistics online, in (2.2) is a consistent estimator of following similar arguments as those in Dabrowska (1989). They proved the kernel Nelson–Aalen estimator is uniformly consistent by deriving an exponential probability bound for the tails of distributions of kernel estimates for the conditional survival functions. Because is a linear combination of consistent estimators , , it follows that is also consistent.

In Section S.2. of the Supplementary material available at Biostatistics online, we also show is asymptotically normal with

An analytic version of the estimated covariance above requires computing , where is the cumulative hazard function in (2.2). This computation involves an integral that is not easy to compute analytically, which is why we and others (see Cai and others, 2010) chose to estimate the covariance using the bootstrap approach.

An important point of our estimator is that we assume is finite and less than the sample size. This allows us to divide individuals into subgroups defined by mixture proportions, construct the nonparametric kernel Nelson-Aalen estimator for that subgroup, and recover . When , we would require a different framework than the one proposed here.

2.3. Evaluation of prediction accuracy

To evaluate our model’s predictive performance, we used nonparametric procedures to quantify discrimination (i.e., how well the model distinguishes between individuals who do and do not experience death) and calibration (i.e., agreement between outcomes and predictions).

To assess discrimination, we use the widely familiar area under the receiver operating characteristic curve (AUC) (Heagerty and Zheng, 2005) defined as

For each , the AUC is the probability that a randomly selected individual experiencing the event (i.e., ) will have a higher predicted probability of having the event than a randomly selected individual not experiencing the event (i.e., ). Similar to Parast and others (2011), we will estimate the AUC using inverse probability of censoring weights such that

where is the Kaplan–Meier estimator for the censoring distribution and When there are ties in the estimated values, in the numerator should be replaced with

This accommodation for ties is used in all AUC calculations in our numerical studies.

We also examine prediction error using the Brier Score (BS) (Brier, 1950; Gerds and Schumacher, 2006) which is the mean squared distance between the outcome and the predicted risk, . It is defined as

and can be estimated as below where :

Both the definitions and estimates of AUC and the BS rely on , the unknown population identifier. This means that these quantities can only be calculated if the population identifiers are known, which will be true in the simulation study but not in practice. We cannot compute the AUC and the BS in practice because with the HD application, for example, our estimation procedure will result in with (carrier and non-carrier). Thus, each person will have two predictions, one if they are truly a carrier and one if they are truly a non-carrier. Without knowing their true carrier status, the AUC and BS cannot be calculated since we would have to arbitrarily use one of these two predictions to compute the AUC and the BS.

2.4. Efficient estimation of the marginal distribution functions

We now describe how our estimator can improve efficiency in cases where patient-specific predictions are not the primary goal. For example, families at risk for HD may want to know predicted survival rates for carriers irrespective of covariates (e.g., gender and a proband’s CAG repeat-length). In this case, rather than estimating , the interest is estimating the marginal distribution which ignores covariates.

To estimate , the methods from Wang and others (2012) and Ma and Wang (2014a) could certainly be used to yield unbiased estimates. However, by simply averaging over the covariates in our estimator, we show we can obtain a more efficient estimator for the marginal distributions.

For ease in presentation, we temporarily ignore landmark times, but this can be re-incorporated. Through an argument of conditional expectations, we have that

Therefore, an estimator for the marginal distribution is

(2.3)

where is exactly the th component of our estimator in (2.1). This marginal estimator is thus obtained by forming our estimator in (2.1) at all covariate values , , and then averaging over them. Based on the theoretical results in Parast and others (2014), such averaging yields an unbiased estimate. When the covariates are truly associated with survival, it has improved efficiency relative to a Kaplan–Meier estimator. We therefore expect that the estimator in (2.3) would be unbiased and more efficient than that of Ma and Wang (2014a) which used the Kaplan–Meier estimator (see Section S.3. of the Supplementary material available at Biostatistics online for a theoretical justification of the improvement). Of course, this is only true if the individuals in each subgroup are representative of that subgroup. If not, our estimator and the Kaplan–Meier estimator would not reflect the marginal distribution in the population of interest.

3. Simulation study

3.1. Study design and performance evaluation

We empirically evaluated our proposed estimator in three ways:

Value of incorporating covariates: We evaluated the importance of incorporating covariates by comparing our modified nonparametric kernel Nelson–Aalen estimator (or NPNA for short) to the Kaplan–Meier estimator of Ma and Wang (2014a) (or KM for short) which ignores covariates. Their estimator replaces in (2.1) with , where is a Kaplan–Meier estimator based on data for which , .

Value of landmarking: We evaluated the value of landmarking by comparing our NPNA estimator at to a version that ignores landmarking. The modified version is the NPNA estimator that fixes regardless of the true landmark time . This estimator is denoted as NPNA and is the prediction calculated at baseline. We also compared our results to a “landmark-specific KM” computed as where . Each term in is the Kaplan–Meier estimator of the survival function. The landmark-specific KM adjusts for landmarking but not covariates.

Efficiency gains: We evaluated the efficiency gain of the marginal version of our estimator, denoted as NPNA and computed as in (2.3), to the KM estimator. The marginal estimate evaluates the overall survival to time in each subgroup irrespective of covariates.

For all comparisons, we used two data settings. In Setting 1, the survival distributions were log-normal. For , we set with and to generate different noise levels for each population. Covariate was generated as a Bernoulli(0.5), and from uniform[0,1]. In Setting 2, the survival distributions mimicked those from the HD kin-cohort study (Section 4): survival estimates for carriers were lower than for non-carriers, and decreased with increasing CAG repeat length. For the carrier group, we set for where , is a normalizing constant, and . For the noncarrier group, we set for and for where is a normalizing constant and . Covariate was generated as a Bernoulli(0.5) to mimic gender, and from a uniform[35,55] to mimic the observed CAG repeat lengths.

For both settings, we set sample size , mixture populations, and different mixture proportions: . Subjects were equally likely to have each of the mixture proportions. Censoring times were generated from a uniform distribution that yielded 40% censoring. We simulated 200 data sets and estimated the survival distributions, . For Setting 1, we estimated survival probabilities at , , , and landmark times . For Setting 2, we estimated survival probabilities at , , , and landmark times .

We evaluated estimators in terms of bias, standard error, mean squared error (MSE), and coverage. Bias is reported as absolute average bias and is computed as for where denotes the set of plausible time points; note that must be larger than . We had for Setting 1 and for Setting 2. In addition, denotes the truth so that we are comparing our estimates to the true values, and denotes the average estimate based on 200 simulations. Standard error, MSE, and coverages are reported as averages taken over . Estimated standard errors for both estimators were obtained by resampling among the subjects using 100 bootstrap replicates. These were compared against empirical standard errors, computed as the standard errors of across all 200 simulations at each combination of .

We also evaluated prediction accuracy using the AUC and BS. Importantly, better prediction accuracy is indicated by higher values for the AUC and by lower values for the BS. Evaluating prediction accuracy requires knowing the population identifier which is, of course, known in simulated data (but not in practice). We computed the AUC and BS as described in Section 2.3, plugging in if the th participant was truly from the th group, i.e., using the true value. The computation of AUC and BS was done using cross-validation to ensure that we were evaluating the prediction accuracy on different data than the one used to estimate the survival distributions. Results in all tables were multiplied by 100 as bias and standard errors were considerably small and difficult to see otherwise.

3.2. Results

For all settings, the NPNA estimator consistently demonstrated negligible bias, low MSE, and 95% coverages near the nominal level (Table 1). In contrast, the NPNA estimator, which adjusts for covariates but ignores landmarking, and the KM estimator, which adjusts for landmarking but ignores covariates, had increased bias, higher MSE, and lower coverage probabilities compared to the NPNA estimator. The poor performance of the KM estimator is attributed to its lack of adjustment for influential covariates; see Tables S.1. and S.2. and Figure S.1. of the Supplementary material available at Biostatistics online to see its bias even when estimating survival curves from baseline. The poor performance of the NPNA estimator is due to its failure to adjust for landmarking when needed.

Table 1.

Estimation performance at landmark times of the proposed nonparametric kernel Nelson-Aalen estimator (NPNA) compared to the estimator that ignores landmarking (NPNA estimator assuming , denoted as NPNA), and the Kaplan–Meier estimator that ignores covariate influence (KM) and modified to adjust for landmarking

	KM	NPNA	NPNA	KM	NPNA	NPNA
	Setting 1:
abs bias	7.0813	0.2352	2.9457	3.1558	0.2564	13.9247
emp var	0.0465	0.3472	0.3647	0.0554	0.4489	0.4199
est var	0.0481	0.3732	0.3813	0.0492	0.3681	0.3755
95% cov	10.3333	94.7222	92.7778	68.3333	90.5556	37.7778
MSE	0.5570	0.3741	0.4750	0.1530	0.3690	2.3430
	Setting 1:
abs bias	4.4770	0.3764	11.3412	2.1490	0.2543	24.5883
emp var	0.0523	0.3809	0.3848	0.0487	0.4277	0.4342
est var	0.0538	0.3664	0.4007	0.0500	0.3324	0.3865
95% cov	51.8125	92.3125	54.1250	84.3125	89.8125	3.9375
MSE	0.2680	0.3683	1.7336	0.1015	0.3333	6.4955
	Setting 2:
abs bias	2.0295	0.7752	4.0646	1.8468	1.5081	1.5619
emp var	0.0220	0.1795	0.2064	0.0262	0.1659	0.1889
est var	0.0218	0.1582	0.1773	0.0231	0.1455	0.1594
95% cov	67.7500	93.0000	87.2500	77.2500	89.2500	91.5000
MSE	0.0682	0.1702	0.4119	0.0573	0.1765	0.1890
	Setting 2:
abs bias	1.7798	2.0042	1.5612	2.2796	2.6579	2.1070
emp var	0.0080	0.0562	0.0406	0.0401	0.2249	0.2155
est var	0.0087	0.0521	0.0378	0.0346	0.1957	0.1825
95% cov	52.5000	93.0000	95.0000	79.0000	90.0000	90.0000
MSE	0.0404	0.0922	0.0622	0.0866	0.2664	0.2269

We report average absolute bias (abs bias), empirical variance (emp var), estimated bootstrap variance (est var), 95% coverage, and MSE for at specified and landmark times . Results are averaged over for Setting 1 and in Setting 2 and summarized over 200 simulations with 40% censoring. Sample size is 2000. All results are multiplied by 100.

These deficiencies of the NPNA estimator and KM estimator resulted in lower AUC and higher BS than the NPNA estimator (Table 2). However, only the AUC differences between the NPNA and KM estimators were significantly different. Still, these results indicate that the NPNA and KM estimators do not discriminate or calibrate well.

Table 2.

Prediction accuracy at landmark times of the proposed nonparametric kernel Nelson-Aalen estimator (NPNA) compared to the estimator that ignores landmarking (NPNA estimator assuming , denoted as NPNA), and the Kaplan–Meier estimator that ignores covariate influence (KM) and modified to adjust for landmarking

	KM	NPNA	NPNA	NPNA-KM	NPNA-NPNA
	AUC, Setting 1:
	51.2854	64.4163	61.4384	(7.4368, 18.8251)	(0.6716, 6.6274)
	55.1505	64.5175	62.8209	(5.0543, 13.6799)	(0.3956, 3.7888)
	57.652	65.9349	64.7988	(4.0615, 12.5044)	(0.5791, 2.8514)
	BS, Setting 1:
	13.5879	13.084	15.2423	(0.9247, 0.0831)	(2.8356, 1.4809)
	23.3485	22.0813	23.2354	(2.0813, 0.4531)	(1.7654, 0.5428)
	24.369	22.9809	23.6991	(2.3385, 0.4377)	(1.2936, 0.1428)
	AUC, Setting 1:
	56.718	64.7069	61.1501	(3.0307, 12.9472)	(0.204, 6.9096)
	58.876	65.9604	63.6971	(2.7056, 11.4633)	(0.2317, 4.7584)
	BS, Setting 1:
	19.0458	18.2571	22.9205	(1.5142, 0.0632)	(5.7315, 3.5954)
	23.6235	22.4671	25.2915	(2.1805, 0.1323)	(3.833, 1.816)
	AUC, Setting 2:
	68.672	82.9663	83.0024	(9.9418, 18.6468)	(1.3709, 1.2988)
	69.2237	82.9825	83.0218	(10.1455, 17.372)	(0.5834, 0.5048)
	70.476	83.3167	83.3574	(8.9964, 16.685)	(0.4441, 0.3628)
	BS, Setting 2:
	15.7572	12.7593	12.8608	(4.0232, 1.9726)	(0.5754, 0.3724)
	21.0976	16.6319	16.5867	(5.8972, 3.0343)	(0.2675, 0.3579)
	18.859	15.3791	15.3327	(4.7729, 2.1868)	(0.1358, 0.2287)
	AUC, Setting 2:
	66.1443	78.9501	79.3607	(6.4, 19.2115)	(4.5741, 3.7529)
	66.8676	79.7724	79.9264	(9.0036, 16.806)	(1.3163, 1.0083)
	68.1474	80.6619	80.7738	(8.7062, 16.3229)
	BS, Setting 2:
	9.2491	8.3604	10.3928	(1.5438, 0.2337)	(3.2139, 0.8508)
	20.6777	17.2407	17.3808	(4.8153, 2.0585)	(0.9188, 0.6387)
	20.6045	17.2658	17.1442	(4.6848, 1.9927)	(0.299, 0.5421)

These columns correspond to the 95% bootstrap confidence intervals for the differences between estimators. We report the AUC and the BS for at specified and landmark times . Results are summarized over 200 simulations with 40% censoring. Note that better prediction accuracy is indicated by higher AUC and lower BS. Sample size is 2000. All results are multiplied by 100.

These observations were highly evident in Setting 1 and less so in Setting 2. For Setting 2, the NPNA estimator had lower bias and MSEs compared to the NPNA estimator for (carrier group), but not for (non-carrier group) at . This result is reasonable given that Setting 2 mimics the HD study where knowledge that a non-carrier survived to age or years is uninformative because healthy populations (i.e., non-carriers) do not normally die in the first three decades of life. That is, knowing a non-carrier lived to ages or years does not improve survival predictions, nor prediction accuracy (the AUC and BS for NPNA and NPNA were similar at and ).

Last, when patient-specific predictions are not the primary goal, we show in Table 3 that the marginal version of our estimator, NPNA, and KM have negligible bias when estimating . For this table, the true is the marginal form of the distributions in Settings 1 and 2 obtained by integrating over at . We do observe a slightly higher bias in the NPNA estimator because, before marginalizing over covariates, the NPNA estimator can have small subgroup sizes as it adjusts for covariates. Though the increased bias is relatively small in this simulation study, caution should be used in practice when using this approach to obtain a marginal estimate if there are small subgroup sizes. Results in Table 3 also show that the NPNA estimator has roughly 6–22% higher efficiency than does the KM estimator implying that the NPNA yields more precise estimates. This helps improve power in hypothesis testing when, for example, comparing the marginal survival distributions between carriers and non-carriers.

Table 3.

Marginal estimation results of the proposed nonparametric kernel Nelson-Aalen estimator (NPNA) and the Kaplan–Meier based estimator (KM)

	KM	NPNA	KM	NPNA
	Setting 1:
bias	0.1964	-0.0544	0.1509	-0.2704
emp var	0.0324	0.0332	0.0409	0.0397
est var	0.0332	0.0302	0.0358	0.0331
95% cov	96.5000	94.0000	93.5000	92.5000
MSE	0.0336	0.0303	0.0360	0.0339
rel eff	100.0000	90.9975	100.0000	92.5993
	Setting 1:
bias	-0.0003	-0.5523	0.1674	-0.4324
emp var	0.0485	0.0492	0.0498	0.0495
est var	0.0471	0.0422	0.0439	0.0405
95% cov	96.5000	91.5000	93.0000	89.5000
MSE	0.0471	0.0453	0.0441	0.0424
rel eff	100.0000	89.6270	100.0000	92.3248
	Setting 2:
bias	0.2651	-0.2741	-0.0983	-0.1784
emp var	0.0387	0.0318	0.0232	0.0195
est var	0.0402	0.0323	0.0207	0.0176
95% cov	93.0000	94.0000	92.5000	93.0000
MSE	0.0409	0.0330	0.0208	0.0179
rel eff	100.0000	80.3382	100.0000	84.8547
	Setting 2:
bias	0.0914	-0.7803	-0.1364	-0.3885
emp var	0.0571	0.0439	0.0448	0.0400
est var	0.0523	0.0404	0.0400	0.0329
95% cov	93.0000	93.0000	90.0000	90.0000
MSE	0.0524	0.0465	0.0402	0.0344
rel eff	100.0000	77.3532	100.0000	82.2426

We report bias, empirical variance (emp var), estimated bootstrap variance (est var), 95% coverage and MSE, and relative efficiency (in terms of variance relative to KM) for at specified . Results summarized over 200 simulations with 40% censoring. Improved efficiencies from the NPNA relative to KM are boldfaced. All values are multiplied by 100.

Reported results were from simulated data with a sample size of which is similar to the HD study (Section 4). We ran the analysis at sample size and found that comparative results were similar (see Tables S.3. to S.7. of the Supplementary material available at Biostatistics online).

4. Application to kin-cohort mortality study of HD

4.1. Clinical research problem

We applied our NPNA estimator to a large, multicenter, observational study of HD: the Cooperative Huntington Observational Research Trial (COHORT; Dorsey and Huntington Study Group COHORT Investigators, 2012). COHORT was conducted at 38 sites in the USA, Canada, and Australia from 2005 to 2011, but site information was not available to protect the identity of participants. COHORT collected clinical and genetic data from symptomatic and presymptomatic HD mutation carriers (probands) and their family members. During this time period, probands were systematically interviewed either face-to-face or over the telephone to ascertain neurological disease and mortality information about first-degree relatives. This included relatives who had died prior to the study and those who died during the study period. Relatives were thus “followed” from birth to death or to the end of the study period where if they were still alive, their age of death was censored. The family history interviews were administered by a second person to ensure validity of the information collected.

We analyzed data for the 3291 first-degree relatives of the probands. There were 51% males and 49% females composed of 29% kids, 32% parents, and 39% siblings of the probands. Genotype information for relatives was not available due to resource constraints in obtaining in-person blood samples. Thus, exact knowledge of whether a relative was a carrier or non-carrier of the genetic mutation that causes HD was unknown. But because HD is genetically inheritable, we could compute the probability a relative inherited the gene mutation (see Section S.1. of the Supplementary material available at Biostatistics online for details). We ultimately had distinct mixture probabilities: 64.18% of the population had a mixture proportion of (0.5,0.5), 27.89% of the population had a mixture proportion of (0.97,0.03), and 7.93% of the population had a mixture proportion of (0.25,0.75). We do assume these mixture proportions are correctly estimated. In Section 5, we discuss the impact when the mixture proportions are measured with uncertainty.

Using data from the relatives, we dynamically predicted the survival age () distributions for carrier and non-carrier relatives after adjusting for gender of the relative () and the CAG repeat-length of the relative’s proband (). The relative and his proband may not have the same CAG repeat-length, but because HD is a hereditary disorder, we hypothesize that the proband’s CAG repeat-length could impact the relative’s survival distribution.

Results from this analysis will lead to two predictions of survival for each individual, one if they are truly a carrier and one if they are truly a non-carrier. In genetic counseling sessions, where individuals are interested in learning the impact of HD on their lifestyle, this information can help in three ways. First, for individuals who prefer not to pursue genetic testing, these predictions can help them understand the natural history of the disease based on their specific information. This is because our models personalize the predicted probabilities of survival based the individual’s gender, familial genetic information, and updated mortality information (e.g., if an individual previously visited 5 years ago and is still alive today). This personalization improves the accuracy of mortality rate information given to individuals. Second, hearing these predictions may influence the decision to obtain genetic testing so that an individual can precisely know his/her predicted risk of mortality and begin to seek out care management if needed. Third, results from our models could help recruit participants for future studies. Specifically, results from our models will identify participants who have a specific probability of inheriting the gene mutation and a specific rate of mortality, but their exact gene mutation status is unknown. This information could help recruit participants for an observational study of disease progression and mortality where neither the participants nor clinical investigators know the patients’ gene mutation status to avoid subjective bias. This idea was actually the exact premise for the PHAROS study of HD (Huntington Study Group PHAROS Investigators, 2006) which helped to objectively assess disease progression in adult participants who had a 50% chance of being a carrier.

One possible concern of our analysis is the feasibility of making dynamic predictions of survival for relatives when the baseline is birth and the proband was born after the relative. Consider the following example. Suppose the proband is a child and the relative is a parent, so that at baseline (the parent’s birth), the proband was not even born. The concern is that in this scenario, we would not know the child-proband’s genotype information () and thus could not use it to make dynamic predictions for the parent-relative. But this would never arise in our scenario. We collect data on relatives from the proband. That is, we first sample the child-proband, and then collect mortality information about his parent which is fully available because we can retrospectively collect this information from the child-proband. Thus, we would indeed have the data to assess the mortality information about the parent-relative.

4.2. Results

The proband’s CAG repeat-length impacted the relative’s likelihood of survival (Figure 1). Figure 1 shows the predicted probability that a relative who is a male carrier survives 10 years after having survived to age , years. These probabilities are displayed as a function of the proband’s CAG repeat-length. Results are shown for the NPNA estimator which adjusts for CAG repeat-length and gender, and its marginalized version (NPNA) which does not. Comparing these two helps evaluate the impact of a proband’s CAG repeat-length on the relative’s mortality. For relatives who survived up to or years, the predicted probability of surviving another 10 years did not change as the proband’s CAG repeat-length changed (i.e., the estimates from the NPNA and NONA were similar across CAG repeat-length). Because HD typically onsets in the fourth decade of life (Ross and others, 2014) and death typically occurs 15–20 years after (Rinaldi and others, 2014), it is reasonable to see that CAG repeat-length had little impact on survival during the first 30–50 years of life. However, we found that the probability of survival at older ages depend on the proband’s CAG repeat-length. The probability of survival past 70 years for a male carrier who survived to age 60 and whose proband has 40 CAG repeats is 0.84 (95% CI 0.72–0.96) using the NPNA estimator and 0.59 (95% CI 0.55–0.63) for NPNA.

Fig. 1.

Predicted likelihood of survival 10 years past for relatives who are mutation carrier males in the COHORT study. Results are shown as a function of the proband’s CAG repeat-lengths. Results displayed are for the proposed nonparametric kernel Nelson-Aalen estimator (NPNA) and its marginalized version (NPNA) to evaluate the impact of CAG repeat-length on predicted survival.

Figure 2 illustrates the benefit of incorporating information about survival up to a landmark point when predicting future survival. The curves in Figure 2 display the dynamic probability of survival to age given that an individual has survived to age for years. The curves are left-truncated at because the probability an individual survives to age given he has survived to age only make sense when . Results are shown for relatives who are male carriers and their probands had 46 CAG repeats; results for female relatives were similar. Results are based on estimates from the NPNA estimator. Accounting for survival to through landmarking changes the predicted probability of survival, as would be expected. For example, at baseline () the predicted probability of surviving to age 70 () is 0.19 (95% CI 0.07–0.3), shown by the solid curve. However, if this relative survives to age 60 (), the updated probability that he will survival to age 70 is much higher at 0.5 (95% CI 0.27–0.73), shown by the lightest dashed curve. That is, without the use of a landmark prediction approach, this patient would be told that their probability of surviving to age 70 is 0.19 when in fact, it is 0.50, given that they have already survived to age 60. This difference in predictions can be very meaningful particularly when treatment decisions and/or life decisions are informed by these estimates.

Fig. 2.

Dynamic landmark predictions of survival past age given survival to age in the COHORT study, . Estimates are shown as a function of age (in years) for relatives who are male, mutation carriers whose proband had 46 CAG repeats. Result displayed is from the proposed nonparametric kernel Nelson-Aalen estimator (NPNA).

Figure S.2. of the Supplementary material available at Biostatistics online displays the predicted survival using the NPNA estimator to compare differences between carrier and non-carrier relatives (top-half), and differences between carrier males and females (bottom-half). The probability of survival for non-carriers was nearly one up until age 60, with a steady decrease thereafter. The carriers, however, had steadily decreasing probabilities of survival beginning at age 20 and rapidly decreasing thereafter. These differences underscore the insidious effect of genetic mutation on mortality. Mortality rates appeared similar for males and females (the 95% confidence bands for carrier males and females overlapped; see bottom-half of Figure S.2. of the Supplementary material available at Biostatistics online). The result agrees with earlier studies where gender did not significantly impact the mean survival time of HD patients (Harper, 1996).

5. Discussion

We developed a new nonparametric prediction model for mixture data that incorporates covariates and dynamic landmark prediction, and we showed how it improves prediction accuracy. There are several extensions of our work.

A straightforward extension is the incorporation of clinically important intermediate event information, such as hospitalization. The estimator , , in (2.1) can be adapted to incorporate the timing of intermediate events through . Specifically, let indicate the time of the intermediate event. Similar to the failure time , is subject to censoring and thus we may only observe and . Among individuals in , if then . That is, for those who have survived to and are still under observation at , it is known that and thus if the intermediate event occurred before , then it was observed. Therefore, when we condition on , we can re-define as and thus, will be equal to for those who have not yet experienced the intermediate event by or equal to for those who have experienced the intermediate event by . Estimation of then follows the procedure in Section 2, where and with or without an additional discrete covariate .

Another extension is with competing risks. In the HD study, one could be interested in different causes of death due to cardiovascular disease, pneumonia, or suicide (Sørensen and Fenger, 1992 ). To handle competing risks, we could re-define as a cumulative incidence function: where and denotes the type of event that occurred where and is the number of distinct competing event types. Estimation would focus on a cause-specific version of cumulative hazard in (2.2) and for estimation of each event, individuals experiencing other events would be censored at the event time.

Other extensions apply to the HD example. In the kin-cohort study, mortality outcomes between relatives in the same family are correlated. One could account for this correlation using frailties that adjust for correlations between outcomes from family members. These frailties would be incorporated into our modified nonparametric kernel Nelson–Aalen estimator. We could also explicitly incorporate left truncation of failure times given that survival times for relatives is collected retrospectively and some of these relatives have died prior to the study onset. However, incorporating left truncation requires a new framework beyond the scope of this article.

In practice, it may be of interest to use the estimates obtained to test for differences in predictions at some time or differences in the survival curves up to time between two groups (e.g., carrier vs. non-carriers, or male carriers vs. female carriers). To test at a particular time this difference, e.g., , we could perform a Wald-type test using . Here, indicates carrier and indicates non-carrier and is an estimate of the standard error of the difference between and which we can obtain using our bootstrap samples. To compare survival curves, we could define the function for in some range , and calculate a simultaneous confidence band where is the empirical quantile of where indicates the bootstrap replication.

Our proposed approach does have some limitations. Because we use nonparametric kernel smoothing, we require a relatively large sample size and in particular, there must be adequate sample size within each group. In settings with smaller sample sizes, parametric approaches may be more stable. Second, our proposed estimators were presented assuming covariates and were univariate. Our method can extend to the multivariate case using, for example, a dimension reduction approach whereby a working model is used to reduce the multivariate covariate information into a single scalar summary. After this dimension reduction, our proposed approach could then be directly applied. Third, though we note that our method can incorporate information on time-dependent covariates, where and are updated at the landmark time resulting in and , we do not have access to mixture data with time-dependent covariates and thus, are unable to illustrate this feature. However, this methodological development and the availability of software to implement these methods would allow others with such data to apply these methods. Fourth, the prediction accuracy using AUC and BS requires knowing the population identifier, which is only known in simulated data but not in practice. We thus cannot compute prediction accuracy for our HD study, but results from our simulation study suggest that our proposed estimator indeed has better prediction accuracy than competing methods.

6. Software

All R code is available at https://github.com/tpgarcia/landmix.

Supplementary material

Supplementary material is available at http://biostatistics.oxfordjournals.org.

Funding

The Huntington’s Disease Society of America Human Biology Project Fellowship, the National Institute of Neurological Disorders and Stroke (K01NS099343); and the National Institute of Diabetes and Digestive and Kidney Diseases (R21DK103118).