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fonnesbeck/Survival Example.ipynb

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Survival Example.ipynb
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"source": [
"# Cox regression in PyMC\n",
"\n",
"This example implements the Cox model using the counting process notation introduced by Andersen and Gill (1982) "
]
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"cell_type": "code",
"collapsed": false,
"input": [
"import numpy as np"
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"source": [
"### Leukemia data"
]
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"input": [
"# Survival times in weeks\n",
"obs_t = (1, 1, 2, 2, 3, 4, 4, 5, 5, 8, 8, 8, 8, 11, 11, 12, 12, 15, 17, 22, 23, 6, \n",
" 6, 6, 6, 7, 9, 10, 10, 11, 13, 16, 17, 19, 20, 22, 23, 25, 32, 32, 34, 35)\n",
"# Failure (1) or censored (0)\n",
"fail = (1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, \n",
" 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0)\n",
"# Indicators for placebo or treatment\n",
"Z = (0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, \n",
" 0.5, 0.5, 0.5, 0.5, 0.5, -0.5, -0.5, -0.5, -0.5, -0.5, -0.5, -0.5, -0.5, -0.5, -0.5, \n",
" -0.5, -0.5, -0.5, -0.5, -0.5, -0.5, -0.5, -0.5, -0.5, -0.5, -0.5)\n",
"# Unique failure times\n",
"t = (1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 15, 16, 17, 22, 23, 35)\n",
"N = len(obs_t)\n",
"T = len(t) - 1"
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"language": "python",
"metadata": {
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"slide_type": "subslide"
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"prompt_number": 6
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{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
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"source": [
"Calculate risk set"
]
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{
"cell_type": "code",
"collapsed": false,
"input": [
"Y = np.array([[int(obs >= time) for time in t] for obs in obs_t])"
],
"language": "python",
"metadata": {
"slideshow": {
"slide_type": "fragment"
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"prompt_number": 7
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"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
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"source": [
"Counting process. Jump = 1 if $\\text{obs}_t \\in [ t_j, t_{j+1} )$"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"dN = np.array([[Y[i,j]*(t[j+1] >= obs_t[i])*fail[i] for i in range(N)] for j in range(T)])"
],
"language": "python",
"metadata": {
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"slide_type": "fragment"
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"outputs": [],
"prompt_number": 8
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{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
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"source": [
"Model using Poisson trick:"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"from pymc import Normal, Lambda, Poisson, MCMC\n",
"\n",
"beta0 = Normal('beta0', 0, 0.001, value=np.zeros(T))\n",
"beta1 = Normal('beta1', 0, 0.00001, value=0)\n",
"\n",
"# Poisson trick: independent log-normal hazard increments\n",
"Idt = Lambda('Idt', lambda b0=beta0, b1=beta1: [[Y[i,j]*np.exp(b0[j] + b1*Z[i]) for i in range(N)] for j in range(T)])\n",
"\n",
"# Likelihood\n",
"dN_like = Poisson('dN_like', Idt, value=dN, observed=True)"
],
"language": "python",
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"outputs": [],
"prompt_number": 9
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"source": [
"Initialize MCMC sampler."
]
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"input": [
"M = MCMC([beta0, beta1, Idt, dN_like])"
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"input": [
"M.sample(10000, burn=5000)"
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],
"prompt_number": 13
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"M.trace('beta0')()"
],
"language": "python",
"metadata": {},
"outputs": [
{
"metadata": {},
"output_type": "pyout",
"prompt_number": 14,
"text": [
"array([[-3.45648129, -0.93242907, -3.76504614, ..., -1.65020221,\n",
" -1.04204604, -3.80254949],\n",
" [-3.03129361, -0.94656389, -4.09985302, ..., -1.69601693,\n",
" -0.86306285, -3.34595103],\n",
" [-2.81985925, -0.94104154, -2.65582902, ..., -1.64492766,\n",
" -0.7936265 , -3.71599213],\n",
" ..., \n",
" [-2.23619649, -1.66264799, -3.30862506, ..., -1.91104878,\n",
" -0.5040012 , -1.75395984],\n",
" [-2.03238709, -1.6622854 , -3.5882551 , ..., -1.90309669,\n",
" -0.48624936, -1.88419897],\n",
" [-2.47687504, -1.6455667 , -3.6455275 , ..., -1.95423672,\n",
" -0.62409622, -1.68187115]])"
]
}
],
"prompt_number": 14
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"beta1.summary()"
],
"language": "python",
"metadata": {
"slideshow": {
"slide_type": "fragment"
}
},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"\n",
"beta1:\n",
" \n",
"\tMean SD MC Error 95% HPD interval\n",
"\t------------------------------------------------------------------\n",
"\t1.214 0.254 0.012 [ 0.726 1.769]\n",
"\t\n",
"\t\n",
"\tPosterior quantiles:\n",
"\t\n",
"\t2.5 25 50 75 97.5\n",
"\t |---------------|===============|===============|---------------|\n",
"\t0.72 1.067 1.197 1.374 1.765\n",
"\t\n"
]
}
],
"prompt_number": 15
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"from pymc import Matplot\n",
"\n",
"Matplot.plot(beta1)"
],
"language": "python",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"Plotting beta1\n"
]
}
],
"prompt_number": 16
}
],
"metadata": {}
}
]
}
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