{
"cells": [
{
"cell_type": "markdown",
"id": "35c64959-fd63-49dc-9d48-a2091b7a93a8",
"metadata": {},
"source": [
"# Logit Likelihood"
]
},
{
"cell_type": "markdown",
"id": "adef7c21-52b9-4a61-88d7-d306def166d1",
"metadata": {},
"source": [
"## imports"
]
},
{
"cell_type": "code",
"execution_count": 1,
"id": "774fb99a-2ed5-44e6-b766-863911581c84",
"metadata": {},
"outputs": [],
"source": [
"import matplotlib.pyplot as plt\n",
"\n",
"import math\n",
"\n",
"import numpy as np\n",
"import pandas as pd\n",
"\n",
"from time import time\n",
"\n",
"from sklearn.linear_model import LogisticRegression\n",
"import statsmodels.api as sm"
]
},
{
"cell_type": "markdown",
"id": "c63484c8-a4b8-48a5-991e-20950a393710",
"metadata": {},
"source": [
"## Simulate Data"
]
},
{
"cell_type": "code",
"execution_count": 2,
"id": "654f7247-bd57-47aa-b93e-445be088b639",
"metadata": {},
"outputs": [],
"source": [
"#####################################\n",
"### simulate data\n",
"## p(y=1|x) = F(beta[1] + beta[2]x), y ~ Bern(py1), F(x) = exp(x)/(1+exp(x))\n",
"\n",
"n=500 #sample size\n",
"beta = (1,.5) # true (intercept, slope)\n",
"\n",
"##set seed\n",
"np.random.seed(34)\n",
"\n",
"## draw x ~ N(0,1)\n",
"x = np.random.randn(n)\n",
"\n",
"## sort x to make plotting better\n",
"x = np.sort(x)\n",
"\n",
"## compute prob Y=1|x\n",
"py1 = 1/(1+np.exp(-(beta[0] + beta[1] * x)))\n",
"\n",
"## storage for simulation results\n",
"y = np.zeros(n,dtype='int')\n",
"\n",
"## simulate y and compute py1\n",
"for i in range(n):\n",
" y[i] = np.random.binomial(1,py1[i])\n"
]
},
{
"cell_type": "markdown",
"id": "1bbb0eaf-6a98-4cd1-b25f-0c7da69431bc",
"metadata": {},
"source": [
"## Sumarize and Plot"
]
},
{
"cell_type": "code",
"execution_count": 3,
"id": "38082ce9-bd54-4261-a6dc-dcc99fe150e3",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
" x py1 y\n",
"0 -2.809924 0.400121 0\n",
"1 -2.765588 0.405453 0\n",
"2 -2.705108 0.412763 0\n",
"3 -2.683300 0.415409 0\n",
"4 -2.563626 0.430009 1\n"
]
},
{
"data": {
"text/html": [
"\n",
"\n",
"
\n",
" \n",
" \n",
" | \n",
" x | \n",
" py1 | \n",
" y | \n",
"
\n",
" \n",
" \n",
" \n",
" | count | \n",
" 500.000000 | \n",
" 500.000000 | \n",
" 500.00000 | \n",
"
\n",
" \n",
" | mean | \n",
" -0.026167 | \n",
" 0.718660 | \n",
" 0.75400 | \n",
"
\n",
" \n",
" | std | \n",
" 0.975271 | \n",
" 0.096395 | \n",
" 0.43111 | \n",
"
\n",
" \n",
" | min | \n",
" -2.809924 | \n",
" 0.400121 | \n",
" 0.00000 | \n",
"
\n",
" \n",
" | 25% | \n",
" -0.623134 | \n",
" 0.665618 | \n",
" 1.00000 | \n",
"
\n",
" \n",
" | 50% | \n",
" -0.049129 | \n",
" 0.726202 | \n",
" 1.00000 | \n",
"
\n",
" \n",
" | 75% | \n",
" 0.651016 | \n",
" 0.790097 | \n",
" 1.00000 | \n",
"
\n",
" \n",
" | max | \n",
" 2.479618 | \n",
" 0.903768 | \n",
" 1.00000 | \n",
"
\n",
" \n",
"
\n",
"