ch8_simple_regression.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Introduction to linear regression"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Batter up"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The movie [Moneyball](https://en.wikipedia.org/wiki/Moneyball_(film)) focuses on the \"quest for the secret of success in baseball\". It follows a low-budget team, the Oakland Athletics, who believed that underused statistics, such as a player's ability to get on base, betterpredict the ability to score runs than typical statistics like home runs, RBIs (runs batted in), and batting average. Obtaining players who excelled in these underused statistics turned out to be much more affordable for the team.\n",
    "\n",
    "In this lab we'll be looking at data from all 30 Major League Baseball teams and examining the linear relationship between runs scored in a season and a number of other player statistics. Our aim will be to summarize these relationships both graphically and numerically in order to find which variable, if any, helps us best predict a team's runs scored in a season."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The data"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's load up the data for the 2011 season."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import warnings\n",
    "warnings.filterwarnings(\"ignore\")\n",
    "\n",
    "import numpy as np\n",
    "import pandas as pd\n",
    "import io\n",
    "import requests\n",
    "\n",
    "df_url = 'https://raw.githubusercontent.com/akmand/datasets/master/openintro/mlb11.csv'\n",
    "url_content = requests.get(df_url, verify=False).content\n",
    "mlb11 = pd.read_csv(io.StringIO(url_content.decode('utf-8')))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In addition to runs scored, there are seven traditionally used variables in the data set: at-bats, hits, home runs, batting average, strikeouts, stolen bases, and wins. There are also three newer variables: on-base percentage, slugging percentage, and on-base plus slugging. For the first portion of the analysis we'll consider the seven traditional variables. At the end of the lab, you'll work with the newer variables on your own."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<div class = 'exercise'>\n",
    "<h4>Exercise 1</h4>\n",
    "What type of plot would you use to display the relationship between <code>runs</code> and one of the other numerical variables? Plot this relationship using the variable <code>at_bats</code> as the predictor. Does the relationship look linear? If you knew a team's <code>at_bats</code>, would you be comfortable using a linear model to predict the number of runs?\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If the relationship looks linear, we can quantify the strength of the relationship with the correlation coefficient."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.6106270467206687"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "mlb11['runs'].corr(mlb11['at_bats'])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Sum of squared residuals"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Think back to the way that we described the distribution of a single variable. Recall that we discussed characteristics such as center, spread, and shape. It's also useful to be able to describe the relationship of two numerical variables, such as `runs` and `at_bats` above."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<div class = 'exercise'>\n",
    "<h4>Exercise 2</h4>\n",
    "Looking at your plot from the previous exercise, describe the relationship between these two variables. Make sure to discuss the form, direction, and strength of the relationship as well as any unusual observations.\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Recall that the difference between the observed values and the values predicted by the line are called *residuals*. Note that the data set has 30 observations in total, hence there are 30 residuals."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### ${e}$<sub>i</sub> = ${y}$<sub>i</sub>−$\\bar{y}$<sub>i</sub>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To visualize the residuals of a linear regression, we can use `residplot()` function from `seaborn`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
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      "text/plain": [
       "<Figure size 720x360 with 1 Axes>"
      ]
     },
     "metadata": {
      "image/png": {
       "height": 319,
       "width": 623
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import seaborn as sns\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline \n",
    "%config InlineBackend.figure_format = 'retina'\n",
    "plt.style.use('ggplot')\n",
    "plt.rcParams['figure.figsize'] = (10,5)\n",
    "\n",
    "sns.residplot(mlb11['at_bats'], mlb11['runs'], color = 'red')\n",
    "plt.show();"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The linear model"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In order to determine the best fit line we can use `statsmodel', a very useful module for the estimation of many different statistical models, as well as for conducting statistical tests, and statistical data exploration."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "                            OLS Regression Results                            \n",
      "==============================================================================\n",
      "Dep. Variable:                   runs   R-squared:                       0.373\n",
      "Model:                            OLS   Adj. R-squared:                  0.350\n",
      "Method:                 Least Squares   F-statistic:                     16.65\n",
      "Date:                Sat, 26 Feb 2022   Prob (F-statistic):           0.000339\n",
      "Time:                        19:35:23   Log-Likelihood:                -167.44\n",
      "No. Observations:                  30   AIC:                             338.9\n",
      "Df Residuals:                      28   BIC:                             341.7\n",
      "Df Model:                           1                                         \n",
      "Covariance Type:            nonrobust                                         \n",
      "==============================================================================\n",
      "                 coef    std err          t      P>|t|      [0.025      0.975]\n",
      "------------------------------------------------------------------------------\n",
      "Intercept  -2789.2429    853.696     -3.267      0.003   -4537.959   -1040.526\n",
      "at_bats        0.6305      0.155      4.080      0.000       0.314       0.947\n",
      "==============================================================================\n",
      "Omnibus:                        2.579   Durbin-Watson:                   1.524\n",
      "Prob(Omnibus):                  0.275   Jarque-Bera (JB):                1.559\n",
      "Skew:                           0.544   Prob(JB):                        0.459\n",
      "Kurtosis:                       3.252   Cond. No.                     3.89e+05\n",
      "==============================================================================\n",
      "\n",
      "Notes:\n",
      "[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.\n",
      "[2] The condition number is large, 3.89e+05. This might indicate that there are\n",
      "strong multicollinearity or other numerical problems.\n"
     ]
    }
   ],
   "source": [
    "import statsmodels.api as sm\n",
    "\n",
    "formula_string = \"runs ~ at_bats\"\n",
    "\n",
    "model = sm.formula.ols(formula = formula_string, data = mlb11)\n",
    "model_fitted = model.fit()\n",
    "\n",
    "print(model_fitted.summary())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's print the intercept and slope values."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Intercept = -2789.242885442258\n",
      "Slope = 0.6305499928382836\n"
     ]
    }
   ],
   "source": [
    "print('Intercept =', model_fitted.params[0])\n",
    "print('Slope =', model_fitted.params[1])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Knowing the intercept and slope, we can write down the least squares regression line for the linear model:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### ${y}$ = - 2789.2429 + 0.6305 x ${atbats}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "One last piece of information we will discuss from the summary output is the Multiple R-squared, or more simply, ${R}$<sup>2</sup>. The ${R}$<sup>2</sup> value represents the proportion of variability in the response variable that is explained by the explanatory variable. For this model, 37.3% of the variability in runs is explained by at-bats."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "R-squared = 0.3728653901868063\n"
     ]
    }
   ],
   "source": [
    "print('R-squared =', model_fitted.rsquared)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<div class = 'exercise'>\n",
    "<h4>Exercise 3</h4>\n",
    "    Fit a new model that uses <code>homeruns</code> to predict <code>runs</code>. Using the estimates from the Python output, write the equation of the regression line. What does the slope tell us in the context of the relationship between success of a team and its home runs?\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Prediction and prediction errors"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Just as we used the mean and standard deviation to summarize a single variable, we can summarize the relationship between these two variables by finding the line that best follows their association. Let's plot `at_bats` and `runs` on a scatter plot."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<Figure size 720x360 with 1 Axes>"
      ]
     },
     "metadata": {
      "image/png": {
       "height": 302,
       "width": 598
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = mlb11['at_bats']\n",
    "y = mlb11['runs']\n",
    "\n",
    "y_pred = model_fitted.predict(x)\n",
    "\n",
    "plt.scatter(mlb11['at_bats'], mlb11['runs'])\n",
    "plt.plot(x, y_pred, color = 'orange')\n",
    "plt.show();"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<div class = 'exercise'>\n",
    "<h4>Exercise 4</h4>\n",
    "If a team manager saw the least squares regression line and not the actual data, how many runs would he or she predict for a team with 5,578 at-bats? Is this an overestimate or an underestimate, and by how much? In other words, what is the residual for this prediction?\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Model diagnostics"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To assess whether the linear model is reliable, we need to check for (1) linearity, (2) nearly normal residuals, and (3) constant variability.\n",
    "\n",
    "*Linearity*: You already checked if the relationship between runs and at-bats is linear using a scatterplot. We should also verify this condition with a plot of the residuals vs. at-bats."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<Figure size 720x360 with 1 Axes>"
      ]
     },
     "metadata": {
      "image/png": {
       "height": 322,
       "width": 626
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import seaborn as sns\n",
    "sns.residplot(x, y, color = 'red')\n",
    "plt.xlabel('at_bats', fontsize = 16)\n",
    "plt.ylabel('residuals', fontsize = 16)\n",
    "plt.show();"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<div class = 'exercise'>\n",
    "<h4>Exercise 5</h4>\n",
    "Is there any apparent pattern in the residuals plot? What does this indicate about the linearity of the relationship between runs and at-bats?\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "*Nearly normal residuals*: To check this condition, we can look at a histogram."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 720x360 with 1 Axes>"
      ]
     },
     "metadata": {
      "image/png": {
       "height": 302,
       "width": 585
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "residuals = (y - y_pred)\n",
    "plt.hist(residuals, bins = 8)\n",
    "plt.show();"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "or a normal probability plot of the residuals."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 720x360 with 1 Axes>"
      ]
     },
     "metadata": {
      "image/png": {
       "height": 335,
       "width": 623
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "from scipy.stats import probplot\n",
    "probplot(residuals, plot = plt)\n",
    "plt.show(); "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<div class = 'exercise'>\n",
    "<h4>Exercise 6</h4>\n",
    "Based on the histogram and the normal probability plot, does the nearly normal residuals condition appear to be met?</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "*Constant variability*:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<div class = 'exercise'>\n",
    "<h4>Exercise 7</h4>\n",
    "Based on the plot in (1), does the constant variability condition appear to be met?\n",
    "    </div>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "## On Your Own\n",
    "\n",
    "<ol>\n",
    "    <li>Choose another traditional variable from <code>mlb11</code> that you think might be a good predictor of <code>runs</code>. Produce a scatterplot of the two variables and fit a linear model. At a glance, does there seem to be a linear relationship?</li><br>\n",
    "    <li>How does this relationship compare to the relationship between <code>runs</code> and <code>at_bats</code>? Use the R squared values from the two model summaries to compare. Does your variable seem to predict <code>runs</code> better than <code>at_bats</code>? How can you tell?</li><br>\n",
    "    <li>Now that you can summarize the linear relationship between two variables, investigate the relationships between <code>runs</code> and each of the other five traditional variables. Which variable best predicts <code>runs</code>? Support your conclusion using the graphical and numerical methods we've discussed (for the sake of conciseness, only include output for the best variable, not all five).</li><br>\n",
    "    <li>Now examine the three newer variables. These are the statistics used by the author of <i>Moneyball</i> to predict a teams success. In general, are they more or less effective at predicting runs that the old variables? Explain using appropriate graphical and numerical evidence. Of all ten variables we've analyzed, which seems to be the best predictor of <code>runs</code>? Using the limited (or not so limited) information you know about these baseball statistics, does your result make sense?</li><br>\n",
    "    <li>Check the model diagnostics for the regression model with the variable you decided was the best predictor for <code>runs</code></li>\n",
    "</ol>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<div class = \"license\">\n",
    "This lab was adapted by David Akman and Imran Ture from OpenIntro by Andrew Bray and Mine Çetinkaya-Rundel.\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "***\n",
    "www.featureranking.com"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "language": "python",
   "name": "python3"
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    "name": "ipython",
    "version": 3
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