From disease mapping to archaeology and presence-only modelling

Elena Moltchanova, PhDCanterbury Statistics Day

Disease Mapping. A Bit of History: Besag J (1974) ‘Spatial Interaction and the

Statistical Analysis of Lattice Systems’ JRSS B 36(2) 192-236

Besag J (1975) ‘Statistical Analysis of Non-Lattice Data’ JRSS D 24(3) 179-195

Besag J (1986) ‘On the Statistical Analysis of Dirty Pictures’ JRSS B 48, 259-302

Besag J, York J, and Mollie A (1991) ‘Bayesian image restoration, with two applications in spatial statistics’. Annals of the Institute of Statistical Mathematics 43(1) 1-20

Fig 1. Observed incidence of childhood diabetes (T1DM) in Finland in 1987-1996.

Incidence = number of cases/population at risk*100 000

Click icon to add picture

BYM:

Population at risk or expected counts

)Poisson( ~ iiY

Observed cases

risk

iiiii NX log)log( 0

Background level Area-specific spatial residual

Systematic part

Non-spatial residual

Back to BYM: Conditional AutoRegressive (CAR)

),(~ ,00 iii mN

Areas close together have similar values

Neighborhood Matrix W

BYM model DAG W j h i Ni Yi Xi

Nik Yik Xi

Nik Yik Xi

Applying BYM model to diabetes incidence data:

ObservedEstimated by BYM model

Argeopop project http://www.helsinki.fi/bioscience/argeopop aims to shed new light on the prehistory of the

Finns by integrating evidence from genetic and archeological data within a Bayesian statistical framework.

From Onkamo, P, Kammonen, J, Pesonen P, Sundell, T, Moltchanova E, Oinonen M, Haimila M, Arjas E. “Bayesian Spatiotemporal Analysis of Radiocarbon Dates in Eastern Fennoscandia” Radiocarbon (in press)

9000-6400 BP

From Onkamo, P, Kammonen, J, Pesonen P, Sundell, T, Moltchanova E, Oinonen M, Haimila M, Arjas E. “Bayesian Spatiotemporal Analysis of Radiocarbon Dates in Eastern Fennoscandia” Radiocarbon (in press) www.helsinki.fi/bioscience/argeopop

6400-5100 BP

From Onkamo, P, Kammonen, J, Pesonen P, Sundell, T, Moltchanova E, Oinonen M, Haimila M, Arjas E. “Bayesian Spatiotemporal Analysis of Radiocarbon Dates in Eastern Fennoscandia” Radiocarbon (in press) www.helsinki.fi/bioscience/argeopop

5100-4000 BP

From Onkamo, P, Kammonen, J, Pesonen P, Sundell, T, Moltchanova E, Oinonen M, Haimila M, Arjas E. “Bayesian Spatiotemporal Analysis of Radiocarbon Dates in Eastern Fennoscandia” Radiocarbon (in press) www.helsinki.fi/bioscience/argeopop

4000-3500 BP

From Onkamo, P, Kammonen, J, Pesonen P, Sundell, T, Moltchanova E, Oinonen M, Haimila M, Arjas E. “Bayesian Spatiotemporal Analysis of Radiocarbon Dates in Eastern Fennoscandia” Radiocarbon (in press) www.helsinki.fi/bioscience/argeopop

3500-2500 BP

From Onkamo, P, Kammonen, J, Pesonen P, Sundell, T, Moltchanova E, Oinonen M, Haimila M, Arjas E. “Bayesian Spatiotemporal Analysis of Radiocarbon Dates in Eastern Fennoscandia” Radiocarbon (in press) www.helsinki.fi/bioscience/argeopop

2500-1500 BP

Presence only data…? We only find where we dig We only dig where we’ve found

something Similar to ecological niche modelling?

MaxEnt modelingMaximize

Subject to

Where x[i] is a ‘feature’ i.e. value of the

covariate y[i]=1 for presence and 0 for absence p[i] is (multinomial) probability of presence i=1,…,N areas

BYM model recast:

),multinom(p ~ N:1:1 XY N

Observed distribution of occurrences

probability

Y[i]=1 if there is an observation in area I… and is missing otherwise

X is therefore also missing, with lower limit known

Placing a suitable prior either on X produces an identifiable Bayesian spatial CAR model!

𝑋=∑𝑖=1

𝑁

𝑌 𝑖

Will it work? A very simple example.

Further Work:• Implement multinomial BYM model (MCMC algorithm)

with various spatial autocorrelation structures:• None• CAR prior only• CAR prior + non-spatial residual

• Perform sensitivity analysis

• Compare to MaxEnt performance