Cancer Incidence

Overview


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Transformng 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;

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