K-adaptive partitioning for survival data

Introduction Proposed method Simulation studies Real example Conclusion References

K-Adaptive Partitioning for Survival Data

with an Application to SEER

Soo-Heang Eo

with Sungwan Bang, Seung-Mo Hong, HyungJun Cho

Department of Statistics

Korea University

April 25, 2014


Table of Contents

Introduction

Motivation example

Previous studies

Proposed method

Finding the best split set

Finding optimal number of subgroups

Simulation studies

Simulation design

Simulation results

Real example

Conclusion

References


Cancer staging

cancer staging system that describes the extent of cancer in a

patient’s body. [2]


TNM staging system

The TNM Classification of Malignant Tumors (TNM) is a cancer

staging system that describes the extent of cancer in a patient’s

body. [2]

T describes the size of the tumor and whether it has invaded

nearby tissue,

N describes distant metastasis lymph node (spread of

cancer from one body part to another).

M describes regional lymph nodes that are involved,


Previous studies

• Hilsenbeck and Clark (1996) concerned simulation studies to

examine the effects of number of cutpoints and true marker

prognostic effect size [3].

• Contal and O’Quigley (1999) proposed the asymptotic distribution

of a re-scaled rank statistic [1].

• Lausen et al. (1994) and Hong et al. (2007) employed decision

tree methodology [8, 4].

• Hothorn and Lausen (2003) used maximally selected rank

statistics [6] to find the best cutpoints and extended to CTree [5].

• Ishwaran et al. (2009) adopted the idea of random survival forests

[7].

Most approaches to find cut-off point have been based on concepts from

binary split.


Problem is ...

|

Node 8 Node 9

Node 20 Node 21

Node 11

Node 6 Node 7

n= 36086 n= 7331

n= 4693 n= 3288

n= 4401

n= 6183 n= 3204

med= 115 med= 80

med= 57 med= 47

med= 38

med= 23 med= 13

meta

meta

meta meta

meta

meta

<=5

<=1

<=0 <=3

<=2

<=11

Node 1

Node 2

Node 4 Node 5

Node 10

Node 3

n= 65186

n= 55799

n= 43417 n= 12382

n= 7981

n= 9387

med= 77

med= 95

med= 110 med= 47

med= 53

med= 19

Survival months

Sur

viva

l pro

babi

lity

0.0

0.2

0.4

0.6

0.8

1.0

0 24 48 72 96 120 144 168 192

Node 8 (# of Meta = 0)Node 9 (# of Meta = 1)Node 20 (# of Meta = 2)Node 21 (# of Meta = 3)Node 11 (# of Meta = 4,5)Node 6 (# of Meta = 6,7,...,11)Node 7 (# of Meta = 12,...,80)


Problem is ...

(a) TSK = 148

p<0.0001

(b) TSK = 132

p<0.0001


Our aim is to

• Divide the whole data D into K heterogeneous subgroups

D1, . . . ,DK based on the information of X .

• Overcome the limitation that some subgroups differ

substantially in survival, but others may differ barely or

insignificantly.

• Evaluate multi-way split points simultaneously and find an

optimal set of cutpoints.

• Implement the proposed algorithm into an R package kaps.



Let Ti be a survival time, Ci a censoring status, and Xi be an

ordered covariate for the i th observation. We observe the triples

(Yi , δi ,Xi ) and define

Yi = min(Ti ,Ci ) and δi = I (Ti ≤ Ci ),

which represent the observed response variable and the censoring

indicator, respectively.



Let χ21 be the χ2 statistic with one degree of freedom (df) for

comparing the g th and hth of K subgroups created by a split set

sK when K is given. For a split set sK of D into D1,D2, . . . ,DK ,

the test statistic for a measure of deviance can be defined as

T1(sK ) = min1≤g<h≤K

χ21 for sK ∈ SK , (1)

where SK is a collection of split sets sK generating K disjoint

subgroups.



Then, take s∗K

as the best split set such that

T ∗1(s∗K ) = maxsK∈SK

T1(sK ). (2)

The best split s∗K

is a set of (K − 1) cutpoints which clearly

separate the data D into K disjoint subsets of the data:

D1,D2, . . . ,DK .


Finding the best split set - Illustration

• We assumed that the patients were divided into 3 heterogeneous

groups by the number of lymph nodes (X = {0, 1, . . . , 6}).

• Each group was categorised as D1,D2 and D3.

• 3 pairs of groups (D1vs .D2,D2vs .D3, and D1vs .D3).

• We imagined existing 3 cut-off points candidates, each of which

were composed of two cutpoints; {0,1}, {0,2}, and {1,2}.

• For example, the cutpoints candidate {0, 1} mean that

D1 = {X = 0},D2 = {0 < X ≤ 1}, and D3 = {X > 1}.

• Out of the 3 pairs of the candidates, the smallest test statistic was

selected as a representative statiatic for the candidate.


Finding the best split set - Illustration




Algorithm 1. Finding the best split set for given K

Step 1: Compute chi-squared test statistics χ21 for all possible pairs, g and h, of K

subgroups by sK , where 1 ≤ g < h ≤ K and sK is a split set of (K − 1)

cutpoints generating K disjoint subgroups.

Step 2: Obtain the minimum pairwise statistic T1(sK ) by minimizing χ21 for all

possible pairs, i .e., T1(sK ) = min1≤g<h≤K χ21 for sK ∈ SK ,where SK is a

collection of split sets sK generating K disjoint subsets of the data.

Step 3: Repeat Steps 1 and 2 for all possible split sets SK .

Step 4: Take the best split set s∗K

such that T ∗1(s∗K

) = maxsK ∈SKT1(sK ). When two

or more split sets have the maximum T ∗1 of the minimum pairwise statistics,

choose the best split set with the largest overall statistic T ∗K−1

.


Finding optimal subgroups

• One of the important issues is to determine a reasonable

number of subgroups, i.e. the selection of an optimal K .

• We find an optimal multi-way split at a time for the given

number of subgroups.

• We need to choose only one of a possible number of

subgroups.

• For a data-driven objective choice, we here suggest a

statistical procedure to choose an optimal number of

subgroups.



The minimum pairwise statistics of the permuted data

pK =

R∑r=1

I (T (r )1 (s∗K ) ≥ T ∗1(s∗K ))/R, K = 2, 3, . . . ,

where T(r )1 (s∗

K) is the r th repeated minimum pairwise statistic for

the permuted data.

• The data can be reconstructed by matching their labels after

permuting the labels of X with retaining the labels of (Y , δ).

• When the permuted data are allocated into each subgroup by

s∗K

, there should be no significant differences in survival

among the subgroups.



We choose the largest number to discover as many significantly

different subgroups as possible, given that the corrected p-values

are smaller than or equal to a pre-determined significance level

K = max{K |pcK ≤ α,K = 2, 3, . . .},

where pCK

is the corrected p-values for multiple comparison because

there are (K − 1) comparisons between two adjacent subgroups


Finding optimal subgroups - Illustration

Survival months

Sur

viva

l pro

babi

lity

0.0

0.2

0.4

0.6

0.8

1.0

0 24 48 72 96 120 144 168 192

G1G2

(a) K = 2TSK = 26.37

TS1 = 26.37

pK <0.0001

p1 = <0.0001

Survival months

Sur

viva

l pro

babi

lity

0.0

0.2

0.4

0.6

0.8

1.0

0 24 48 72 96 120 144 168 192

G1G2G3

(b) K = 3TSK = 36.80

TS1 = 7.2

pK <0.0001

p1 = 0.0073

Survival months

Sur

viva

l pro

babi

lity

0.0

0.2

0.4

0.6

0.8

1.0

0 24 48 72 96 120 144 168 192

G1G2G3G4

(c) K = 4TSK = 38.00

TS1 = 1.895

pK <0.0001

p1 = 0.1686


Finding optimal subgroups - Illustration

meta

Sur

viva

l mon

ths

0 10 20 30

1248

7296

120

144

168

192

Event

Censored

Survival months

Sur

viva

l pro

babi

lity

0.0

0.2

0.4

0.6

0.8

1.0

0 24 48 72 96 120 144 168 192

G1

G2

G3

K

p-va

lues

of X

k

2 3 4

0.0

0.1

0.2

0.3

0.4

0.5

K

p-va

lues

of X

1

2 3 4

0.0

0.2

0.4

0.6

0.8

1.0


Algorithm 2. Selecting the optimal number of subgroups

Step 1: Find s∗K

and T1(s∗K

) with the raw data for each K using Algorithm 1.

Step 2: Construct the permuted data by permuting the labels of X whilst retaining

the labels of (Y , δ).

Step 3: Allocate the permuted data into each subgroup by s∗K

.

Step 4: Obtain the minimum pairwise statistic T(r )1 (s∗

K) for the permuted data.

Step 5: Repeat steps 2 to 4 R times, and then obtain

T(1)1 (s∗

K),T (2)

1 (s∗K

), . . . ,T (R)1 (s∗

K).

Step 6: Compute the permutation p-value pK for each K , i .e.,

pK =∑R

r=1 I (T (r )1 (s∗

K) ≥ T1(s∗

K))/R, K = 2, 3, . . . .

Step 7: Correct the permutation p-value pK by correcting for multiple comparisons,

e.g ., corrected p-value pcK= pK /(K − 1),K = 2, 3, . . ..

Step 8: Select the largest K when the corrected p-values are less than or equal to α,

i .e., K = max{K | pcK≤ α,K = 2, 3, . . .}.


Simulation

Simulation study: kaps vs. binary split (rpart and ctree)

Design:

• survival time Ti ∼ exp(λi )

• censoring time Ci ∼ exp(c0)

• response Yi = min(Ti ,Ci ) and δi = I (Ti ≤ Ci ), where i = 1, 2, . . . ,n

• a prognostic factor Xi ∼ DU (1, 20)

• KAPS vs. Survival CART [9] and CTREE [5]

• training and test samples: 200 observations each with censoring rates of

15% or 30%

• repeated 100 times independently


Setting 1 - stepwise model

We consider the following stepwise model (SM) defining parameter

λi as follows.

λi =

0.02, Xi ≤ 7,

0.04, 7 < Xi ≤ 14,

0.08, 14 < Xi ,

(3)

This model has three different hazard rates that are distinguished

by two cutpoints 7 and 14.


Setting 2 - linear model

In addition, we consider the following linear model (LM) defining

parameter λi as follows.

λi = 0.1Xi . (4)

In this model, λi depends on Xi linearly.

It follows that Yi depends on Xi nonlinearly.

This model has a number of different hazard rates.


Results of the stepwise model (SM)

Selection of cut-off points when K is known

Cutpoint

Pro

porti

on (%

)

0 5 10 15 20

010

2030

4050

60 kaps

Cutpoint

Pro

porti

on (%

)

0 5 10 15 20

010

2030

4050

60 ctree

Cutpoint

Pro

porti

on (%

)

0 5 10 15 20

010

2030

4050

60 rpart

0 5 10 15 20

05

1015

20

C1

C2

kaps

0 5 10 15 20

05

1015

20

C1

C2

ctree

0 5 10 15 20

05

1015

20

C1

C2

rpart


Results of the stepwise model (SM)

Selection of the optimal number of subgroups (K = 3)

Number of subgroups

Pro

porti

on (%

)0

2040

6080

1 2 3 4 1 2 3 4 1 2 3 4

rpart ctree kaps

(a) SM model with CR = 15%

Number of subgroups

Pro

porti

on (%

)0

2040

6080

1 2 3 4 1 2 3 4 1 2 3 4

rpart ctree kaps

(b) SM model with CR = 30%


Results of the linear model (LM)

Selection of cut-off points when K is unknown

Model K Method CR = 15% CR = 30%

Overall Pairwise Overall Pairwise

rpart 43.57 (1.17) 43.57 (1.17) 38.73 (1.03) 38.73 (1.03)

2 ctree 38.94 (1.16) 38.94 (1.16) 36.29 (1.01) 36.29 (1.01)

kaps 44.52 (1.15) 44.52 (1.15) 38.74 (1.11) 38.74 (1.11)

rpart 48.07 (1.47) 6.87 (0.50) 43.21 (1.11) 6.32 (0.44)

LM 3 ctree 55.64 (1.39) 12.94 (1.14) 47.99 (1.40) 8.33 (0.64)

kaps 54.96 (1.39) 13.83 (0.63) 47.95 (1.36) 11.33 (0.59)

rpart 59.60 (1.59) 2.59 (0.26) 53.01 (1.33) 2.30 (0.22)

4 ctree 59.82 (1.35) 3.17 (0.37) 52.05 (1.34) 2.10 (0.21)

kaps 61.27 (1.39) 3.22 (0.24) 53.48 (1.34) 2.70 (0.23)


Results of the linear model (LM)

Selection of the optimal number of subgroups (K =?)

Number of subgroups

Pro

porti

on (%

)0

1020

3040

5060

70

2 3 4 5 2 3 4 5 2 3 4 5

rpart ctree kaps

(a) LM model with CR = 15%

Number of subgroups

Pro

porti

on (%

)0

2040

6080

2 3 4 5 2 3 4 5 2 3 4 5

rpart ctree kaps

(b) LM model with CR = 30%


Surveillance, Epidemiology, and End Results Program


Surveillance, Epidemiology, and End Results Program

• Population-based database by NCI and CDC

• Collects data on cancer cases from various locations and

sources throughout the US

• Data collection began in 1973 with a limited amount of

registries and continues to expand to include even more areas

and demographics today


Colon cancer in SEER database


Colon cancer analysis

Table: Numbers of metastasis lymph nodes for the staging systems by

AJCC, kaps, ctree, and rpart

Subgroup AJCC rpart ctree kaps

Subgroup 1 0 0 0 0

Subgroup 2 1 1,2,3 1 1

Subgroup 3 2,3 4 ∼ 10 2,3 2,3

Subgroup 4 4,5,6 ≥ 11 4,5 4,5,6

Subgroup 5 ≥7 — 6,7,8 7,8,9,10

Subgroup 6 — — ≥9 ≥ 11

Min. pairwise statistic 131.23 932.30 78.35 131.23

Corresponding pair (2, 3) (3, 4) (3, 4) (2, 3)



Survival months

Sur

viva

l pro

babi

lity

0.0

0.2

0.4

0.6

0.8

1.0

0 24 48 72 96 120 144 168 192

Subgroup 1 (No LN)Subgroup 2 (1 LN)Subgroup 3 (2-3 LNs)Subgroup 4 (4-6 LNs)Subgroup 5 (7-10 LNs)Subgroup 6 (>=11 LNs)

(a) kaps

Survival months

Sur

viva

l pro

babi

lity

0.0

0.2

0.4

0.6

0.8

1.0

0 24 48 72 96 120 144 168 192

Subgroup 1 (No LN)Subgroup 2 (1 LN)Subgroup 3 (2-3 LNs)Subgroup 4 (4-6 LNs)Subgroup 5 (>=7 LNs)

(b) AJCC



Survival months

Sur

viva

l pro

babi

lity

0.0

0.2

0.4

0.6

0.8

1.0

0 24 48 72 96 120 144 168 192


(a) kaps

Survival months

Sur

viva

l pro

babi

lity

0.0

0.2

0.4

0.6

0.8

1.0

0 24 48 72 96 120 144 168 192

Subgroup 1 (No LN)Subgroup 2 (1-3 LNs)Subgroup 3 (4-10 LNs)Subgroup 4 (>=11 LNs)

(b) rpart



Survival months

Sur

viva

l pro

babi

lity

0.0

0.2

0.4

0.6

0.8

1.0

0 24 48 72 96 120 144 168 192


(a) kaps

Survival months

Sur

viva

l pro

babi

lity

0.0

0.2

0.4

0.6

0.8

1.0

0 24 48 72 96 120 144 168 192


(b) ctree


Conclusion

We propose a multi-way partitioning algorithm for survival data,

• Divide the data into K heterogeneous subgroups based on the

information of a prognostic factor.

• Generate only fairly well-separated subgroups rather than a

mixture of extremely poorly and well-separated subgroups.

• Identify a multi-way partition which maximizes the minimum

of the pairwise test statistics among subgroups.

• Utilize a permutation test to consists of tow or more

cutpoints.

• Implement an add-on R package into kaps available at CRAN.


Conclusion

• HJ Kang*, S-H Eo*, SC Kim, KM Park, YJ Lee, SK Lee, E Yu, H Cho,

and S-M Hong (2014). Increased Number of Metastatic Lymph Nodes in

Adenocarcinoma of the Ampulla of Vater as a Prognostic Factor: A

Proposal of New Nodal Classification, Surgery, 155:1, 74-84. (*:co-first

author)

• S-H Eo, S-M Hong, and H Cho (2014). K -adaptive partitioning for

survival data: the kaps add-on package for R, arXiv, 1306.4015v2.

• S-H Eo, S-M Hong, and H Cho. K -adaptive partitioning for survival

data, submitted to Statistics in Medicine.

• Park et al. Survival effect of tumor size and extrapancreatic extension in

surgically resected pancreas cancer patients: Proposal for improved T

classification, submitted to Surgery.


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K-adaptive partitioning for survival data

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