Modeling Survival of Hemodialysis Patients

Christos Argyropoulos

Impetus for the project

• Dialysis patients have a high background mortality rate and multiple comorbidities

• Discrepant findings of observational studies and randomized trials in HD & PD of the dose of dialysis v.s. comorbid conditions/age

• Randomized trial results are counterintuitive

• Interpretations of the discrepant results are not entirely satisfactory

Aims and Objectives

• Aim 1 : quantify the absolute and relative contribution of “under-dialysis” to high mortality rate

• Aim 2 : explore novel statistical approaches to quantify underdialysis

• Aim 3: explain the discrepancies noted between epidemiological studies and randomized trials

Hypotheses

• Hypothesis 1: Dialysis dose is important

• Hypothesis 2: Kinetic modeling constructs may be used to quantify underdialysis

• Hypothesis 3: Discrepant findings are due to mismatches/incompatibilities between two types of models:– Kinetic models of a single dialysis session

– Statistical models for survival analysis in a population of patients

Hypothesis 3

• Fact: Kinetic equations do not include a term for clinical outcome, while statistical approaches to survival analysis do not include a term for the molecular fluxes during a single dialysis session

• Challenge: Put a term for molecular flux in the statistical model

• Question: What is out there in terms of statistical models for survival analysis

Starting Point:How does dialysis work?

• Dialysis removes “stuff”

• The “stuff” is mix of poorly characterized molecules with systemic ill effects

• Accumulation of “stuff” → Accumulation of toxic effects → Uremic manifestations (“residual syndrome”) → Increased likelihood of “treatment failure” (NCDS terminology for underdialysis)

The “missing link” between kinetics and statistics

Accumulation → Failure

(“kinetics”) → (“statistics”)

• One does not need to know what is the nature of the “stuff” that accumulates

• It suffices to know that the mechanism of failure is due to the (cumulative over time) accumulation of toxic effects

• The statistical model for survival (failure) should be capable of describing this “additive damage” mechanism

Basic Concepts In Survival Analysis

• Analyzes the time to the occurrence of some event (“failure”)

• Survivor function S(t): probability of an individual surviving until time t

• Hazard function h(t): probability that an individual will “fail” at time t, given that he or she has survived until that time

• Cumulative hazard function H(t): the Area Under the Curve of h(t)

Basic Concepts In Survival Analysis

• The three functions are inter-related:

0

( ) exp[ ( )], ( ) ( )t

S t H t H t h t dt • There are two main ways to analyze life-time data:

– Model the effects of covariates upon the hazard function (biomedical & epidemiological research)– Model the effects of covariates directly upon the time to failure (reliability engineering applications)

The Proportional Hazard Model

• Introduced in 1972 by Sir David Cox as a tool to model failure time data in both contexts

• Revolutionized analysis of outcomes data (initial applications were in oncology)

• Invokes a multiplicative effect of covariates on the hazard (probability) of the failure

• The covariate effect may be estimated without specifying a survival distribution (“semi-parametrically”) if the corresponding hazards are proportional

The Proportional Hazard Model

• Semi-parametric ≠ assumption free

•The physical interpretation of the model was also examined in the paper

The Proportional Hazard Model

It seems that the mathematics of the proportional hazard model cannot represent conditions of cumulative “shocks”

A mathematical proof of this assertion was given 4 years later in a Russian journal (a translation may be downloaded from the SIAM website).

Bagdonavicius, V. B., A Statistical Test of a Model of Additive Accumulation of Damage, Theory of Probability and its Applications, 1979, 23: 385-390.

The Accelerated Failure Time Model

• Cox and Oakes (and others) discussed alternative approaches based on Accelerated Failure Time models in their 1984 textbook (“Analysis of Survival Data”)

• These models can represent mechanisms of failure due to cumulative exposures (“additive damage”)

• Rarely used in outcomes research (oncology is an exception) because they are “parametric” and they make assumptions.

The Accelerated Failure Time Model

• In the AFT model, covariates accelerate or decelerate time until failure

• At the group level, covariates shift the median survival to greater values or smaller numerical values

• The interpretation of covariate effects is very intuitive. It is based on the acceleration factor

• AF is the ratio of medians for patients exposed to different levels (or doses of dialysis)

Implications of using a PH instead of an AFT

• D.R. Cox and many others looked into the effects of using the PH model when an AFT is the “correct” one (and vice versa)

• Literature spans > 20 publications over 20 years in journals of basic and applied statistics

• AFTMs are more robust w.r.t measurement noise, missing covariates, individual heterogeneity and frailty

• The parametric form they assume allow one to infer mechanisms of failure (this is an active area of research by statistical engineers)

Implications of using a PH instead of an AFT

• Every test or statistical procedure effects a decomposition of the raw data into signal and noise terms.

• Sakett put it succinctly:

• Mis-specified PHs will inflate the estimate for the noise and reduce the signal estimate

• Unless big sample sizes are used, tests of significance will lose power (big time)

" "Signal

Confidence SampleSizeNoise

Sakett. Canadian Medical Association Journal, 2001, 165:

1226-1237

A case study : CHOICE

• CHOICE: a national, prospective cohort study of incident in-center hemodialysis and peritoneal dialysis patients.

• Enrolled 1041 patients from October 1995 to June 1998 from dialysis clinics across the US All study participants were enrolled a median of 45 days from initiation of chronic dialysis (98% within 4 months).

• Enrolled patients were followed up until December 31st 2004.

Analysis of CHOICE I

• Fitted proportional hazard and accelerated failure time models to measure the effects of increasing age, CHF, worse comorbidity indices, assigned cause of ESRD, albumin, physical functioning (PF/SF-36), sex, and Kt/V

• Also run stratified analyses w.r.t sex and non-linear models (for BMI)

Analysis of CHOICE I

• Fitted proportional hazard and accelerated failure time models to measure the effects of increasing age, CHF, worse comorbidity indices, assigned cause of ESRD, albumin, physical functioning (PF/SF-36), sex, and Kt/V

• Also run stratified analyses w.r.t sex and non-linear models (for BMI). Total # of models = 6

Analysis of CHOICE II

• All six models defined equivalent prognostic indices. This means they assigned high probabilities of dying for patients who died sooner than the others

• The proportional hazard models (6 in total), yielded the exact same effects for all covariates with the exception of Kt/V.

• The hazard ratios were equal (up to 2 decimal points!)

• Briefly: older, white patients, with multiple comorbidities, and low albumin die

Quo Vadis Kt/V?

How is the illusion created?

• The theory says that one cannot use PH models to study cumulative effects (“additive damage”)

• The data say so too – one can run goodness of fit tests in the dataset as suggested by D.R. Cox in 1972

• Terry Therneau (chief biostatistician at the Mayo clinic) associated a picture with such tests in 1994

• Incidentally these tests were used in the recent NEMJ paper regarding lung transplants for pediatric patients with Cystic Fibrosis

How does the hazard of underdialysis unfold over time?

• The hazard is non-monotonic, first increasing and then non-decreasing

• The simplest AFT with this behaviour is the log-normal distribution

• If the PH held we would expect a straight line

• The p-value for non-proportional hazards for the Kt/V (model) is 0.03 (0.01)

The AFTM CHOICE

AF 95% CI p-value

Age 0.98 0.973-0.987 <0.001

Male 1.07 0.876-1.295 0.530*Race

African American 1.4 1.149-1.711 0.001†

Other 1.63 1.124-2.365 0.010†Cause of renal failure

Hypertension 0.91 0.720-1.145 0.414‡

Glomerulonephritis 0.99 0.745-1.316 0.945‡

Other 1.15 0.897-1.469 0.273‡Comorbidity

ICED 2 0.72 0.576-0.906 0.005 ICED 3 0.56 0.441-0.709 <0.001Heart Failure 0.79 0.650-0.949 0.012Body Mass Index 1.01 0.995-1.024 0.206Physical Functioning 1.01 1.002-1.009 0.001Age 1.68 1.307-2.169 <0.001

Kt/V 1.42 1.018-1.981 0.039

Log – Normal AFTM

Older, underdialyzed, white patients, with multiple comorbidities, poor physical functioning and low albumin dieAll six AFTMs consistently detected underdialysis (low Kt/V) along with the other predictors (there was a moderate effect on p-value size there as well)

Graphical Assessment of Goodness of Fit

• PH model tracks initial and late (uncertain) part of the survival function

• AFTM tracks the central (larger segment) of the mortality curve

• Clearly both models are approximations

• Early and late deaths probably reflect extremely vulnerable (very sick?) or resilient individuals. The former do not die from underdialysis, the latter are more “resistant” than the average patient to it

Sum – up & directions for further research

• Failure to understand what the survival models can and cannot do will lead to “statistical illusions” and paradoxical or conflicting results

• Randomized studies DO NOT protect us from statistical illusions

• Have to use models that take into account the physical mechanisms of failure

• Physiology and medicine should guide the choice of statistical procedures

• Paradoxical results should prompt us to look into the statistical models (the way a basic science researcher will check his reagents to make sure they have not expired when a band fails to appear or disappear)

Sum – up & directions for further research

• Erroneous (mis-specified) statistical models can confound the results of our analyses

• The greater success of mis-specified models lies in their apparent failure (they prompt us to look why they failed!)

• In this case they forced me to read the literature (“EBM applied to statistics”)

Sum – up & directions for further research

• There is considerable variation that is not explained by our simple log-normal AFT

• Understanding how the non-monotonic arises is necessary to explain the early/late part of the survival curve

• May help us reduce early deaths by smoothing the transition from CKD5→ESRD

• May help us understand why some people survive (metabolic analyses in long term survivors looking for actual uremic toxins)

Putting the Kinetic Term into The Survival Model

/log /

( )Kt V

OtherFactors

T AF Kt V

AF OtherFactors error

/( / )Kt VT Exp AF Kt V

Acknowledgements I

• Mark Unruh MD, MSc for endless discussions, manuscript revisions, and am email exchanges over the last 10 months

• Laura Plantinga MSc , Josef Coresh MD PhD, Nancy Fink MPH, Neil Powe MD MPH MBA for the access to the CHOICE data

• Baxter Extramural Grant Program “Renal Discoveries” for funding part of this research

Acknowledgements II

• The R Development Core Team for the statistical environment (R) used to run these analyses

• The JSTOR project for putting at the WWW (in text searchable format!) the archives of the major statistical journals

• Sir David Cox for pointing out certain key references, tests and procedures for the proportional hazard model