{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## GEE nested covariance structure simulation study\n",
    "\n",
    "This notebook is a simulation study that illustrates and evaluates the performance of the GEE nested covariance structure.\n",
    "\n",
    "A nested covariance structure is based on a nested sequence of groups, or \"levels\".  The top level in the hierarchy is defined by the `groups` argument to GEE.  Subsequent levels are defined by the `dep_data` argument to GEE."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2021-02-02T06:51:37.532460Z",
     "iopub.status.busy": "2021-02-02T06:51:37.525267Z",
     "iopub.status.idle": "2021-02-02T06:51:38.627270Z",
     "shell.execute_reply": "2021-02-02T06:51:38.625962Z"
    }
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import pandas as pd\n",
    "import statsmodels.api as sm"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Set the number of covariates."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2021-02-02T06:51:38.632227Z",
     "iopub.status.busy": "2021-02-02T06:51:38.630810Z",
     "iopub.status.idle": "2021-02-02T06:51:38.636719Z",
     "shell.execute_reply": "2021-02-02T06:51:38.637807Z"
    }
   },
   "outputs": [],
   "source": [
    "p = 5"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "These parameters define the population variance for each level of grouping."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2021-02-02T06:51:38.642265Z",
     "iopub.status.busy": "2021-02-02T06:51:38.640963Z",
     "iopub.status.idle": "2021-02-02T06:51:38.646375Z",
     "shell.execute_reply": "2021-02-02T06:51:38.647283Z"
    }
   },
   "outputs": [],
   "source": [
    "groups_var = 1\n",
    "level1_var = 2\n",
    "level2_var = 3\n",
    "resid_var = 4"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Set the number of groups"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2021-02-02T06:51:38.652537Z",
     "iopub.status.busy": "2021-02-02T06:51:38.650235Z",
     "iopub.status.idle": "2021-02-02T06:51:38.655516Z",
     "shell.execute_reply": "2021-02-02T06:51:38.656743Z"
    }
   },
   "outputs": [],
   "source": [
    "n_groups = 100"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Set the number of observations at each level of grouping.  Here, everything is balanced, i.e. within a level every group has the same size."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2021-02-02T06:51:38.661455Z",
     "iopub.status.busy": "2021-02-02T06:51:38.660704Z",
     "iopub.status.idle": "2021-02-02T06:51:38.664906Z",
     "shell.execute_reply": "2021-02-02T06:51:38.665506Z"
    }
   },
   "outputs": [],
   "source": [
    "group_size = 20\n",
    "level1_size = 10\n",
    "level2_size = 5"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Calculate the total sample size."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2021-02-02T06:51:38.670657Z",
     "iopub.status.busy": "2021-02-02T06:51:38.669867Z",
     "iopub.status.idle": "2021-02-02T06:51:38.673205Z",
     "shell.execute_reply": "2021-02-02T06:51:38.673808Z"
    }
   },
   "outputs": [],
   "source": [
    "n = n_groups * group_size * level1_size * level2_size"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Construct the design matrix."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2021-02-02T06:51:38.682564Z",
     "iopub.status.busy": "2021-02-02T06:51:38.681819Z",
     "iopub.status.idle": "2021-02-02T06:51:38.712859Z",
     "shell.execute_reply": "2021-02-02T06:51:38.713897Z"
    }
   },
   "outputs": [],
   "source": [
    "xmat = np.random.normal(size=(n, p))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Construct labels showing which group each observation belongs to at each level."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2021-02-02T06:51:38.718542Z",
     "iopub.status.busy": "2021-02-02T06:51:38.717050Z",
     "iopub.status.idle": "2021-02-02T06:51:38.766871Z",
     "shell.execute_reply": "2021-02-02T06:51:38.768040Z"
    }
   },
   "outputs": [],
   "source": [
    "groups_ix = np.kron(np.arange(n // group_size), np.ones(group_size)).astype(np.int)\n",
    "level1_ix = np.kron(np.arange(n // level1_size), np.ones(level1_size)).astype(np.int)\n",
    "level2_ix = np.kron(np.arange(n // level2_size), np.ones(level2_size)).astype(np.int)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Simulate the random effects."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2021-02-02T06:51:38.772795Z",
     "iopub.status.busy": "2021-02-02T06:51:38.771440Z",
     "iopub.status.idle": "2021-02-02T06:51:38.780682Z",
     "shell.execute_reply": "2021-02-02T06:51:38.781786Z"
    }
   },
   "outputs": [],
   "source": [
    "groups_re = np.sqrt(groups_var) * np.random.normal(size=n // group_size)\n",
    "level1_re = np.sqrt(level1_var) * np.random.normal(size=n // level1_size)\n",
    "level2_re = np.sqrt(level2_var) * np.random.normal(size=n // level2_size)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Simulate the response variable."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2021-02-02T06:51:38.786719Z",
     "iopub.status.busy": "2021-02-02T06:51:38.785192Z",
     "iopub.status.idle": "2021-02-02T06:51:38.799242Z",
     "shell.execute_reply": "2021-02-02T06:51:38.800364Z"
    }
   },
   "outputs": [],
   "source": [
    "y = groups_re[groups_ix] + level1_re[level1_ix] + level2_re[level2_ix]\n",
    "y += np.sqrt(resid_var) * np.random.normal(size=n)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Put everything into a dataframe."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2021-02-02T06:51:38.805317Z",
     "iopub.status.busy": "2021-02-02T06:51:38.803814Z",
     "iopub.status.idle": "2021-02-02T06:51:38.817918Z",
     "shell.execute_reply": "2021-02-02T06:51:38.818840Z"
    }
   },
   "outputs": [],
   "source": [
    "df = pd.DataFrame(xmat, columns=[\"x%d\" % j for j in range(p)])\n",
    "df[\"y\"] = y + xmat[:, 0] - xmat[:, 3]\n",
    "df[\"groups_ix\"] = groups_ix\n",
    "df[\"level1_ix\"] = level1_ix\n",
    "df[\"level2_ix\"] = level2_ix"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Fit the model."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2021-02-02T06:51:38.823245Z",
     "iopub.status.busy": "2021-02-02T06:51:38.821937Z",
     "iopub.status.idle": "2021-02-02T06:51:49.832878Z",
     "shell.execute_reply": "2021-02-02T06:51:49.833795Z"
    }
   },
   "outputs": [],
   "source": [
    "cs = sm.cov_struct.Nested()\n",
    "dep_fml = \"0 + level1_ix + level2_ix\"\n",
    "m = sm.GEE.from_formula(\"y ~ x0 + x1 + x2 + x3 + x4\", cov_struct=cs,\n",
    "                        dep_data=dep_fml, groups=\"groups_ix\", data=df)\n",
    "r = m.fit()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The estimated covariance parameters should be similar to `groups_var`, `level1_var`, etc. as defined above."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2021-02-02T06:51:49.844362Z",
     "iopub.status.busy": "2021-02-02T06:51:49.843575Z",
     "iopub.status.idle": "2021-02-02T06:51:49.861332Z",
     "shell.execute_reply": "2021-02-02T06:51:49.862419Z"
    }
   },
   "outputs": [
    {
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       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>Variance</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>groups_ix</th>\n",
       "      <td>0.946292</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>level1_ix</th>\n",
       "      <td>2.084635</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>level2_ix</th>\n",
       "      <td>3.022204</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Residual</th>\n",
       "      <td>4.009961</td>\n",
       "    </tr>\n",
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       "           Variance\n",
       "groups_ix  0.946292\n",
       "level1_ix  2.084635\n",
       "level2_ix  3.022204\n",
       "Residual   4.009961"
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