A Spatial Scan Statistic for Survival Data

1
A Spatial Scan Statistic for Survival Data

• Lan Huang, Dep Statistics, Univ Connecticut
• Martin Kulldorff, Harvard Medical School
• David Gregorio, Dep Community Medicine, Univ
  Connecticut

2
Motivation and Background

• What is the geographical distribution of
  prostate cancer survival in Connecticut?
• Are there geographical clusters with
  exceptionally short or long survival?

3
Survival Data

• For each person
• Time of diagnosis.
• Whether dead or censored
• Time until death/censoring
• Residential geographical coordinates
• Age
• etc

4
Motivation and Background

• Spatial scan-statistics with Bernoulli and
  Poisson models are designed for count data.
• Length of survival is continuous data.
• Survival data is often censored.

5

• Solution
• Spatial Scan Statistic using an
• Exponential Probability Model

6
Methodology

• Exponential model based spatial statistic

Spatial scan-statistic
Ha ?in ?out
Exponential likelihood
H0 ?in ?out
Permutation test
distribution
Stat inference Hypothesis test
Detect a significant cluster
7
Methods Evaluation

• Location of 610 Connecticut prostate cancer
  patients diagnosed in 1984.
• 47 patients in southwest Connecticut constitute a
  cluster with shorter survival (cluster radius
  8.65 km)
• Each of the 610 patients assigned a random
  survival or censoring time using different
  distributions inside and outside the cluster

8
Model Evaluation
610 individuals
47
563
?in
?out
?diff
-

1
9
Exponential
3
7
Non-cen
Gamma
10
5
5
random
censored
Log-normal
fixed
3
7
9
1
9
individuals inside the true cluster ,
successfully detected for the simulated datasets
without censoring
s
P-valuelt0.05
?diff
10
individuals inside the true cluster ,
successfully detected for censored datasets with
fixed censoring time
s
P-valuelt0.05
?diff
11
individuals inside the true cluster ,
successfully detected for censored datasets with
random censoring time
s
P-valuelt0.05
?diff
12
Model Evaluation

• Exponential model is robust, since the
  exponential based scan statistic is able to
  reject the null hypothesis with a low p-value
  when the distribution difference is moderate or
  large, no matter the distribution and censoring
  mechanism.

13
Application to Prostate Cancer Data

• Between 1984 and 1995, the Connecticut Tumor
  registry recorded 22612 invasive prostate cancer
  incidence cases among the population-at-risk
  (roughly 1.2 million males 20 years old in
  1990).
• 19061 records available after data cleaning.
• Follow-up through December 2000.
• 10308 had died and 8753 were censored.

14
Significant clusters using exponential model
15
Application to Prostate Cancer Data
cluster In cluster In cluster RR LLR P
cluster death indivi RR LLR P
Short survival 1 646 938 1.45 41.88 0.001
Short survival 2 2154 3706 1.13 19.06 0.001
Short survival 3 33 36 3.26 16.13 0.003
Long survival 4 661 1445 0.75 31.83 0.001
Long survival 5 200 529 0.65 22.24 0.001
Long survival 6 37 114 12.11 12.11 0.015
16
Covariate Adjustment

• Younger patients may live longer
• Geographical variation in histology or stage

17
Significant clusters after age-adjustment
18
Discuss

• Exponential model works well for censored and
  non-censored survival data from difference
  distribution, but probably no do well for all
  continuous variables, like data that is
  approximated normally distributed.
• The statistical inference is valid even though
  the survival times are not exponentially
  distributed because of the permutation based test
  procedure.

19
Discussion

• The covariate adjustment method here is based on
  the exponential model, assuming a constant
  hazard. It could be extended to non-constant
  hazard with several levels, or as a function of
  survival time associated with different kind of
  models.
• It could be extends to a space-time scan
  statistic when time series data are available.
• It could also be extended to create a
  scan-statistic with elliptical or other cluster
  shapes.
• Unfortunatly, no statistical software available.