Model comparison

STA 210 - Summer 2022

Author

Yunran Chen

Welcome

Announcements

HW 2 posted today, due Sunday
Lab 3 posted today, due Friday

Review on Exam 1 Part 1

Exam 1 Part 1

Interpretation on coefficients

Notes

Topics

ANOVA for Multiple Linear Regression and sum of squares
Comparing models with \(R^2\) vs. \(R^2_{adj}\)
Comparing models with AIC and BIC
Occam’s razor and parsimony
Overfitting and spending our data

Computational setup

# load packages
library(tidyverse)
library(tidymodels)
library(patchwork)
library(knitr)
library(kableExtra)

# set default theme and larger font size for ggplot2
ggplot2::theme_set(ggplot2::theme_minimal(base_size = 20))

Introduction

Data: Restaurant tips

Which variables help us predict the amount customers tip at a restaurant?

# A tibble: 169 × 4
     Tip Party Meal   Age   
   <dbl> <dbl> <chr>  <chr> 
 1  2.99     1 Dinner Yadult
 2  2        1 Dinner Yadult
 3  5        1 Dinner SenCit
 4  4        3 Dinner Middle
 5 10.3      2 Dinner SenCit
 6  4.85     2 Dinner Middle
 7  5        4 Dinner Yadult
 8  4        3 Dinner Middle
 9  5        2 Dinner Middle
10  1.58     1 Dinner SenCit
# … with 159 more rows

Variables

Predictors:

Party: Number of people in the party
Meal: Time of day (Lunch, Dinner, Late Night)
Age: Age category of person paying the bill (Yadult, Middle, SenCit)

Outcome: Tip: Amount of tip

Outcome: `Tip`

Predictors

Relevel categorical predictors

tips <- tips %>%
  mutate(
    Meal = fct_relevel(Meal, "Lunch", "Dinner", "Late Night"),
    Age  = fct_relevel(Age, "Yadult", "Middle", "SenCit")
  )

Predictors, again

Outcome vs. predictors

Fit and summarize model

tip_fit <- linear_reg() %>%
  set_engine("lm") %>%
  fit(Tip ~ Party + Age, data = tips)

tidy(tip_fit, conf.int = TRUE) %>%
  kable(digits = 3)

term	estimate	std.error	statistic	p.value	conf.low	conf.high
(Intercept)	-0.170	0.366	-0.465	0.643	-0.893	0.553
Party	1.837	0.124	14.758	0.000	1.591	2.083
AgeMiddle	1.009	0.408	2.475	0.014	0.204	1.813
AgeSenCit	1.388	0.485	2.862	0.005	0.430	2.345

Is this the best model to explain variation in tips?

Another model summary

term	df	sumsq	meansq	statistic	p.value
Party	1	1188.64	1188.64	285.71	0.00
Age	2	38.03	19.01	4.57	0.01
Residuals	165	686.44	4.16	NA	NA

Analysis of variance (ANOVA)

Main Idea: Decompose the total variation on the outcome into:
- the variation that can be explained by the each of the variables in the model
- the variation that can’t be explained by the model (left in the residuals)
If the variation that can be explained by the variables in the model is greater than the variation in the residuals, this signals that the model might be “valuable” (at least one of the \(\beta\)s not equal to 0)

ANOVA output

anova(tip_fit$fit) %>%
  tidy() %>%
  kable(digits = 2)

term	df	sumsq	meansq	statistic	p.value
Party	1	1188.64	1188.64	285.71	0.00
Age	2	38.03	19.01	4.57	0.01
Residuals	165	686.44	4.16	NA	NA

ANOVA output, with totals

term	df	sumsq	meansq	statistic	p.value
Party	1	1188.64	1188.64	285.71	0
Age	2	38.03	19.01	4.57	0.01
Residuals	165	686.44	4.16
Total	168	1913.11

Sum of squares

term	df	sumsq
Party	1	1188.64
Age	2	38.03
Residuals	165	686.44
Total	168	1913.11

\(SS_{Total}\): Total sum of squares, variability of outcome, \(\sum_{i = 1}^n (y_i - \bar{y})^2\)
\(SS_{Error}\): Residual sum of squares, variability of residuals, \(\sum_{i = 1}^n (y_i - \hat{y})^2\)
\(SS_{Model} = SS_{Total} - SS_{Error}\): Variability explained by the model

R-squared, \(R^2\)

Recall: \(R^2\) is the proportion of the variation in the response variable explained by the regression model.

\[ R^2 = \frac{SS_{Model}}{SS_{Total}} = 1 - \frac{SS_{Error}}{SS_{Total}} = 1 - \frac{686.44}{1913.11} = 0.641 \]

glance(tip_fit)

# A tibble: 1 × 12
  r.squared adj.r.squared sigma statistic  p.value    df logLik   AIC   BIC
      <dbl>         <dbl> <dbl>     <dbl>    <dbl> <dbl>  <dbl> <dbl> <dbl>
1     0.641         0.635  2.04      98.3 1.56e-36     3  -358.  726.  742.
# … with 3 more variables: deviance <dbl>, df.residual <int>, nobs <int>

Model comparison

R-squared, \(R^2\)

\(R^2\) will always increase as we add more variables to the model + If we add enough variables, we can always achieve \(R^2=100\%\)
If we only use \(R^2\) to choose a best fit model, we will be prone to choose the model with the most predictor variables

Adjusted \(R^2\)

Adjusted \(R^2\): measure that includes a penalty for unnecessary predictor variables
Similar to \(R^2\), it is a measure of the amount of variation in the response that is explained by the regression model
Differs from \(R^2\) by using the mean squares rather than sums of squares and therefore adjusting for the number of predictor variables

\(R^2\) and Adjusted \(R^2\)

\[R^2 = \frac{SS_{Model}}{SS_{Total}} = 1 - \frac{SS_{Error}}{SS_{Total}}\]

\[R^2_{adj} = 1 - \frac{SS_{Error}/(n-p-1)}{SS_{Total}/(n-1)}\]

Using \(R^2\) and Adjusted \(R^2\)

Adjusted \(R^2\) can be used as a quick assessment to compare the fit of multiple models; however, it should not be the only assessment!
Use \(R^2\) when describing the relationship between the response and predictor variables

Comparing models with \(R^2_{adj}\)

tip_fit_1 <- linear_reg() %>%
  set_engine("lm") %>%
  fit(Tip ~ Party + 
            Age + 
            Meal,
      
      data = tips)

glance(tip_fit_1) %>% 
  select(r.squared, adj.r.squared)

# A tibble: 1 × 2
  r.squared adj.r.squared
      <dbl>         <dbl>
1     0.674         0.664

tip_fit_2 <- linear_reg() %>%
  set_engine("lm") %>%
  fit(Tip ~ Party + 
            Age + 
            Meal + 
            Day, 
      data = tips)

glance(tip_fit_2) %>% 
  select(r.squared, adj.r.squared)

# A tibble: 1 × 2
  r.squared adj.r.squared
      <dbl>         <dbl>
1     0.683         0.662

AIC & BIC

Estimators of prediction error and relative quality of models:

Akaike’s Information Criterion (AIC): \[AIC = n\log(SS_\text{Error}) - n \log(n) + 2(p+1)\]

Schwarz’s Bayesian Information Criterion (BIC): \[BIC = n\log(SS_\text{Error}) - n\log(n) + log(n)\times(p+1)\]

AIC & BIC

\[ \begin{aligned} & AIC = \color{blue}{n\log(SS_\text{Error})} - n \log(n) + 2(p+1) \\ & BIC = \color{blue}{n\log(SS_\text{Error})} - n\log(n) + \log(n)\times(p+1) \end{aligned} \]

First Term: Decreases as p increases

AIC & BIC

\[ \begin{aligned} & AIC = n\log(SS_\text{Error}) - \color{blue}{n \log(n)} + 2(p+1) \\ & BIC = n\log(SS_\text{Error}) - \color{blue}{n\log(n)} + \log(n)\times(p+1) \end{aligned} \]

Second Term: Fixed for a given sample size n

AIC & BIC

\[ \begin{aligned} & AIC = n\log(SS_\text{Error}) - n\log(n) + \color{blue}{2(p+1)} \\ & BIC = n\log(SS_\text{Error}) - n\log(n) + \color{blue}{\log(n)\times(p+1)} \end{aligned} \]

Third Term: Increases as p increases

Using AIC & BIC

\[ \begin{aligned} & AIC = n\log(SS_{Error}) - n \log(n) + \color{red}{2(p+1)} \\ & BIC = n\log(SS_{Error}) - n\log(n) + \color{red}{\log(n)\times(p+1)} \end{aligned} \]

Choose model with the smaller value of AIC or BIC
If \(n \geq 8\), the penalty for BIC is larger than that of AIC, so BIC tends to favor more parsimonious models (i.e. models with fewer terms)

Comparing models with AIC and BIC

tip_fit_1 <- linear_reg() %>%
  set_engine("lm") %>%
  fit(Tip ~ Party + 
            Age + 
            Meal,
      
      data = tips)

glance(tip_fit_1) %>% 
  select(AIC, BIC)

# A tibble: 1 × 2
    AIC   BIC
  <dbl> <dbl>
1  714.  736.

tip_fit_2 <- linear_reg() %>%
  set_engine("lm") %>%
  fit(Tip ~ Party + 
            Age + 
            Meal + 
            Day, 
      data = tips)

glance(tip_fit_2) %>% 
  select(AIC, BIC)

# A tibble: 1 × 2
    AIC   BIC
  <dbl> <dbl>
1  720.  757.

Commonalities between criteria

\(R^2_{adj}\), AIC, and BIC all apply a penalty for more predictors
The penalty for added model complexity attempts to strike a balance between underfitting (too few predictors in the model) and overfitting (too many predictors in the model)
Goal: Parsimony

Parsimony and Occam’s razor

The principle of parsimony is attributed to William of Occam (early 14th-century English nominalist philosopher), who insisted that, given a set of equally good explanations for a given phenomenon, the correct explanation is the simplest explanation¹
Called Occam’s razor because he “shaved” his explanations down to the bare minimum
Parsimony in modeling:
- models should have as few parameters as possible
- linear models should be preferred to non-linear models
- experiments relying on few assumptions should be preferred to those relying on many
- models should be pared down until they are minimal adequate
- simple explanations should be preferred to complex explanations

In pursuit of Occam’s razor

Occam’s razor states that among competing hypotheses that predict equally well, the one with the fewest assumptions should be selected
Model selection follows this principle
We only want to add another variable to the model if the addition of that variable brings something valuable in terms of predictive power to the model
In other words, we prefer the simplest best model, i.e. parsimonious model

Alternate views

Sometimes a simple model will outperform a more complex model . . . Nevertheless, I believe that deliberately limiting the complexity of the model is not fruitful when the problem is evidently complex. Instead, if a simple model is found that outperforms some particular complex model, the appropriate response is to define a different complex model that captures whatever aspect of the problem led to the simple model performing well.

Radford Neal - Bayesian Learning for Neural Networks²

Other concerns with our approach

All criteria we considered for model comparison require making predictions for our data and then uses the prediction error (\(SS_{Error}\)) somewhere in the formula
But we’re making prediction for the data we used to build the model (estimate the coefficients), which can lead to overfitting
Instead we should
- split our data into testing and training sets
- “train” the model on the training data and pick a few models we’re genuinely considering as potentially good models
- test those models on the testing set

Recap

ANOVA for Multiple Linear Regression and sum of squares
Comparing models with \(R^2\) vs. \(R^2_{adj}\)
Comparing models with AIC and BIC
Occam’s razor and parsimony
Overfitting and spending our data

Footnotes

Source: The R Book by Michael J. Crawley.↩︎
Suggested blog post: Occam by Andrew Gelman↩︎

Welcome

Announcements

Review on Exam 1 Part 1

Interpretation on coefficients

Topics

Computational setup

Introduction

Data: Restaurant tips

Variables

Outcome: Tip

Predictors

Relevel categorical predictors

Predictors, again

Outcome vs. predictors

Fit and summarize model

Another model summary

Analysis of variance (ANOVA)

Analysis of variance (ANOVA)

ANOVA output

ANOVA output, with totals

Sum of squares

R-squared, \(R^2\)

Model comparison

R-squared, \(R^2\)

Adjusted \(R^2\)

\(R^2\) and Adjusted \(R^2\)

Using \(R^2\) and Adjusted \(R^2\)

Comparing models with \(R^2_{adj}\)

AIC & BIC

AIC & BIC

AIC & BIC

AIC & BIC

Using AIC & BIC

Comparing models with AIC and BIC

Commonalities between criteria

Parsimony and Occam’s razor

In pursuit of Occam’s razor

Alternate views

Other concerns with our approach

Recap

Footnotes

Outcome: `Tip`