# Chapter 4 Modeling Basics in `R`

**TODO:** Instead of specifically considering regression, change the focus of this chapter to modeling, with regression as an example.

This chapter will recap the basics of performing regression analyses in `R`

. For more detailed coverage, see Applied Statistics with `R`

.

We will use the Advertising data associated with Introduction to Statistical Learning.

After loading data into `R`

, our first step should **always** be to inspect the data. We will start by simply printing some observations in order to understand the basic structure of the data.

```
## # A tibble: 200 x 4
## TV Radio Newspaper Sales
## <dbl> <dbl> <dbl> <dbl>
## 1 230. 37.8 69.2 22.1
## 2 44.5 39.3 45.1 10.4
## 3 17.2 45.9 69.3 9.3
## 4 152. 41.3 58.5 18.5
## 5 181. 10.8 58.4 12.9
## 6 8.7 48.9 75 7.2
## 7 57.5 32.8 23.5 11.8
## 8 120. 19.6 11.6 13.2
## 9 8.6 2.1 1 4.8
## 10 200. 2.6 21.2 10.6
## # … with 190 more rows
```

Because the data was read using `read_csv()`

, `Advertising`

is a tibble. We see that there are a total of 200 observations and 4 variables, each of which is numeric. (Specifically double-precision vectors, but more importantly they are numbers.) For the purpose of this analysis, `Sales`

will be the **response variable**. That is, we seek to understand the relationship between `Sales`

, and the **predictor variables**: `TV`

, `Radio`

, and `Newspaper`

.

## 4.1 Visualization for Regression

After investigating the structure of the data, the next step should be to visualize the data. Since we have only numeric variables, we should consider **scatter plots**.

We could do so for any individual predictor.

```
plot(Sales ~ TV, data = Advertising, col = "dodgerblue", pch = 20, cex = 1.5,
main = "Sales vs Television Advertising")
```

The `pairs()`

function is a useful way to quickly visualize a number of scatter plots.

Often, we will be most interested in only the relationship between each predictor and the response. For this, we can use the `featurePlot()`

function from the `caret`

package. (We will use the `caret`

package more and more frequently as we introduce new topics.)

```
library(caret)
featurePlot(x = Advertising[ , c("TV", "Radio", "Newspaper")], y = Advertising$Sales)
```

We see that there is a clear increase in `Sales`

as `Radio`

or `TV`

are increased. The relationship between `Sales`

and `Newspaper`

is less clear. How all of the predictors work together is also unclear, as there is some obvious correlation between `Radio`

and `TV`

. To investigate further, we will need to model the data.

## 4.2 The `lm()`

Function

The following code fits an additive **linear model** with `Sales`

as the response and each remaining variable as a predictor. Note, by not using `attach()`

and instead specifying the `data =`

argument, we are able to specify this model without using each of the variable names directly.

```
mod_1 = lm(Sales ~ ., data = Advertising)
# mod_1 = lm(Sales ~ TV + Radio + Newspaper, data = Advertising)
```

Note that the commented line is equivalent to the line that is run, but we will often use the `response ~ .`

syntax when possible.

## 4.3 Hypothesis Testing

The `summary()`

function will return a large amount of useful information about a model fit using `lm()`

. Much of it will be helpful for hypothesis testing including individual tests about each predictor, as well as the significance of the regression test.

```
##
## Call:
## lm(formula = Sales ~ ., data = Advertising)
##
## Residuals:
## Min 1Q Median 3Q Max
## -8.8277 -0.8908 0.2418 1.1893 2.8292
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 2.938889 0.311908 9.422 <2e-16 ***
## TV 0.045765 0.001395 32.809 <2e-16 ***
## Radio 0.188530 0.008611 21.893 <2e-16 ***
## Newspaper -0.001037 0.005871 -0.177 0.86
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.686 on 196 degrees of freedom
## Multiple R-squared: 0.8972, Adjusted R-squared: 0.8956
## F-statistic: 570.3 on 3 and 196 DF, p-value: < 2.2e-16
```

The `anova()`

function is useful for comparing two models. Here we compare the full additive model, `mod_1`

, to a reduced model `mod_0`

. Essentially we are testing for the significance of the `Newspaper`

variable in the additive model.

```
## Analysis of Variance Table
##
## Model 1: Sales ~ TV + Radio
## Model 2: Sales ~ TV + Radio + Newspaper
## Res.Df RSS Df Sum of Sq F Pr(>F)
## 1 197 556.91
## 2 196 556.83 1 0.088717 0.0312 0.8599
```

Note that hypothesis testing is *not* our focus, so we omit many details.

## 4.4 Prediction

The `predict()`

function is an extremely versatile function, for, prediction. When used on the result of a model fit using `lm()`

it will, by default, return predictions for each of the data points used to fit the model. (Here, we limit the printed result to the first 10.)

```
## 1 2 3 4 5 6 7 8
## 20.523974 12.337855 12.307671 17.597830 13.188672 12.478348 11.729760 12.122953
## 9 10
## 3.727341 12.550849
```

Note that the effect of the `predict()`

function is dependent on the input to the function. Here, we are supplying as the first argument a model object of class `lm`

. Because of this, `predict()`

then runs the `predict.lm()`

function. Thus, we should use `?predict.lm()`

for details.

We could also specify new data, which should be a data frame or tibble with the same column names as the predictors.

We can then use the `predict()`

function for point estimates, confidence intervals, and prediction intervals.

Using only the first two arguments, `R`

will simply return a point estimate, that is, the “predicted value,” \(\hat{y}\).

```
## 1
## 17.34375
```

If we specify an additional argument `interval`

with a value of `"confidence"`

, `R`

will return a 95% confidence interval for the mean response at the specified point. Note that here `R`

also gives the point estimate as `fit`

.

```
## fit lwr upr
## 1 17.34375 16.77654 17.91096
```

Lastly, we can alter the level using the `level`

argument. Here we report a prediction interval instead of a confidence interval.

```
## fit lwr upr
## 1 17.34375 12.89612 21.79138
```

## 4.5 Unusual Observations

`R`

provides several functions for obtaining metrics related to unusual observations.

`resid()`

provides the residual for each observation`hatvalues()`

gives the leverage of each observation`rstudent()`

give the studentized residual for each observation`cooks.distance()`

calculates the influence of each observation

```
## 1 2 3 4 5 6
## 1.57602559 -1.93785482 -3.00767078 0.90217049 -0.28867186 -5.27834763
## 7 8 9 10
## 0.07024005 1.07704683 1.07265914 -1.95084872
```

```
## 1 2 3 4 5 6
## 0.025202848 0.019418228 0.039226158 0.016609666 0.023508833 0.047481074
## 7 8 9 10
## 0.014435091 0.009184456 0.030714427 0.017147645
```

```
## 1 2 3 4 5 6
## 0.94680369 -1.16207937 -1.83138947 0.53877383 -0.17288663 -3.28803309
## 7 8 9 10
## 0.04186991 0.64099269 0.64544184 -1.16856434
```

```
## 1 2 3 4 5 6
## 5.797287e-03 6.673622e-03 3.382760e-02 1.230165e-03 1.807925e-04 1.283058e-01
## 7 8 9 10
## 6.452021e-06 9.550237e-04 3.310088e-03 5.945006e-03
```

## 4.6 Adding Complexity

We have a number of ways to add complexity to a linear model, even allowing a linear model to be used to model non-linear relationships.

### 4.6.1 Interactions

Interactions can be introduced to the `lm()`

procedure in a number of ways.

We can use the `:`

operator to introduce a single interaction of interest.

```
## (Intercept) TV Radio Newspaper TV:Newspaper
## 3.8730824491 0.0392939602 0.1901312252 -0.0320449675 0.0002016962
```

The `response ~ . ^ k`

syntax can be used to model all `k`

-way interactions. (As well as the appropriate lower order terms.) Here we fit a model with all two-way interactions, and the lower order main effects.

```
## (Intercept) TV Radio Newspaper TV:Radio
## 6.460158e+00 2.032710e-02 2.292919e-02 1.703394e-02 1.139280e-03
## TV:Newspaper Radio:Newspaper
## -7.971435e-05 -1.095976e-04
```

The `*`

operator can be used to specify all interactions of a certain order, as well as all lower order terms according to the usual hierarchy. Here we see a three-way interaction and all lower order terms.

```
## (Intercept) TV Radio Newspaper
## 6.555887e+00 1.971030e-02 1.962160e-02 1.310565e-02
## TV:Radio TV:Newspaper Radio:Newspaper TV:Radio:Newspaper
## 1.161523e-03 -5.545501e-05 9.062944e-06 -7.609955e-07
```

Note that, we have only been dealing with numeric predictors. **Categorical predictors** are often recorded as **factor** variables in `R`

.

```
library(tibble)
cat_pred = tibble(
x1 = factor(c(rep("A", 10), rep("B", 10), rep("C", 10))),
x2 = runif(n = 30),
y = rnorm(n = 30)
)
cat_pred
```

```
## # A tibble: 30 x 3
## x1 x2 y
## <fct> <dbl> <dbl>
## 1 A 0.262 -0.0343
## 2 A 0.115 -0.0178
## 3 A 0.399 0.461
## 4 A 0.160 0.0463
## 5 A 0.816 -0.0817
## 6 A 0.515 -0.646
## 7 A 0.162 0.981
## 8 A 0.432 0.753
## 9 A 0.0166 -1.03
## 10 A 0.0274 1.72
## # … with 20 more rows
```

Notice that in this simple simulated tibble, we have coerced `x1`

to be a factor variable, although this is not strictly necessary since the variable took values `A`

, `B`

, and `C`

. When using `lm()`

, even if not a factor, `R`

would have treated `x1`

as such. Coercion to factor is more important if a categorical variable is coded for example as `1`

, `2`

and `3`

. Otherwise it is treated as numeric, which creates a difference in the regression model.

The following two models illustrate the effect of factor variables on linear models.

```
## (Intercept) x1B x1C x2
## 0.35326468 0.22856423 0.05949584 -0.47329142
```

```
## (Intercept) x1B x1C x2 x1B:x2 x1C:x2
## 0.38520381 0.07113332 0.20209042 -0.58327064 0.35868402 -0.19520729
```

### 4.6.2 Polynomials

Polynomial terms can be specified using the inhibit function `I()`

or through the `poly()`

function. Note that these two methods produce different coefficients, but the same residuals! This is due to the `poly()`

function using orthogonal polynomials by default.

```
## (Intercept) TV I(TV^2)
## 6.114120e+00 6.726593e-02 -6.846934e-05
```

```
## (Intercept) poly(TV, degree = 2)1 poly(TV, degree = 2)2
## 14.022500 57.572721 -6.228802
```

`## [1] TRUE`

Polynomials and interactions can be mixed to create even more complex models.

```
mod_7 = lm(Sales ~ . ^ 2 + poly(TV, degree = 3), data = Advertising)
# mod_7 = lm(Sales ~ . ^ 2 + I(TV ^ 2) + I(TV ^ 3), data = Advertising)
coef(mod_7)
```

```
## (Intercept) TV Radio
## 6.206394e+00 2.092726e-02 3.766579e-02
## Newspaper poly(TV, degree = 3)1 poly(TV, degree = 3)2
## 1.405289e-02 NA -9.925605e+00
## poly(TV, degree = 3)3 TV:Radio TV:Newspaper
## 5.309590e+00 1.082074e-03 -5.690107e-05
## Radio:Newspaper
## -9.924992e-05
```

Notice here that `R`

ignores the first order term from `poly(TV, degree = 3)`

as it is already in the model. We could consider using the commented line instead.

### 4.6.3 Transformations

Note that we could also create more complex models, which allow for non-linearity, using transformations. Be aware, when doing so to the response variable, that this will affect the units of said variable. You may need to un-transform to compare to non-transformed models.

```
mod_8 = lm(log(Sales) ~ ., data = Advertising)
sqrt(mean(resid(mod_8) ^ 2)) # incorrect RMSE for Model 8
```

`## [1] 0.1849483`

`## [1] 0.4813215`

`## [1] 1.023205`

## 4.7 `rmarkdown`

The `rmarkdown`

file for this chapter can be found **here**. The file was created using `R`

version 4.0.2. The following packages (and their dependencies) were loaded in this file:

`## [1] "tibble" "caret" "ggplot2" "lattice" "readr"`