2024-02-12
Model Selection
Building a model
Selecting variables
Prediction vs interpretation
Comparing potential models
Model Fitting
Find best fit line
Using OLS in this class
Parameter estimation
Categorical covariates
Interactions
Model Evaluation
Model Use (Inference)
Bad option for visualization:
ggplot(gapm2, aes(x = four_regions, y = LifeExpectancyYrs)) +
geom_point() +
labs(x = "World region",
y = "Country life expectancy (years)",
title = "Life expectancy vs. world region") +
theme(axis.title = element_text(size = 20),
axis.text = element_text(size = 20),
title = element_text(size = 20))
Good option for visualization:
ggplot(gapm2, aes(x = four_regions, y = LifeExpectancyYrs)) +
geom_jitter(size = 1, alpha = .6, width = 0.2) +
stat_summary(fun = mean, geom = "point", size = 8, shape = 18) +
labs(x = "World region",
y = "Country life expectancy (years)",
title = "Life expectancy vs. world region",
caption = "Diamonds = region averages") +
theme(axis.title = element_text(size = 20),
axis.text = element_text(size = 20),
title = element_text(size = 20))
Good option for visualization:
ggplot(gapm2, aes(x = four_regions, y = LifeExpectancyYrs)) +
geom_boxplot() +
labs(x = "World region",
y = "Country life expectancy (years)",
title = "Life expectancy vs. world region") +
theme(axis.title = element_text(size = 20),
axis.text = element_text(size = 20),
title = element_text(size = 20))
When using a categorical covariate/predictor (that is not ordered),
Instead we model the means of the outcome
In 511, we used Kruskal-Wallis test and our ANOVA table to test if groups means were statistically different from one another
We can do this using linear models AND we can include other variable in the model
We want to create a function that can map each region to life expectancy
If in Africa: \(\widehat{LE} = 61.32\)
If in the Americas: \(\widehat{LE} = 74.64\)
If in Asia: \(\widehat{LE} = 71.70\)
If in Europe: \(\widehat{LE} = 77.61\)
Can we make one equation for \(\widehat{LE}\) by putting the “if” statements within the equation?
\[\begin{aligned} \widehat{\textrm{LE}} = & 61.32 \cdot I(\text{Africa}) + 74.64 \cdot I(\text{Americas}) + \\ &71.7 \cdot I(\text{Asia}) + 77.61 \cdot I(\text{Europe}) \end{aligned}\]
However, a linear regression equation still requires an intercept!
\[\begin{aligned} \widehat{\textrm{LE}} = & 61.32 + 13.32 \cdot I(\text{Americas}) + \\ &10.38 \cdot I(\text{Asia}) + 16.29 \cdot I(\text{Europe}) \\ \widehat{\textrm{LE}} = & \widehat\beta_0 + \widehat\beta_1 \cdot I(\text{Americas}) + \\ & \widehat\beta_2 \cdot I(\text{Asia}) + \widehat\beta_3 \cdot I(\text{Europe}) \end{aligned}\]
\[\widehat{\textrm{LE}} = \widehat\beta_0 + \widehat\beta_1 \cdot I(\text{Americas}) + \widehat\beta_2 \cdot I(\text{Asia}) + \widehat\beta_3 \cdot I(\text{Europe})\]
| World region | Regression equation for WR | Average Life Expectancy for WR |
|---|---|---|
| Africa | \(\begin{aligned} \widehat{\textrm{LE}} = &\widehat\beta_0 + \widehat\beta_1 \cdot 0 + \\ & \widehat\beta_2 \cdot 0 + \widehat\beta_3 \cdot 0 \end{aligned}\) | \(\widehat{\textrm{LE}} = \widehat\beta_0\) |
| Americas | \(\begin{aligned} \widehat{\textrm{LE}} = &\widehat\beta_0 + \widehat\beta_1 \cdot 1+ \\ & \widehat\beta_2 \cdot 0 + \widehat\beta_3 \cdot 0 \end{aligned}\) | \(\widehat{\textrm{LE}} = \widehat\beta_0 + \widehat\beta_1\) |
| Asia | \(\begin{aligned} \widehat{\textrm{LE}} = &\widehat\beta_0 + \widehat\beta_1 \cdot 0 + \\ & \widehat\beta_2 \cdot 1 + \widehat\beta_3 \cdot 0 \end{aligned}\) | \(\widehat{\textrm{LE}} = \widehat\beta_0 + \widehat\beta_2\) |
| Europe | \(\begin{aligned} \widehat{\textrm{LE}} = &\widehat\beta_0 + \widehat\beta_1 \cdot 0 + \\ & \widehat\beta_2 \cdot 0 + \widehat\beta_3 \cdot 1 \end{aligned}\) | \(\widehat{\textrm{LE}} = \widehat\beta_0 + \widehat\beta_3\) |
Similar to as before:
Observed values \(y\) are the values in the dataset
Fitted values \(\widehat{y}\) are the values that fall on the best-fit line for a specific value of x are the means of the outcome stratified by the categorical predictor’s levels
Residuals \(y - \widehat{y}\) are the differences between the two
A few changes needed:
Make the income levels readable
ggplot(gapm2, aes(x = income_levels, y = LifeExpectancyYrs)) +
geom_jitter(size = 1, alpha = .6, width = 0.2) +
stat_summary(fun = mean, geom = "point", size = 8, shape = 18) +
labs(x = "Income levels",
y = "Country life expectancy (years)",
title = "Life expectancy vs. income levels",
caption = "Diamonds = Income level averages") +
theme(axis.title = element_text(size = 20),
axis.text = element_text(size = 20),
title = element_text(size = 20),
axis.text.x=element_text(angle = 20, vjust = 1, hjust=1))
Using visual assessment, linearity holds for our income levels
We can use the ordinal encoding for income levels
