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

Motivation and Background

What is the geographical distribution of prostate cancer survival in Connecticut?

Are there geographical clusters with exceptionally short or long survival?

Survival Data

For each person:

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

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.

Solution

Spatial Scan Statistic using an

Exponential Probability Model

Methodology• Exponential model based spatial statistic

H0: θin= θout

Ha: θin θoutExponential likelihood

Spatial scan-statistic

distribution

Permutation test

Stat inference Hypothesis testDetect a

significant cluster

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

Model Evaluation

Exponential

Gamma

Log-normal

θin θout

1

5

3

7

9

10

θdiff

1

3

5

7

9

Non-cen

censoredrandom

fixed

610 individuals

47563

- =

#individuals inside the true cluster , successfully detected for the simulated datasets without censoring

0

5101520253035404550

1 3 5 7 9

expgammalog-nor

θdiff

P-value<0.05

s

#individuals inside the true cluster , successfully detected for censored datasets with fixed censoring time

0

5101520253035404550

1 3 5 7 9

expgammalog-nor

θdiff

P-value<0.05

s

#individuals inside the true cluster , successfully detected for censored datasets with random censoring time

0

5101520253035404550

1 3 5 7 9

expgammalog-nor

P-value<0.05

θdiff

s

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.

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.

Significant clusters using exponential model

cluster In cluster RR LLR P

#death #indivi

Short

survival

1 646 938 1.45 41.88 0.001

2 2154 3706 1.13 19.06 0.001

3 33 36 3.26 16.13 0.003

Long

survival

4 661 1445 0.75 31.83 0.001

5 200 529 0.65 22.24 0.001

6 37 114 12.11 12.11 0.015

Application to Prostate Cancer Data

Covariate Adjustment

• Younger patients may live longer

• Geographical variation in histology or stage

Significant clusters after age-adjustment

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.

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.