Confidence_Intervals_Differences_Population_Parameters.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Confidence Intervals\n",
    "\n",
    "\n",
    "This tutorial is going to demonstrate how to load data, clean/manipulate a dataset, and construct a confidence interval for the difference between two population proportions and means.\n",
    "\n",
    "We will use the 2015-2016 wave of the NHANES data for our analysis.\n",
    "\n",
    "*Note: We have provided a notebook that includes more analysis, with examples of confidence intervals for one population proportions and means, in addition to the analysis I will show you in this tutorial.  I highly recommend checking it out!\n",
    "\n",
    "__For our population proportions, we will analyze the difference of proportion between female and male smokers.  The column that specifies smoker and non-smoker is `\"SMQ020\"` in our dataset.__\n",
    "\n",
    "__For our population means, we will analyze the difference of mean of body mass index within our female and male populations.  The column that includes the body mass index value is `\"BMXBMI\"`.__\n",
    "\n",
    "Additionally, the gender is specified in the column `\"RIAGENDR\"`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import pandas as pd\n",
    "import numpy as np\n",
    "import matplotlib\n",
    "matplotlib.use('Agg')\n",
    "import seaborn as sns\n",
    "%matplotlib inline\n",
    "import matplotlib.pyplot as plt\n",
    "import statsmodels.api as sm"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>SEQN</th>\n",
       "      <th>ALQ101</th>\n",
       "      <th>ALQ110</th>\n",
       "      <th>ALQ130</th>\n",
       "      <th>SMQ020</th>\n",
       "      <th>RIAGENDR</th>\n",
       "      <th>RIDAGEYR</th>\n",
       "      <th>RIDRETH1</th>\n",
       "      <th>DMDCITZN</th>\n",
       "      <th>DMDEDUC2</th>\n",
       "      <th>...</th>\n",
       "      <th>BPXSY2</th>\n",
       "      <th>BPXDI2</th>\n",
       "      <th>BMXWT</th>\n",
       "      <th>BMXHT</th>\n",
       "      <th>BMXBMI</th>\n",
       "      <th>BMXLEG</th>\n",
       "      <th>BMXARML</th>\n",
       "      <th>BMXARMC</th>\n",
       "      <th>BMXWAIST</th>\n",
       "      <th>HIQ210</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>83732</td>\n",
       "      <td>1.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>1.0</td>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "      <td>62</td>\n",
       "      <td>3</td>\n",
       "      <td>1.0</td>\n",
       "      <td>5.0</td>\n",
       "      <td>...</td>\n",
       "      <td>124.0</td>\n",
       "      <td>64.0</td>\n",
       "      <td>94.8</td>\n",
       "      <td>184.5</td>\n",
       "      <td>27.8</td>\n",
       "      <td>43.3</td>\n",
       "      <td>43.6</td>\n",
       "      <td>35.9</td>\n",
       "      <td>101.1</td>\n",
       "      <td>2.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>83733</td>\n",
       "      <td>1.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>6.0</td>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "      <td>53</td>\n",
       "      <td>3</td>\n",
       "      <td>2.0</td>\n",
       "      <td>3.0</td>\n",
       "      <td>...</td>\n",
       "      <td>140.0</td>\n",
       "      <td>88.0</td>\n",
       "      <td>90.4</td>\n",
       "      <td>171.4</td>\n",
       "      <td>30.8</td>\n",
       "      <td>38.0</td>\n",
       "      <td>40.0</td>\n",
       "      <td>33.2</td>\n",
       "      <td>107.9</td>\n",
       "      <td>NaN</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>83734</td>\n",
       "      <td>1.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "      <td>78</td>\n",
       "      <td>3</td>\n",
       "      <td>1.0</td>\n",
       "      <td>3.0</td>\n",
       "      <td>...</td>\n",
       "      <td>132.0</td>\n",
       "      <td>44.0</td>\n",
       "      <td>83.4</td>\n",
       "      <td>170.1</td>\n",
       "      <td>28.8</td>\n",
       "      <td>35.6</td>\n",
       "      <td>37.0</td>\n",
       "      <td>31.0</td>\n",
       "      <td>116.5</td>\n",
       "      <td>2.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>83735</td>\n",
       "      <td>2.0</td>\n",
       "      <td>1.0</td>\n",
       "      <td>1.0</td>\n",
       "      <td>2</td>\n",
       "      <td>2</td>\n",
       "      <td>56</td>\n",
       "      <td>3</td>\n",
       "      <td>1.0</td>\n",
       "      <td>5.0</td>\n",
       "      <td>...</td>\n",
       "      <td>134.0</td>\n",
       "      <td>68.0</td>\n",
       "      <td>109.8</td>\n",
       "      <td>160.9</td>\n",
       "      <td>42.4</td>\n",
       "      <td>38.5</td>\n",
       "      <td>37.7</td>\n",
       "      <td>38.3</td>\n",
       "      <td>110.1</td>\n",
       "      <td>2.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>83736</td>\n",
       "      <td>2.0</td>\n",
       "      <td>1.0</td>\n",
       "      <td>1.0</td>\n",
       "      <td>2</td>\n",
       "      <td>2</td>\n",
       "      <td>42</td>\n",
       "      <td>4</td>\n",
       "      <td>1.0</td>\n",
       "      <td>4.0</td>\n",
       "      <td>...</td>\n",
       "      <td>114.0</td>\n",
       "      <td>54.0</td>\n",
       "      <td>55.2</td>\n",
       "      <td>164.9</td>\n",
       "      <td>20.3</td>\n",
       "      <td>37.4</td>\n",
       "      <td>36.0</td>\n",
       "      <td>27.2</td>\n",
       "      <td>80.4</td>\n",
       "      <td>2.0</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>5 rows × 28 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "    SEQN  ALQ101  ALQ110  ALQ130  SMQ020  RIAGENDR  RIDAGEYR  RIDRETH1  \\\n",
       "0  83732     1.0     NaN     1.0       1         1        62         3   \n",
       "1  83733     1.0     NaN     6.0       1         1        53         3   \n",
       "2  83734     1.0     NaN     NaN       1         1        78         3   \n",
       "3  83735     2.0     1.0     1.0       2         2        56         3   \n",
       "4  83736     2.0     1.0     1.0       2         2        42         4   \n",
       "\n",
       "   DMDCITZN  DMDEDUC2  ...  BPXSY2  BPXDI2  BMXWT  BMXHT  BMXBMI  BMXLEG  \\\n",
       "0       1.0       5.0  ...   124.0    64.0   94.8  184.5    27.8    43.3   \n",
       "1       2.0       3.0  ...   140.0    88.0   90.4  171.4    30.8    38.0   \n",
       "2       1.0       3.0  ...   132.0    44.0   83.4  170.1    28.8    35.6   \n",
       "3       1.0       5.0  ...   134.0    68.0  109.8  160.9    42.4    38.5   \n",
       "4       1.0       4.0  ...   114.0    54.0   55.2  164.9    20.3    37.4   \n",
       "\n",
       "   BMXARML  BMXARMC  BMXWAIST  HIQ210  \n",
       "0     43.6     35.9     101.1     2.0  \n",
       "1     40.0     33.2     107.9     NaN  \n",
       "2     37.0     31.0     116.5     2.0  \n",
       "3     37.7     38.3     110.1     2.0  \n",
       "4     36.0     27.2      80.4     2.0  \n",
       "\n",
       "[5 rows x 28 columns]"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "url = \"nhanes_2015_2016.csv\"\n",
    "da = pd.read_csv(url)\n",
    "da.head()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Investigating and Cleaning Data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0    Yes\n",
       "1    Yes\n",
       "2    Yes\n",
       "3     No\n",
       "4     No\n",
       "5     No\n",
       "6    Yes\n",
       "7     No\n",
       "8     No\n",
       "9     No\n",
       "Name: SMQ020x, dtype: object"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Recode SMQ020 from 1/2 to Yes/No into new variable SMQ020x\n",
    "da[\"SMQ020x\"] = da.SMQ020.replace({1: \"Yes\", 2: \"No\", 7: np.nan, 9: np.nan})\n",
    "da[\"SMQ020x\"].head(10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0      Male\n",
       "1      Male\n",
       "2      Male\n",
       "3    Female\n",
       "4    Female\n",
       "5    Female\n",
       "6      Male\n",
       "7    Female\n",
       "8      Male\n",
       "9      Male\n",
       "Name: RIAGENDRx, dtype: object"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Recode RIAGENDR from 1/2 to Male/Female into new variable RIAGENDRx\n",
    "da[\"RIAGENDRx\"] = da.RIAGENDR.replace({1: \"Male\", 2: \"Female\"})\n",
    "da[\"RIAGENDRx\"].head(10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th>RIAGENDRx</th>\n",
       "      <th>Female</th>\n",
       "      <th>Male</th>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>SMQ020x</th>\n",
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       "      <th>No</th>\n",
       "      <td>2066</td>\n",
       "      <td>1340</td>\n",
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       "      <th>Yes</th>\n",
       "      <td>906</td>\n",
       "      <td>1413</td>\n",
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      "text/plain": [
       "RIAGENDRx  Female  Male\n",
       "SMQ020x                \n",
       "No           2066  1340\n",
       "Yes           906  1413"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dx = da[[\"SMQ020x\", \"RIAGENDRx\"]].dropna()\n",
    "pd.crosstab(dx.SMQ020x, dx.RIAGENDRx)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0    1\n",
       "1    1\n",
       "2    1\n",
       "3    0\n",
       "4    0\n",
       "5    0\n",
       "6    1\n",
       "7    0\n",
       "8    0\n",
       "9    0\n",
       "Name: SMQ020x, dtype: int64"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Recode SMQ020x from Yes/No to 1/0 into existing variable SMQ020x\n",
    "dx[\"SMQ020x\"] = dx.SMQ020x.replace({\"Yes\": 1, \"No\": 0})\n",
    "dx[\"SMQ020x\"].head(10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
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       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>Proportion</th>\n",
       "      <th>Total n</th>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>RIAGENDRx</th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
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       "  <tbody>\n",
       "    <tr>\n",
       "      <th>Female</th>\n",
       "      <td>0.304845</td>\n",
       "      <td>2972</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Male</th>\n",
       "      <td>0.513258</td>\n",
       "      <td>2753</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
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      ],
      "text/plain": [
       "           Proportion  Total n\n",
       "RIAGENDRx                     \n",
       "Female       0.304845     2972\n",
       "Male         0.513258     2753"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dz = dx.groupby(\"RIAGENDRx\").agg({\"SMQ020x\": [np.mean, np.size]})\n",
    "dz.columns = [\"Proportion\", \"Total n\"]\n",
    "dz"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Constructing Confidence Intervals\n",
    "\n",
    "Now that we have the population proportions of male and female smokers, we can begin to calculate confidence intervals.  From lecture, we know that the equation is as follows:\n",
    "\n",
    "$$Best\\ Estimate \\pm Margin\\ of\\ Error$$\n",
    "\n",
    "Where the *Best Estimate* is the **observed population proportion or mean** from the sample and the *Margin of Error* is the **t-multiplier**.\n",
    "\n",
    "The equation to create a 95% confidence interval can also be shown as:\n",
    "\n",
    "$$Population\\ Proportion\\ or\\ Mean\\ \\pm (t-multiplier *\\ Standard\\ Error)$$\n",
    "\n",
    "The Standard Error (SE) is calculated differenly for population proportion and mean:\n",
    "\n",
    "$$Standard\\ Error \\ for\\ Population\\ Proportion = \\sqrt{\\frac{Population\\ Proportion * (1 - Population\\ Proportion)}{Number\\ Of\\ Observations}}$$\n",
    "\n",
    "$$Standard\\ Error \\ for\\ Mean = \\frac{Standard\\ Deviation}{\\sqrt{Number\\ Of\\ Observations}}$$\n",
    "\n",
    "Lastly, the standard error for difference of population proportions and means is:\n",
    "\n",
    "$$Standard\\ Error\\ for\\ Difference \\ of\\ Two\\ Population\\ Proportions\\ Or\\ Means = \\sqrt{(SE_{\\ 1})^2 + (SE_{\\ 2})^2}$$\n",
    "\n",
    "#### Difference of Two Population Proportions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.00844415041930423"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#Calculate population proportion for female smokers\n",
    "p = .304845\n",
    "n = 2972\n",
    "se_female = np.sqrt(p * (1 - p)/n) #𝑆𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝐸𝑟𝑟𝑜𝑟 𝑓𝑜𝑟 female\n",
    "se_female"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.009526078787008965"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#Calculate population proportion for male smokers\n",
    "p = .513258\n",
    "n = 2753\n",
    "se_male = np.sqrt(p * (1 - p)/ n) #𝑆𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝐸𝑟𝑟𝑜𝑟 𝑓𝑜𝑟 male\n",
    "se_male"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.012729880335656654"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#Standard error for difference of population proportions\n",
    "se_diff = np.sqrt(se_female**2 + se_male**2)\n",
    "se_diff"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(-0.23336356545788706, -0.18346243454211297)"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#Standard Error Difference between two population proiportion\n",
    "d = .304845 - .513258\n",
    "\n",
    "lcb = d - 1.96 * se_diff #lowerbound Confidence interval\n",
    "ucb = d + 1.96 * se_diff #uperbound Confidence interval\n",
    "(lcb, ucb)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Difference of Two Population Means"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0    27.8\n",
       "1    30.8\n",
       "2    28.8\n",
       "3    42.4\n",
       "4    20.3\n",
       "Name: BMXBMI, dtype: float64"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "da[\"BMXBMI\"].head()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
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       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr>\n",
       "      <th></th>\n",
       "      <th colspan=\"3\" halign=\"left\">BMXBMI</th>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th></th>\n",
       "      <th>mean</th>\n",
       "      <th>std</th>\n",
       "      <th>size</th>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>RIAGENDRx</th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>Female</th>\n",
       "      <td>29.939946</td>\n",
       "      <td>7.753319</td>\n",
       "      <td>2976</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Male</th>\n",
       "      <td>28.778072</td>\n",
       "      <td>6.252568</td>\n",
       "      <td>2759</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "              BMXBMI                \n",
       "                mean       std  size\n",
       "RIAGENDRx                           \n",
       "Female     29.939946  7.753319  2976\n",
       "Male       28.778072  6.252568  2759"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "da.groupby(\"RIAGENDRx\").agg({\"BMXBMI\": [np.mean, np.std, np.size]})"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0.14212523289878048, 0.11903716451870151)"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#Calculate Standard Error of Mean for female\n",
    "sem_female = 7.753319 / np.sqrt(2976)\n",
    "\n",
    "#Calculate Standard Error of Mean for male\n",
    "sem_male = 6.252568 / np.sqrt(2759)\n",
    "(sem_female, sem_male)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.18538993598139303"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#𝑆𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝐸𝑟𝑟𝑜𝑟 𝑓𝑜𝑟 𝐷𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 𝑜𝑓 𝑇𝑤𝑜 𝑀𝑒𝑎𝑛𝑠\n",
    "sem_diff = np.sqrt(sem_female**2 + sem_male**2)\n",
    "sem_diff"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0.798509725476467, 1.5252382745235278)"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "d = 29.939946 - 28.778072\n",
    "\n",
    "lcb = d - 1.96 * sem_diff\n",
    "ucb = d + 1.96 * sem_diff\n",
    "(lcb, ucb)"
   ]
  }
 ],
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