An Ensemble Survival Model for Estimating Relative …February 21, 2014 TOPIC: AN ENSEMBLE SURVIVAL MODEL FOR ESTIMATING RELATIVE RESIDUAL LONGEVITY FOLLOWING STROKE: APPLICATION TO

February 21, 2014 TOPIC: AN ENSEMBLE SURVIVAL MODEL FOR ESTIMATING RELATIVE RESIDUAL LONGEVITY FOLLOWING STROKE: APPLICATION TO MORTALITY DATA IN THE

CHRONIC DIALYSIS POPULATION

Copyright © 2014 by Milind A. Phadnis. All rights reserved. “Manucript in Submission” Page 1

An Ensemble Survival Model for Estimating

Relative Residual Longevity following Stroke: Application to Mortality Data in the Chronic Dialysis Population

- Milind A. Phadnis

Acknowledgements:

James B. Wetmore

Theresa I. Shireman

Edward F. Ellerbeck

Sally K. Rigler

Jonathan D. Mahnken




OUTLINE: Research question Initial assessments for model building A state transition model approach:

Using a semi-Markov model with additive hazards The Generalized Gamma (GG) distribution and the

concept of Relative Times (RT) Step-wise model building approach to estimate:

[1] Residual Life Lost (RLL) due to stroke [2] Relative (Non-proportional) Hazards

Results




RESEARCH QUESTION:

Data Description:

A large cohort with 69371 patients representing the chronic dialysis population in the time period 2000-2005 who were eligible for receiving both Medicare and Medicaid was constructed by linking the United States Renal Data System (USRDS) to Medicaid claims data1.

Research team developed clinical algorithms that allowed identification of incident hemorrhagic and ischemic strokes from Medicare claims data and as described in Wetmore et al.2,3

Follow-up for this cohort began upon dialysis initiation plus 90 days (as is

standard with analyses of USRDS data). Subjects were followed until death (outcome of interest) and subjects were right censored in the case they lost their Medicare or Medicaid coverage, received a kidney transplant, or were otherwise lost to follow-up (e.g., follow-up ended on 31 December 2005).




RESEARCH QUESTION:

Problem Statement:

Both hemorrhagic (N=534) and ischemic (N=2391) strokes are hypothesized to confer substantial risk for mortality in dialysis patients. Various patient level factors and comorbidities may affect the mortality risk. Since stroke can occur at any time in the observation window (cohort entry time to death or censoring time; maximum 69 months), it is a time-dependent covariate. Main interest is to estimate residual longevity of strokers as compared to non-strokers keeping in mind the potential effect on mortality of [1] Stroke type [2] Time at which stroke occurs [3] Time spent since stroke.

Note: Atrial fibrillation is a 0-1 time-dependent covariate; other covariates are considered to be time-independent covariates.




INITIAL ASSESSMENTS: Step 1. Fit Kaplan-Meier curve for Time-to-Death after stroke stratifying by stroke type

Figure 1 Risk of mortality due to the two stroke types is different (p-value < 0.0001)

# Alive Start 6m 12m 18m 24m 30m 36m 42m 48m 54m 60m

Hemorrhagic 534 133 84 64 47 34 19 11 5 1 0

Ischemic 2381 1166 747 482 308 186 126 78 40 19 9

Hemorrhagic

Ischemic




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TimeAfterStroke

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STRATA: SYear=0-1 SYear=1-2 SYear=2-3 SYear=3+

Step 2. Fit Kaplan-Meier curve for ‘Time-to-Death after stroke’ stratifying by ‘year of stroke occurrence’ for both stroke types.

Ischemic (p-value=0.40) Hemorrhagic (p-value=0.53)

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STRATA: SYear=0-1 SYear=1-2 SYear=2-3 SYear=3+

0 10 20 30 40 50 60 70

Time After Stroke 0 10 20 30 40 50 60

Time After Stroke

1.00 0.75 0.50 0.25 0.00

1.00 0.75 0.50 0.25 0.00

Stroke Year = 1 Stroke Year = 2 Stroke year = 3 Stroke Year = 4

Stroke Year = 1 Stroke Year = 2 Stroke year = 3 Stroke Year = 4

Figure 2 KM curves for ‘Time-to-Death’ stratified by year

S U

R V

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A L




Step 3. Try a Cox PH model4 treating stroke as a time-dependent covariate

(1) Here, are the time independent covariates age, gender, …, other comorbidities and are the associated regression coefficients

is the time-dependent indicator variable for Stroke (1 if stroke occurs, else 0); is the regression coefficient associated with

is the time-dependent term representing a functional interaction form

between stroke and time; is the regression coefficient associated with Two most commonly used functional forms for are:

or Some other functional forms can also be tried. Note: As time of stroke occurrence does not seem to affect mortality, we can replace

by where = time of stroke




Which functional form for should we use? We can try different functional forms but we need a way to diagnostically assess which one is a good fit. Approach: 1. One way is to predict the survival profile of strokers using whichever functional

form we used. Then plug s = 0 (i.e. stroke occurs right at the time of entry into cohort).

2. If our functional form provided a good model fit, the survival profile of strokers so obtained must match (approximately) the KM curves of Figure 2

3. Unfortunately, both and fail miserably

here (plots not shown). 4. Many different functional forms were tried (including piece-wise modeling

approaches). But predicted survival profiles were way off target. What is happening here?




Some well-known results5:

1. Using that is, using to model effect of stroke yields the same relative risk function as a Gompertz model with different values of shape parameter in the two groups (stroker and non-stroker)

2. Using that is, to model effect of stroke yields the same relative risk function as a Weibull model with different values of shape parameter in the two groups (stroker and non-stroker)

As discussed earlier, both these approaches are not adequate for the problem at hand. So how can we proceed here?




A STATE TRANSITION MODEL APPROACH: One way to approach the problem is to use a State-dependent covariate approach i.e. we can exploit the fact that a single time-dependent covariate model can be easily expressed as a single intermediate state transition model in the following way6: Figure 3. Single state transition model for mortality with ‘Stroke’ as intermediate state

State 1 Entry into

Cohort Time = 0

State 2 Stroke

Time = s

State 3 Death

Time = t




Under the Markov assumption (what happens to a patient in a particular state depends only on the fact that he is in that state and not on the history preceding it), it can be shown that the transition hazard for moving from

state i to state j can be modeled as6:

(2)

Here, represents some baseline hazard attributable to state 3,

represents the vector of regression coefficients for the other risk factors, is the vector of other risk factors,

represents the difference in covariate effects of these other risk factors for death after having stroke, compared to before having stroke, and represents the estimate of the stroke effect.




But in our case a Markov model may not be adequate.

Semi-Markov Model: Here we relax the Markov assumption by allowing the sojourn times as covariates to depend on the history of the process only through the present state and the time since entry of that state.

Extended Semi-Markov Model: Here we even allow the sojourn times as covariates to depend on the history of the process through the time at which earlier states have been entered as well as time since entry of those states. From the KM curves of Figure 1 and Figure 2, a semi-Markov model seems appropriate in our case. So we will try to incorporate (t-s) somewhere in equation #2.




Also note that: 1. We should be able to generate a survival profile for both strokers and non-strokers.

Only then we can estimate and hence compare residual longevity for both groups.

2. Since the modeling is with respect to the hazard functions, we need to be able to estimate the baseline hazard from which we can generate baseline survival curve (for non-strokers) and then adjust these curves for the effect of the other covariates.

3. Final model should pass the diagnostic check mentioned on Slide #8.

To accomplish all these things at one time:

1. Incorporate in equation #2, the fact that “occurrence of a stroke” drastically

alters survival profile of patients (perhaps even changing the underlying parametric distribution)

2. We target a multi-parameter family of distributions.




Semi-Markov Model with Additive Hazards:

To achieve the objectives mentioned above, keeping in mind Figure 3, we need to combine the proposed semi-Markov hazard part of the model (to account for the contribution of t-s) with an additive hazards extension (to account for the fact that stroke is time-dependent i.e. strokers are non-strokers before they experience a stroke; but once stroke occurs their survival experience changes drastically).

Thus our model is:

(3)

Thus for those with no strokes, and resulting in:

(4)

and for those with strokes, and resulting in:

(5)




Here, is some baseline hazard associated with the transition from State 1

directly to State 3. However, for those who experience a stroke at time s, the corresponding part of the baseline hazard is only because after the stroke they

experience a change in their overall hazard owing to the effect of stroke, measured as a function of time spent since stroke. This incremental hazard is given by and it can be modeled in two ways, the discussion of which is postponed to the next section. In the next section, we show how we can use the three parameter Generalized Gamma (GG) distribution to identify and and

how the properties of this distribution can be used to conceptually do the calculations related to Residual Life Lost (RLL) due to each stroke type using the concept of Relative Times (RT).




THE GENERALIZED GAMMA (GG) DISTRIBUTION:

The GG distribution7 is a three parameter family with location , scale and shape parameters that generalizes the two parameter gamma distribution3. Its density function is given by:

(6)

Special cases:

gives the two parameter gamma distribution gives the standard gamma distribution for fixed values of

gives the Weibull distribution gives the exponential distribution

gives the lognormal distribution gives the inverse Weibull distribution

gives the inverse gamma distribution gives the ammag distribution

gives the inverse ammag distribution




Thus, we have the flexibility to model different types of hazard functions such as8:

1. Increasing hazard from 0 to 2. Increasing hazard from a constant c to 3. Decreasing hazard from to 0 4. Decreasing hazard from to a constant c 5. Bath tub curve 6. Arc shaped hazard

Quantiles of the GG distribution:

As the interest of the project is to model residual longevity, we are interested in looking at quantiles of the GG distribution. It can be shown that8:

(7) where is the pth quantile of the distribution and is the

logarithm of this pth quantile.




Thus, The location parameter acts as a multiplier of time and governs the values of the median for fixed values of and The scale parameter determines the interquartile ratio for fixed value of and independently of The shape parameter determines the quantiles of the standard gamma distribution.

and determine the type of hazard function.




Concept of Relative Times RT(p):

The time by which p % of the population experience an event can lead to a statistic called ‘Relative times (RT)’ which can be used to compare survival profiles of patients in the two groups8 (strokers vs non-strokers):

Thus can be interpreted as:

“The time required for p % of individuals in the stroke group to experience death is RT(p) fold times the time required for p % of individuals in the non-stroke group to experience death.” If and denote two different sets of GG parameter values, then

(8)




Thus

1. If and then we have a conventional AFT model resulting in non-proportional hazards but proportional RT covariates affect

only.

2. If only then we have a model that results in non-proportional hazards and non-proportional RT(p) covariates affect both and .

3. Full generalization is obtained by having covariates affect all three

parameters.

4. Reduced models are possible for sub-families of the GG distribution




STEP-WISE APPROACH TO ESTIMATE PARAMETERS: With reference to the model in (3) and the specifics discussed in the sub-sections above, following step-wise approach can be utilized to answer the research question:

1. Obtain baseline hazard for pre-stroke times by fitting a Cox PH model

given in (4) with right-censoring also occurring at the time of stroke (when applicable). The Cox PH framework also allows estimation of risk due to AF, which is a time-dependent covariate.

2. To model the baseline hazard, estimate parameters for the most parsimonious subfamily of GG distribution for these pre-stroke times (including subjects with no strokes prior to death).

3. Then generate baseline survival curve for pre-stroke times using the general relation




4. Generate a survival profile for pre-stroke times using the average values of all

other covariates X, using the relation

5. For any time s (of interest) for occurrence of a stroke, calculate Adjust

this for the effect of all other covariates X.

6. The hazard for those that had strokes as a function of ‘time since stroke’ can be obtained in two ways.

(a) The first way is to have all the other covariates affect in the AFT model of (7) and then estimate parameters of the most parsimonious subfamily of the GG distribution. Then generate survival curves for those that had strokes as a function of t-s, and augment them to the survival curve for those that had not had a stroke at time of stroke = s. These survival curves can be generated for either particular values of covariates (such as African American males >= 65 years, say) or for an average profile of patients in the cohort as done in step 4 of this list.




(b) The second way is to consider such

that only the baseline hazard as a function of ‘time since stroke’ is

due to the GG distribution and is the increment vector of regression coefficients corresponding to the other covariates that indicates whether or not the

effect of these risk factors on mortality change due to the occurrence of a stroke. However, it should be noted that the PH assumption needs to be verified here, failing which this approach cannot be adopted.

7. Residual longevity following a stroke can be readily calculated in terms of the pth percentile (for median, p = 0.5) survival time after the occurrence of a stroke at time s from step 6 (a or b) above. With reference to Figure 4, this corresponds to calculating:

(9)

8. Residual longevity absent any strokes for any time s can be calculated by evaluating the pth percentile of survival time from the conditional survival

function .




With reference to Figure 4, this corresponds to calculating

(10)

9. can be used to calculate the relative pth percentile of time required for absent a stroke to experience death as compared to when a stroke occurs. Alternatively, the pth percentile of RLL due to stroke, a concept more appealing in practice to the clinicians can be calculated using the relation given by

(11)

Specifically, and may prove to be quite informative in explaining the skew in the RLL due to stroke. These calculations have been graphically depicted in Figure 4 shown on the next slide.

10. Evaluation of the model fit can be performed using the graphical diagnostic checking procedure mentioned on Slide #8. Software used: SAS (PROC PHREG, PROC LIFEREG, PROC NLMIXED)




Figure 4 Calculating residual longevity following a stroke following time of stroke occurrence (s)

Time of stroke = s

Snonstr (s)

Snonstr({1-p}s)

RLL(p)

In our dataset, following

parametric distributions

were observed:

[1] Non-strokers: Weibull

[2] Hemorrhagic: GG

[3] Ischemic: Lognormal




RESULTS: Table 1. Modeling results using equation (4) for [A] no or pre-stroke times and (5) for times following [B] hemorrhagic and [C] ischemic strokes

Risk Factors (Covariates) [A] Non-strokers (Cox PH) [B] Hemorrhagic Strokers (AFT extension) [C] Ischemic Strokers (AFT extension)

HR Estimate

99% CI p-value Exponentiated Location Estimate

99% CI p-value Exponentiated Location Estimate

99% CI p-value

Intercept - - - 0.113 0.009, 1.497 0.0297 20.918 5.288, 82.756 <.0001

Race: Black vs White 0.779 0.750, 0.809 <.0001 1.292 0.862, 1.937 0.1035 1.750 1.355, 2.589 <.0001

Race: Hispanic vsWhite 0.673 0.641, 0.707 <.0001 1.469 0.902, 2.390 0.0421 1.802 1.306, 2.485 <.0001

Race: Others vs White 0.585 0.541, 0.633 <.0001 1.328 0.676, 2.609 0.2801 2.673 1.507, 4.743 <.0001

Age 1.030 1.028, 1.031 <.0001 1.003 0.992, 1.014 0.4794 0.959 -0.053, -0.033 <.0001

Sex: F vs M 0.944 0.912, 0.977 <.0001 0.861 0.603, 1.230 0.2796 1.068 -0.178, 0.309 0.4886

NoAmbulation: No vs Yes 0.686 0.639, 0.736 <.0001 0.821 0.291, 2.317 0.6242 1.082 -0.416, 0.575 0.6804

NoTransfer: No vs Yes 0.808 0.726-0.899 <.0001 2.752 0.309, 24.498 0.2329 1.41 -0.420, 1.108 0.2461

Dialysis: In-center vs Self 0.962 0.887, 1.044 0.2232 0.808 0.353, 1.847 0.5057 1.208 -0.317, 0.695 0.3360

CardiacFailure: No vs Yes 0.887 0.840, 0.936 <.0001 0.791 0.439, 1.423 0.3031 1.485 0.026, 0.765 0.0059

PreviousStroke: No vs Yes 0.875 0.832, 0.920 <.0001 0.887 0.550, 1.432 0.5200 1.059 -0.257, 0.371 0.6388

VascularDisease: No vs Yes 0.898 0.857, 0.941 <.0001 0.994 0.587, 1.684 0.9781 0.946 -0.374, 0.262 0.6504

Diabetes: No vs Yes 0.979 0.936, 1.024 0.2248 0.744 0.478, 1.158 0.0852 1.16 -0.169, 0.465 0.2289

CAD: No vs Yes 0.959 0.921, 0.999 0.0078 1.234 0.809, 1.882 0.1998 1.201 -0.080, 0.446 0.0734

Hypertension: No vs Yes 1.323 1.266, 1.382 <.0001 0.952 0.594, 1.526 0.7869 0.814 -0.534, 0.123 0.1064

BMI: Normal vs (<25) 0.764 0.725, 0.806 <.0001 1.082 0.666, 1.756 0.6761 1.024 -0.375, 0.421 0.8796

BMI: (25-30) vs (<25) 0.644 0.609, 0.681 <.0001 1.031 0.593, 1.791 0.8886 1.152 -0.261, 0.544 0.3656

BMInew: 30+ vs (<25) 0.592 0.560, 0.626 <.0001 1.317 0.767, 2.264 0.1898 1.296 -0.146, 0.664 0.0992

Epmloyed: Yes vs No 0.588 0.491, 0.703 <.0001 0.464 0.155, 1.383 0.0700 0.48 -1.815, 0.347 0.0801

Smoker: No vs Yes 0.915 0.854, 0.980 0.0009 1.167 0.651, 2.091 0.4962 1.145 -0.372, 0.643 0.4908

SubstanceAbuser: No vs Yes 0.822 0.742, 0.910 <.0001 1.068 0.536, 2.129 0.8061 1.207 -0.673, 1.049 0.5739

LiuIndex*: (3-6) vs (<3) 0.812 0.744, 0.887 <.0001 1.619 0.678, 3.862 0.1538 0.917 -0.680, 0.506 0.7063

LiuIndex*: (> 6) vs (<3) 0.954 0.897, 1.014 0.0464 1.451 0.768, 2.741 0.1316 1.013 -0.402, 0.428 0.9358

Atrial Fibrillation:Yes vs No Atrial Fibrillation*Time

1.181 1.006

1.098, 1.269 1.003, 1.009

<.0001 <.0001

Scale, = 1.258

Shape, = -2.281

1.065, 1.485 -2.853, -1.710

Scale, = 1.883 1.793, 1.978




Table 2 Comparison of hemorrhagic stroke versus no stroke in terms of RT(p) and RLLmonths(p)

p

s = 0 s = 12 months s = 24 months

RT [99%CI]

RLL [99%CI]

RT [99%CI]

RLL [99%CI]

RT [99%CI]

RLL [99%CI]

0.05 19.993 [16.633-23.354]

2.291 [2.198-2.198]

25.204 [20.311-30.096]

2.919 [2.602-3.236]

26.290 [20.100-32.480]

3.050 [2.514-3.586]

0.10 32.432 [26.739-38.124]

5.115 [4.960-5.271]

38.588 [31.531-45.645]

6.117 [5.745-6.489]

40.447 [32.515-48.380]

6.420 [5.812-7.027]

0.15 40.711 [33.082-48.340]

8.251 [8.039-8.464]

46.523 [37.637-55.409]

9.459 [9.027-9.891]

49.208 [39.489-58.926]

10.017 [9.330-10.705]

0.20 45.885 [36.720-55.051]

11.684 [11.413-11.955]

51.842 [41.372-62.312]

13.235 [12.730-13.740]

53.791 [42.708-64.874]

13.743 [12.967-14.518]

0.25 48.550 [38.259-58.842]

15.425 [15.090-15.760]

54.025 [42.487-65.563]

17.201 [16.613-17.789]

56.056 [43.927-68.185]

17.860 [16.981-18.738]

0.30 49.127 [38.112-60.143]

19.496 [19.087-19.904]

53.743 [41.625-65.861]

21.365 [20.683-22.048]

56.023 [43.274-68.771]

22.289 [21.294-23.283]

0.35 47.959 [36.593-59.324]

23.930 [23.433-24.427]

51.989 [39.336-64.642]

25.984 [24.421-27.547]

53.819 [40.921-66.717]

26.917 [25.793-28.040]

0.40 45.350 [33.971-56.729]

28.772 [28.168-29.376]

48.784 [36.500-61.067]

31.000 [30.073-31.926]

50.461 [37.688-63.233]

32.088 [30.812-33.364]

0.45 41.591 [30.490-52.692]

34.075 [33.338-34.811]

44.444 [32.549-56.334]

36.469 [35.386-37.553]

- -

0.50 36.966 [26.397-47.536]

39.903 [39.000-40.807]

39.271 [28.017-50.525]

42.461 [41.188-43.733]

- -

0.55 31.760 [21.942-41.577]

46.335 [45.214-47.457]

- - - -

0.60 26.253 [17.385-35.122]

53.453 [52.029-54.878]

- - - -

Sample based average estimate of median RLL due to hemorrhagic stroke = 40.688 months; RT(0.5)=0.049




Table 3 Comparison of ischemic stroke versus no stroke in terms of RT(p) and RLLmonths(p)

p

s = 0 s = 12 months s = 24 months

RT [99%CI]

RLL [99%CI]

RT [99%CI]

RLL [99%CI]

RT [99%CI]

RLL [99%CI]

0.05 5.186 [4.260-6.112]

1.946 [1.825-2.067]

6.538 [5.209-7.866]

2.575 [2.248-2.901]

6.819 [5.162-8.476]

2.706 [2.164-3.248]

0.10 5.727 [4.829-6.625]

4.356 [4.148-4.565]

6.814 [5.689-7.938]

5.358 [4.961-5.755]

7.142 [5.860-8.425]

5.661 [5.034-6.284]

0.15 5.785 [4.944-6.626]

6.997 [6.701-7.293]

6.611 [5.619-7.602]

8.205 [7.726-8.683]

6.992 [5.886-8.098]

8.763 [8.045-9.480]

0.20 5.660 [4.876-6.444]

9.834 [9.442-10.227]

6.394 [5.488-7.300]

11.385 [10.805-11.965]

6.635 [5.657-7.613]

11.892 [11.067-12.718]

0.25 5.447 [4.715-6.179]

12.858 [12.353-13.363]

6.061 [5.232-6.891]

14.634 [13.935-15.333]

6.289 [5.401-7.178]

15.293 [14.337-16.249]

0.30 5.188 [4.502-5.875]

16.065 [15.424-16.706]

5.676 [4.913-6.439]

17.935 [17.093-18.777]

5.916 [5.100-6.732]

18.858 [17.747-19.968]

0.35 4.903 [4.257-5.549]

19.456 [18.647-20.264]

5.315 [4.605-6.025]

21.509 [20.490-22.528]

5.502 [4.752-6.253]

22.442 [21.150-23.734]

0.40 4.604 [3.994-5.213]

23.030 [22.011-24.049]

4.952 [4.289-5.616]

25.258 [24.020-26.496]

5.122 [4.423-5.582]

26.346 [24.828-27.863]

0.45 4.296 [3.719-4.872]

26.786 [25.500-28.072]

4.590 [3.968-5.213]

29.181 [27.669-30.693]

- -

0.50 3.982 [3.437-4.528]

30.715 [29.084-32.345]

4.231 [3.646-4.816]

33.272 [31.411-35.132]

- -

0.55 3.667 [3.151-4.182]

34.793 [32.713-36.874]

- - - -

0.60 3.349 [2.862-3.835]

38.975 [36.293-41.656]

- - - -

Sample based average estimate of median RLL due to ischemic stroke = 34.572 months; RT(0.5)=0.195




Figure 5. Plotting the HR of death for Strokers versus Non-strokers as a function of 'Time-After-Stroke'




References 1 United States Renal Data System. USRDS 2006 Annual Data Report: Atlas of end-stage renal

disease in the United States, National Institutes of Health, National Institute of Diabetes and Digestive and Kidney Disease, (2006)

2 Wetmore, J.B., Ellerbeck, E.F., Mahnken, J.D., Phadnis, M., Rigler, S.K., Spertus, P.J., et al.:

Stroke and the ‘stroke belt’ in dialysis: contribution of patient characteristics to ischemic stroke

rate and its geographic variation. Journal of American Society of Nephrology 24(12), 2053-2061

(2013)

3 Wetmore, J.B., Phadnis, M.A., Mahnken, J.D., Ellerbeck, E.F., Rigler, S.K., Zhou, X.: Race,

ethnicity, and state-by-state geographic variation in hemorrhagic stroke in dialysis patients.

Clinical Journal of the American Society of Nephrology doi:10.2215/CJN.06980713 (2013)

4 Cox, D.R.: Regression models and life tables (with discussion). Journal of the Royal Statistical

Society B 34, 187-220 (1972)

5 Hougaard, P.: Analysis of Multivariate Survival Data. Springer, New York (2001)

6 Putter, H., Fiocco, M., Geskus, R.B.: Tutorial in biostatistics: Competing risks and multi-state

models. Statistics in Medicine 26, 2389-2430 (2007)

7 Stacy, E.W., Mihram, G.A.: Parameter estimation for a generalized gamma distribution.

Technometrics 7(3), 349-358 (1965)

8 Cox, C., Chu, H., Schneider M.F., Munoz, A.: Parametric survival analysis and taxonomy of

hazard functions for the generalized gamma distribution. Statistics in Medicine 26, 4352-4374

(2007)




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An Ensemble Survival Model for Estimating Relative …February 21, 2014 TOPIC: AN ENSEMBLE SURVIVAL MODEL FOR ESTIMATING RELATIVE RESIDUAL LONGEVITY FOLLOWING STROKE: APPLICATION TO