6.2 Car Sales

We have a dataset on 100 cars offered for sale at a car dealership. The variables are


Car sales: dataset
Variable Descripción Valores
Car ID Identification code 1 - 100
Price Sale Price of the car 000s Eur
Age Age of the car, months
PinkSlip Certificate of Title 1: No, 2: Yes
Sold Car sold? 1: No, 2: Yes


Venta de Vehículos: Primeras 12 observaciones
Car ID 1 2 3 4 5 6 7 8 9 10 11 12
Price 1 9 0 3 10 2 4 2 2 5 5 2
Odometer 30 20 170 68 12 88 3 41 21 74 41 121
Age 28 40 58 12 3 23 4 13 5 10 62 20
PinkSlip 1 1 0 1 0 0 1 1 1 1 0 1
Sold 1 0 1 1 0 0 0 1 1 1 0 1
Nota: http://www.zstatistics.com/


The dataset:

Venta de Vehículos: Resumen de Datos
No Variable Stats / Values Freqs (% of Valid) Graph Missing
1 Car ID
[integer]		 Mean (sd) : 50.5 (29)
min < med < max:
1 < 50.5 < 100
IQR (CV) : 49.5 (0.6)		 100 distinct values
(Integer sequence)		 0
(0.0%)
2 Price
[numeric]		 Mean (sd) : 5.2 (5.1)
min < med < max:
0.5 < 4 < 34.5
IQR (CV) : 5.5 (1)		 29 distinct values 0
(0.0%)
3 Odometer
[numeric]		 Mean (sd) : 60.1 (76.7)
min < med < max:
0.2 < 30.5 < 452.5
IQR (CV) : 63 (1.3)		 100 distinct values 0
(0.0%)
4 Age
[integer]		 Mean (sd) : 20.2 (16.1)
min < med < max:
1 < 15 < 90
IQR (CV) : 19 (0.8)		 39 distinct values 0
(0.0%)
5 PinkSlip
[integer]		 Min : 0
Mean : 0.8
Max : 1		 0 : 23 (23.0%)
1 : 77 (77.0%)		 0
(0.0%)
6 Sold
[integer]		 Min : 0
Mean : 0.7
Max : 1		 0 : 35 (35.0%)
1 : 65 (65.0%)		 0
(0.0%)


\[ \operatorname{\widehat{Sold}} = 0.8 - 0.03(\operatorname{Price}) \]


Let’s think of the fitted line as estimating the chance of being sold:

\[{\pi_i= Prob(\operatorname{Sold}=1)} = \beta_0 + \beta_1 \operatorname{Price}_i + \epsilon_i\]


What would be the probability of being sold of a car costing 45k euros? -> We need to transform/modify the dependent variable.


\[\dfrac{\pi_i}{1-\pi_i}={\dfrac{Prob(\operatorname{Sold}=1)}{1- Prob(\operatorname{Sold}=1)}} = \beta_0 +\beta_1 \operatorname{Price}_i + \epsilon_i\]

Evitamos obtener probabilidad negativa, la distribución es muy asimétrica (no Normal) -> Necesitamos transformar/modificar la variable dependiente.


Using logs:

\[\log \Bigl ( \dfrac{\pi_i}{1-\pi_i} \Bigr ) =\log \Bigl ( \dfrac{Prob(\operatorname{Sold}=1)}{1- Prob(\operatorname{Sold}=1)} \Bigr ) = \beta_0 +\beta_1\operatorname{Price}_i + \epsilon_i \]


The Binomial Logistic Regression is given by

\[\operatorname{logit}(\pi_i)= \log \Bigl ( \dfrac{\pi_i}{1-\pi_i} \Bigr ) = \beta_0 +\beta_1x_{1i}+\ldots+\beta_kx_{ki}\]

A model used to predict the probability of a certain class, given a set of independent variables.

  • Binomial: la variable dependiente es binaria, \(\pi_i = Prob(y_i=1)\)

  • Logistic: uses log-odds and the logit function

  • \(\beta_0, \beta_1, \ldots, \beta_k\) are the parameters

  • \(x_1, \ldots, x_k\) are the independent variables or predictors


Foe example:

\[\log \Bigl ( \dfrac{\pi_i}{1-\pi_i} \Bigr ) = \operatorname{logit} \Bigl [ Prob(\operatorname{Sold}=1) \Bigr ] = \beta_0 +\beta_1\operatorname{Price}_i\]

Observations 100
Dependent variable Sold
Type Generalized linear model
Family binomial
Link logit
χ²(1) 9.454
Pseudo-R² (Cragg-Uhler) 0.124
Pseudo-R² (McFadden) 0.073
AIC 124.036
BIC 129.246
Est. S.E. z val. p
(Intercept) 1.386 0.356 3.894 0.000
Price -0.143 0.053 -2.695 0.007
Standard errors: MLE

\[ \log\left[ \frac { \widehat{P( \operatorname{Sold} = \operatorname{1} )} }{ 1 - \widehat{P( \operatorname{Sold} = \operatorname{1} )} } \right] = 1.39 - 0.14(\operatorname{Price}) \]


What does -0.143 mean in the estimated model?

\[\widehat{\operatorname{logit}(\pi_i)} =1.386 -0.143 \operatorname{Price}_i\]

  • For each unit increase in \(\operatorname{Price}\), \(\operatorname{logit}(\pi)\) decreases in 0.143 units

  • What about \(\pi\)?


From \(\operatorname{logit}(\pi_i)\) to \(\pi_i\):

  • we have:

\[ \operatorname{logit}(\pi) = \log \Bigl ( \dfrac{\pi}{1-\pi} \Bigr )=\beta_0 +\beta_1 \operatorname{Price}\]

  • then:

\[\pi = \dfrac{e^{\beta_0 +\beta_1 \operatorname{Price}}}{1+e^{\beta_0 +\beta_1 \operatorname{Price}}}\] - O alternativamente (más fácil):

\[\pi = \dfrac{e^{\operatorname{logit}(\pi) }}{1+e^{\operatorname{logit}(\pi) }}\]

In the example:


The coefficients determine the curve:

Multiple Logistic Regression

Assume that the \(operatorname{logit}\) transformation of the dependent variable has a linear relationship with a set of independent variables.


Let us include one more variable in the model.

\[\log \Bigl ( \dfrac{\pi_i}{1-\pi_i} \Bigr ) =\log \Bigl ( \dfrac{Prob(\operatorname{Sold_i}=1)}{1- Prob(\operatorname{Sold_i}=1)} \Bigr ) = \beta_0 +\beta_1\operatorname{Price}_i +\beta_2\operatorname{Pink Slip}_i\]

Observations 100
Dependent variable Sold
Type Generalized linear model
Family binomial
Link logit
χ²(2) 18.407
Pseudo-R² (Cragg-Uhler) 0.232
Pseudo-R² (McFadden) 0.142
AIC 117.083
BIC 124.898
Est. S.E. z val. p
(Intercept) 0.396 0.480 0.824 0.410
Price -0.173 0.057 -3.044 0.002
PinkSlip 1.555 0.531 2.926 0.003
Standard errors: MLE

\[ \log\left[ \frac { \widehat{P( \operatorname{Sold} = \operatorname{1} )} }{ 1 - \widehat{P( \operatorname{Sold} = \operatorname{1} )} } \right] = 0.4 - 0.17(\operatorname{Price}) + 1.55(\operatorname{PinkSlip}) \]


Let’s estimate \(\pi\)


Parameter’s interpretation:

The coefficients of the logistic regression estimate the change in log-odds of the dependent variable given a one-unit increase in the independent variable.

Coeficientes Estimados
Coeficiente 2.5 % 97.5 %
(Intercept) 0.396 -0.552 1.354
Price -0.173 -0.295 -0.071
PinkSlip 1.555 0.533 2.632

  • \(\beta_1=-0.173\) : If the price increases by 1000 euros, the log-odds of selling the car decreases, in mean, by 0.173, holding the rest constant.

  • \(\beta_2= 1.555\) : If the car has Pink Slip, the log-odds of selling the car increase, on average, by 1,555, holding the rest constant.

If we apply $exp to the coefficients, we can interpret them as odds-ratios.

Coeficientes Estimados
OR 2.5 % 97.5 %
(Intercept) 1.486 0.576 3.872
Price 0.841 0.745 0.931
PinkSlip 4.734 1.704 13.903

  • \(\exp(\beta_1)=0.84\): If Price increases by one unit, the odds of being sold (versus not being sold) increases by a factor of 0.84.

  • \(\exp(\beta_2)=4.73\): If Pink Slip increases by one unit, the odds of being sold (vs. not being sold) increase by a factor of 4.73.


  • \(\beta\) represents the effect of \(x\) on log-odds.

  • \(\operatorname{exp}(\beta)\) represents the effect of \(x\) on odds-ratio.