Cancer Incidence

Overview


This example utilizes the following dataset.

copy

The cases column represents the number of observed cancer cases in the given region.

Transforming the Data


The first step is to transform the data to use One Hot Encoding 


let data2 = data.map(p=>{
  let record = {
    u1:0,
    u2:0,
    u3:0,
    u4:0,
    u5:0,
    u6:0,
    u7:0,
    u8:0,
    e:0
  };

  record.population = p.population;
  record.cases = p.cases;
  if(p.age === '15-24') record.u1 = 1;
  if(p.age === '25-34') record.u2 = 1;
  if(p.age === '35-44') record.u3 = 1;
  if(p.age === '45-54') record.u4 = 1;
  if(p.age === '55-64') record.u5 = 1;
  if(p.age === '65-74') record.u6 = 1;
  if(p.age === '75-84') record.u7 = 1;
  if(p.age === '85+') record.u8 = 1;
  if(p.city === 'Lemur City') record.e=1;
  return record;
});

Maximum Likelihood


Next, we assume that the mean parameter to the Poisson distribution can be modeled as
{% \mu_t = population \times exp[\beta_0 + \beta_1 u_1 + \beta_2 u_2 + ... + \beta_7 u_7] %}
Then we use newtons algorithm is used to find the maximum likelihood values of {% \beta_0,...,\beta_7 %} 


let op = await import('/lib/numeric/optimization/v1.0.0/newton.mjs');

let f = function(b0, b1, b2, b3, b4, b5, b6, b7, b9){
  let logL = pl.logLikelihood(data2, item=>item.cases, item=>{
    let eta = b0+b1*item.u1 + b2*item.u2 + b3*item.u3 + b4*item.u4 + b5*item.u5 + b6*item.u6 + b7*item.u7 +  b9*item.e;
    return item.population * Math.exp(eta);
  });
  return -1*logL;
}
    
let iterator = op.iterator(f, [0.1,0.1,0.1,0.1,0.1,0.1,0.1,0.1,0.1,0.1],{
  speed:op.linesearch.bisection(f)
});

for(let i=0;i<30;i++){
	iterator.iterate();
}
let result = iterator.x;

Poisson Regression


The above maximum Likelihood code is encapsulated in the Poisson Regression Library. 


let op = await import('/lib/glm/v1.0.0/poisson.mjs');
let iterator = pl.newton(data2,'cases', ['u1','u2','u3','u4','u5','u6','u7','e'],{
	exposure:'population',
	linesearch:'bisection'
});

for(let i=0;i<30;i++){
	iterator.iterate();
}
let result = iterator.x;