{ "cells": [ { "cell_type": "code", "execution_count": 1, "id": "d8d691bb", "metadata": { "tags": [ "remove-cell" ] }, "outputs": [ { "name": "stderr", "output_type": "stream", "text": [ "Registered S3 methods overwritten by 'ggplot2':\n", " method from \n", " [.quosures rlang\n", " c.quosures rlang\n", " print.quosures rlang\n" ] }, { "name": "stderr", "output_type": "stream", "text": [ "Registered S3 method overwritten by 'rvest':\n", " method from\n", " read_xml.response xml2\n" ] }, { "name": "stderr", "output_type": "stream", "text": [ "-- Attaching packages --------------------------------------- tidyverse 1.2.1 --\n" ] }, { "name": "stderr", "output_type": "stream", "text": [ "v ggplot2 3.1.1 v purrr 0.3.2 \n", "v tibble 2.1.1 v dplyr 0.8.0.1\n", "v tidyr 0.8.3 v stringr 1.4.0 \n", "v readr 1.3.1 v forcats 0.4.0 \n" ] }, { "name": "stderr", "output_type": "stream", "text": [ "-- Conflicts ------------------------------------------ tidyverse_conflicts() --\n", "x dplyr::filter() masks stats::filter()\n", "x dplyr::lag() masks stats::lag()\n" ] }, { "name": "stderr", "output_type": "stream", "text": [ "Parsed with column specification:\n", "cols(\n", " ID = col_double(),\n", " Name = col_character(),\n", " Gender = col_character(),\n", " Age = col_double(),\n", " Rating = col_double(),\n", " Degree = col_character(),\n", " Start_Date = col_character(),\n", " Retired = col_logical(),\n", " Division = col_character(),\n", " Salary = col_character()\n", ")\n" ] } ], "source": [ "library(tidyverse)\n", "employees <- read_csv(\"../_build/data/employee_data.csv\")\n", "employees$Salary <- parse_number(employees$Salary)\n", "employees$Start_Date <- parse_date(employees$Start_Date, format = \"%m/%d/%Y\")\n", "degreeLevels <- c(\"High School\", \"Associate's\", \"Bachelor's\", \"Master's\", \"Ph.D\")\n", "employees$Degree <- parse_factor(employees$Degree, levels = degreeLevels, ordered = TRUE)\n", "modelAge <- lm(Salary ~ Age, data=employees)" ] }, { "cell_type": "markdown", "id": "b2ca3c74", "metadata": {}, "source": [ "# Understanding Our Regression Model\n", "\n", "Our least squares estimates $b_0$ and $b_1$ are estimates of the true theoretical values $\\beta_0$ and $\\beta_1$ respectively. Since the regression line is calculated using sample data, we can construct confidence intervals and perform hypothesis tests on our estimates. The `confint()` function reports 95\\% confidence intervals for the regression coefficients.\n", "\n", "```{admonition} Syntax\n", "`confint(model)`\n", "```" ] }, { "cell_type": "code", "execution_count": 2, "id": "ad2fd323", "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "\n", "\n", "\t\n", "\t\n", "\n", "
2.5 %97.5 %
(Intercept)58390.19775877.208
Age 1834.364 2219.292
\n" ], "text/latex": [ "\\begin{tabular}{r|ll}\n", " & 2.5 \\% & 97.5 \\%\\\\\n", "\\hline\n", "\t(Intercept) & 58390.197 & 75877.208\\\\\n", "\tAge & 1834.364 & 2219.292\\\\\n", "\\end{tabular}\n" ], "text/markdown": [ "\n", "| | 2.5 % | 97.5 % |\n", "|---|---|---|\n", "| (Intercept) | 58390.197 | 75877.208 |\n", "| Age | 1834.364 | 2219.292 |\n", "\n" ], "text/plain": [ " 2.5 % 97.5 % \n", "(Intercept) 58390.197 75877.208\n", "Age 1834.364 2219.292" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "confint(modelAge)" ] }, { "cell_type": "markdown", "id": "bdbf74b2", "metadata": {}, "source": [ "We can obtain more detailed output using the `summary()` command.\n", "\n", "```{admonition} Syntax\n", "`summary(model)`\n", "```" ] }, { "cell_type": "code", "execution_count": 3, "id": "9d410420", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "\n", "Call:\n", "lm(formula = Salary ~ Age, data = employees)\n", "\n", "Residuals:\n", " Min 1Q Median 3Q Max \n", "-103272 -21766 2428 23138 90680 \n", "\n", "Coefficients:\n", " Estimate Std. Error t value Pr(>|t|) \n", "(Intercept) 67133.70 4455.17 15.07 <2e-16 ***\n", "Age 2026.83 98.07 20.67 <2e-16 ***\n", "---\n", "Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1\n", "\n", "Residual standard error: 32630 on 918 degrees of freedom\n", " (80 observations deleted due to missingness)\n", "Multiple R-squared: 0.3175,\tAdjusted R-squared: 0.3168 \n", "F-statistic: 427.1 on 1 and 918 DF, p-value: < 2.2e-16\n" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "summary(modelAge)" ] }, { "cell_type": "markdown", "id": "7d37c26c", "metadata": {}, "source": [ "Our guess for $\\beta_1$ is \\$2,026.83, but because we are basing this estimate on a data sample, we could be wrong. We can say with 95% confidence that the true value of the slope (denoted $\\beta_1$) is between \\$1,834.36 and \\$2,219.29. The good news is the confidence interval is all positive, indicating a positive relationship between `Salary` and `Age`. If zero were in this interval we would not be able to conclude that there was a positive relationship between `Salary` and Age.\n", "\n", "By applying the `summary()` command to our regression we can see p-values listed for each coefficient. The hypothesis test being done here is \"does the true slope equal 0?\", and \"does the true intercept equal 0?\". In symbols we are testing for the slope:\n", "\n", "+ $H_o: \\beta_1 = 0$ \n", "+ $H_a: \\beta_1 \\ne 0$\n", "\n", "A small p-value (typically below 0.05) means we reject the null hypothesis and can conclude there is likely a linear relationship between the $X$ and $Y$ variables; simply put, a small p-value indicates we need $X$ in the model. \n", "\n", "The p-value for the intercept corresponds to the hypothesis test asking, \"does the true intercept equal 0?\" We often ignore the p-value of the intercept when the intercept is difficult to interpret.\n", "\n", "## The R-Squared ($R^2$)\n", "\n", "From our regression summary we can also see the **multiple R-squared** (denoted $R^2$) and the **adjusted R-squared** (denoted $R_a^2$). $R^2$ is a goodness-of-fit measure and always falls in the interval 0 to 1. In essence, we do regression to try to understand how changes in the $X$ variable explain what is happening to a $Y$ variable. If $Y$ didn't vary when $X$ changes, it would be very boring to explain, so in essence we are trying to explain the variation in $Y$ using $X$. The $R^2$ tells us the percentage of the variation in $Y$ explained by $X$, with higher being better. In our regression of `Salary` against `Age`, the $R^2$ equals `0.3175`; from this we conclude that “in our model, 31.75% of the variation in income is explained by age.”\n", "\n", "One issue with $R^2$ is that it can go up simply by adding more $X$ variables to a model, so $R_a^2$ has a penalty factor related to the number of $X$ variables in the model. The adjusted R-squared, which in our example equals 0.3168, is always preferable to report of these two measures. However, neither $R^2$ nor $R_a^2$ indicates anything about predictability, so another measure described below is more useful to consider.\n", "\n", "```{warning}\n", "When a linear regression model has more than one independent variable, report $R_a^2$ instead of $R^2$.\n", "```\n", "\n", "## Estimate of Noise and Prediction\n", "\n", "When we look at the scatter plot with our regression model, we see noise in the model, that is, we see a line plus variation (noise) around the line. Regression modeling assumes that the noise $\\epsilon$ is normally distributed with mean 0 and variance $\\sigma^2$. The value $\\sigma^2$ is key when using a regression model for prediction, since it tells us how accurate our predictions are. The estimate of $\\sigma$ from our sample data is denoted $s_e$ and is called the **residual standard error**. \n", "\n", "This residual standard error is important because approximate 95% prediction intervals can be found using the formula:\n", "\n", "$$predicted\\; y =\\hat{y} = (b_0+b_1(x))\\pm 1.96(s_e)$$\n", "\n", "To understand this, let's examine $s_e$ from the regression, which equals \\$32,630. The predicted salary of someone 40 years old is approximately: \n", "$$\\$67,133.70 + \\$2,026.83(40) = \\$148,206.9.$$ \n", "\n", "But we know it is not an exact relationship, so it would be nice to put lower and upper bounds on this prediction. Using $s_e$, we can say we are roughly 95\\% confident that the predicted salary for someone 40 years old is in the interval: \n", "\n", "$$\\$148,206.9 \\pm 1.96(\\$32,630) = (\\$84,252.1 ; \\$212,161.7)$$\n", "\n", "A more formal prediction interval can be found in R using the `predict()` function. Note that the formula shown above is an approximation and that this function's results are more precise, so the intervals differ slightly." ] }, { "cell_type": "code", "execution_count": 4, "id": "17b5b63e", "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "\n", "\n", "\t\n", "\n", "
fitlwrupr
148206.884124.55212289.1
\n" ], "text/latex": [ "\\begin{tabular}{r|lll}\n", " fit & lwr & upr\\\\\n", "\\hline\n", "\t 148206.8 & 84124.55 & 212289.1\\\\\n", "\\end{tabular}\n" ], "text/markdown": [ "\n", "| fit | lwr | upr |\n", "|---|---|---|\n", "| 148206.8 | 84124.55 | 212289.1 |\n", "\n" ], "text/plain": [ " fit lwr upr \n", "1 148206.8 84124.55 212289.1" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "newData <- data.frame(Age = 40)\n", "predict(modelAge, newData, interval = \"predict\")" ] }, { "cell_type": "markdown", "id": "16700372", "metadata": {}, "source": [ "A rough rule of thumb when looking at regression models is to look at $s_e$ then double it; your 95% prediction interval will be roughly $\\hat{y} \\pm 2(s_e)$ (for this estimate 2 is rounded up from 1.96). You can use this as a quick guide to decide if you think your model is acceptable.\n", "\n", "```{warning}\n", "$R^2$ does not inform about a linear regression model's predictive ability, so one must also report $s_e$.\n", "```\n", "\n", "## (§) Basic Diagnostic Plots\n", "\n", ":::{note}\n", "This section is optional, and will not be covered in the DSM course. For the assigned reading move on to the next section, [5.4. Multiple Linear Regression](multiple_linear_regression.html#multiple-linear-regression). \n", ":::\n", "\n", "When doing regression analysis, we assume a linear relationship between the $X$ and $Y$ variable. We also assume the noise in the model follows a normal distribution. Both of these assumptions may be checked with separate **diagnostic plots**.\n", "\n", "The residuals $e_i$, also called the **errors**, are the observed ($y_i$) minus the predicted ($\\hat{y_i}$) values for each row in our data set. The most basic plot is a residual versus fitted (predicted) value plot. This plot is easily created in R as shown below, and if the modeling assumptions are correct should show a random scatter of points. Note that we pass our regression model into the same `plot()` function we have seen before, but set the `which` argument to `1` in order to get a residual vs. fitted plot." ] }, { "cell_type": "code", "execution_count": 5, "id": "e936027e", "metadata": {}, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAA0gAAANICAMAAADKOT/pAAAANlBMVEUAAABNTU1oaGh8fHyM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\n+D/vS6O4XUi68QAAAABJRU5ErkJggg==", "text/plain": [ "Plot with title \"\"" ] }, "metadata": { "filenames": { "image/png": "C:\\Users\\phamilton\\Documents\\Projects\\dsm\\dsm_website\\_build\\jupyter_execute\\05_linear_regression\\understanding_reg_8_0.png" } }, "output_type": "display_data" } ], "source": [ "plot(modelAge, which=1)" ] }, { "cell_type": "markdown", "id": "d78a9394", "metadata": {}, "source": [ "The red line shown in the plot is a local linear smooth used to capture the trend of the points. We see no relationship between the residuals and the fitted values, which must be true if the regression assumptions hold.\n", "\n", "To contrast with the ideal plot, consider some simulated data of a non-linear relationship between $X$ and $Y$, which we don't know so we fit a line. As the plots below indicate, there is an unusual relationship between the residuals and fitted values that would indicate we should investigate this model more fully.\n", "\n", "```{figure} ../_build/html/_images/unusual_resid_fitted.png\n", "---\n", "height: 400px\n", "align: center\n", "name: unusual_resid\n", "---\n", "Unusual Residuals v. Fitted Plot\n", "```\n", "\n", "The normality assumption of the noise in the model may be checked by either looking at a histogram or a **qq-plot** of the error terms. We can extract an atomic vector of the residuals from our model using the `residuals()` function:\n", "\n", "```{admonition} Syntax\n", "`residuals(model)`\n", "```\n", "\n", "By passing these residuals into `hist()`, we can observe the histogram of the residuals to check whether they are normally-distributed:" ] }, { "cell_type": "code", "execution_count": 6, "id": "7a88a56a", "metadata": {}, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAA0gAAANICAMAAADKOT/pAAAAMFBMVEUAAABNTU1oaGh8fHyM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\nAxAJwABEAjAAkQAMQCQAAxAJwIB/wHzgDiZ8yI4AAAAASUVORK5CYII=", "text/plain": [ "Plot with title \"Histogram of residuals(modelAge)\"" ] }, "metadata": { "filenames": { "image/png": "C:\\Users\\phamilton\\Documents\\Projects\\dsm\\dsm_website\\_build\\jupyter_execute\\05_linear_regression\\understanding_reg_10_0.png" } }, "output_type": "display_data" } ], "source": [ "hist(residuals(modelAge))" ] }, { "cell_type": "markdown", "id": "dac622dc", "metadata": {}, "source": [ "Based on this plot, the residuals of our model appear to be relatively normal. Another important diagnostic plot to look at is the **qq-plot**, which also helps us visually inspect whether our residuals are normally distributed. A qq-plot plots the residuals from our model against a straight line. If the plotted residuals fall mostly on the straight line, we conclude that they are most likely normally-distributed. If they deviate from the straight line, we conclude they might not be normally-distributed.\n", "\n", "To create the qq-plot, we can pass our model into the `plot()` function and set the `which` parameter equal to `2`:" ] }, { "cell_type": "code", "execution_count": 7, "id": "8ff63fa6", "metadata": {}, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAA0gAAANICAMAAADKOT/pAAAAM1BMVEUAAABNTU1oaGh8fHx/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\nIwAAAABJRU5ErkJggg==", "text/plain": [ "Plot with title \"\"" ] }, "metadata": { "filenames": { "image/png": "C:\\Users\\phamilton\\Documents\\Projects\\dsm\\dsm_website\\_build\\jupyter_execute\\05_linear_regression\\understanding_reg_12_0.png" } }, "output_type": "display_data" } ], "source": [ "plot(modelAge, which = 2)" ] }, { "cell_type": "markdown", "id": "d3964313", "metadata": {}, "source": [ "Consistent with the histogram of the residuals, the qq-plot shows that the residuals are approximately normal (*i.e.*, they mostly fall on the straight line).\n", "\n", "```{warning}\n", "Always verify that the model residuals are normally distributed.\n", "```" ] } ], "metadata": { "jupytext": { "cell_metadata_filter": "-all", "formats": "md:myst", "text_representation": { "extension": ".md", "format_name": "myst", "format_version": 0.13, "jupytext_version": "1.11.5" } }, "kernelspec": { "display_name": "R", "language": "R", "name": "ir" }, "language_info": { "codemirror_mode": "r", "file_extension": ".r", "mimetype": "text/x-r-source", "name": "R", "pygments_lexer": "r", "version": "3.6.1" }, "source_map": [ 16, 25, 35, 37, 45, 47, 87, 90, 108, 110, 133, 135, 141, 143 ] }, "nbformat": 4, "nbformat_minor": 5 }