30.3.1 A single-variable model for the Mario Kart data

Let’s fit a linear regression model with the game’s condition as a predictor of auction price. Before we start let’s change cond and stock_photo into factors.

mariokart <- mariokart %>%
  mutate(cond=factor(cond),stock_photo=factor(stock_photo))

Next let’s summarize the data.

inspect(mariokart)

## 
## categorical variables:  
##          name     class levels   n missing
## 1          id character    143 143       0
## 2        cond    factor      2 143       0
## 3     ship_sp character      8 143       0
## 4 stock_photo    factor      2 143       0
## 5       title character     80 142       1
##                                    distribution
## 1 110439174663 (0.7%) ...                      
## 2 used (58.7%), new (41.3%)                    
## 3 standard (23.1%), upsGround (21.7%) ...      
## 4 yes (73.4%), no (26.6%)                      
## 5  (%) ...                                     
## 
## quantitative variables:  
##             name   class   min      Q1 median      Q3       max         mean
## ...1    duration numeric  1.00   1.000    3.0    7.00     10.00     3.769231
## ...2      n_bids numeric  1.00  10.000   14.0   17.00     29.00    13.538462
## ...3    start_pr numeric  0.01   0.990    1.0   10.00     69.95     8.777203
## ...4     ship_pr numeric  0.00   0.000    3.0    4.00     25.51     3.143706
## ...5    total_pr numeric 28.98  41.175   46.5   53.99    326.51    49.880490
## ...6 seller_rate numeric  0.00 109.000  820.0 4858.00 270144.00 15898.419580
## ...7      wheels numeric  0.00   0.000    1.0    2.00      4.00     1.146853
##                sd   n missing
## ...1 2.585693e+00 143       0
## ...2 5.878786e+00 143       0
## ...3 1.506745e+01 143       0
## ...4 3.213179e+00 143       0
## ...5 2.568856e+01 143       0
## ...6 5.184032e+04 143       0
## ...7 8.471829e-01 143       0

Finally, let’s plot the data.

mariokart %>% 
  gf_boxplot(total_pr~cond) %>%
  gf_theme(theme_bw()) %>%
  gf_labs(title="Ebay Auction Prices",x="Condition", y="Total Price")

Figure 30.1: Total price of Mario Kart on Ebay for each condition.

We have several outliers that may impact our analysis, Figure 30.1.

Now let’s build the model.

mario_mod <- lm(total_pr~cond,data=mariokart)

summary(mario_mod)

## 
## Call:
## lm(formula = total_pr ~ cond, data = mariokart)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -18.168  -7.771  -3.148   1.857 279.362 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   53.771      3.329  16.153   <2e-16 ***
## condused      -6.623      4.343  -1.525     0.13    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 25.57 on 141 degrees of freedom
## Multiple R-squared:  0.01622,    Adjusted R-squared:  0.009244 
## F-statistic: 2.325 on 1 and 141 DF,  p-value: 0.1296

The model may be written as

\[ \hat{\text{totalprice}} = 53.771 - 6.623 \times \text{condused} \]

A scatterplot for price versus game condition is shown in Figure 30.2. Since the predictor is binary, the scatterplot is not appropriate but we will look at it for reference.

mariokart %>%
  gf_point(total_pr~cond) %>%
  gf_theme(theme_classic()) %>%
  gf_labs(title="Ebay Auction Prices",x="Condition", y="Total Price")

Figure 30.2: Scatterplot of total price of Mario Kart on Ebay versus condition.

The largest outlier probably is significantly impacting the relationship in the model. If we find the mean and median for the two groups, we will see this.

mariokart %>%
  group_by(cond) %>%
  summarize(xbar=mean(total_pr), stand_dev=sd(total_pr),xmedian=median(total_pr))

## # A tibble: 2 × 4
##   cond   xbar stand_dev xmedian
##   <fct> <dbl>     <dbl>   <dbl>
## 1 new    53.8      7.44    54.0
## 2 used   47.1     32.7     42.8

It appears that used items have a right skewed distribution where their average is higher because of at least one of the outliers.

There are at least two outliers in the plot. Let’s gather more information about them.

mariokart %>%
  filter(total_pr > 100)

## # A tibble: 2 × 12
##   id         duration n_bids cond  start_pr ship_pr total_pr ship_sp seller_rate
##   <chr>         <dbl>  <dbl> <fct>    <dbl>   <dbl>    <dbl> <chr>         <dbl>
## 1 110439174…        7     22 used      1       25.5     327. parcel          115
## 2 130335427…        3     27 used      6.95     4       118. parcel           41
## # … with 3 more variables: stock_photo <fct>, wheels <dbl>, title <chr>

If you look at the variable title there were additional items in the sale for these two observations. Let’s remove those two outliers and run the model again. Note that the reason we are removing them is not because they are annoying us and messing up our model. It is because we don’t think they are representative of the population of interest. Figure 30.3 is a boxplot of the data with the outliers dropped.

mariokart_new <- mariokart %>%
  filter(total_pr <= 100) %>% 
  select(total_pr,cond,stock_photo,duration,wheels)

summary(mariokart_new)

##     total_pr       cond    stock_photo    duration          wheels     
##  Min.   :28.98   new :59   no : 36     Min.   : 1.000   Min.   :0.000  
##  1st Qu.:41.00   used:82   yes:105     1st Qu.: 1.000   1st Qu.:0.000  
##  Median :46.03                         Median : 3.000   Median :1.000  
##  Mean   :47.43                         Mean   : 3.752   Mean   :1.149  
##  3rd Qu.:53.99                         3rd Qu.: 7.000   3rd Qu.:2.000  
##  Max.   :75.00                         Max.   :10.000   Max.   :4.000

mariokart_new %>% 
  gf_boxplot(total_pr~cond) %>%
  gf_theme(theme_bw()) %>%
  gf_labs(title="Ebay Auction Prices",subtitle="Outliers removed",x="Condition", y="Total Price")

Figure 30.3: Boxplot of total price and condition with outliers removed.

mario_mod2 <- lm(total_pr~cond,data=mariokart_new)

summary(mario_mod2)

## 
## Call:
## lm(formula = total_pr ~ cond, data = mariokart_new)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -13.8911  -5.8311   0.1289   4.1289  22.1489 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  53.7707     0.9596  56.034  < 2e-16 ***
## condused    -10.8996     1.2583  -8.662 1.06e-14 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 7.371 on 139 degrees of freedom
## Multiple R-squared:  0.3506, Adjusted R-squared:  0.3459 
## F-statistic: 75.03 on 1 and 139 DF,  p-value: 1.056e-14

Notice how much the residual standard error has decreased and likewise the \(R\)-squared has increased.

The model may be written as:

\[ \hat{total price} = 53.771 - 10.90 \times condused \]

Now we see that the average price for a used items is $10.90 less than the average of new items.

Exercise:
Does the linear model seem reasonable? Which assumptions should you check?

The model does seem reasonable in the sense that the assumptions on the errors is plausible. The residuals indicate some skewness to the right which may be driven predominantly by the skewness in the new items, Figure 30.4.

plot(mario_mod2,2)

Figure 30.4: Check of normality using quantile-quantile plot.

The normality assumption is somewhat suspect but we have more than 100 data points so the short tails of the distribution are not a concern. The shape of this curve indicates a positive skew.

plot(mario_mod2,3)

Figure 30.5: Residual plot to assess equal variance assumption.

From Figure 30.5, equal variance seems reasonable.

plot(mario_mod2,5)

Figure 30.6: Residual plot for checking leverage points.

No high leverage points, Figure 30.6.

No need to check linearity, we only have two different values for the explanatory variable.

Example: Interpretation
Interpret the coefficient for the game’s condition in the model. Is this coefficient significantly different from 0?

Note that cond is a two-level categorical variable and the reference level is new. So - 10.90 means that the model predicts an extra $10.90 on average for those games that are new versus those that are used. Examining the regression output, we can see that the p-value for cond is very close to zero, indicating there is strong evidence that the coefficient is different from zero when using this simple one-variable model.

30.3.2 Including and assessing many variables in a model

Sometimes there are underlying structures or relationships between predictor variables. For instance, new games sold on Ebay tend to come with more Wii wheels, which may have led to higher prices for those auctions. We would like to fit a model that includes all potentially important variables simultaneously. This would help us evaluate the relationship between a predictor variable and the outcome while controlling for the potential influence of other variables. This is the strategy used in multiple regression. While we remain cautious about making any causal interpretations using multiple regression, such models are a common first step in providing evidence of a causal connection.

We want to construct a model that accounts for not only the game condition, but simultaneously accounts for three other variables: stock_photo, duration, and wheels. This model can be represented as:

\[ \widehat{\text{totalprice}} = \beta_0 + \beta_1 \times \text{cond} + \beta_2 \times \text{stockphoto} + \beta_3 \times \text{duration} + \beta_4 \times \text{wheels} \]

or:

\[\begin{equation} \hat{y} = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \beta_3 x_3 + \beta_4 x_4 \tag{30.1} \end{equation}\]

In Equation (30.1), \(y\) represents the total price, \(x_1\) indicates whether the game is new, \(x_2\) indicates whether a stock photo was used, \(x_3\) is the duration of the auction, and \(x_4\) is the number of Wii wheels included with the game. Just as with the single predictor case, a multiple regression model may be missing important components or it might not precisely represent the relationship between the outcome and the available explanatory variables. While no model is perfect, we wish to explore the possibility that this model may fit the data reasonably well.

We estimate the parameters \(\beta_0\), \(\beta_1\), …, \(\beta_4\) in the same way as we did in the case of a single predictor. We select \(b_0\), \(b_1\), …, \(b_4\) that minimize the sum of the squared residuals:

\[ \text{SSE} = e_1^2 + e_2^2 + \dots + e_{141}^2 = \sum_{i=1}^{141} e_i^2 = \sum_{i=1}^{141} \left(y_i - \hat{y}_i\right)^2 \]

In our problem, there are 141 residuals, one for each observation. We use a computer to minimize the sum and compute point estimates.

mario_mod_multi <- lm(total_pr~., data=mariokart_new)

The formula total_pr~. uses a dot. This means we want to use all the predictors. We could have also used the following code:

mario_mod_multi <- lm(total_pr~cond+stock_photo+duration+wheels, data=mariokart_new)

Recall, the + symbol does not mean to literally add the predictors together. It is not a mathematical operation but a formula operation that means to include the predictor.

You can view a summary of the model using the summmary() function.

summary(mario_mod_multi)

## 
## Call:
## lm(formula = total_pr ~ ., data = mariokart_new)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -11.3788  -2.9854  -0.9654   2.6915  14.0346 
## 
## Coefficients:
##                Estimate Std. Error t value Pr(>|t|)    
## (Intercept)    41.34153    1.71167  24.153  < 2e-16 ***
## condused       -5.13056    1.05112  -4.881 2.91e-06 ***
## stock_photoyes  1.08031    1.05682   1.022    0.308    
## duration       -0.02681    0.19041  -0.141    0.888    
## wheels          7.28518    0.55469  13.134  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 4.901 on 136 degrees of freedom
## Multiple R-squared:  0.719,  Adjusted R-squared:  0.7108 
## F-statistic: 87.01 on 4 and 136 DF,  p-value: < 2.2e-16

Which we can summarize in a tibble using the broom package.

Using this output, Table 30.1, we identify the point estimates \(b_i\) of each \(\beta_i\), just as we did in the one-predictor case.

Multiple regression model
A multiple regression model is a linear model with many predictors.

In general, we write the model as

\[ \hat{y} = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \cdots + \beta_k x_k %+ \epsilon \]

when there are \(k\) predictors. We often estimate the \(\beta_i\) parameters using a computer.

Exercise: Write out the the multiple regression model using the point estimates from regression output. How many predictors are there in this model?⁹⁷

Exercise:
What does \(\beta_4\), the coefficient of variable \(x_4\) (Wii wheels), represent? What is the point estimate of \(\beta_4\)?⁹⁸

Exercise:
Compute the residual of the first observation in the dataframe using the regression equation.

mario_mod_multi$residuals[1]

##        1 
## 1.923402

The broom package has a function augment() that will calculate the predicted and residuals.

library(broom)

augment(mario_mod_multi) %>%
  head(1)

## # A tibble: 1 × 11
##   total_pr cond  stock_photo duration wheels .fitted .resid   .hat .sigma
##      <dbl> <fct> <fct>          <dbl>  <dbl>   <dbl>  <dbl>  <dbl>  <dbl>
## 1     51.6 new   yes                3      1    49.6   1.92 0.0215   4.92
## # … with 2 more variables: .cooksd <dbl>, .std.resid <dbl>

\(e_i = y_i - \hat{y_i} = 51.55 - 49.62 = 1.93\)

Example:
We estimated a coefficient for cond as \(b_1 = - 10.90\) with a standard error of \(SE_{b_1} = 1.26\) when using simple linear regression. Why might there be a difference between that estimate and the one in the multiple regression setting?

If we examined the data carefully, we would see that some predictors are correlated. For instance, when we estimated the connection of the outcome total_pr and predictor cond using simple linear regression, we were unable to control for other variables like the number of Wii wheels included in the auction. That model was biased by the confounding variable wheels. When we use both variables, this particular underlying and unintentional bias is reduced or eliminated (though bias from other confounding variables may still remain).

The previous example describes a common issue in multiple regression: correlation among predictor variables. We say the two predictor variables are collinear (pronounced as co-linear) when they are correlated, and this collinearity complicates model estimation. While it is impossible to prevent collinearity from arising in observational data, experiments are usually designed to prevent predictors from being collinear.

Exercise:
The estimated value of the intercept is 41.34, and one might be tempted to make some interpretation of this coefficient, such as, it is the model’s predicted price when each of the variables take a value of zero: the game is new, the primary image is not a stock photo, the auction duration is zero days, and there are no wheels included. Is there any value gained by making this interpretation?⁹⁹

30.3.3 Inference

From the printout of the model summary, we can see that both the stock_photo and duration variables are not significantly different from zero. Thus we may want to drop them from the model. In a machine learning course, you explore different ways to determine the best model.

Likewise, we could generate confidence intervals for the coefficients:

confint(mario_mod_multi)

##                     2.5 %     97.5 %
## (Intercept)    37.9566036 44.7264601
## condused       -7.2092253 -3.0519030
## stock_photoyes -1.0096225  3.1702442
## duration       -0.4033592  0.3497442
## wheels          6.1882392  8.3821165

This confirms that the stock_photo and duration may not have an impact on total price.

30.3.4 Adjusted \(R^2\) as a better estimate of explained variance

We first used \(R^2\) in simple linear regression to determine the amount of variability, we used sum of squares and not mean squared errors, in the response that was explained by the model: \[ R^2 = 1 - \frac{\text{sum of squares of residuals}}{\text{sum of squares of the outcome}} \] This equation remains valid in the multiple regression framework, but a small enhancement can often be even more informative.

Exercise: The variance of the residuals for the model is \(4.901^2\), and the variance of the total price in all the auctions is 83.06. Estimate the \(R^2\) for this model.¹⁰⁰

To get the \(R^2\) we need the sum of squares and not variance, so we multiply by the appropriate degrees of freedom.

1-(24.0198*136)/(83.05864*140)

## [1] 0.7190717

summary(mario_mod_multi)$r.squared

## [1] 0.7190261

This strategy for estimating \(R^2\) is acceptable when there is just a single variable. However, it becomes less helpful when there are many variables. The regular \(R^2\) is actually a biased estimate of the amount of variability explained by the model. To get a better estimate, we use the adjusted \(R^2\).

Adjusted \(\mathbf{R^2}\) as a tool for model assessment:
The adjusted \(\mathbf{R^2}\) is computed as: \[ R_{adj}^{2} = 1-\frac{\text{sum of squares of residuals} / (n-k-1)}{\text{sum of squares of the outcome} / (n-1)} \] where \(n\) is the number of cases used to fit the model and \(k\) is the number of predictor variables in the model.

Because \(k\) is never negative, the adjusted \(R^2\) will be smaller – often times just a little smaller – than the unadjusted \(R^2\). The reasoning behind the adjusted \(R^2\) lies in the degrees of freedom associated with each variance.¹⁰¹

Exercise:
Suppose you added another predictor to the model, but the variance of the errors didn’t go down. What would happen to the \(R^2\)? What would happen to the adjusted \(R^2\)?¹⁰²

Again, in a machine learning course, you will spend more time on how to select models. Using internal metrics of performance such as p-values or adjusted \(R\) squared are one way but using external measures of predictive performance such as cross validation or hold out sets will be introduced.

30.3.5 Reduced model

Now let’s drop duration from the model and compare to our previous model:

mario_mod_multi2 <- lm(total_pr~cond+stock_photo+wheels, data=mariokart_new)

And the summary:

summary(mario_mod_multi2)

## 
## Call:
## lm(formula = total_pr ~ cond + stock_photo + wheels, data = mariokart_new)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -11.454  -2.959  -0.949   2.712  14.061 
## 
## Coefficients:
##                Estimate Std. Error t value Pr(>|t|)    
## (Intercept)     41.2245     1.4911  27.648  < 2e-16 ***
## condused        -5.1763     0.9961  -5.196 7.21e-07 ***
## stock_photoyes   1.1177     1.0192   1.097    0.275    
## wheels           7.2984     0.5448  13.397  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 4.884 on 137 degrees of freedom
## Multiple R-squared:  0.719,  Adjusted R-squared:  0.7128 
## F-statistic: 116.8 on 3 and 137 DF,  p-value: < 2.2e-16

As a reminder, the previous model summary is:

summary(mario_mod_multi)

## 
## Call:
## lm(formula = total_pr ~ ., data = mariokart_new)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -11.3788  -2.9854  -0.9654   2.6915  14.0346 
## 
## Coefficients:
##                Estimate Std. Error t value Pr(>|t|)    
## (Intercept)    41.34153    1.71167  24.153  < 2e-16 ***
## condused       -5.13056    1.05112  -4.881 2.91e-06 ***
## stock_photoyes  1.08031    1.05682   1.022    0.308    
## duration       -0.02681    0.19041  -0.141    0.888    
## wheels          7.28518    0.55469  13.134  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 4.901 on 136 degrees of freedom
## Multiple R-squared:  0.719,  Adjusted R-squared:  0.7108 
## F-statistic: 87.01 on 4 and 136 DF,  p-value: < 2.2e-16

Notice that the adjusted \(R^2\) improved by dropping duration. Finally, let’s drop stock_photo.

mario_mod_multi3 <- lm(total_pr~cond+wheels, data=mariokart_new)

summary(mario_mod_multi3)

## 
## Call:
## lm(formula = total_pr ~ cond + wheels, data = mariokart_new)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -11.0078  -3.0754  -0.8254   2.9822  14.1646 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  42.3698     1.0651  39.780  < 2e-16 ***
## condused     -5.5848     0.9245  -6.041 1.35e-08 ***
## wheels        7.2328     0.5419  13.347  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 4.887 on 138 degrees of freedom
## Multiple R-squared:  0.7165, Adjusted R-squared:  0.7124 
## F-statistic: 174.4 on 2 and 138 DF,  p-value: < 2.2e-16

Though the adjusted \(R^2\) dropped a little, it is only in the fourth decimal place and thus essentially the same value. We therefore will go with this model.

30.3.6 Confidence and prediction intervals

Let’s suppose we want to predict the average total price for a Mario Kart sale with 2 wheels and in new condition. We can again use the predict() function.

predict(mario_mod_multi3,newdata=data.frame(cond="new",wheels=2),interval = "confidence")

##        fit      lwr      upr
## 1 56.83544 55.49789 58.17299

We are 95% confident that the average price of a Mario Kart sale for a new item with 2 wheels will be between 55.50 and 58.17.

Exercise: Find and interpret the prediction interval for a new Mario Kart with 2 wheels.

predict(mario_mod_multi3,newdata=data.frame(cond="new",wheels=2),interval = "prediction")

##        fit      lwr      upr
## 1 56.83544 47.07941 66.59147

We are 95% confident that the price of a Mario Kart sale for a new item with 2 wheels will be between 47.07 and 66.59.

30.3.7 Diagnostics

The diagnostics for the model are similar to what we did in a previous lesson. Nothing in these plots gives us concern; however, there is one leverage point, Figure 30.7.

plot(mario_mod_multi3)

Figure 30.7: Diagnostic residual plots for multiple regression model.

30.4.1 Summary

In this chapter we have extended the linear regression model by allowing multiple predictors. This allows us to account for confounding variables and make more sophisticated models. The interpretation and evaluation of the model changes.