Tutorial/STA130_TUT_7.ipynb

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    "# STA130 Tutorial 7: Simple Linear Regression\n",
    "\n",
    "The ol' $y=mx+b$, now fancily written like...\n",
    "\n",
    "$$Y_i = \\beta_0 + \\beta_1 x_i + \\epsilon_i \\quad\\quad \\text{ with an } \\epsilon_i \\sim \\mathcal N\\left(0, \\sigma^2\\right) \\text{ assumption}$$\n",
    "\n",
    "to allow us to do statistical analysis stuffs...\n",
    "\n",
    "| |\n",
    "|-|\n",
    "| ![](https://blogs.sas.com/content/iml/files/2015/09/GLM_normal_identity.png) | \n",
    "\n",
    "<!--\n",
    "**Today’s agenda:**<br><br> $\\bullet$ Q&A/vocabulary list <br><br>$\\bullet$ Group Discussion <br><br>  $\\bullet$ Writing prompt <br><br>$\\bullet$ Work on Project <br> &nbsp;&nbsp; (time permitting) <br><br><br><br><br><br> |  -->"
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    "# Correlation: Mini Lecture  (<10 minutes)  \n",
    "\n",
    "- Non-Linear Associations $\\color{gray}{\\text{(draw on board)}}$\n",
    "- Linear (and Approximately Linear) Associations $\\color{gray}{\\text{(draw on board)}}$\n",
    "    - ***Correlation*** measures **LINEAR** Association Strength $\\color{gray}{\\text{(demonstrate)}}$ \n",
    "    - **\"Correlation is not causation\"**\n",
    "    \n",
    "$\\color{gray}{\\text{Get some practice with summation and index notation:}}$\n",
    "\n",
    "\\begin{align}\n",
    "r_{xy} & = {} \\require{cancel}\\frac{Cov(x,y)}{s_x s_y} = \\frac{\\frac{1}{n-1} \\sum_{i=1}^n (x_i-\\bar x)(y_i-\\bar y)}{s_x s_y}\\\\ & = {}  \\frac{\\sum_{i=1}^n (x_i-\\bar x)(y_i-\\bar y)\\xcancel{/(n-1)}}{\\sqrt{\\sum_{i=1}^n (x_i-\\bar x)^2\\xcancel{/(n-1)}} \\sqrt{\\sum_{i=1}^n (y_i-\\bar y)^2\\xcancel{/(n-1)}} }\n",
    "\\end{align}\n",
    "\n",
    "<sub><sup>At this stage you should know what correlation is, and be able to do the calculations specified by the formula; but, it's not likely to be the case that you'll get a deep understanding of what correlation is just based on the mathematical formula.<sub><sup>"
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   "source": [
    "# Correlation: Code Reminder ($\\approx$1 minute)  "
   ]
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       "array([[ 1.25238567,  0.05022546],\n",
       "       [ 0.62084642,  0.61061207],\n",
       "       [ 0.35101574,  0.45284337],\n",
       "       [-2.48735342, -0.66397188],\n",
       "       [-1.39770941, -1.36251024]])"
      ]
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     "execution_count": 6,
     "metadata": {},
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   "source": [
    "from scipy import stats\n",
    "some_bivariate_distribution = stats.multivariate_normal([0.5, -0.2], [[2.0, 0.3], [0.3, 0.5]])\n",
    "n = 100\n",
    "some_bivariate_sample_n100 = some_bivariate_distribution.rvs(n)\n",
    "some_bivariate_sample_n100[:5]\n",
    "#import plotly.express as px\n",
    "#px.scatter(x=some_bivariate_sample_n100[:,0], y=some_bivariate_sample_n100[:,1])"
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       "array([[1.        , 0.36055768],\n",
       "       [0.36055768, 1.        ]])"
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   "source": [
    "import numpy as np\n",
    "np.corrcoef(some_bivariate_sample_n100[:,0], some_bivariate_sample_n100[:,1])"
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       "      <th>x</th>\n",
       "      <td>1.000000</td>\n",
       "      <td>0.360558</td>\n",
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       "      <td>1.000000</td>\n",
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       "          x         y\n",
       "x  1.000000  0.360558\n",
       "y  0.360558  1.000000"
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   "source": [
    "import pandas as pd\n",
    "pd.DataFrame(some_bivariate_sample_n100, columns=['x','y']).corr()"
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    "# Think you've got it all figured out? (<5 minutes)\n",
    "\n",
    "| | |\n",
    "|-|-|\n",
    "|Let's play a little game then...<br><br>CLICK ME if you dare:<br>https://istics.net/Correlations/<br><br><br>Can you guess which plot<br>corresponds to which<br>correlation coefficient?<br><br><br><br><br><br><br><br><br>|<img src=\"im/7/guess_corr.png\" style=\"width:800px\">|\n"
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    "## Think you've got it all figured out? (<5 minutes)\n",
    "\n",
    "|x is more strongly associated with y<br><br>in which plot?<br><br><br><br><br><br>[Double Click for Answers]|<img src=\"im/7/Anscombes_quartet.png\" style=\"width:800px\">|\n",
    "|-|-|\n",
    "| | |\n",
    "\n",
    "<!-- x is strongly associated with y in three of these figures -->\n",
    "<!-- the last figure has a single data point outlier\n",
    "     which is the only thing suggesting a strong association -->\n",
    "<!-- x is LINEARLY assocaited with y in two of the figures -->\n",
    "<!-- one of which has an outlier which might be erronious -->\n",
    "<!-- x is STRONGLY NONLINEARLY assocaited with y in one figure -->\n"
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    "## Reference Figures for what we're talking about  next\n",
    "\n",
    "*Just go to the next slide and refer back to this slide if helpful*\n",
    "\n",
    "| | |\n",
    "|-|-|\n",
    "|<img src=\"https://mobiledevmemo.com/wp-content/uploads/2012/10/w5449egf111.gif\" alt=\"\" style=\"width: 450px;\"/>|<img src=\"https://sphweb.bumc.bu.edu/otlt/MPH-Modules/BS/R/R5_Correlation-Regression/Illustration%20of%20Simple%20Linear%20Regression.gif\" alt=\"\" style=\"width: 450px;\"/>|\n",
    "| | |\n",
    "|<img src=\"https://saylordotorg.github.io/text_introductory-statistics/section_14/88a6e0919d8617c025826c1e187ad591.jpg\" alt=\"\" style=\"width: 450px;\"/>|<img src=\"https://i.stack.imgur.com/sVUby.png\" alt=\"\" style=\"width: 450px;\"/>| \n"
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    "## Simple Linear Regression: Mini Lecture (<10 minutes) \n",
    "\n",
    "- (Continuous) Outcome / Response / Dependent variable: $\\;\\; Y_i$\n",
    "   \n",
    "- Explanatory/Independent variable / Covariate / Feature / Predictor: $\\;\\; x_i$\n",
    "  \n",
    "- Simple Linear Regression Coefficients/Parameters: **Intercept and Slope**\n",
    "\n",
    "  $$\\beta_0 + \\beta_1 x_i$$ \n",
    "   \n",
    "- Noise/Error\n",
    "\n",
    "  $$Y_i = \\beta_0 + \\beta_1 x_i + \\epsilon_i$$ \n",
    "  \n",
    "- $\\color{gray}{\\text{FYI: Simple Linear Regression (is just a Normal Distribution Model)}}$\n",
    "\n",
    "  $${\\begin{align*}\n",
    "  Y_i & = {} \\beta_0 + \\beta_1 x_i + \\epsilon_i \\text{ with } \\epsilon_i \\sim \\mathcal N(0, \\sigma)\\\\ & \\color{gray}{\\text{ which can also just be written as }}\\\\\n",
    "  Y_i & \\sim {} \\mathcal N(\\beta_0 + \\beta_1 x_i + \\epsilon_i, \\sigma)\n",
    "  \\end{align*}}$$\n",
    "  \n"
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   "source": [
    "### Fitted Simple Linear Regression Model: Mini Lecture (<10 minutes) \n",
    "\n",
    "- Fitted Regression Line $\\color{gray}{\\text{which will have $\\hat \\beta_1 = r_{xy}\\frac{s_y}{s_x}$ and $\\hat \\beta_0 = \\bar y - \\beta_1 \\bar x$}}$\n",
    "\n",
    "  $$\\hat y_i = \\hat \\beta_0 + \\hat \\beta_1 x_i$$ \n",
    "  \n",
    "  $$\\text{IS NOT}$$\n",
    "\n",
    "  $$Y_i = \\beta_0 + \\beta_1 x_i \\color{gray}{+ \\epsilon_i \\text{ with } \\epsilon_i \\sim N(0, \\sigma)}$$\n",
    "\n",
    " $$\\text{but why?}$$\n",
    "\n",
    "- Residuals \n",
    "\n",
    "  $$\\hat \\epsilon_i = y_i - \\hat y_i = y_i - \\hat \\beta_0 + \\hat \\beta_1 x_i $$\n",
    "\n",
    "  $$\\text{ARE NOT Noise/Error terms $\\epsilon_i \\quad... \\quad$ but why?}$$"
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   "source": [
    "### Starbucks Dataset: Introduction ($\\approx$1 minute) [Click \"down\" next]\n",
    "\n",
    "The `starbucks.csv` dataset contains calorie and carbohydrate information (in grams) for  Starbucks food menu items"
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   "source": [
    "import pandas as pd; starbucks = pd.read_csv(\"data/7/starbucks.csv\", encoding=\"ISO-8859-1\")\n",
    "import plotly.express as px; fig=px.scatter(starbucks, x=\"carb\", y=\"calories\", trendline='ols'); fig.update_layout(title=\"Calories versus Carbs for Starbucks food menu items\")"
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   "source": [
    "### Starbucks Dataset: Questions (< 10 minutes) [Click \"down\" next]\n",
    "\n",
    "- What would you guess the correlation to be? \n",
    "- What's you guess for the slope of the linear association? [*Hint: \"rise over run\"*]\n",
    "- What about for the equation of the linear association? [*Hint: How to do this...?*]"
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   "source": [
    "import pandas as pd; starbucks = pd.read_csv(\"data/7/starbucks.csv\", encoding=\"ISO-8859-1\")\n",
    "import plotly.express as px; fig=px.scatter(starbucks, x=\"carb\", y=\"calories\", trendline='ols'); fig.update_layout(title=\"Calories versus Carbs for Starbucks food menu items\")\n",
    "# hover over the figure for the answers! "
   ]
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   "id": "985c2b42",
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   "source": [
    "### Starbucks Dataset: Questions (<5 minutes) [Click \"down\" next]\n",
    "\n",
    "$$E[\\text{calories}] \\approx 146 + 4.3 \\text{carb}$$\n",
    "\n",
    "- ***Interpret the linear model:***<br>How much to calories change on average per unit change in carbohydrates?\n",
    "- ***Use the linear model:***<br>How many calories are expected if a Starbucks menu item has 50 calories?\n",
    "    - *Check your equation-based guess against the line on the previous slide*\n",
    "- ***Analyze the linear model*** **[ASK Q AND GO TO ANSWER ON NEXT SLIDE]**:<br>What's the ***null hypothesis*** for the $\\beta_{\\text{carb}}$ coefficient of this model and what does a ***hypothesis test*** suggest about a `carb`-`calories` association?"
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       "<table class=\"simpletable\">\n",
       "<tr>\n",
       "      <td></td>         <th>coef</th>     <th>std err</th>      <th>t</th>      <th>P>|t|</th>  <th>[0.025</th>    <th>0.975]</th>  \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Intercept</th> <td>  146.0204</td> <td>   25.919</td> <td>    5.634</td> <td> 0.000</td> <td>   94.388</td> <td>  197.653</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>carb</th>      <td>    4.2971</td> <td>    0.542</td> <td>    7.923</td> <td> 0.000</td> <td>    3.217</td> <td>    5.378</td>\n",
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   "source": [
    "import pandas as pd; starbucks = pd.read_csv(\"starbucks.csv\", encoding=\"ISO-8859-1\")\n",
    "import statsmodels.formula.api as smf\n",
    "SLR_model = smf.ols(formula='calories ~ 1 + carb', data=starbucks)\n",
    "SLR_model.fit().summary().tables[-2]"
   ]
  },
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   "cell_type": "markdown",
   "id": "e247f4c3",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
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   },
   "source": [
    "### Starbucks Dataset: Mini Lecture (<5 minutes) [Click \"down\" next]\n",
    "\n",
    "<br>\n",
    "Under the $\\;\\;E[\\text{calories}] \\approx \\beta_0 + \\beta_\\text{carb}\\text{carb}\\;\\;$ Simple Linear Regression Model\n",
    "\n",
    "\\begin{align*}\n",
    "H_0 & : {} \\text{There is NO linear association bewteen carb and calories} \\\\\n",
    "H_1 & : {} \\text{There is SOME linear association bewteen carb and calories}\n",
    "\\end{align*}\n",
    "\n",
    "$$\\color{gray}{\n",
    "\\begin{align*}\n",
    "H_0 & : {} \\beta_{\\text{carb}} = 0 \\quad \\text{for the $\\;\\;E[\\text{calories}] \\approx \\beta_0 + \\beta_\\text{carb}\\text{carb}\\;\\;$ SLR model} \\\\\n",
    "H_1 & : {} \\beta_{\\text{carb}} \\neq 0 \\quad \\text{for the $\\;\\;E[\\text{calories}] \\approx \\beta_0 + \\beta_\\text{carb}\\text{carb}\\;\\;$ SLR model}\n",
    "\\end{align*}}$$\n",
    "\n",
    "\n",
    "> The ***p-value*** on the previous slide indicates very strong statistical evidence against the ***null hypothesis***, which suggests that there indeed does appear to be a LINEAR association between the `carb` and `calories` variables"
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   "cell_type": "markdown",
   "id": "b270f58a",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
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   },
   "source": [
    "### Starbucks Dataset: Reminder (<1 minute)\n",
    "\n",
    "The homework introduced you to the assumptions of linear regression \n",
    "\n",
    "  - Normality, homoskedasticity, independence, and <br> \n",
    "    correctness of Linear form (and $x_i$ measured exactly without error)\n",
    "      - [all of which are important knowledge for future courses!]\n",
    "\n",
    "\n",
    "entailed in the simple linear regression specification\n",
    "\n",
    "$$Y_i ∼ \\mathcal{N}\\left(\\beta_0 + \\beta_1x_i, \\sigma^2\\right)$$\n",
    "\n",
    "> The computation of the p-value used to provide a measure of evidence against the **null hypothesis** is based on the correctness of these assumptions.<br><br>*so hopefully they're fairly accurate for the starbucks data...*<br>**but we'll not focus on this for now...**\n",
    "\n",
    "  "
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   "id": "79ebc7ce",
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
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   },
   "source": [
    "## OLS [Ordinary Least Squares]: Mini Lecture (SKIP AS THERE WILL NOT BE ENOUGH TIME TO COVER THIS...) \n",
    "\n",
    "- Ordingary Least Squares fit/estimator $\\big(\\hat \\beta_1 = r_{xy}\\frac{s_y}{s_x}$ and $\\hat \\beta_0 = \\bar y - \\beta_1 \\bar x\\big)$\n",
    "\n",
    "|<img src=\"https://learningstatisticswithr.com/book/lsr_files/figure-html/regression3a-1.png\" alt=\"\" style=\"width: 600px;\"/>|OLS just means choosing coefficients<br>$\\beta_0$ and $\\beta_1$<br>which will minimize<br>the sum of the squared residuals<br><br>through the optimization problem: <br><br>\\begin{align} & {}\\min_{\\hat \\beta_0, \\hat \\beta_1} \\sum_{i=1}^n \\epsilon_i^2 = \\min_{\\hat \\beta_0, \\hat \\beta_1} \\sum_{i=1}^n (y_i-\\hat y_i)^2 \\\\ = & {} \\min_{\\hat \\beta_0, \\hat  \\beta_1} \\sum_{i=1}^n (y_i-\\hat \\beta_0 - \\hat\\beta_1 x_i)^2\\end{align}|\n",
    "|:-:| |\n",
    "|||\n",
    "\n",
    "\n",
    "- Measure of model fit ***coefficient of determination*** $R^2 = r_{y\\hat y}^2 = 1- \\frac{\\sum_{i=1}^n \\epsilon_i^2}{\\sum_{i=1}^n (y_i-\\bar y)^2}$\n",
    "    - **\"Proportion of Variation Explained\"**"
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   "id": "bdcf9c61",
   "metadata": {
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   },
   "source": [
    "### Ordinary Least Squares (OLS):  Demo (SKIP AS THERE WILL NOT BE ENOUGH TIME TO COVER THIS...) [Click \"down\" next]"
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   "source": [
    "import pandas as pd; starbucks = pd.read_csv(\"starbucks.csv\", encoding=\"ISO-8859-1\")\n",
    "#import statsmodels.api as sm\n",
    "#Y = starbucks.calories; X = broadway.carb; X = sm.add_constant(X); model = sm.OLS(Y,X)\n",
    "import statsmodels.formula.api as smf\n",
    "SLR_model = smf.ols(formula='calories ~ 1 + carb', data=starbucks)\n",
    "SLR_model_fit = SLR_model.fit()\n",
    "import plotly.express as px; import plotly.graph_objects as go\n",
    "fig = px.scatter(starbucks, x=\"carb\", y=\"calories\", trendline='ols'); fig.update_layout(title=\"OLS line fits go through points so the sum of the squared residuals (the sum of all red line squared distances) is minimized\"); fig.add_annotation(x=57, y=200, text='(As seen on the next slide)', showarrow=False, yshift=0); fig.add_annotation(x=57, y=175, text='The \"Proportion of Variation Explained\" is 45.6%:', showarrow=False, yshift=0); fig.add_annotation(x=57, y=150, text='How well does the linear regression line', showarrow=False, yshift=0); fig.add_annotation(x=57, y=125, text='seem to capture the relationship', showarrow=False, yshift=0); fig.add_annotation(x=57, y=100, text='between carb and calories?', showarrow=False, yshift=0); fig.update_annotations(font_size=20)\n",
    "for y,yhat,x in zip(SLR_model_fit.predict(), starbucks.calories.values, starbucks.carb.values):\n",
    "    fig.add_trace(go.Scatter(x=[x,x], y=[yhat, y], mode=\"lines\", line=go.scatter.Line(color=\"red\"), showlegend=False))\n",
    "fig.show()"
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   "id": "e38750d6",
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "### Ordinary Least Squares (OLS): Code Demo Confirmations (SKIP AS THERE WILL NOT BE ENOUGH TIME TO COVER THIS...)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "125346a5",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Model Fit Results\n",
      "Intercept    146.020432\n",
      "carb           4.297084\n",
      "dtype: float64\n",
      "Confirmed with Formulas\n",
      "146.02043171319244\n",
      "4.297084445176319\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "(0.4556236754779174, 0.45562367547791766, 0.45562367547791727)"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
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   "source": [
    "x, y, yhat = starbucks.carb.values, starbucks.calories.values, SLR_model_fit.predict()\n",
    "\n",
    "# confirming coefficient calculations\n",
    "print(\"Model Fit Results\", SLR_model_fit.params, sep='\\n')\n",
    "beta_1_hat = np.corrcoef(x,y)[0,1]*y.std(ddof=1)/x.std(ddof=1) # known analytical formula\n",
    "beta_0_hat = y.mean() - beta_1_hat*x.mean() # known analytical formula\n",
    "print(\"Confirmed with Formulas\", beta_0_hat, beta_1_hat, sep='\\n')\n",
    "\n",
    "# correlation between y and yhat\n",
    "r_y_yhat = np.corrcoef(y, yhat)[0,1]\n",
    "# model residuals\n",
    "residuals = y - yhat\n",
    "# R-squared calculation\n",
    "sum_of_squared_residuals = ((residuals)**2).sum()\n",
    "total_variance = ((y - y.mean())**2).sum()\n",
    "R_squared = 1 - sum_of_squared_residuals/total_variance\n",
    "\n",
    "# All \"Proportion of Variation Explained\" Calculations Confirmed as well:\n",
    "R_squared, r_y_yhat**2, SLR_model_fit.rsquared"
   ]
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  {
   "cell_type": "markdown",
   "id": "98f29366",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### SLR Extension: Using Indicator Variables (<3 minutes) [Click \"down\" next]\n",
    "\n",
    "Do you think we can use the simple linear regression framework to peform a hypothesis test of the difference between two groups? **How could we do this?**"
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box\"]},\"legend\":{\"title\":{\"text\":\"type\"},\"tracegroupgap\":0},\"margin\":{\"t\":60},\"boxmode\":\"overlay\"},                        {\"responsive\": true}                    ).then(function(){\n",
       "                            \n",
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       "            Plotly.purge(gd);\n",
       "            observer.disconnect();\n",
       "        }}\n",
       "}});\n",
       "\n",
       "// Listen for the removal of the full notebook cells\n",
       "var notebookContainer = gd.closest('#notebook-container');\n",
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       "                        })                };                });            </script>        </div>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Give a linear model specfication for the data subset below...\n",
    "import plotly.express as px\n",
    "starbucks_subset = starbucks[((starbucks.type == \"sandwich\") | (starbucks.type == \"bistro box\"))]\n",
    "px.box(starbucks_subset, y=\"type\", x=\"calories\", color=\"type\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "537f701f",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### SLR Extension: Using Indicator Variables (<7 minutes) [Click \"down\" next]\n",
    "\n",
    "$$E[\\text{calories}] = \\beta_0 + \\beta_{\\text{sandwich}} I(\\text{type = \"sandwich\"})$$\n",
    "\n",
    "- What's the predicted number of calories for bistro boxes?\n",
    "- What's the predicted number of calories for sandwiches?\n",
    "- What's the predicted difference in the number of calories for these two?\n",
    "- What's the ***null hypothesis*** tested below and what does the ***hypothesis test*** result suggest about an actual difference in the number of calories between bistro boxes and sandwiches? **[ASK Q AND MOVE ON TO THE NEXT SLIDE]**\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "a81ec92d",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"simpletable\">\n",
       "<tr>\n",
       "          <td></td>            <th>coef</th>     <th>std err</th>      <th>t</th>      <th>P>|t|</th>  <th>[0.025</th>    <th>0.975]</th>  \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Intercept</th>        <td>  377.5000</td> <td>   17.846</td> <td>   21.153</td> <td> 0.000</td> <td>  338.946</td> <td>  416.054</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>type[T.sandwich]</th> <td>   18.2143</td> <td>   26.124</td> <td>    0.697</td> <td> 0.498</td> <td>  -38.223</td> <td>   74.651</td>\n",
       "</tr>\n",
       "</table>"
      ],
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       "<class 'statsmodels.iolib.table.SimpleTable'>"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import pandas as pd; starbucks = pd.read_csv(\"starbucks.csv\", encoding=\"ISO-8859-1\")\n",
    "starbucks_subset = starbucks[((starbucks.type == \"sandwich\") | (starbucks.type == \"bistro box\"))]\n",
    "\n",
    "import statsmodels.formula.api as smf; import warnings; warnings.simplefilter('ignore', category=UserWarning)\n",
    "SLR_model = smf.ols(formula='calories ~ 1 + type', data=starbucks_subset)\n",
    "SLR_model.fit().summary().tables[1]"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c3bce2ce",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### SLR Extension: Using Indicator Variables ($\\approx$1 minute) [Click \"down\" next]\n",
    "\n",
    "Here's what's actually happening here...\n",
    "\n",
    "$$E[\\text{calories}] \\approx \\underset{\\beta_0}{377.5} + \\underset{\\beta_{\\text{sandwich}}}{18.2} \\underbrace{\\times\\;\\;  I(\\text{type = \"sandwich\"})}_{\\text{$1$ if type = \"sandwich\"; 0, otherwise}}$$\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "d86feaf9",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>const</th>\n",
       "      <th>sandwich</th>\n",
       "      <th>type</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>47</th>\n",
       "      <td>1.0</td>\n",
       "      <td>0</td>\n",
       "      <td>bistro box</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>48</th>\n",
       "      <td>1.0</td>\n",
       "      <td>0</td>\n",
       "      <td>bistro box</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>67</th>\n",
       "      <td>1.0</td>\n",
       "      <td>1</td>\n",
       "      <td>sandwich</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>68</th>\n",
       "      <td>1.0</td>\n",
       "      <td>1</td>\n",
       "      <td>sandwich</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "    const  sandwich        type\n",
       "47    1.0         0  bistro box\n",
       "48    1.0         0  bistro box\n",
       "67    1.0         1    sandwich\n",
       "68    1.0         1    sandwich"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import pandas as pd; starbucks = pd.read_csv(\"data/7/starbucks.csv\", encoding=\"ISO-8859-1\")\n",
    "starbucks_subset = starbucks[((starbucks.type == \"sandwich\") | (starbucks.type == \"bistro box\"))]\n",
    "import statsmodels.api as sm; import warnings; warnings.simplefilter('ignore', category=UserWarning)\n",
    "Y = starbucks_subset.calories; X = (starbucks_subset.type == 'sandwich').rename('sandwich').astype(int)\n",
    "X = sm.add_constant(X); pd.concat([X[6:10], starbucks_subset[['type']][6:10]],axis=1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "a4134afe",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"simpletable\">\n",
       "<tr>\n",
       "      <td></td>        <th>coef</th>     <th>std err</th>      <th>t</th>      <th>P>|t|</th>  <th>[0.025</th>    <th>0.975]</th>  \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>const</th>    <td>  377.5000</td> <td>   17.846</td> <td>   21.153</td> <td> 0.000</td> <td>  338.946</td> <td>  416.054</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>sandwich</th> <td>   18.2143</td> <td>   26.124</td> <td>    0.697</td> <td> 0.498</td> <td>  -38.223</td> <td>   74.651</td>\n",
       "</tr>\n",
       "</table>"
      ],
      "text/latex": [
       "\\begin{center}\n",
       "\\begin{tabular}{lcccccc}\n",
       "\\toprule\n",
       "                  & \\textbf{coef} & \\textbf{std err} & \\textbf{t} & \\textbf{P$> |$t$|$} & \\textbf{[0.025} & \\textbf{0.975]}  \\\\\n",
       "\\midrule\n",
       "\\textbf{const}    &     377.5000  &       17.846     &    21.153  &         0.000        &      338.946    &      416.054     \\\\\n",
       "\\textbf{sandwich} &      18.2143  &       26.124     &     0.697  &         0.498        &      -38.223    &       74.651     \\\\\n",
       "\\bottomrule\n",
       "\\end{tabular}\n",
       "\\end{center}"
      ],
      "text/plain": [
       "<class 'statsmodels.iolib.table.SimpleTable'>"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sm.OLS(Y,X).fit().summary().tables[1]"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a298beb9",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### SLR Extension: Using Indicator Variables (<5 minutes) \n",
    "\n",
    "<br>\n",
    "For the Simple Linear Regression Model\n",
    "\n",
    "$$\\;\\;E[\\text{calories}] = \\beta_0 + \\beta_{\\text{sandwich}} I(\\text{type = \"sandwich\"})\\;\\;$$\n",
    "\n",
    "\\begin{align*}\n",
    "H_0 & : {} \\text{NO difference in expected calories for \"sandwich\" and \"bistro box\"} \\\\\n",
    "H_1 & : {} \\text{SOME difference in expected calories for \"sandwich\" and \"bistro box\"}\n",
    "\\end{align*}\n",
    "\n",
    "$$\\color{gray}{\n",
    "\\begin{align*}\n",
    "H_0 & : {} \\beta_{\\text{sandwich}} = 0 \\quad \\text{for $\\;\\;E[\\text{calories}] = \\beta_0 + \\beta_{\\text{sandwich}} I(\\text{type = \"sandwich\"})$} \\\\\n",
    "H_1 & : {} \\beta_{\\text{sandwich}} \\neq 0 \\quad \\text{for $\\;\\;E[\\text{calories}] = \\beta_0 + \\beta_{\\text{sandwich}} I(\\text{type = \"sandwich\"})$}\n",
    "\\end{align*}}$$\n",
    "\n",
    "\n",
    "> The ***p-value*** on the previous slide indicates no statistical evidence against the ***null hypothesis***, which suggests that there does not appear to be a difference in average calaores for \"sandwich\" and \"bistro box\" Starbucks menu items...\n",
    "> - depends on the simple linear regression assumptions, though...\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "76967ee8",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Tutorial Activity: Quiz (10 minutes)\n",
    "\n",
    "> - Hand in written answers to TA at the end of the quiz\n",
    "> - Question credit for attempting to provide an answer: answers will not be reviewed in detail during marking\n",
    "\n",
    "0. What's your name?\n",
    "1. Explain to calculate predicted values from a Linear Model Regression.\n",
    "2. Explain how to use a \"0/1\" indicator in a Linear Model Regression.\n",
    "3. Write the *null hypothesis* for a Simple Linear Regression $\\beta_1$ coefficient.\n",
    "4. What's the difference in the interpretation of a *null hypothesis* for the $\\beta_1$ coefficient in a Simple Linear Regression model if it's for a continuous variable versus if it's for a an indicator variable?\n",
    "5. Explain how Linear Model Regression is a \"Normal distribution model\".\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fac8f934",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Tutorial Activity: Quiz Review (<5 minutes):\n",
    "\n",
    "1. Use the estimated the coefficients $\\hat \\beta_0 + \\hat \\beta_1 x_i$ from a fit model and plug in the $x_i$ value and calculate\n",
    "\n",
    "2. A \"0/1\" indicator predictor variable gets used exactly like a continuous predictor variable: just plug the \"0/1\" value and calculate\n",
    "\n",
    "3. $H_0 : \\beta_1=0$\n",
    "\n",
    "4. There is no linear association between the outcome and predictor variables; versus, there is no difference between the groups defined by the indicator variable\n",
    "\n",
    "5. The full model specification is $Y_i = \\beta_0 + \\hat \\beta_1 x_i + \\epsilon_i$ where the **error terms** $\\epsilon_i$ are assumed to be destributed as $\\mathcal{N} \\left(0,\\sigma^2\\right)$; or, $Y_i \\sim \\mathcal{N} \\left(\\beta_0 + \\hat \\beta_1 x_i, \\sigma^2\\right)$\n",
    "\n"
   ]
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    "## Tutorial Assignment *[next click is down not right]*\n",
    "\n",
    "- Submit your work for the assignment through Quercus\n",
    "\n",
    "#### Marking is based on clarity and correctness of your written submission as judged from the perspective of a nonstatistical audience\n",
    "\n",
    "- Don't spend more than 60 minutes on this assignment (unless really needed...)    \n",
    "\n",
    "    - Aim for something close to 200 to 500 words\n",
    "    - Grammar is *not* the main focus of the assessment, but it is important that you communicate in a clear and professional manner; so, \n",
    "    - Use full sentences (without slang or emojis) \n",
    "    \n",
    "\n",
    "\n"
   ]
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    "## Tutorial Assignment (complete at home if needed)\n",
    "\n",
    "Congratulations! You've just been hired as the first statistician for a startup company (Mr. BadaBing's) that makes side walk chalk. You were hired because the owners are looking to add more credibility to their work by testing if their side walk chalk is significantly better than their competitors (Mr. BingBong's). This is based on whether children enjoy their free time more with Mr. BadaBing's side walk chalk or Mr. BingBong's. Each child enrolled in the study will received either your side walk chalk or your competitors. An adult in the household reported how much enjoyment their children got from playing with the chalk on a scale from 1 to 100.\n",
    "\n",
    "The big boss at your company (Sunny \"Gweeny\" Lang) has heard about how her competitors use linear regression for their own studies and wants you to do the same kind of thing; however, the Sunny does not actually know what linear regression is. Therefore, you need to craft an email explaining to Miss Lang what linear regression is, and whether it would be appropriate to use it for the proposed analysis. You should write out a hypothetical linear regression equation for the experiment and define what each part of the equation is in simple terms. Don't be afraid to use technical statistical terms, but be sure to explain their meaning in simple and understandable ways that would help nonstatistical audience made sense of what you're taking about.\n",
    "\n"
   ]
  }
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