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Threshold RegressionThreshold Regression for Time-to-event Data
Mei-Ling Ting Lee
University of Maryland, College Park
OutlineOut e
• The proportional hazards (PH) model is widelyThe proportional hazards (PH) model is widely used in analyzing time-to-event data. It requires however a strong PH assumptionrequires, however, a strong PH assumption.
• Threshold regression (TR) is an alternative model without the PH assumption.
• Brief introduction of the TR model• Brief introduction of the TR model.• Extensions and Examples• Connections between TR and PH models
Hazard Function:The instantaneous failure rate
Pr( | )t T t t T t≤ < + Δ ≥0
Pr( | )( ) limt
t T t t T th ttΔ →
≤ < + Δ ≥=
ΔtΔ
( )( ) f th t =( )( )S t
The Proportional Hazards (PH) Modelp ( )For any covariates x1 and x2 , the PH model
assumes that the hazard functions areassumes that the hazard functions are proportional to a baseline hazard h0(t).
1 0 1( | ) ( ) e x p ( )( | ) ( ) ( )
h t x h t xh t h t
ββ
=0 22( | ) ( ) e x p ( )h t x h t x β=
1 11 2
22
( | ) e x p ( ) e x p [ ( ) ]( | ) e x p ( )
h t x x x xh t x x
β ββ
−= =
The hazard ratio of the PH model has a simple e a a d a o o e ode as a s p eform that does not depend on time t .
“Hazard rate is really an elusive concept, especially when one tries to interpret its shape considered as a function of time”.
Aalen, Gjessing (2001), Understanding the shape of the hazard rate: a process point of view Statisticalthe hazard rate: a process point of view, Statistical Science, 16, 1-22.
A l B d Gj i (2008) S i l dAalen, Borgan, and Gjessing (2008). Survival and Event History Analysis: a Process Point of view. S iSpringer.
Consequences of the PH AssumptionConsequences of the PH AssumptionAs a consequence of the proportional hazardsAs a consequence of the proportional hazards
assumption, the survivorship functions of the two groups must be paralleltwo groups must be parallel.
Example 1Ti t i f ti f kid di l i ti t ithTime to infection of kidney dialysis patients with
different catheterization procedures(Nahman et al 1992 Klein & Moesberger 2003)(Nahman et al 1992, Klein & Moesberger 2003)
• Surgical group: 43 patients utilized aSurgical group: 43 patients utilized a surgically placed catheter
• Percutaneous group: 76 patients utilized a percutaneous placement of their catheterp p
Th t ti i d fi d b th ti tThe event time is defined by the time to cutaneous exit-site infection.
Example 1K l M i E ti t PH C M d lKaplan-Meier Estimate versus PH Cox Model
Example 1
G li d G Threshold RegressionGeneralized Gamma Threshold Regression
9
Threshold Regression Model can estimate hazard ratios at different time points
ti l 2 ti l 20
xi: threg,lny0(i.group) mu(i.group) failure(infection) /**/ time(time) hr(group) timevalue(2) graph(hz)
xi: threg,lny0(i.group) mu(i.group) failure(infection) /**/ time(time) hr(group) timevalue(20) graph(hz)
timevalue=2 timevalue=20
06.0
8.1
Hazard Rate
06.0
8.1
Hazard Rate
.02
.04
.0h(
t)
.02
.04
.0h(
t)
0
0 10 20 302Calendar Time
reference _Igroup_2==1
0
0 10 20 30Calendar Time
reference _Igroup_2==1
P ti l H d C M d l l ti t t t h d ti
stset time, failure(infection)xi: stcox i.group
Proportional Hazard Cox Model can only estimate a constant hazard ratio
Example 2Example 2Time to progression of disease for patients with
limited Stage II or IIIA ovarian carcinomalimited Stage II or IIIA ovarian carcinoma (Fleming et al., 1980; Hess, 1995)
Low-grade tumor group:15 patients with low grade or well differentiated15 patients with low-grade or well-differentiated tumors
High-grade tumor group:20 patients with high grade or undifferentiated20 patients with high-grade or undifferentiated tumors
12
Example 2Kaplan-Meier Estimates
13
Example 2
Generalized Gamma Threshold RegressionThreshold
14
Example 3Example 3Time to death for patients with bile duct cancer p
(Fleming et al., 1980; Lawless, 2003)
Control group:25 patients without treatment p
Treatment group:22 patients with a combined treatment of radiation and the drug 5-fluorouracil (5-FU)g ( )
15
Example 3
Kaplan-Meier Estimates
16
Example 3
Generalized Gamma Threshold Regression
17
Threshold Regression is based on gthe First-hitting-time (FHT) Model
Modeling event times by a stochastic g yprocess reaching a boundary
Threshold Regression is based on First hitting time Methodology :First-hitting-time Methodology :
E l E i i• Example: EngineeringEquipment fails when its cumulative wear first reaches a failure threshold.failure threshold.
• Example: Health and Medicine A i di f h di h di i f h hA patient dies of heart disease when condition of the heart deteriorates to a critical state.
• Example: Social SciencesDivorce occurs when tensions in a marriage reach a breaking pointpoint.
A review article:Modeling Event Times by Stochastic Process Reaching a BoundaryModeling Event Times by Stochastic Process Reaching a Boundary (Lee & Whitmore 2006, Statistical Sciences)
y0
P S l thProcess Y(t)
Sample path
S0
0
time t
First hitting time S of a fixed boundary at level zero for a stochastic process of interest Y(t)
y0
Process Sample paths Y(t)
0
Y(t)
S L0
Time tTime t
Two sample paths of a stochastic process of interest:(1) O th i ‘f il ’ t fi t hitti ti S(1) One path experiences ‘failure’ at first hitting time S
(2) One path is ‘surviving’ at end of follow up at time L
First Hitting Time (FHT) Models
{Y(t)} : the stochastic process of interest
Β : the threshold or boundary set
First hitting time S defined by
S = inf { t : Y(t) ∈ B }
Examples of First-hitting-time (FHT) Models:
• Wiener diffusion to a fixed boundaryP f lti l l til d thProgress of multiple myeloma until death
R l t fi d t f• Renewal process to a fixed count of renewal events
Ti t th th il ti iTime to the nth epileptic seizure
Semi Marko process to an absorbing• Semi-Markov process to an absorbing state
Multi state model for disease with death asMulti-state model for disease with death as an absorbing state
First Hitting Time (FHT) Modelsg ( )
• The first hitting time (FHT) model describes many• The first hitting time (FHT) model describes many time-to-event applications
• The stochastic process of interest {Y(t)} may represent the latent (unobservable) health status of arepresent the latent (unobservable) health status of a subject.
• The threshold constitutes the critical level of the process that triggers the failure event (e gprocess that triggers the failure event (e.g., symptomatic cancer, death). The event occurs when health status {Y(t)} first decreases to the zerohealth status {Y(t)} first decreases to the zero threshold.
Parameters for the FHT Model
Model parameters for a latent process Y(t) :
• Process parameters: θ = (μ, σ2), where μp (μ, ), μis the mean drift and σ2 is the variance
• Baseline level of process: Y(0) = y0
• Because Y(t) is latent, we set σ2 = 1.( ) ,
Threshold Regression:FHT Model with Regression Structureg
• Including covariates to FHT model parameters g pθ = (μ, σ2) using link functions, where σ2 = 1 .
• Mean drift parameters: μ = g1( x1, x2, …)
• Baseline level: y(0) = g2 (x1, x2, …)
A variety of link functions g have been considered:Linear combinations of covariates X1,…, XpLinear combinations of covariates X1,…, XpPolynomial combinations of X1, …, XpSemi-parametric regression splines
26Penalized regression splinesRandom effects and Bayesian models
Likelihood Inference for the FHT Model
The likelihood contribution of each sample subject is asThe likelihood contribution of each sample subject is as follows.
• If the subject fails at S=s:
f (s | y0, μ) = Pr [ first-hitting-time in (s, s+ds) ]
• If the subject survives beyond time L: j y
1- F (L | y0 ,μ) = Pr [ no first-hitting-time before L ]( | y0 μ) [ g ]
( ) ( ) ( ) ( ){ }0 0 01
ln , ln , 1 ln , .n
i i i ii
L x d f t x d F t xθ θ θ=
= + −∑where
is the failure indicator for subjectid i
( )
is the failure indicator for subject is a censored survival time if subject fails
i
i i i
d it t s i=
and denote the FHT f F p.d.f and complementary c.d.f.
Extensions of Threshold Regression (TR)
1. Process Y(t): Wiener process, gamma process, etc
2. Boundary: straight lines or curves
3. Time scale: calendar time, running or analytical time
Example 4Comparing TR and PH for two possible scenarios
Scenario 1: A favorable surgical intervention might double the health level of a patient (relative todouble the health level of a patient (relative to control) at the outset of the study but leaves the rate of decline of health unchanged on both studyrate of decline of health unchanged on both study arms.
Scenario 2: Drug treatment for chronic disease may g yslow the rate of decline in health (relative to control) but leaves the initial health level )unchanged on both study arms.
30
Scenario 1 Scenario 2
2 2
T t tProcessLevel
ProcessLevel 1
-1
1
Treatment
T t ty y-1 -1 -0.5Control
Treatment
Control
Time Time
Control
Time Time
Example 4Consider a simulated randomized clinical trial with two
study arms and n=200 subjects on each arm. (control: arm=0; treatment: arm=1). The TR model is a Wiener diffusion model.
Regression links for the TR modelln (y ) = α + α armln (y0) = α0 + α1 arm
μ = β0 + β1 arm
In Scenario 1: α = ln (2) = 0 6931 β =0In Scenario 1: α1 = ln (2) = 0.6931, β1=0In Scenario 2: α1 = ln (1) = 0, β1= 0.5
32
Example 4 (simulation)
The TR model:ln (y0) = α0 + α1 arm
β + β armμ = β0 + β1 armThe mean survival time on any arm is given by
E(S)= y0 / |μ|
Both scenarios have the same mean of 1 i l th t l d1 year survival on the control arm and 2 years on the treatment arm.
33
y
A Comparison of TR and PH Regression Results
Example 4: Modeling Results for Scenario 1In scenario 1, the PH assumption is untenable.
The analyst would then need to proceed to some other model.
If the TR model is considered, the table shows a significant positive treatment effect for ln y butsignificant positive treatment effect for ln y0 but an insignificant effect for μ (as expected).
The significant coefficient for α1 translates into ti t d (0 7752) 2 2 lti li fan estimated exp(0.7752)=2.2 multiplier for
initial health level y0 (note that the true value is 2 i h i i l )2, i.e. the treatment arm is twice as large at y0)
Example 4: Modeling Results for Scenario 2I i 2 h PH i i blIn scenario 2, the PH assumption is tenable. The analyst using PH regression would discover a
significant hazard ratio of 0.5581, indicating that the treatment is about 56% of the control risk. However, the question of how the treatment is acting on thethe question of how the treatment is acting on the subject is not known from the PH model.
If the TR model is considered, as expected, the table h i ifi t iti t t t ff t f b tshows a significant positive treatment effect for μ but
an insignificant effect for ln y0 (). Th i ifi t ffi i t f β t l t i tThe significant coefficient for β1 translates into an
estimated 0.4282 multiplier in the value of μ (the treatment is about half of the control)
36
treatment is about half of the control).
22
Scenario 1
1.2
1.6
1.2
1.6
Control hazardTreatment hazard
.4.8
.4.8
Log-hazard ratio
0
0 2 4 6 8 10
0
0 5 10 Time (years) Time (years)
22
Scenario 2
1.2
1.6
1.2
1.6
Control hazard
Treatment hazardLog-hazard ratio
0.4
.8
0.4
.8
0
0 5 10
0
0 5 10 Time (years) Time (years)
Connection of the TR model with the PH model
Most survival distributions are hitting time distributions for stochastic processes (the basic TR model)
Families of PH functions can be generated by varying time scales or boundaries of a TR modelvarying time scales or boundaries of a TR model
Th f il f PH f ti b d dThe same family of PH functions can be produced by different TR mode
Connection of the TR model with PH model
Families of PH functions can be generated by g y1. Varying process Y and time scales of a TR model 2. Varying process Y and boundaries of a TR model. y g p
Hence, the same family of PH functions can beHence, the same family of PH functions can be produced by different TR models.
For details, seeM-LT Lee, Whitmore GA. Proportional hazards and , p
threshold regression: their theoretical and practical connections. Lifetime Data Analysis (2010).
39
Benefits of the TR ModelBenefits of the TR Model
Probing the causal forces behind a hazard function i l th hil dis always a worthwhile endeavor.
A TR model represents more fundamental knowledge about the underlying science than its corresponding PH model.
These insights would deepen the investigator’s understanding of the case application.understanding of the case application.
Whitmore, G. A. (1986). First-passage-time models for duration data: Regression structures and competingduration data: Regression structures and competing risks. The Statistician 35 207–219.
Aalen O.O. and Gjessing H.K. (2001). Understanding the shape of the hazard rate: a process point of view. Statistical Science, 16: 1-22.
Lee M L T and G A Whitmore (2006) ThresholdLee, M.-L. T. and G. A. Whitmore (2006). Threshold regression for survival analysis: modeling event times by a stochastic process reaching a boundary.times by a stochastic process reaching a boundary. Statistical Sciences.
Aalen O.O., Borgon O, and Gjessing H.K (2008). Survival and Event History Analysis: A process Point of View Springer
41
Point of View. Springer.
• Yu Z, Tu W, Lee M-LT (2009). A Semiparametric ( )Threshold Regression Analysis of Sexually Transmitted Infections in Adolescent Women, Statistics in Medicine.
• Pennell M, GA Whitmore, Lee M-LT (2009). Bayesian random effects threshold regression with application torandom effects threshold regression with application to survival data with nonproportional hazards, Biostatistics.
• Lee M-LT, Whitmore GA (2010). Proportional hazards and threshold regression: their theoretical and practical
ti Lif ti D t A l iconnections. Lifetime Data Analysis.
• Lee M-LT Whitmore GA Rosner B (2010) ThresholdLee M LT, Whitmore GA, Rosner B. (2010) Threshold regression for survival data with time-varying covariates. Statistics in Medicine .
Z=0 Z=1 Z=2
Nested boundaries
Z=0 Z=1 Z=2
ProcessLevel
y
Representative sample path for z=2
Time
.4.3
Z=0 Z=1 Z=2
.3.2
Hazard
.1
Hazardh(t)
.10
0 10 20 30 400 10 20 30 40xTime t
y(t|z)Boundary b(t|z)Boundary b(t|z)
ProcessLevel
y
s tTime0
0
y(t|z)
Overshooty(s|z)-b(s|z)
Boundary
Boundaryb(s|z)Process
LevelBoundary
b(t|z)y
s tTime
• Aalen O.O. and Gjessing H.K. (2001). Understanding the shape of the hazard rate: a process point of view. p p pStatistical Science, 16: 1-22.
• Lee, M.-L. T. and G. A. Whitmore (2006). Threshold regression for survival analysis: modeling event times by a stochastic process reaching a boundary. Statistical SciencesStatistical Sciences.
• Aalen O.O., Borgon O, and Gjessing H.K (2008). Survival and Event History Analysis: A process Point of y y pView. Springer.
• Yu Z, Tu W, Lee M-LT (2009). A Semiparametric Th h ld R i A l i f S ll T itt dThreshold Regression Analysis of Sexually Transmitted Infections in Adolescent Women, Statistics in Medicine(in press)(in press).
• Pennell M, GA Whitmore, Lee M-LT (2009). Bayesian random effects threshold regression with application g ppto survival data with nonproportional hazards, Biostatistics, (In Press).
The Connection ofThreshold Regression Models
With Cox Proportional Hazard ModelsWith Cox Proportional Hazard Models
Mei-Ling Ting Lee, University of MarylandAlex Whitmore, McGill University
Lifetime Data Analysis 2010
Relation of Cox Proportional Hazards andRelation of Cox Proportional Hazards and Threshold Regression
The class of survival functions that are first-hitting-time (FHT) distributions includes mosthitting time (FHT) distributions includes most distributions of practical interest
Almost any family of proportional hazard (PH) functions can be generated from an FHT contextfunctions can be generated from an FHT context by an appropriate choice of process time or boundary for the stochastic process defining theboundary for the stochastic process defining the FHT.
C i b id d i lCox regression can be considered as a special case of threshold regression (TR) for the class
f FHT di t ib ti th t h th ti lof FHT distributions that have the proportional hazard property
Suitable methods allow TR to replicate Cox PH i ith ith fi d ti iregression with either fixed or time-varying
covariates
Generating PH Functions in an FHT ContextGenerating PH Functions in an FHT Context
Three building blocks for threshold regressionThree building blocks for threshold regression
1 A stochastic process that describes the1. A stochastic process that describes the evolution of the subject's underlying health statestate
2. A critical level, condition or boundary (the threshold) that triggers the event of interestthreshold) that triggers the event of interest when it is reached by the process for the first time (the FHT)time (the FHT)
3. A time scale on which the process unfolds.
Generating PH Functions in an FHT ContextGenerating PH Functions in an FHT Context
The three building blocks of TR give manyThe three building blocks of TR give many combinations of conditions that can
t th f il f Cgenerate the same family of Cox proportional hazard functions:
We now show two methods of construction of a PH family from an FHT context.of a PH family from an FHT context.
1. Generating PH by Varying Process Timeg y y gLet denote a process time function, defined as a function of calendar time t,,conditional on covariate z.
Let the survival function for FHT context be denoted by: y
We have the following two correspondences:g p
Varying process time:
Generating PH by Varying Process Time g y y g
Equating the two right-hand sides and solvingEquating the two right hand sides and solving for gives process time functions that generate a family of PH functionsgenerate a family of PH functions.
Examples: Poisson process:p p
A Poisson process with unit intensity has theA Poisson process with unit intensity has the following exponential survival time distribution:distribution:
Solving for the process time function gives the following Observe that the process timethe following. Observe that the process time is proportional to the cumulative baseline hazard function H (t)hazard function H0(t).
Examples: Brownian motion process:p pA Brownian motion process with a unit variance,
starting at x0 , has the following survival time g 0 , gdistribution:
where Φ denotes the standard normal c.d.f.
Solving for the process time function gives:
2. Generating PH by Varying Boundariesg y y gA family of PH functions can be generated by varying the boundary function.y g y
Example: Brownian motion processp p
The following identity links the boundary b(t|z)to the c.d.f. of survival time F(t|z) for a given covariate vector z in standard Brownian motion starting at zero:
Generating PH by Varying Boundariesg y y g
Replacing F(t|z) by the c.d.f. for each member of a p g ( | ) yPH family and solving for boundary function b(t|z) gives a set of nested boundaries that ( | ) ggenerate FHTs with proportional hazard functions.
Example:p
The next figure shows a nested set of three gboundaries for standard Brownian motion that correspond to three levels of a fixed covariate z.p
Covariate z might represent, for example, level g p , p ,of occupational exposure to a carcinogen
The figure shows an illustrative sample path for a case where exposure level z=2 generates a p gsurvival time S.
b0 b1 b2
10
z=0Nested boundaries
0
z=1
z=2z=2
-10 Simulated path x(t) for z=2
x0 20 40Sfor z=2
Three boundaries in Brownian motion that correspond to three levels of covariate zthree levels of covariate z
Example:p
The next figure shows the hazard functions for gsurvival time corresponding to the preceding three boundaries for Brownian motion.
The boundaries and process have been chosen pso the hazard functions are, in fact, proportional for the three levels of fixed covariate z.
haz0 haz1 haz2
.3
z=2h(t).2 z=1
0
( )
.1
z=0
0
x0 20 40
Hazard functions generated by the FHT model. The functions are proportional for the three levels of covariate z.p p
The threg package in R
Install threg package in RInstall threg package in R
• Open R in Windows
• From menu “Packages ‐> Install package(s)From menu Packages > Install package(s) from local zip files...”, select the file "threg 1 0 1 zip" and click "Open“threg_1.0.1.zip and click Open .
• Type the following command in R prompt to load this package:
> library(threg)> library(threg)
• Now you can use the threg package in R.
An exampleAn example
• The following example can be found in the help document of the threg package, by p g p g ytyping the following command in R prompt:
> ?threg> ?threg
Themyeloma datasetThe myeloma dataset*The myeloma dataset is a simulated dataset with 49 observations.
Variable Name Description
time survival timetime survival time
fail censoring indicator variable ‐ 0 indicates a right censored time and 1 indicates an observed event timeobserved event time.
age age of the corresponding subject. It is a continuous variable.
gender gender of the corresponding subject. It is a categorical variable.
treat treatment type of the correspondingtreat treatment type of the corresponding subject. It is a categorical variable with 4 levels.
Load themyeloma datasetLoad the myeloma dataset
l d h d " l “ i h k• load the data "myeloma“ in the package:> data("myeloma")( y )
• Transform the "gender" and "treat" variables into factor variables "f gender“ and "f treat"into factor variables f.gender and f.treat .> f.gender<‐factor(myeloma$gender)f f ( l $ )> f.treat<‐factor(myeloma$treat)
• bind the two new factor variables "f.gender" gand "f.treat" to the dataset "myeloma".> myeloma<‐cbind(myeloma f gender f treat)> myeloma< cbind(myeloma,f.gender,f.treat)
Fit a threshold regression model by using threg function
Fit th h ld i d l b i d• Fit a threshold regression model, by using age and f.gender to predict parameter lny0 in the threshold regression model, and by using age, f.gender and g , y g g , gf.treat to predict parameter mu in the threshold regression model:th (d t l> threg(data=myeloma,
+ lny0=age+f.gender,+ mu age+f gender+f treat+ mu=age+f.gender+f.treat,+ time=time,+ failure=fail+ failure=fail+ )
OutputOutput(regression coefficient estimates, log‐likelihood value and AIC value)
$coef est$coef.estCoef. Est. Std. Err. z P>|z|
lny0: (Intercept) 3.2982293 1.2223463 2.698 0.0070lny0: age ‐0.0141177 0.0169046 ‐0.835 0.4037l f d ( )lny0: f.gender(1) 0.0779063 0.3553153 0.219 0.8267mu: (Intercept) ‐4.6616101 4.9709192 ‐0.938 0.3482mu: age 0.0172552 0.0707793 0.244 0.8072mu: f.gender(1) ‐0.1134031 1.3306915 ‐0.085 0.9323mu: f.treat(1) 0.1710347 0.2071231 0.826 0.4088mu: f.treat(2) 0.0941807 0.2110082 0.446 0.6556mu: f.treat(4) ‐0.7977747 0.6781792 ‐1.176 0.2396
$log.likelihood[1] ‐29.91943
$AIC$AIC[1] 77.83886
Hazard ratio calculationHazard ratio calculation
Fit th th h ld i d l i• Fit the same threshold regression model as previous, and calculate the hazard ratios for the factor variable "f.treat", at a fixed time point 3, for a specified , p , pscenario: age=50, gender=0:> threg(data=myeloma,
l 0 f d+ lny0=age+f.gender,+ mu=age+f.gender+f.treat,+ time=time+ time=time,+ failure=fail,+ hr=f.treat,+ timevalue=3,+ scenario=age(50)+f.gender(0)
)+ )
OutputOutput(hazard ratios)
( t t itt d)(some output omitted)$scenario[1] "f.gender = 0 age=50 at time = 3"
$hazard.ratioHaz. Ratio
f.treat(1) 0.6416700f.treat(2) 0.7891791( )f.treat(4) 4.1874754
Plot of hazard functionsPlot of hazard functions
G t th l id ti t d h d f ti• Generate the overlaid estimated hazard functions corresponding to different levels of "treat", for a specified scenario: age=50, gender=0:p g , g> threg(data=myeloma,+ lny0=age+f.gender,+ mu=age+f.gender+f.treat,+ time=time,+ failure=fail+ failure=fail,+ hr=f.treat,+ timevalue=3,,+ scenario=age(50)+f.gender(0),+ graph="hz"+ )
OutputOutput(estimated hazard functions )
Plot of survival functionsPlot of survival functions
G t th l id ti t d i l f ti• Generate the overlaid estimated survival functions corresponding to different levels of "treat", for a specified scenario: age=50, gender=0:p g , g> threg(data=myeloma,+ lny0=age+f.gender,+ mu=age+f.gender+f.treat,+ time=time,+ failure=fail+ failure=fail,+ hr=f.treat,+ timevalue=3,,+ scenario=age(50)+f.gender(0),+ graph=“sv"+ )
OutputOutput(estimated survival functions )
Plot of probability density functionsPlot of probability density functions
G t th l id ti t d b bilit d it• Generate the overlaid estimated probability density functions corresponding to different levels of "treat", for a specified scenario: age=50, gender=0:p g , g> threg(data=myeloma,+ lny0=age+f.gender,+ mu=age+f.gender+f.treat,+ time=time,+ failure=fail+ failure=fail,+ hr=f.treat,+ timevalue=3,,+ scenario=age(50)+f.gender(0),+ graph=“ds"+ )
OutputOutput(estimated probability density functions )
