Welcome and Chapter 1
Textbook
Beyond Multiple Linear Regression by Paul Roback and Julie Legler
- Available online
- Hard copies available for purchase
Assumptions for linear regression
Linearity: Linear relationship between mean of the response \(Y\) and the predictor \(X\)
Independence: No connection between how any two points lie above or below the regression line
Normality: Response, \(Y\), follows a normal distribution at each level of the predictor, \(X\) (indicated by red curves)
Equal variance: Variance (or standard deviation) of the response, \(Y\), is equal for all levels of the predictor, \(X\)
Data Analysis Life Cycle
Plots for univariate EDA
p1 <- ggplot(data = derby, aes(x = speed)) +
geom_histogram(fill = "forestgreen", color = "black") +
labs(x = "Winning speed (ft/s)", y = "Count")
p2 <- ggplot(data = derby, aes(x = starters)) +
geom_histogram(fill = "forestgreen", color = "black") +
labs(x = "Starters", y = "Count")
p3 <- ggplot(data = derby, aes(x = condition)) +
geom_bar(fill = "forestgreen", color = "black", aes(x = ))
p1 + (p2 / p3) +
plot_annotation(title = "Univariate data analysis")Plots for bivariate EDA
p4 <- ggplot(data = derby, aes(x = starters, y = speed)) +
geom_point() +
labs(x = "Starters", y = "Speed (ft / s)")
p5 <- ggplot(data = derby, aes(x = year, y = speed)) +
geom_point() +
labs(x = "Year", y = "Speed (ft / s)")
p6 <- ggplot(data = derby, aes(x = condition, y = speed)) +
geom_boxplot(fill = "forestgreen", color = "black") +
labs(x = "Conditions", y = "Speed (ft / s)")
(p4 + p5) + p6 +
plot_annotation(title = "Bivariate data analysis")Scatterplot matrix
A scatterplot matrix helps quickly visualize relationships between many variable pairs. They are particularly useful to identify potentially correlated predictors.
Plots for multivariate EDA
Plot the relationship between the response and a predictor based on levels of another predictor to assess potential interactions.
#library(viridis)
ggplot(data = derby, aes(x = year, y = speed, color = condition,
shape = condition, linetype = condition)) +
geom_point() +
geom_smooth(method = "lm", se = FALSE, aes(linetype = condition)) +
labs(x = "Year", y = "Speed (ft/s)", color = "Condition",
title = "Speed vs. year",
subtitle = "by track condition") +
guides(lty = FALSE, shape = FALSE) +
scale_color_viridis_d(end = 0.9)Model 1: Check model assumptions
Model 2: Add quadratic effect for year?
Model 2: Check model assumptions
Model 3: Include interaction term?
Recall from the EDA…
library(viridis)
ggplot(data = derby, aes(x = year, y = speed, color = condition,
shape = condition, linetype = condition)) +
geom_point() +
geom_smooth(method = "lm", se = FALSE, aes(linetype = condition)) +
labs(x = "Year", y = "Speed (ft/s)", color = "Condition",
title = "Speed vs. year",
subtitle = "by track condition") +
guides(lty = FALSE, shape = FALSE) +
scale_color_viridis_d(end = 0.9)Model 3: Add interaction term
\[\begin{aligned}\widehat{speed} = & 52.387 - 0.003 ~ starters + 0.020 ~ yearnew - 1.070 ~ good - 2.183 ~ slow \\ &+0.012 ~ yearnew \times good + 0.012 ~ yearnew \times slow \end{aligned}\]
| term | estimate | std.error | statistic | p.value |
|---|---|---|---|---|
| (Intercept) | 52.387 | 0.200 | 262.350 | 0.000 |
| starters | -0.003 | 0.016 | -0.189 | 0.850 |
| yearnew | 0.020 | 0.003 | 7.576 | 0.000 |
| conditiongood | -1.070 | 0.423 | -2.527 | 0.013 |
| conditionslow | -2.183 | 0.270 | -8.097 | 0.000 |
| yearnew:conditiongood | 0.012 | 0.008 | 1.598 | 0.113 |
| yearnew:conditionslow | 0.012 | 0.004 | 2.866 | 0.005 |
Inference for regression
Use least squares regression to get the estimates \(\hat{\beta}_0\), \(\hat{\beta}_1\), and \(\hat{\sigma}^2\)
\(\hat{\sigma}\) is the regression standard error
. . .
\[\hat{\sigma} = \sqrt{\frac{\sum_{i=1}^n(y_i - \hat{y}_i)^2}{n - p - 1}} = \sqrt{\frac{\sum_{i=1}^n e_i^2}{n-p-1}}\]
