Linear Regression and ANOVA shaken and stirred (Part 2)

In the first part of this entry I did show some mtcars examples, where am can be useful to explain ANOVA as its observations are defined as:

$$ am_i = \begin{cases}1 &\text{ if car } i \text{ is manual} \cr 0 &\text{ if car } i \text{ is automatic}\end{cases} $$

Now I'll show another example to continue the last example from Part 1 and I'll move to something involved more variables.

ANOVA with one dummy variable

Consider a model where the outcome is mpg and the design matrix is \(\renewcommand{\vec}[1]{\boldsymbol{#1}} \newcommand{\R}{\mathbb{R}} X = (\vec{1} \: \vec{x}_2)\).

From the last entry let

$$ x_2 = \begin{cases}1 &\text{ if car } i \text{ is automatic} \cr 0 &\text{ if car } i \text{ is manual}\end{cases} $$

This will lead to this estimate:

$$ \hat{\vec{\beta}} = \begin{bmatrix}\bar{y}_1 \cr \bar{y}_2 - \bar{y}_1\end{bmatrix} $$

Fitting the model gives:

y = mtcars$mpg
x1 = mtcars$am
x2 = ifelse(x1 == 1, 0, 1)

fit = lm(y ~ x2)
summary(fit)
Call:
lm(formula = y ~ x2)

Residuals:
    Min      1Q  Median      3Q     Max 
-9.3923 -3.0923 -0.2974  3.2439  9.5077 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)   24.392      1.360  17.941  < 2e-16 ***
x2            -7.245      1.764  -4.106 0.000285 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 4.902 on 30 degrees of freedom
Multiple R-squared:  0.3598,    Adjusted R-squared:  0.3385 
F-statistic: 16.86 on 1 and 30 DF,  p-value: 0.000285

So to see the relationship between the estimates and the group means I need additional steps:

x0 = rep(1,length(y))
X = cbind(x0,x2)
beta = solve(t(X)%*%X) %*% (t(X)%*%y)
beta
        [,1]
x0 24.392308
x2 -7.244939

I did obtain the same estimates with lm command so now I calculate the group means:

x1 = ifelse(x1 == 0, NA, x1)
x2 = ifelse(x2 == 0, NA, x2)

m1 = mean(y*x1, na.rm = TRUE)
m2 = mean(y*x2, na.rm = TRUE)

beta0 = m1
beta2 = m2-m1

beta0;beta2
[1] 24.39231
[1] -7.244939

In this case this means that the slope for the two groups is the same but the intercept is different, and therefore exists a negative effect of automatic transmission on miles per gallon in average terms.

Again I'll verify the equivalency between lm and aov in this particular case:

y = mtcars$mpg
x1 = mtcars$am
x2 = ifelse(x1 == 1, 0, 1)

fit2 = aov(y ~ x2)
fit2$coefficients
(Intercept)          x2 
  24.392308   -7.244939 

I can calculate the residuals by hand:

mean_mpg = mean(mtcars$mpg)
fitted_mpg = fit3$coefficients[1] + fit3$coefficients[2]*mtcars$am
observed_mpg = mtcars$mpg

TSS = sum((observed_mpg - mean_mpg)^2) 
ESS = sum((fitted_mpg - mean_mpg)^2)
RSS = sum((observed_mpg - fitted_mpg)^2)

TSS;ESS;RSS
[1] 1126.047
[1] 405.1506
[1] 720.8966

Here its verified that \(TSS = ESS + RSS\) but aside from that I can extract information from aov:

summary(fit2)
            Df Sum Sq Mean Sq F value   Pr(>F)    
x2           1  405.2   405.2   16.86 0.000285 ***
Residuals   30  720.9    24.0                     
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

And check that, as expected, \(ESS\) is the variance explained by x2.

I also can run ANOVA over lm with:

anova(fit)
Analysis of Variance Table

Response: y
          Df Sum Sq Mean Sq F value   Pr(>F)    
x2         1 405.15  405.15   16.86 0.000285 ***
Residuals 30 720.90   24.03                     
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

The table provides information on the effect of x2 over y. In this case the null hypothesis is rejected because of the large F-value and the associated p-values.

Considering a 0.05 significance threshold I can say, with 95% of confidence, that the regression slope is statistically different from zero or that there is a difference in group means between automatic and manual transmission.

ANOVA with three dummy variables

Now let's explore something more complex than am. Reading the documentation I wonder if cyl has an impact on mpg so I explore that variable:

str(mtcars)
'data.frame':   32 obs. of  11 variables:
 $ mpg : num  21 21 22.8 21.4 18.7 18.1 14.3 24.4 22.8 19.2 ...
 $ cyl : num  6 6 4 6 8 6 8 4 4 6 ...
 $ disp: num  160 160 108 258 360 ...
 $ hp  : num  110 110 93 110 175 105 245 62 95 123 ...
 $ drat: num  3.9 3.9 3.85 3.08 3.15 2.76 3.21 3.69 3.92 3.92 ...
 $ wt  : num  2.62 2.88 2.32 3.21 3.44 ...
 $ qsec: num  16.5 17 18.6 19.4 17 ...
 $ vs  : num  0 0 1 1 0 1 0 1 1 1 ...
 $ am  : num  1 1 1 0 0 0 0 0 0 0 ...
 $ gear: num  4 4 4 3 3 3 3 4 4 4 ...
 $ carb: num  4 4 1 1 2 1 4 2 2 4 ...
unique(mtcars$cyl)
[1] 6 4 8

One (wrong) possibility is to write:

y = mtcars$mpg
x1 = mtcars$cyl; x1 = ifelse(x1 == 4, 1, 0)
x2 = mtcars$cyl; x2 = ifelse(x1 == 6, 1, 0)
x3 = mtcars$cyl; x3 = ifelse(x1 == 8, 1, 0)

fit = lm(y ~ x1 + x2 + x3)
summary(fit)
Call:
lm(formula = y ~ x1 + x2 + x3)

Residuals:
    Min      1Q  Median      3Q     Max 
-6.2476 -2.2846 -0.4556  2.6774  7.2364 

Coefficients: (2 not defined because of singularities)
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  16.6476     0.7987  20.844  < 2e-16 ***
x1           10.0160     1.3622   7.353 3.44e-08 ***
x2                NA         NA      NA       NA    
x3                NA         NA      NA       NA    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 3.66 on 30 degrees of freedom
Multiple R-squared:  0.6431,    Adjusted R-squared:  0.6312 
F-statistic: 54.06 on 1 and 30 DF,  p-value: 3.436e-08

Here the NAs mean there are variables that are linearly related to the other variables (e.g. the variable pointed with NA is an average of one or more of the rest of the variables, like \(x_2 = 2x_1 + x_3\) or another linear combination), then there's no unique solution to the regression without dropping variables.

My model will include these variables:

$$ x_2 = \begin{cases}1 &\text{ if car } i \text{ has 6 cylinders} \cr 0 &\text{ otherwise }\end{cases} \quad \quad x_3 = \begin{cases}1 &\text{ if car } i \text{ has 8 cylinders} \cr 0 &\text{ otherwise }\end{cases} $$

In this particular case regression coefficients are given by this estimate:

$$ \hat{\vec{\beta}} = \begin{bmatrix}\bar{y}_1 \cr \bar{y}_2 \end{bmatrix} $$

But R has a command called as.factor() that is useful in these cases and also can save you some lines of code in other cases:

fit2 = lm(mpg ~ as.factor(cyl), data = mtcars)
summary(fit2)
Call:
lm(formula = mpg ~ as.factor(cyl), data = mtcars)

Residuals:
    Min      1Q  Median      3Q     Max 
-5.2636 -1.8357  0.0286  1.3893  7.2364 

Coefficients:
                Estimate Std. Error t value Pr(>|t|)    
(Intercept)      26.6636     0.9718  27.437  < 2e-16 ***
as.factor(cyl)6  -6.9208     1.5583  -4.441 0.000119 ***
as.factor(cyl)8 -11.5636     1.2986  -8.905 8.57e-10 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 3.223 on 29 degrees of freedom
Multiple R-squared:  0.7325,    Adjusted R-squared:  0.714 
F-statistic:  39.7 on 2 and 29 DF,  p-value: 4.979e-09

The aov version of this is:

fit3 = aov(mpg ~ as.factor(cyl), data = mtcars)
TukeyHSD(fit3)
  Tukey multiple comparisons of means
    95% family-wise confidence level

Fit: aov(formula = mpg ~ as.factor(cyl), data = mtcars)

$`as.factor(cyl)`
          diff        lwr        upr     p adj
6-4  -6.920779 -10.769350 -3.0722086 0.0003424
8-4 -11.563636 -14.770779 -8.3564942 0.0000000
8-6  -4.642857  -8.327583 -0.9581313 0.0112287

As I said many times in this entry, ANOVA is linear regression. Interpreting the coefficients is up to you.