Midterm Homework AFS
Maulana Ahmad Fahrezi
Student ID: 114035115
2026-04-13
Chapter 2 - Auto Dataset
Understanding the Dataset
The first step is to examine the dataset and identify the type of
each variable. We load the dataset and inspect its structure using
str() to determine whether variables are quantitative or
qualitative.
library(ISLR2)
library(ggplot2)
library(GGally)
library(dplyr)
library(tidyr)
library(reshape2)
PALETTE <- c("#1b9e77", "#d95f02", "#7570b3")
# Load dataset
Auto <- Auto
# Check structure
str(Auto)## 'data.frame': 392 obs. of 9 variables:
## $ mpg : num 18 15 18 16 17 15 14 14 14 15 ...
## $ cylinders : int 8 8 8 8 8 8 8 8 8 8 ...
## $ displacement: num 307 350 318 304 302 429 454 440 455 390 ...
## $ horsepower : int 130 165 150 150 140 198 220 215 225 190 ...
## $ weight : int 3504 3693 3436 3433 3449 4341 4354 4312 4425 3850 ...
## $ acceleration: num 12 11.5 11 12 10.5 10 9 8.5 10 8.5 ...
## $ year : int 70 70 70 70 70 70 70 70 70 70 ...
## $ origin : int 1 1 1 1 1 1 1 1 1 1 ...
## $ name : Factor w/ 304 levels "amc ambassador brougham",..: 49 36 231 14 161 141 54 223 241 2 ...
## - attr(*, "na.action")= 'omit' Named int [1:5] 33 127 331 337 355
## ..- attr(*, "names")= chr [1:5] "33" "127" "331" "337" ...From the structure output, we can observe the data types of each variable, which will help us distinguish between quantitative and qualitative variables.
(a) Quantitative vs Qualitative Variables
In this step, we identify which variables are quantitative (numerical) and which are qualitative (categorical). This distinction is important for determining the appropriate statistical methods and visualizations.
sapply(Auto, class)## mpg cylinders displacement horsepower weight acceleration
## "numeric" "integer" "numeric" "integer" "integer" "numeric"
## year origin name
## "integer" "integer" "factor"Based on the data structure:
- Quantitative variables: mpg, cylinders,
displacement, horsepower, weight, acceleration, year
- Qualitative variables: origin, name
Quantitative variables are numerical and measurable, while qualitative variables represent categories or labels.
(b) Range of Quantitative Variables
Next, we compute the range (minimum and maximum values) for each quantitative variable. This helps us understand the spread and scale of the data.
quant_vars <- Auto[, -which(names(Auto) %in% c("origin", "name"))]
sapply(quant_vars, range)## mpg cylinders displacement horsepower weight acceleration year
## [1,] 9.0 3 68 46 1613 8.0 70
## [2,] 46.6 8 455 230 5140 24.8 82The range shows the minimum and maximum values of each
variable.
For example, mpg varies across a wide range, indicating differences in
fuel efficiency among cars.
This information is useful for understanding variability and detecting potential outliers.
(c) Mean and Standard Deviation
To further summarize the data, we compute the mean and standard deviation for each quantitative variable. The mean represents the central tendency, while the standard deviation indicates the variability of the data.
# Mean
sapply(quant_vars, mean)## mpg cylinders displacement horsepower weight acceleration
## 23.445918 5.471939 194.411990 104.469388 2977.584184 15.541327
## year
## 75.979592# Standard deviation
sapply(quant_vars, sd)## mpg cylinders displacement horsepower weight acceleration
## 7.805007 1.705783 104.644004 38.491160 849.402560 2.758864
## year
## 3.683737The mean provides the average value of each variable, while the standard deviation measures how spread out the values are. Variables with higher standard deviation exhibit greater variability, indicating less consistency in their values.
(d) Removing Observations 10–85
In this step, we remove observations from index 10 to 85 and recompute the summary statistics. This allows us to examine how sensitive the data is to changes in the sample.
Auto_new <- Auto[-(10:85), ]
quant_vars_new <- Auto_new[, -which(names(Auto_new) %in% c("origin", "name"))]
# Range
sapply(quant_vars_new, range)## mpg cylinders displacement horsepower weight acceleration year
## [1,] 11.0 3 68 46 1649 8.5 70
## [2,] 46.6 8 455 230 4997 24.8 82# Mean
sapply(quant_vars_new, mean)## mpg cylinders displacement horsepower weight acceleration
## 24.404430 5.373418 187.240506 100.721519 2935.971519 15.726899
## year
## 77.145570# Standard deviation
sapply(quant_vars_new, sd)## mpg cylinders displacement horsepower weight acceleration
## 7.867283 1.654179 99.678367 35.708853 811.300208 2.693721
## year
## 3.106217After removing a subset of observations, the summary statistics change. This indicates that the removed data had an influence on the overall distribution. Such differences highlight the importance of data consistency and sample selection in statistical analysis.
(e) Graphical Analysis (Scatterplot Matrix)
To explore relationships between variables, we create a scatterplot matrix. This visualization helps identify correlations, patterns, and potential trends between predictors.
pairs(Auto[, -which(names(Auto) %in% c("name"))],
main = "Scatterplot Matrix - Auto Dataset")
Visualization – Scatterplot Matrix with Origin
ggpairs(
Auto[, c("mpg","displacement","horsepower","weight","acceleration")],
aes(color = factor(Auto$origin), alpha = 0.3),
upper = list(continuous = wrap("cor", size = 3)),
diag = list(continuous = wrap("densityDiag"))
) +
scale_color_manual(values = PALETTE)
This enhanced visualization shows how relationships vary across different origins. Clustering patterns indicate that vehicle characteristics differ by region.
From the scatterplot matrix, several relationships can be observed:
- mpg shows a strong negative relationship with weight and
displacement
- mpg tends to increase with year
- horsepower is negatively correlated with mpg
These patterns suggest that newer and lighter cars tend to have better fuel efficiency.
(f) Predicting mpg
Finally, based on the graphical analysis, we identify which variables are most useful for predicting mpg. Based on the observed relationships, the most useful predictors for mpg are:
- weight (strong negative relationship)
- horsepower (negative relationship)
- displacement (negative relationship)
- year (positive relationship)
These variables are likely to be significant predictors in a regression model for mpg.
Visualization – Non-linear Patterns
Auto %>%
select(mpg, displacement, horsepower, weight, acceleration) %>%
pivot_longer(-mpg) %>%
ggplot(aes(value, mpg)) +
geom_point(alpha = 0.3) +
geom_smooth(method = "loess", color = "black") +
geom_smooth(method = "lm", linetype = "dashed", color = "red") +
facet_wrap(~name, scales = "free_x")
The LOESS curves reveal that some relationships are not strictly linear, suggesting that transformations may improve model performance.
Chapter 3 - Multiple Linear Regression on Auto Dataset
(a) Scatterplot Matrix
To begin the analysis, we explore the relationships between variables using a scatterplot matrix. This helps identify potential correlations and patterns among predictors.
pairs(Auto, main = "Scatterplot Matrix - Auto Dataset")
The scatterplot matrix provides an overview of relationships between variables.
Several clear patterns can be observed: - mpg shows a strong negative
relationship with weight
- mpg decreases as horsepower increases
- mpg tends to increase with year
These patterns indicate that fuel efficiency is influenced by vehicle weight, engine power, and manufacturing year.
This step is important as it provides an initial understanding before building a regression model.
(b) Correlation Matrix
Next, we compute the correlation matrix to quantify the strength and
direction of relationships between numerical variables. The variable
name is excluded because it is qualitative.
cor(Auto[, -which(names(Auto) == "name")])## mpg cylinders displacement horsepower weight
## mpg 1.0000000 -0.7776175 -0.8051269 -0.7784268 -0.8322442
## cylinders -0.7776175 1.0000000 0.9508233 0.8429834 0.8975273
## displacement -0.8051269 0.9508233 1.0000000 0.8972570 0.9329944
## horsepower -0.7784268 0.8429834 0.8972570 1.0000000 0.8645377
## weight -0.8322442 0.8975273 0.9329944 0.8645377 1.0000000
## acceleration 0.4233285 -0.5046834 -0.5438005 -0.6891955 -0.4168392
## year 0.5805410 -0.3456474 -0.3698552 -0.4163615 -0.3091199
## origin 0.5652088 -0.5689316 -0.6145351 -0.4551715 -0.5850054
## acceleration year origin
## mpg 0.4233285 0.5805410 0.5652088
## cylinders -0.5046834 -0.3456474 -0.5689316
## displacement -0.5438005 -0.3698552 -0.6145351
## horsepower -0.6891955 -0.4163615 -0.4551715
## weight -0.4168392 -0.3091199 -0.5850054
## acceleration 1.0000000 0.2903161 0.2127458
## year 0.2903161 1.0000000 0.1815277
## origin 0.2127458 0.1815277 1.0000000The correlation matrix confirms strong relationships observed earlier:
- mpg is negatively correlated with weight, horsepower, and
displacement
- Some predictors are also highly correlated with each other, indicating potential multicollinearity
This information is important when building regression models.
Visualization – Correlation Heatmap
cor_mat <- cor(Auto[, -which(names(Auto) == "name")])
melted <- melt(cor_mat)
ggplot(melted, aes(Var1, Var2, fill = value)) +
geom_tile() +
scale_fill_gradient2(low = "blue", high = "red", mid = "white") +
theme(axis.text.x = element_text(angle = 45, hjust = 1))
The heatmap highlights strong relationships and potential multicollinearity.
(c) Multiple Linear Regression Model
We now fit a multiple linear regression model with mpg as the response variable and all other variables (except name) as predictors. The summary output provides statistical significance, coefficients, and model fit.
model <- lm(mpg ~ . - name, data = Auto)
summary(model)##
## Call:
## lm(formula = mpg ~ . - name, data = Auto)
##
## Residuals:
## Min 1Q Median 3Q Max
## -9.5903 -2.1565 -0.1169 1.8690 13.0604
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -17.218435 4.644294 -3.707 0.00024 ***
## cylinders -0.493376 0.323282 -1.526 0.12780
## displacement 0.019896 0.007515 2.647 0.00844 **
## horsepower -0.016951 0.013787 -1.230 0.21963
## weight -0.006474 0.000652 -9.929 < 2e-16 ***
## acceleration 0.080576 0.098845 0.815 0.41548
## year 0.750773 0.050973 14.729 < 2e-16 ***
## origin 1.426141 0.278136 5.127 4.67e-07 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3.328 on 384 degrees of freedom
## Multiple R-squared: 0.8215, Adjusted R-squared: 0.8182
## F-statistic: 252.4 on 7 and 384 DF, p-value: < 2.2e-16The regression output provides key statistical information:
Overall Model Significance
The F-statistic with a very small p-value indicates that the model is statistically significant.
This means that at least one predictor is related to mpg.Significant Predictors
Variables with p-values less than 0.05 are considered significant.
Common significant predictors include:- weight
- year
- horsepower (often borderline or significant depending on output)
- weight
Interpretation of Coefficients
Each coefficient represents the expected change in mpg for a one-unit increase in the predictor, holding other variables constant.For example:
- A negative coefficient for weight means that heavier cars tend to
have lower mpg
- A positive coefficient for year suggests that newer cars have higher mpg
- A negative coefficient for weight means that heavier cars tend to
have lower mpg
Numerically, this implies that even a small increase in weight can lead to a noticeable decrease in fuel efficiency.
Visualization – Coefficient Plot
coef_df <- data.frame(
variable = names(coef(model))[-1],
value = coef(model)[-1]
)
ggplot(coef_df, aes(reorder(variable, value), value)) +
geom_col(fill = "steelblue") +
coord_flip()
This plot shows the relative importance of each predictor.
(d) Diagnostic Plots
To evaluate the model, we generate diagnostic plots. These plots help identify issues such as non-linearity, outliers, and high-leverage observations.
par(mfrow = c(2, 2))
plot(model)
The diagnostic plots provide insight into model performance:
- Residuals vs Fitted: may indicate slight
non-linearity
- Normal Q-Q: checks whether residuals follow a
normal distribution
- Scale-Location: assesses constant variance
(homoscedasticity)
- Residuals vs Leverage: identifies influential observations
Some points may appear as outliers or high-leverage observations, which can significantly influence the regression results.
This suggests that the model may benefit from refinement.
Visualization – Residual Plot (ggplot)
ggplot(data.frame(fitted = fitted(model), resid = resid(model)),
aes(fitted, resid)) +
geom_point(alpha = 0.4) +
geom_smooth(method = "loess", color = "red")
(e) Interaction Effects
We now include interaction terms in the regression model to examine whether combinations of variables improve the model.
model_interaction <- lm(mpg ~ . * . - name, data = Auto)
summary(model_interaction)##
## Call:
## lm(formula = mpg ~ . * . - name, data = Auto)
##
## Residuals:
## ALL 392 residuals are 0: no residual degrees of freedom!
##
## Coefficients: (1737 not defined because of singularities)
## Estimate Std. Error
## (Intercept) 1.922e+03 NaN
## cylinders -3.623e+00 NaN
## displacement -5.886e+00 NaN
## horsepower 6.561e+00 NaN
## weight -8.695e-01 NaN
## acceleration -7.249e+00 NaN
## year -2.886e+00 NaN
## origin -4.071e+02 NaN
## cylinders:displacement -4.268e-02 NaN
## cylinders:horsepower -3.296e+00 NaN
## cylinders:weight 1.907e-01 NaN
## cylinders:acceleration -2.330e+00 NaN
## cylinders:year -5.977e-01 NaN
## cylinders:origin -1.693e+01 NaN
## cylinders:nameamc ambassador dpl -1.746e+01 NaN
## cylinders:nameamc ambassador sst 4.226e+00 NaN
## cylinders:nameamc concord -6.261e+03 NaN
## cylinders:nameamc concord d/l 6.765e+01 NaN
## cylinders:nameamc concord dl 6 1.246e+02 NaN
## cylinders:nameamc gremlin -2.450e+02 NaN
## cylinders:nameamc hornet -1.006e+02 NaN
## cylinders:nameamc hornet sportabout (sw) 9.069e+01 NaN
## cylinders:nameamc matador -2.139e+02 NaN
## cylinders:nameamc matador (sw) 3.623e+01 NaN
## cylinders:nameamc pacer 1.234e+02 NaN
## cylinders:nameamc pacer d/l 1.666e+02 NaN
## cylinders:nameamc rebel sst 4.123e+00 NaN
## cylinders:nameamc spirit dl -8.820e+01 NaN
## cylinders:nameaudi 100 ls -1.757e+01 NaN
## cylinders:nameaudi 100ls 1.302e+03 NaN
## cylinders:nameaudi 4000 -5.636e+00 NaN
## cylinders:nameaudi 5000 -3.694e+01 NaN
## cylinders:nameaudi 5000s (diesel) 1.332e+01 NaN
## cylinders:nameaudi fox -1.127e+01 NaN
## cylinders:namebmw 2002 -3.613e+01 NaN
## cylinders:namebmw 320i -4.842e+01 NaN
## cylinders:namebuick century -7.578e+01 NaN
## cylinders:namebuick century 350 -1.325e+01 NaN
## cylinders:namebuick century limited -5.823e+01 NaN
## cylinders:namebuick century luxus (sw) 1.073e+02 NaN
## cylinders:namebuick century special 6.514e+01 NaN
## cylinders:namebuick electra 225 custom -5.590e+01 NaN
## cylinders:namebuick estate wagon (sw) 8.692e+02 NaN
## cylinders:namebuick lesabre custom 7.999e+01 NaN
## cylinders:namebuick opel isuzu deluxe -1.032e+02 NaN
## cylinders:namebuick regal sport coupe (turbo) -1.662e+02 NaN
## cylinders:namebuick skyhawk 4.033e+01 NaN
## cylinders:namebuick skylark -4.056e+03 NaN
## cylinders:namebuick skylark 320 1.985e+01 NaN
## cylinders:namebuick skylark limited -3.669e+01 NaN
## cylinders:namecadillac eldorado 1.810e+02 NaN
## cylinders:namecadillac seville -2.933e+01 NaN
## cylinders:namecapri ii -7.621e+01 NaN
## cylinders:namechevroelt chevelle malibu 1.402e+02 NaN
## cylinders:namechevrolet bel air 1.203e+02 NaN
## cylinders:namechevrolet camaro -3.424e+01 NaN
## cylinders:namechevrolet caprice classic -1.678e+02 NaN
## cylinders:namechevrolet cavalier -1.423e+02 NaN
## cylinders:namechevrolet cavalier 2-door -1.324e+02 NaN
## cylinders:namechevrolet cavalier wagon -1.443e+02 NaN
## cylinders:namechevrolet chevelle concours (sw) 7.834e+01 NaN
## cylinders:namechevrolet chevelle malibu -6.833e+02 NaN
## cylinders:namechevrolet chevelle malibu classic -1.260e+03 NaN
## cylinders:namechevrolet chevette -9.377e+01 NaN
## cylinders:namechevrolet citation -1.329e+02 NaN
## cylinders:namechevrolet concours 9.686e+01 NaN
## cylinders:namechevrolet impala 1.495e+02 NaN
## cylinders:namechevrolet malibu -2.291e+02 NaN
## cylinders:namechevrolet malibu classic (sw) 1.449e+01 NaN
## cylinders:namechevrolet monte carlo 1.602e+02 NaN
## cylinders:namechevrolet monte carlo landau 1.016e+02 NaN
## cylinders:namechevrolet monte carlo s 1.082e+02 NaN
## cylinders:namechevrolet monza 2+2 4.831e+01 NaN
## cylinders:namechevrolet nova 4.156e+02 NaN
## cylinders:namechevrolet nova custom 1.323e+02 NaN
## cylinders:namechevrolet vega 2.595e+02 NaN
## cylinders:namechevrolet vega (sw) 6.594e+00 NaN
## cylinders:namechevrolet vega 2300 -6.960e+01 NaN
## cylinders:namechevrolet woody -7.301e+01 NaN
## cylinders:namechevy c10 1.065e+02 NaN
## cylinders:namechevy c20 -1.867e+02 NaN
## cylinders:namechevy s-10 -1.011e+02 NaN
## cylinders:namechrysler cordoba 8.655e+00 NaN
## cylinders:namechrysler lebaron medallion -3.520e+01 NaN
## cylinders:namechrysler lebaron salon 1.274e+02 NaN
## cylinders:namechrysler lebaron town @ country (sw) 1.012e+02 NaN
## cylinders:namechrysler new yorker brougham -3.527e+01 NaN
## cylinders:namechrysler newport royal 1.093e+01 NaN
## cylinders:namedatsun 1200 1.402e+02 NaN
## cylinders:namedatsun 200-sx 1.099e+02 NaN
## cylinders:namedatsun 200sx 1.020e+02 NaN
## cylinders:namedatsun 210 1.381e+02 NaN
## cylinders:namedatsun 210 mpg 1.253e+02 NaN
## cylinders:namedatsun 280-zx 4.594e+01 NaN
## cylinders:namedatsun 310 1.206e+02 NaN
## cylinders:namedatsun 310 gx 1.230e+02 NaN
## cylinders:namedatsun 510 1.135e+02 NaN
## cylinders:namedatsun 510 (sw) 1.148e+02 NaN
## cylinders:namedatsun 510 hatchback 1.160e+02 NaN
## cylinders:namedatsun 610 1.106e+02 NaN
## cylinders:namedatsun 710 1.638e+02 NaN
## cylinders:namedatsun 810 5.629e+01 NaN
## cylinders:namedatsun 810 maxima 4.102e+01 NaN
## cylinders:namedatsun b-210 1.242e+02 NaN
## cylinders:namedatsun b210 1.257e+02 NaN
## cylinders:namedatsun b210 gx 1.245e+02 NaN
## cylinders:namedatsun f-10 hatchback 1.257e+02 NaN
## cylinders:namedatsun pl510 1.225e+02 NaN
## cylinders:namedodge aries se -6.127e+01 NaN
## cylinders:namedodge aries wagon (sw) -3.408e+01 NaN
## cylinders:namedodge aspen 4.462e+02 NaN
## cylinders:namedodge aspen 6 3.069e+01 NaN
## cylinders:namedodge aspen se 7.927e+01 NaN
## cylinders:namedodge challenger se 4.132e+01 NaN
## cylinders:namedodge charger 2.2 -5.998e+01 NaN
## cylinders:namedodge colt -8.046e+01 NaN
## cylinders:namedodge colt (sw) -1.329e+02 NaN
## cylinders:namedodge colt hardtop -1.327e+02 NaN
## cylinders:namedodge colt hatchback custom -1.198e+02 NaN
## cylinders:namedodge colt m/m -1.351e+02 NaN
## cylinders:namedodge coronet brougham 2.750e+01 NaN
## cylinders:namedodge coronet custom 2.581e+01 NaN
## cylinders:namedodge coronet custom (sw) 2.950e+01 NaN
## cylinders:namedodge d100 2.463e+01 NaN
## cylinders:namedodge d200 -2.062e+02 NaN
## cylinders:namedodge dart custom 2.463e+01 NaN
## cylinders:namedodge diplomat 5.934e+01 NaN
## cylinders:namedodge magnum xe 6.271e+01 NaN
## cylinders:namedodge monaco (sw) 5.076e+01 NaN
## cylinders:namedodge monaco brougham 4.510e+01 NaN
## cylinders:namedodge omni -1.073e+02 NaN
## cylinders:namedodge rampage -6.071e+01 NaN
## cylinders:namedodge st. regis 7.716e+01 NaN
## cylinders:namefiat 124 sport coupe -2.371e+01 NaN
## cylinders:namefiat 124 tc 2.375e+01 NaN
## cylinders:namefiat 124b -9.276e+00 NaN
## cylinders:namefiat 128 6.750e+00 NaN
## cylinders:namefiat 131 -1.308e+01 NaN
## cylinders:namefiat strada custom -2.552e+00 NaN
## cylinders:namefiat x1.9 -1.044e+01 NaN
## cylinders:nameford country 1.526e+02 NaN
## cylinders:nameford country squire (sw) 5.304e+01 NaN
## cylinders:nameford escort 2h -9.693e+01 NaN
## cylinders:nameford escort 4w -9.434e+01 NaN
## cylinders:nameford f108 6.411e+01 NaN
## cylinders:nameford f250 -1.731e+02 NaN
## cylinders:nameford fairmont -5.963e+01 NaN
## cylinders:nameford fairmont (auto) 5.252e+01 NaN
## cylinders:nameford fairmont (man) -6.029e+01 NaN
## cylinders:nameford fairmont 4 -5.985e+01 NaN
## cylinders:nameford fairmont futura -8.140e+01 NaN
## cylinders:nameford fiesta -9.033e+01 NaN
## cylinders:nameford futura 3.663e+01 NaN
## cylinders:nameford galaxie 500 2.587e+02 NaN
## cylinders:nameford gran torino -6.621e+01 NaN
## cylinders:nameford gran torino (sw) 3.341e+01 NaN
## cylinders:nameford granada 1.533e+02 NaN
## cylinders:nameford granada ghia 2.543e+02 NaN
## cylinders:nameford granada gl 4.402e+01 NaN
## cylinders:nameford granada l 3.251e+01 NaN
## cylinders:nameford ltd 8.850e+02 NaN
## cylinders:nameford ltd landau 6.569e+01 NaN
## cylinders:nameford maverick -4.586e+01 NaN
## cylinders:nameford mustang 1.713e+02 NaN
## cylinders:nameford mustang gl -5.476e+01 NaN
## cylinders:nameford mustang ii 6.124e+01 NaN
## cylinders:nameford mustang ii 2+2 -6.319e+01 NaN
## cylinders:nameford pinto -1.439e+02 NaN
## cylinders:nameford pinto (sw) -9.928e+01 NaN
## cylinders:nameford pinto runabout -9.826e+01 NaN
## cylinders:nameford ranger -8.619e+01 NaN
## cylinders:nameford thunderbird 1.008e+02 NaN
## cylinders:nameford torino 3.174e+01 NaN
## cylinders:nameford torino 500 1.840e+02 NaN
## cylinders:namehi 1200d -1.826e+02 NaN
## cylinders:namehonda accord 1.838e+02 NaN
## cylinders:namehonda accord cvcc 1.268e+02 NaN
## cylinders:namehonda accord lx 1.227e+02 NaN
## cylinders:namehonda civic 1.672e+02 NaN
## cylinders:namehonda civic (auto) 1.218e+02 NaN
## cylinders:namehonda civic 1300 1.227e+02 NaN
## cylinders:namehonda civic 1500 gl 1.265e+02 NaN
## cylinders:namehonda civic cvcc 1.030e+02 NaN
## cylinders:namehonda prelude 1.223e+02 NaN
## cylinders:namemaxda glc deluxe 1.198e+02 NaN
## cylinders:namemaxda rx3 1.583e+02 NaN
## cylinders:namemazda 626 3.225e+02 NaN
## cylinders:namemazda glc 1.225e+02 NaN
## cylinders:namemazda glc 4 1.228e+02 NaN
## cylinders:namemazda glc custom 1.240e+02 NaN
## cylinders:namemazda glc custom l 1.246e+02 NaN
## cylinders:namemazda glc deluxe 1.174e+02 NaN
## cylinders:namemazda rx-4 1.037e+02 NaN
## cylinders:namemazda rx-7 gs 1.358e+02 NaN
## cylinders:namemazda rx2 coupe 1.453e+02 NaN
## cylinders:namemercedes benz 300d 1.507e+02 NaN
## cylinders:namemercedes-benz 240d 1.616e+02 NaN
## cylinders:namemercedes-benz 280s -7.800e+01 NaN
## cylinders:namemercury capri 2000 -9.729e+01 NaN
## cylinders:namemercury capri v6 -7.291e+01 NaN
## cylinders:namemercury cougar brougham 6.712e+01 NaN
## cylinders:namemercury grand marquis 1.320e+02 NaN
## cylinders:namemercury lynx l -1.070e+02 NaN
## cylinders:namemercury marquis -2.072e+01 NaN
## cylinders:namemercury marquis brougham 5.080e+01 NaN
## cylinders:namemercury monarch 2.715e+02 NaN
## cylinders:namemercury monarch ghia 3.524e+01 NaN
## cylinders:namemercury zephyr 5.495e+01 NaN
## cylinders:namemercury zephyr 6 5.553e+01 NaN
## cylinders:namenissan stanza xe 1.234e+02 NaN
## cylinders:nameoldsmobile cutlass ciera (diesel) 2.053e+02 NaN
## cylinders:nameoldsmobile cutlass ls 2.516e+02 NaN
## cylinders:nameoldsmobile cutlass salon brougham 3.773e+02 NaN
## cylinders:nameoldsmobile cutlass supreme 4.149e+01 NaN
## cylinders:nameoldsmobile delta 88 royale 5.661e+01 NaN
## cylinders:nameoldsmobile omega -3.402e+01 NaN
## cylinders:nameoldsmobile omega brougham -7.095e+01 NaN
## cylinders:nameoldsmobile starfire sx -7.861e+00 NaN
## cylinders:nameoldsmobile vista cruiser -2.913e+01 NaN
## cylinders:nameopel 1900 1.306e+02 NaN
## cylinders:nameopel manta -1.862e+02 NaN
## cylinders:namepeugeot 304 -1.102e+01 NaN
## cylinders:namepeugeot 504 3.709e+02 NaN
## cylinders:namepeugeot 504 (sw) 6.915e+00 NaN
## cylinders:namepeugeot 505s turbo diesel 8.189e+01 NaN
## cylinders:namepeugeot 604sl -9.202e+01 NaN
## cylinders:nameplymouth 'cuda 340 2.645e+01 NaN
## cylinders:nameplymouth arrow gs -1.307e+02 NaN
## cylinders:nameplymouth champ -1.131e+02 NaN
## cylinders:nameplymouth cricket -1.144e+02 NaN
## cylinders:nameplymouth custom suburb 3.776e+01 NaN
## cylinders:nameplymouth duster 2.081e+01 NaN
## cylinders:nameplymouth fury 1.076e+02 NaN
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## horsepower:namedatsun 210 mpg NA NA
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## origin:namemazda rx-4 NA NA
## origin:namemazda rx-7 gs NA NA
## origin:namemazda rx2 coupe NA NA
## origin:namemercedes benz 300d NA NA
## origin:namemercedes-benz 240d NA NA
## origin:namemercedes-benz 280s NA NA
## origin:namemercury capri 2000 NA NA
## origin:namemercury capri v6 NA NA
## origin:namemercury cougar brougham NA NA
## origin:namemercury grand marquis NA NA
## origin:namemercury lynx l NA NA
## origin:namemercury marquis NA NA
## origin:namemercury marquis brougham NA NA
## origin:namemercury monarch NA NA
## origin:namemercury monarch ghia NA NA
## origin:namemercury zephyr NA NA
## origin:namemercury zephyr 6 NA NA
## origin:namenissan stanza xe NA NA
## origin:nameoldsmobile cutlass ciera (diesel) NA NA
## origin:nameoldsmobile cutlass ls NA NA
## origin:nameoldsmobile cutlass salon brougham NA NA
## origin:nameoldsmobile cutlass supreme NA NA
## origin:nameoldsmobile delta 88 royale NA NA
## origin:nameoldsmobile omega NA NA
## origin:nameoldsmobile omega brougham NA NA
## origin:nameoldsmobile starfire sx NA NA
## origin:nameoldsmobile vista cruiser NA NA
## origin:nameopel 1900 NA NA
## origin:nameopel manta NA NA
## origin:namepeugeot 304 NA NA
## origin:namepeugeot 504 NA NA
## origin:namepeugeot 504 (sw) NA NA
## origin:namepeugeot 505s turbo diesel NA NA
## origin:namepeugeot 604sl NA NA
## origin:nameplymouth 'cuda 340 NA NA
## origin:nameplymouth arrow gs NA NA
## origin:nameplymouth champ NA NA
## origin:nameplymouth cricket NA NA
## origin:nameplymouth custom suburb NA NA
## origin:nameplymouth duster NA NA
## origin:nameplymouth fury NA NA
## origin:nameplymouth fury gran sedan NA NA
## origin:nameplymouth fury iii NA NA
## origin:nameplymouth grand fury NA NA
## origin:nameplymouth horizon NA NA
## origin:nameplymouth horizon 4 NA NA
## origin:nameplymouth horizon miser NA NA
## origin:nameplymouth horizon tc3 NA NA
## origin:nameplymouth reliant NA NA
## origin:nameplymouth sapporo NA NA
## origin:nameplymouth satellite NA NA
## origin:nameplymouth satellite custom NA NA
## origin:nameplymouth satellite custom (sw) NA NA
## origin:nameplymouth satellite sebring NA NA
## origin:nameplymouth valiant NA NA
## origin:nameplymouth valiant custom NA NA
## origin:nameplymouth volare NA NA
## origin:nameplymouth volare custom NA NA
## origin:nameplymouth volare premier v8 NA NA
## origin:namepontiac astro NA NA
## origin:namepontiac catalina NA NA
## origin:namepontiac catalina brougham NA NA
## origin:namepontiac firebird NA NA
## origin:namepontiac grand prix NA NA
## origin:namepontiac grand prix lj NA NA
## origin:namepontiac j2000 se hatchback NA NA
## origin:namepontiac lemans v6 NA NA
## origin:namepontiac phoenix NA NA
## origin:namepontiac phoenix lj NA NA
## origin:namepontiac safari (sw) NA NA
## origin:namepontiac sunbird coupe NA NA
## origin:namepontiac ventura sj NA NA
## origin:namerenault 12 (sw) NA NA
## origin:namerenault 12tl NA NA
## origin:namerenault 5 gtl NA NA
## origin:namesaab 99e NA NA
## origin:namesaab 99gle NA NA
## origin:namesaab 99le NA NA
## origin:namesubaru NA NA
## origin:namesubaru dl NA NA
## origin:nametoyota carina NA NA
## origin:nametoyota celica gt NA NA
## origin:nametoyota celica gt liftback NA NA
## origin:nametoyota corolla NA NA
## origin:nametoyota corolla 1200 NA NA
## origin:nametoyota corolla 1600 (sw) NA NA
## origin:nametoyota corolla liftback NA NA
## origin:nametoyota corolla tercel NA NA
## origin:nametoyota corona NA NA
## origin:nametoyota corona hardtop NA NA
## origin:nametoyota corona liftback NA NA
## origin:nametoyota corona mark ii NA NA
## origin:nametoyota cressida NA NA
## origin:nametoyota mark ii NA NA
## origin:nametoyota starlet NA NA
## origin:nametoyota tercel NA NA
## origin:nametoyouta corona mark ii (sw) NA NA
## origin:nametriumph tr7 coupe NA NA
## origin:namevokswagen rabbit NA NA
## origin:namevolkswagen 1131 deluxe sedan NA NA
## origin:namevolkswagen 411 (sw) NA NA
## origin:namevolkswagen dasher NA NA
## origin:namevolkswagen jetta NA NA
## origin:namevolkswagen model 111 NA NA
## origin:namevolkswagen rabbit NA NA
## origin:namevolkswagen rabbit custom NA NA
## origin:namevolkswagen rabbit custom diesel NA NA
## origin:namevolkswagen rabbit l NA NA
## origin:namevolkswagen scirocco NA NA
## origin:namevolkswagen super beetle NA NA
## origin:namevolkswagen type 3 NA NA
## origin:namevolvo 144ea NA NA
## origin:namevolvo 145e (sw) NA NA
## origin:namevolvo 244dl NA NA
## origin:namevolvo 245 NA NA
## origin:namevolvo 264gl NA NA
## origin:namevolvo diesel NA NA
## origin:namevw dasher (diesel) NA NA
## origin:namevw pickup NA NA
## origin:namevw rabbit NA NA
## origin:namevw rabbit c (diesel) NA NA
## origin:namevw rabbit custom NA NA
## t value Pr(>|t|)
## (Intercept) NaN NaN
## cylinders NaN NaN
## displacement NaN NaN
## horsepower NaN NaN
## weight NaN NaN
## acceleration NaN NaN
## year NaN NaN
## origin NaN NaN
## cylinders:displacement NaN NaN
## cylinders:horsepower NaN NaN
## cylinders:weight NaN NaN
## cylinders:acceleration NaN NaN
## cylinders:year NaN NaN
## cylinders:origin NaN NaN
## cylinders:nameamc ambassador dpl NaN NaN
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## cylinders:nameamc concord d/l NaN NaN
## cylinders:nameamc concord dl 6 NaN NaN
## cylinders:nameamc gremlin NaN NaN
## cylinders:nameamc hornet NaN NaN
## cylinders:nameamc hornet sportabout (sw) NaN NaN
## cylinders:nameamc matador NaN NaN
## cylinders:nameamc matador (sw) NaN NaN
## cylinders:nameamc pacer NaN NaN
## cylinders:nameamc pacer d/l NaN NaN
## cylinders:nameamc rebel sst NaN NaN
## cylinders:nameamc spirit dl NaN NaN
## cylinders:nameaudi 100 ls NaN NaN
## cylinders:nameaudi 100ls NaN NaN
## cylinders:nameaudi 4000 NaN NaN
## cylinders:nameaudi 5000 NaN NaN
## cylinders:nameaudi 5000s (diesel) NaN NaN
## cylinders:nameaudi fox NaN NaN
## cylinders:namebmw 2002 NaN NaN
## cylinders:namebmw 320i NaN NaN
## cylinders:namebuick century NaN NaN
## cylinders:namebuick century 350 NaN NaN
## cylinders:namebuick century limited NaN NaN
## cylinders:namebuick century luxus (sw) NaN NaN
## cylinders:namebuick century special NaN NaN
## cylinders:namebuick electra 225 custom NaN NaN
## cylinders:namebuick estate wagon (sw) NaN NaN
## cylinders:namebuick lesabre custom NaN NaN
## cylinders:namebuick opel isuzu deluxe NaN NaN
## cylinders:namebuick regal sport coupe (turbo) NaN NaN
## cylinders:namebuick skyhawk NaN NaN
## cylinders:namebuick skylark NaN NaN
## cylinders:namebuick skylark 320 NaN NaN
## cylinders:namebuick skylark limited NaN NaN
## cylinders:namecadillac eldorado NaN NaN
## cylinders:namecadillac seville NaN NaN
## cylinders:namecapri ii NaN NaN
## cylinders:namechevroelt chevelle malibu NaN NaN
## cylinders:namechevrolet bel air NaN NaN
## cylinders:namechevrolet camaro NaN NaN
## cylinders:namechevrolet caprice classic NaN NaN
## cylinders:namechevrolet cavalier NaN NaN
## cylinders:namechevrolet cavalier 2-door NaN NaN
## cylinders:namechevrolet cavalier wagon NaN NaN
## cylinders:namechevrolet chevelle concours (sw) NaN NaN
## cylinders:namechevrolet chevelle malibu NaN NaN
## cylinders:namechevrolet chevelle malibu classic NaN NaN
## cylinders:namechevrolet chevette NaN NaN
## cylinders:namechevrolet citation NaN NaN
## cylinders:namechevrolet concours NaN NaN
## cylinders:namechevrolet impala NaN NaN
## cylinders:namechevrolet malibu NaN NaN
## cylinders:namechevrolet malibu classic (sw) NaN NaN
## cylinders:namechevrolet monte carlo NaN NaN
## cylinders:namechevrolet monte carlo landau NaN NaN
## cylinders:namechevrolet monte carlo s NaN NaN
## cylinders:namechevrolet monza 2+2 NaN NaN
## cylinders:namechevrolet nova NaN NaN
## cylinders:namechevrolet nova custom NaN NaN
## cylinders:namechevrolet vega NaN NaN
## cylinders:namechevrolet vega (sw) NaN NaN
## cylinders:namechevrolet vega 2300 NaN NaN
## cylinders:namechevrolet woody NaN NaN
## cylinders:namechevy c10 NaN NaN
## cylinders:namechevy c20 NaN NaN
## cylinders:namechevy s-10 NaN NaN
## cylinders:namechrysler cordoba NaN NaN
## cylinders:namechrysler lebaron medallion NaN NaN
## cylinders:namechrysler lebaron salon NaN NaN
## cylinders:namechrysler lebaron town @ country (sw) NaN NaN
## cylinders:namechrysler new yorker brougham NaN NaN
## cylinders:namechrysler newport royal NaN NaN
## cylinders:namedatsun 1200 NaN NaN
## cylinders:namedatsun 200-sx NaN NaN
## cylinders:namedatsun 200sx NaN NaN
## cylinders:namedatsun 210 NaN NaN
## cylinders:namedatsun 210 mpg NaN NaN
## cylinders:namedatsun 280-zx NaN NaN
## cylinders:namedatsun 310 NaN NaN
## cylinders:namedatsun 310 gx NaN NaN
## cylinders:namedatsun 510 NaN NaN
## cylinders:namedatsun 510 (sw) NaN NaN
## cylinders:namedatsun 510 hatchback NaN NaN
## cylinders:namedatsun 610 NaN NaN
## cylinders:namedatsun 710 NaN NaN
## cylinders:namedatsun 810 NaN NaN
## cylinders:namedatsun 810 maxima NaN NaN
## cylinders:namedatsun b-210 NaN NaN
## cylinders:namedatsun b210 NaN NaN
## cylinders:namedatsun b210 gx NaN NaN
## cylinders:namedatsun f-10 hatchback NaN NaN
## cylinders:namedatsun pl510 NaN NaN
## cylinders:namedodge aries se NaN NaN
## cylinders:namedodge aries wagon (sw) NaN NaN
## cylinders:namedodge aspen NaN NaN
## cylinders:namedodge aspen 6 NaN NaN
## cylinders:namedodge aspen se NaN NaN
## cylinders:namedodge challenger se NaN NaN
## cylinders:namedodge charger 2.2 NaN NaN
## cylinders:namedodge colt NaN NaN
## cylinders:namedodge colt (sw) NaN NaN
## cylinders:namedodge colt hardtop NaN NaN
## cylinders:namedodge colt hatchback custom NaN NaN
## cylinders:namedodge colt m/m NaN NaN
## cylinders:namedodge coronet brougham NaN NaN
## cylinders:namedodge coronet custom NaN NaN
## cylinders:namedodge coronet custom (sw) NaN NaN
## cylinders:namedodge d100 NaN NaN
## cylinders:namedodge d200 NaN NaN
## cylinders:namedodge dart custom NaN NaN
## cylinders:namedodge diplomat NaN NaN
## cylinders:namedodge magnum xe NaN NaN
## cylinders:namedodge monaco (sw) NaN NaN
## cylinders:namedodge monaco brougham NaN NaN
## cylinders:namedodge omni NaN NaN
## cylinders:namedodge rampage NaN NaN
## cylinders:namedodge st. regis NaN NaN
## cylinders:namefiat 124 sport coupe NaN NaN
## cylinders:namefiat 124 tc NaN NaN
## cylinders:namefiat 124b NaN NaN
## cylinders:namefiat 128 NaN NaN
## cylinders:namefiat 131 NaN NaN
## cylinders:namefiat strada custom NaN NaN
## cylinders:namefiat x1.9 NaN NaN
## cylinders:nameford country NaN NaN
## cylinders:nameford country squire (sw) NaN NaN
## cylinders:nameford escort 2h NaN NaN
## cylinders:nameford escort 4w NaN NaN
## cylinders:nameford f108 NaN NaN
## cylinders:nameford f250 NaN NaN
## cylinders:nameford fairmont NaN NaN
## cylinders:nameford fairmont (auto) NaN NaN
## cylinders:nameford fairmont (man) NaN NaN
## cylinders:nameford fairmont 4 NaN NaN
## cylinders:nameford fairmont futura NaN NaN
## cylinders:nameford fiesta NaN NaN
## cylinders:nameford futura NaN NaN
## cylinders:nameford galaxie 500 NaN NaN
## cylinders:nameford gran torino NaN NaN
## cylinders:nameford gran torino (sw) NaN NaN
## cylinders:nameford granada NaN NaN
## cylinders:nameford granada ghia NaN NaN
## cylinders:nameford granada gl NaN NaN
## cylinders:nameford granada l NaN NaN
## cylinders:nameford ltd NaN NaN
## cylinders:nameford ltd landau NaN NaN
## cylinders:nameford maverick NaN NaN
## cylinders:nameford mustang NaN NaN
## cylinders:nameford mustang gl NaN NaN
## cylinders:nameford mustang ii NaN NaN
## cylinders:nameford mustang ii 2+2 NaN NaN
## cylinders:nameford pinto NaN NaN
## cylinders:nameford pinto (sw) NaN NaN
## cylinders:nameford pinto runabout NaN NaN
## cylinders:nameford ranger NaN NaN
## cylinders:nameford thunderbird NaN NaN
## cylinders:nameford torino NaN NaN
## cylinders:nameford torino 500 NaN NaN
## cylinders:namehi 1200d NaN NaN
## cylinders:namehonda accord NaN NaN
## cylinders:namehonda accord cvcc NaN NaN
## cylinders:namehonda accord lx NaN NaN
## cylinders:namehonda civic NaN NaN
## cylinders:namehonda civic (auto) NaN NaN
## cylinders:namehonda civic 1300 NaN NaN
## cylinders:namehonda civic 1500 gl NaN NaN
## cylinders:namehonda civic cvcc NaN NaN
## cylinders:namehonda prelude NaN NaN
## cylinders:namemaxda glc deluxe NaN NaN
## cylinders:namemaxda rx3 NaN NaN
## cylinders:namemazda 626 NaN NaN
## cylinders:namemazda glc NaN NaN
## cylinders:namemazda glc 4 NaN NaN
## cylinders:namemazda glc custom NaN NaN
## cylinders:namemazda glc custom l NaN NaN
## cylinders:namemazda glc deluxe NaN NaN
## cylinders:namemazda rx-4 NaN NaN
## cylinders:namemazda rx-7 gs NaN NaN
## cylinders:namemazda rx2 coupe NaN NaN
## cylinders:namemercedes benz 300d NaN NaN
## cylinders:namemercedes-benz 240d NaN NaN
## cylinders:namemercedes-benz 280s NaN NaN
## cylinders:namemercury capri 2000 NaN NaN
## cylinders:namemercury capri v6 NaN NaN
## cylinders:namemercury cougar brougham NaN NaN
## cylinders:namemercury grand marquis NaN NaN
## cylinders:namemercury lynx l NaN NaN
## cylinders:namemercury marquis NaN NaN
## cylinders:namemercury marquis brougham NaN NaN
## cylinders:namemercury monarch NaN NaN
## cylinders:namemercury monarch ghia NaN NaN
## cylinders:namemercury zephyr NaN NaN
## cylinders:namemercury zephyr 6 NaN NaN
## cylinders:namenissan stanza xe NaN NaN
## cylinders:nameoldsmobile cutlass ciera (diesel) NaN NaN
## cylinders:nameoldsmobile cutlass ls NaN NaN
## cylinders:nameoldsmobile cutlass salon brougham NaN NaN
## cylinders:nameoldsmobile cutlass supreme NaN NaN
## cylinders:nameoldsmobile delta 88 royale NaN NaN
## cylinders:nameoldsmobile omega NaN NaN
## cylinders:nameoldsmobile omega brougham NaN NaN
## cylinders:nameoldsmobile starfire sx NaN NaN
## cylinders:nameoldsmobile vista cruiser NaN NaN
## cylinders:nameopel 1900 NaN NaN
## cylinders:nameopel manta NaN NaN
## cylinders:namepeugeot 304 NaN NaN
## cylinders:namepeugeot 504 NaN NaN
## cylinders:namepeugeot 504 (sw) NaN NaN
## cylinders:namepeugeot 505s turbo diesel NaN NaN
## cylinders:namepeugeot 604sl NaN NaN
## cylinders:nameplymouth 'cuda 340 NaN NaN
## cylinders:nameplymouth arrow gs NaN NaN
## cylinders:nameplymouth champ NaN NaN
## cylinders:nameplymouth cricket NaN NaN
## cylinders:nameplymouth custom suburb NaN NaN
## cylinders:nameplymouth duster NaN NaN
## cylinders:nameplymouth fury NaN NaN
## cylinders:nameplymouth fury gran sedan NaN NaN
## cylinders:nameplymouth fury iii NaN NaN
## cylinders:nameplymouth grand fury NaN NaN
## cylinders:nameplymouth horizon NaN NaN
## cylinders:nameplymouth horizon 4 NaN NaN
## cylinders:nameplymouth horizon miser NaN NaN
## cylinders:nameplymouth horizon tc3 NaN NaN
## cylinders:nameplymouth reliant NaN NaN
## cylinders:nameplymouth sapporo NaN NaN
## cylinders:nameplymouth satellite NaN NaN
## cylinders:nameplymouth satellite custom NaN NaN
## cylinders:nameplymouth satellite custom (sw) NaN NaN
## cylinders:nameplymouth satellite sebring NaN NaN
## cylinders:nameplymouth valiant NaN NaN
## cylinders:nameplymouth valiant custom NaN NaN
## cylinders:nameplymouth volare NaN NaN
## cylinders:nameplymouth volare custom NaN NaN
## cylinders:nameplymouth volare premier v8 NaN NaN
## cylinders:namepontiac astro NaN NaN
## cylinders:namepontiac catalina NaN NaN
## cylinders:namepontiac catalina brougham NaN NaN
## cylinders:namepontiac firebird NaN NaN
## cylinders:namepontiac grand prix NaN NaN
## cylinders:namepontiac grand prix lj NaN NaN
## cylinders:namepontiac j2000 se hatchback NaN NaN
## cylinders:namepontiac lemans v6 NaN NaN
## cylinders:namepontiac phoenix NaN NaN
## cylinders:namepontiac phoenix lj NaN NaN
## cylinders:namepontiac safari (sw) NaN NaN
## cylinders:namepontiac sunbird coupe NaN NaN
## cylinders:namepontiac ventura sj NaN NaN
## cylinders:namerenault 12 (sw) NaN NaN
## cylinders:namerenault 12tl NaN NaN
## cylinders:namerenault 5 gtl NaN NaN
## cylinders:namesaab 99e NaN NaN
## cylinders:namesaab 99gle NaN NaN
## cylinders:namesaab 99le NaN NaN
## cylinders:namesubaru NaN NaN
## cylinders:namesubaru dl NaN NaN
## cylinders:nametoyota carina NaN NaN
## cylinders:nametoyota celica gt NaN NaN
## cylinders:nametoyota celica gt liftback NaN NaN
## cylinders:nametoyota corolla NaN NaN
## cylinders:nametoyota corolla 1200 NaN NaN
## cylinders:nametoyota corolla 1600 (sw) NaN NaN
## cylinders:nametoyota corolla liftback NaN NaN
## cylinders:nametoyota corolla tercel NaN NaN
## cylinders:nametoyota corona NaN NaN
## cylinders:nametoyota corona hardtop NaN NaN
## cylinders:nametoyota corona liftback NaN NaN
## cylinders:nametoyota corona mark ii NaN NaN
## cylinders:nametoyota cressida NaN NaN
## cylinders:nametoyota mark ii NaN NaN
## cylinders:nametoyota starlet NaN NaN
## cylinders:nametoyota tercel NaN NaN
## cylinders:nametoyouta corona mark ii (sw) NaN NaN
## cylinders:nametriumph tr7 coupe NaN NaN
## cylinders:namevokswagen rabbit NaN NaN
## cylinders:namevolkswagen 1131 deluxe sedan NaN NaN
## cylinders:namevolkswagen 411 (sw) NaN NaN
## cylinders:namevolkswagen dasher NaN NaN
## cylinders:namevolkswagen jetta NaN NaN
## cylinders:namevolkswagen model 111 NaN NaN
## cylinders:namevolkswagen rabbit NaN NaN
## cylinders:namevolkswagen rabbit custom NaN NaN
## cylinders:namevolkswagen rabbit custom diesel NaN NaN
## cylinders:namevolkswagen rabbit l NaN NaN
## cylinders:namevolkswagen scirocco NaN NaN
## cylinders:namevolkswagen super beetle NaN NaN
## cylinders:namevolkswagen type 3 NaN NaN
## cylinders:namevolvo 144ea NaN NaN
## cylinders:namevolvo 145e (sw) NaN NaN
## cylinders:namevolvo 244dl NaN NaN
## cylinders:namevolvo 245 NaN NaN
## cylinders:namevolvo 264gl NaN NaN
## cylinders:namevolvo diesel NA NA
## cylinders:namevw dasher (diesel) NaN NaN
## cylinders:namevw pickup NaN NaN
## cylinders:namevw rabbit NaN NaN
## cylinders:namevw rabbit c (diesel) NaN NaN
## cylinders:namevw rabbit custom NA NA
## displacement:horsepower NaN NaN
## displacement:weight NaN NaN
## displacement:acceleration NaN NaN
## displacement:year NaN NaN
## displacement:origin NaN NaN
## displacement:nameamc ambassador dpl NA NA
## displacement:nameamc ambassador sst NA NA
## displacement:nameamc concord NaN NaN
## displacement:nameamc concord d/l NA NA
## displacement:nameamc concord dl 6 NA NA
## displacement:nameamc gremlin NaN NaN
## displacement:nameamc hornet NaN NaN
## displacement:nameamc hornet sportabout (sw) NA NA
## displacement:nameamc matador NaN NaN
## displacement:nameamc matador (sw) NA NA
## displacement:nameamc pacer NA NA
## displacement:nameamc pacer d/l NA NA
## displacement:nameamc rebel sst NA NA
## displacement:nameamc spirit dl NA NA
## displacement:nameaudi 100 ls NA NA
## displacement:nameaudi 100ls NaN NaN
## displacement:nameaudi 4000 NA NA
## displacement:nameaudi 5000 NA NA
## displacement:nameaudi 5000s (diesel) NA NA
## displacement:nameaudi fox NA NA
## displacement:namebmw 2002 NA NA
## displacement:namebmw 320i NA NA
## displacement:namebuick century NA NA
## displacement:namebuick century 350 NA NA
## displacement:namebuick century limited NA NA
## displacement:namebuick century luxus (sw) NA NA
## displacement:namebuick century special NA NA
## displacement:namebuick electra 225 custom NA NA
## displacement:namebuick estate wagon (sw) NaN NaN
## displacement:namebuick lesabre custom NA NA
## displacement:namebuick opel isuzu deluxe NA NA
## displacement:namebuick regal sport coupe (turbo) NA NA
## displacement:namebuick skyhawk NA NA
## displacement:namebuick skylark NaN NaN
## displacement:namebuick skylark 320 NA NA
## displacement:namebuick skylark limited NA NA
## displacement:namecadillac eldorado NA NA
## displacement:namecadillac seville NA NA
## displacement:namecapri ii NA NA
## displacement:namechevroelt chevelle malibu NA NA
## displacement:namechevrolet bel air NA NA
## displacement:namechevrolet camaro NA NA
## displacement:namechevrolet caprice classic NaN NaN
## displacement:namechevrolet cavalier NA NA
## displacement:namechevrolet cavalier 2-door NA NA
## displacement:namechevrolet cavalier wagon NA NA
## displacement:namechevrolet chevelle concours (sw) NA NA
## displacement:namechevrolet chevelle malibu NaN NaN
## displacement:namechevrolet chevelle malibu classic NaN NaN
## displacement:namechevrolet chevette NaN NaN
## displacement:namechevrolet citation NaN NaN
## displacement:namechevrolet concours NA NA
## displacement:namechevrolet impala NaN NaN
## displacement:namechevrolet malibu NaN NaN
## displacement:namechevrolet malibu classic (sw) NA NA
## displacement:namechevrolet monte carlo NA NA
## displacement:namechevrolet monte carlo landau NaN NaN
## displacement:namechevrolet monte carlo s NA NA
## displacement:namechevrolet monza 2+2 NA NA
## displacement:namechevrolet nova NA NA
## displacement:namechevrolet nova custom NA NA
## displacement:namechevrolet vega NA NA
## displacement:namechevrolet vega (sw) NA NA
## displacement:namechevrolet vega 2300 NA NA
## displacement:namechevrolet woody NA NA
## displacement:namechevy c10 NA NA
## displacement:namechevy c20 NA NA
## displacement:namechevy s-10 NA NA
## displacement:namechrysler cordoba NA NA
## displacement:namechrysler lebaron medallion NA NA
## displacement:namechrysler lebaron salon NA NA
## displacement:namechrysler lebaron town @ country (sw) NA NA
## displacement:namechrysler new yorker brougham NA NA
## displacement:namechrysler newport royal NA NA
## displacement:namedatsun 1200 NA NA
## displacement:namedatsun 200-sx NA NA
## displacement:namedatsun 200sx NA NA
## displacement:namedatsun 210 NA NA
## displacement:namedatsun 210 mpg NA NA
## displacement:namedatsun 280-zx NA NA
## displacement:namedatsun 310 NA NA
## displacement:namedatsun 310 gx NA NA
## displacement:namedatsun 510 NA NA
## displacement:namedatsun 510 (sw) NA NA
## displacement:namedatsun 510 hatchback NA NA
## displacement:namedatsun 610 NA NA
## displacement:namedatsun 710 NaN NaN
## displacement:namedatsun 810 NA NA
## displacement:namedatsun 810 maxima NA NA
## displacement:namedatsun b-210 NA NA
## displacement:namedatsun b210 NA NA
## displacement:namedatsun b210 gx NA NA
## displacement:namedatsun f-10 hatchback NA NA
## displacement:namedatsun pl510 NA NA
## displacement:namedodge aries se NA NA
## displacement:namedodge aries wagon (sw) NA NA
## displacement:namedodge aspen NA NA
## displacement:namedodge aspen 6 NA NA
## displacement:namedodge aspen se NA NA
## displacement:namedodge challenger se NA NA
## displacement:namedodge charger 2.2 NA NA
## displacement:namedodge colt NaN NaN
## displacement:namedodge colt (sw) NA NA
## displacement:namedodge colt hardtop NA NA
## displacement:namedodge colt hatchback custom NA NA
## displacement:namedodge colt m/m NA NA
## displacement:namedodge coronet brougham NA NA
## displacement:namedodge coronet custom NA NA
## displacement:namedodge coronet custom (sw) NA NA
## displacement:namedodge d100 NA NA
## displacement:namedodge d200 NA NA
## displacement:namedodge dart custom NA NA
## displacement:namedodge diplomat NA NA
## displacement:namedodge magnum xe NA NA
## displacement:namedodge monaco (sw) NA NA
## displacement:namedodge monaco brougham NA NA
## displacement:namedodge omni NA NA
## displacement:namedodge rampage NA NA
## displacement:namedodge st. regis NA NA
## displacement:namefiat 124 sport coupe NA NA
## displacement:namefiat 124 tc NA NA
## displacement:namefiat 124b NA NA
## displacement:namefiat 128 NaN NaN
## displacement:namefiat 131 NA NA
## displacement:namefiat strada custom NA NA
## displacement:namefiat x1.9 NA NA
## displacement:nameford country NA NA
## displacement:nameford country squire (sw) NaN NaN
## displacement:nameford escort 2h NA NA
## displacement:nameford escort 4w NA NA
## displacement:nameford f108 NA NA
## displacement:nameford f250 NA NA
## displacement:nameford fairmont NA NA
## displacement:nameford fairmont (auto) NA NA
## displacement:nameford fairmont (man) NA NA
## displacement:nameford fairmont 4 NA NA
## displacement:nameford fairmont futura NA NA
## displacement:nameford fiesta NA NA
## displacement:nameford futura NA NA
## displacement:nameford galaxie 500 NaN NaN
## displacement:nameford gran torino NaN NaN
## displacement:nameford gran torino (sw) NA NA
## displacement:nameford granada NA NA
## displacement:nameford granada ghia NA NA
## displacement:nameford granada gl NA NA
## displacement:nameford granada l NA NA
## displacement:nameford ltd NA NA
## displacement:nameford ltd landau NA NA
## displacement:nameford maverick NaN NaN
## displacement:nameford mustang NA NA
## displacement:nameford mustang gl NA NA
## displacement:nameford mustang ii NA NA
## displacement:nameford mustang ii 2+2 NA NA
## displacement:nameford pinto NaN NaN
## displacement:nameford pinto (sw) NA NA
## displacement:nameford pinto runabout NA NA
## displacement:nameford ranger NA NA
## displacement:nameford thunderbird NA NA
## displacement:nameford torino NA NA
## displacement:nameford torino 500 NA NA
## displacement:namehi 1200d NA NA
## displacement:namehonda accord NA NA
## displacement:namehonda accord cvcc NA NA
## displacement:namehonda accord lx NA NA
## displacement:namehonda civic NaN NaN
## displacement:namehonda civic (auto) NA NA
## displacement:namehonda civic 1300 NA NA
## displacement:namehonda civic 1500 gl NA NA
## displacement:namehonda civic cvcc NA NA
## displacement:namehonda prelude NA NA
## displacement:namemaxda glc deluxe NA NA
## displacement:namemaxda rx3 NA NA
## displacement:namemazda 626 NA NA
## displacement:namemazda glc NA NA
## displacement:namemazda glc 4 NA NA
## displacement:namemazda glc custom NA NA
## displacement:namemazda glc custom l NA NA
## displacement:namemazda glc deluxe NA NA
## displacement:namemazda rx-4 NA NA
## displacement:namemazda rx-7 gs NA NA
## displacement:namemazda rx2 coupe NA NA
## displacement:namemercedes benz 300d NA NA
## displacement:namemercedes-benz 240d NA NA
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## year:namemercury zephyr NA NA
## year:namemercury zephyr 6 NA NA
## year:namenissan stanza xe NA NA
## year:nameoldsmobile cutlass ciera (diesel) NA NA
## year:nameoldsmobile cutlass ls NA NA
## year:nameoldsmobile cutlass salon brougham NA NA
## year:nameoldsmobile cutlass supreme NA NA
## year:nameoldsmobile delta 88 royale NA NA
## year:nameoldsmobile omega NA NA
## year:nameoldsmobile omega brougham NA NA
## year:nameoldsmobile starfire sx NA NA
## year:nameoldsmobile vista cruiser NA NA
## year:nameopel 1900 NA NA
## year:nameopel manta NA NA
## year:namepeugeot 304 NA NA
## year:namepeugeot 504 NA NA
## year:namepeugeot 504 (sw) NA NA
## year:namepeugeot 505s turbo diesel NA NA
## year:namepeugeot 604sl NA NA
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## year:nameplymouth arrow gs NA NA
## year:nameplymouth champ NA NA
## year:nameplymouth cricket NA NA
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## year:nameplymouth duster NA NA
## year:nameplymouth fury NA NA
## year:nameplymouth fury gran sedan NA NA
## year:nameplymouth fury iii NA NA
## year:nameplymouth grand fury NA NA
## year:nameplymouth horizon NA NA
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## year:nameplymouth reliant NA NA
## year:nameplymouth sapporo NA NA
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## year:namepontiac phoenix NA NA
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## year:namepontiac safari (sw) NA NA
## year:namepontiac sunbird coupe NA NA
## year:namepontiac ventura sj NA NA
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## year:namerenault 12tl NA NA
## year:namerenault 5 gtl NA NA
## year:namesaab 99e NA NA
## year:namesaab 99gle NA NA
## year:namesaab 99le NA NA
## year:namesubaru NA NA
## year:namesubaru dl NA NA
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## year:nametoyota celica gt NA NA
## year:nametoyota celica gt liftback NA NA
## year:nametoyota corolla NA NA
## year:nametoyota corolla 1200 NA NA
## year:nametoyota corolla 1600 (sw) NA NA
## year:nametoyota corolla liftback NA NA
## year:nametoyota corolla tercel NA NA
## year:nametoyota corona NA NA
## year:nametoyota corona hardtop NA NA
## year:nametoyota corona liftback NA NA
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## year:nametoyota cressida NA NA
## year:nametoyota mark ii NA NA
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## year:namevolkswagen jetta NA NA
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## year:namevolkswagen rabbit NA NA
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## year:namevolkswagen scirocco NA NA
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## year:namevolvo 245 NA NA
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## origin:namechrysler lebaron medallion NA NA
## origin:namechrysler lebaron salon NA NA
## origin:namechrysler lebaron town @ country (sw) NA NA
## origin:namechrysler new yorker brougham NA NA
## origin:namechrysler newport royal NA NA
## origin:namedatsun 1200 NA NA
## origin:namedatsun 200-sx NA NA
## origin:namedatsun 200sx NA NA
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## origin:namedatsun 210 mpg NA NA
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## origin:namedatsun 510 hatchback NA NA
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## origin:nameoldsmobile delta 88 royale NA NA
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## origin:nameoldsmobile starfire sx NA NA
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## origin:nameopel 1900 NA NA
## origin:nameopel manta NA NA
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## origin:nameplymouth cricket NA NA
## origin:nameplymouth custom suburb NA NA
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## origin:nameplymouth fury gran sedan NA NA
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## origin:namepontiac phoenix NA NA
## origin:namepontiac phoenix lj NA NA
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## origin:namepontiac sunbird coupe NA NA
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## origin:namerenault 12tl NA NA
## origin:namerenault 5 gtl NA NA
## origin:namesaab 99e NA NA
## origin:namesaab 99gle NA NA
## origin:namesaab 99le NA NA
## origin:namesubaru NA NA
## origin:namesubaru dl NA NA
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## origin:nametoyota corona liftback NA NA
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## origin:namevolkswagen dasher NA NA
## origin:namevolkswagen jetta NA NA
## origin:namevolkswagen model 111 NA NA
## origin:namevolkswagen rabbit NA NA
## origin:namevolkswagen rabbit custom NA NA
## origin:namevolkswagen rabbit custom diesel NA NA
## origin:namevolkswagen rabbit l NA NA
## origin:namevolkswagen scirocco NA NA
## origin:namevolkswagen super beetle NA NA
## origin:namevolkswagen type 3 NA NA
## origin:namevolvo 144ea NA NA
## origin:namevolvo 145e (sw) NA NA
## origin:namevolvo 244dl NA NA
## origin:namevolvo 245 NA NA
## origin:namevolvo 264gl NA NA
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## origin:namevw dasher (diesel) NA NA
## origin:namevw pickup NA NA
## origin:namevw rabbit NA NA
## origin:namevw rabbit c (diesel) NA NA
## origin:namevw rabbit custom NA NA
##
## Residual standard error: NaN on 0 degrees of freedom
## Multiple R-squared: 1, Adjusted R-squared: NaN
## F-statistic: NaN on 391 and 0 DF, p-value: NABy including interaction terms, the model can capture relationships between variables.
Some interaction terms may become statistically significant, indicating that the effect of one variable depends on another.
However: - The model becomes more complex
- Interpretation becomes more difficult
This step shows that interactions can improve flexibility, but must be used carefully.
(f) Transformations
Finally, we apply transformations to variables to improve model performance. Common transformations include logarithmic, square root, and polynomial terms.
model_transform <- lm(log(mpg) ~ horsepower + I(horsepower^2) + weight + year, data = Auto)
summary(model_transform)##
## Call:
## lm(formula = log(mpg) ~ horsepower + I(horsepower^2) + weight +
## year, data = Auto)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.38454 -0.06569 0.00547 0.07036 0.34537
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.995e+00 1.479e-01 13.484 < 2e-16 ***
## horsepower -6.717e-03 1.030e-03 -6.523 2.15e-10 ***
## I(horsepower^2) 2.088e-05 3.496e-06 5.971 5.33e-09 ***
## weight -2.441e-04 1.491e-05 -16.365 < 2e-16 ***
## year 2.992e-02 1.771e-03 16.890 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1166 on 387 degrees of freedom
## Multiple R-squared: 0.8837, Adjusted R-squared: 0.8825
## F-statistic: 734.9 on 4 and 387 DF, p-value: < 2.2e-16Transformations help address non-linear relationships:
- Log transformation stabilizes variance
- Polynomial terms capture curved relationships
For example: - The squared term of horsepower allows the model to represent non-linear effects of engine power on fuel efficiency
If the transformed model shows improved fit (higher R-squared, better residual behavior), it suggests that the relationship is not purely linear.
This highlights the importance of exploring alternative model forms.
Visualization – Non-linear Relationship
ggplot(Auto, aes(horsepower, mpg)) +
geom_point(alpha = 0.3) +
geom_smooth(method = "loess") +
geom_smooth(method = "lm", linetype = "dashed", color = "red")
Chapter 3 – Question 15 (Boston Dataset)
(a) Fit Multiple Linear Regression Model
We begin by fitting a multiple linear regression model using all
predictors to explain the response variable crim (per
capita crime rate). This model helps us understand how all variables
jointly influence crime rate.
library(ISLR2)
# Load dataset
Boston <- Boston
# Fit model
model_full <- lm(crim ~ ., data = Boston)
summary(model_full)##
## Call:
## lm(formula = crim ~ ., data = Boston)
##
## Residuals:
## Min 1Q Median 3Q Max
## -8.534 -2.248 -0.348 1.087 73.923
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 13.7783938 7.0818258 1.946 0.052271 .
## zn 0.0457100 0.0187903 2.433 0.015344 *
## indus -0.0583501 0.0836351 -0.698 0.485709
## chas -0.8253776 1.1833963 -0.697 0.485841
## nox -9.9575865 5.2898242 -1.882 0.060370 .
## rm 0.6289107 0.6070924 1.036 0.300738
## age -0.0008483 0.0179482 -0.047 0.962323
## dis -1.0122467 0.2824676 -3.584 0.000373 ***
## rad 0.6124653 0.0875358 6.997 8.59e-12 ***
## tax -0.0037756 0.0051723 -0.730 0.465757
## ptratio -0.3040728 0.1863598 -1.632 0.103393
## lstat 0.1388006 0.0757213 1.833 0.067398 .
## medv -0.2200564 0.0598240 -3.678 0.000261 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 6.46 on 493 degrees of freedom
## Multiple R-squared: 0.4493, Adjusted R-squared: 0.4359
## F-statistic: 33.52 on 12 and 493 DF, p-value: < 2.2e-16The regression output provides important information:
- The F-statistic with a very small p-value indicates that the model
is statistically significant
- This means that at least one predictor is related to the crime rate
Several variables may appear significant, suggesting they have a meaningful impact on crime levels.
However, not all predictors contribute equally, and some may not be statistically significant.
(b) Simple Linear Regression (One Predictor at a Time)
Next, we fit simple linear regression models using each predictor individually to examine their isolated relationship with crime rate.
simple_models <- list()
for (var in names(Boston)[-1]) {
formula <- as.formula(paste("crim ~", var))
simple_models[[var]] <- summary(lm(formula, data = Boston))
}
simple_models## $zn
##
## Call:
## lm(formula = formula, data = Boston)
##
## Residuals:
## Min 1Q Median 3Q Max
## -4.429 -4.222 -2.620 1.250 84.523
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 4.45369 0.41722 10.675 < 2e-16 ***
## zn -0.07393 0.01609 -4.594 5.51e-06 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 8.435 on 504 degrees of freedom
## Multiple R-squared: 0.04019, Adjusted R-squared: 0.03828
## F-statistic: 21.1 on 1 and 504 DF, p-value: 5.506e-06
##
##
## $indus
##
## Call:
## lm(formula = formula, data = Boston)
##
## Residuals:
## Min 1Q Median 3Q Max
## -11.972 -2.698 -0.736 0.712 81.813
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -2.06374 0.66723 -3.093 0.00209 **
## indus 0.50978 0.05102 9.991 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 7.866 on 504 degrees of freedom
## Multiple R-squared: 0.1653, Adjusted R-squared: 0.1637
## F-statistic: 99.82 on 1 and 504 DF, p-value: < 2.2e-16
##
##
## $chas
##
## Call:
## lm(formula = formula, data = Boston)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.738 -3.661 -3.435 0.018 85.232
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 3.7444 0.3961 9.453 <2e-16 ***
## chas -1.8928 1.5061 -1.257 0.209
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 8.597 on 504 degrees of freedom
## Multiple R-squared: 0.003124, Adjusted R-squared: 0.001146
## F-statistic: 1.579 on 1 and 504 DF, p-value: 0.2094
##
##
## $nox
##
## Call:
## lm(formula = formula, data = Boston)
##
## Residuals:
## Min 1Q Median 3Q Max
## -12.371 -2.738 -0.974 0.559 81.728
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -13.720 1.699 -8.073 5.08e-15 ***
## nox 31.249 2.999 10.419 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 7.81 on 504 degrees of freedom
## Multiple R-squared: 0.1772, Adjusted R-squared: 0.1756
## F-statistic: 108.6 on 1 and 504 DF, p-value: < 2.2e-16
##
##
## $rm
##
## Call:
## lm(formula = formula, data = Boston)
##
## Residuals:
## Min 1Q Median 3Q Max
## -6.604 -3.952 -2.654 0.989 87.197
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 20.482 3.365 6.088 2.27e-09 ***
## rm -2.684 0.532 -5.045 6.35e-07 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 8.401 on 504 degrees of freedom
## Multiple R-squared: 0.04807, Adjusted R-squared: 0.04618
## F-statistic: 25.45 on 1 and 504 DF, p-value: 6.347e-07
##
##
## $age
##
## Call:
## lm(formula = formula, data = Boston)
##
## Residuals:
## Min 1Q Median 3Q Max
## -6.789 -4.257 -1.230 1.527 82.849
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -3.77791 0.94398 -4.002 7.22e-05 ***
## age 0.10779 0.01274 8.463 2.85e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 8.057 on 504 degrees of freedom
## Multiple R-squared: 0.1244, Adjusted R-squared: 0.1227
## F-statistic: 71.62 on 1 and 504 DF, p-value: 2.855e-16
##
##
## $dis
##
## Call:
## lm(formula = formula, data = Boston)
##
## Residuals:
## Min 1Q Median 3Q Max
## -6.708 -4.134 -1.527 1.516 81.674
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 9.4993 0.7304 13.006 <2e-16 ***
## dis -1.5509 0.1683 -9.213 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 7.965 on 504 degrees of freedom
## Multiple R-squared: 0.1441, Adjusted R-squared: 0.1425
## F-statistic: 84.89 on 1 and 504 DF, p-value: < 2.2e-16
##
##
## $rad
##
## Call:
## lm(formula = formula, data = Boston)
##
## Residuals:
## Min 1Q Median 3Q Max
## -10.164 -1.381 -0.141 0.660 76.433
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -2.28716 0.44348 -5.157 3.61e-07 ***
## rad 0.61791 0.03433 17.998 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 6.718 on 504 degrees of freedom
## Multiple R-squared: 0.3913, Adjusted R-squared: 0.39
## F-statistic: 323.9 on 1 and 504 DF, p-value: < 2.2e-16
##
##
## $tax
##
## Call:
## lm(formula = formula, data = Boston)
##
## Residuals:
## Min 1Q Median 3Q Max
## -12.513 -2.738 -0.194 1.065 77.696
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -8.528369 0.815809 -10.45 <2e-16 ***
## tax 0.029742 0.001847 16.10 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 6.997 on 504 degrees of freedom
## Multiple R-squared: 0.3396, Adjusted R-squared: 0.3383
## F-statistic: 259.2 on 1 and 504 DF, p-value: < 2.2e-16
##
##
## $ptratio
##
## Call:
## lm(formula = formula, data = Boston)
##
## Residuals:
## Min 1Q Median 3Q Max
## -7.654 -3.985 -1.912 1.825 83.353
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -17.6469 3.1473 -5.607 3.40e-08 ***
## ptratio 1.1520 0.1694 6.801 2.94e-11 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 8.24 on 504 degrees of freedom
## Multiple R-squared: 0.08407, Adjusted R-squared: 0.08225
## F-statistic: 46.26 on 1 and 504 DF, p-value: 2.943e-11
##
##
## $lstat
##
## Call:
## lm(formula = formula, data = Boston)
##
## Residuals:
## Min 1Q Median 3Q Max
## -13.925 -2.822 -0.664 1.079 82.862
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -3.33054 0.69376 -4.801 2.09e-06 ***
## lstat 0.54880 0.04776 11.491 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 7.664 on 504 degrees of freedom
## Multiple R-squared: 0.2076, Adjusted R-squared: 0.206
## F-statistic: 132 on 1 and 504 DF, p-value: < 2.2e-16
##
##
## $medv
##
## Call:
## lm(formula = formula, data = Boston)
##
## Residuals:
## Min 1Q Median 3Q Max
## -9.071 -4.022 -2.343 1.298 80.957
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 11.79654 0.93419 12.63 <2e-16 ***
## medv -0.36316 0.03839 -9.46 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 7.934 on 504 degrees of freedom
## Multiple R-squared: 0.1508, Adjusted R-squared: 0.1491
## F-statistic: 89.49 on 1 and 504 DF, p-value: < 2.2e-16Each predictor shows a different level of influence when considered individually. Some variables appear significant in isolation, suggesting a strong relationship with crime rate. However, these results do not account for interactions or correlations with other predictors.
(c) Compare Simple vs Multiple Regression
We now compare coefficients from simple regression models with those from the multiple regression model to understand how relationships change when controlling for other variables.
# Simple coefficients
simple_coefs <- sapply(names(Boston)[-1], function(var) {
coef(lm(as.formula(paste("crim ~", var)), data = Boston))[2]
})
# Multiple regression coefficients
multi_coefs <- coef(model_full)[-1]
# Combine into dataframe
comparison <- data.frame(
Variable = names(simple_coefs),
Simple = simple_coefs,
Multiple = multi_coefs[names(simple_coefs)]
)
comparisonThe coefficients differ between simple and multiple regression models.
This occurs because:
- In simple regression, each predictor is evaluated
independently
- In multiple regression, predictors are evaluated while controlling for others
As a result: - A variable may appear important in isolation
- But become less significant when other variables are included
This is a key indication of multicollinearity.
Visualization – Coefficient Comparison
ggplot(comparison, aes(Simple, Multiple)) +
geom_point() +
geom_abline(slope = 1, intercept = 0, color = "red")
(d) Visualization of Coefficients
To better understand the differences, we visualize the coefficients from both models.
plot(simple_coefs, multi_coefs,
xlab = "Simple Regression Coefficients",
ylab = "Multiple Regression Coefficients",
main = "Simple vs Multiple Regression Coefficients")
abline(0, 1, col = "red")
The plot shows how coefficients differ between the two models:
- Points near the diagonal line indicate similar estimates
- Points far from the line indicate large differences
These differences highlight the impact of multicollinearity and interactions between predictors.
This visualization helps us understand how relationships change when multiple variables are considered together.
(e) Non-linear Relationships
Finally, we investigate whether there are non-linear relationships between predictors and the response variable by including polynomial terms.
nonlinear_models <- list()
for (var in names(Boston)[-1]) {
formula <- as.formula(paste("crim ~", var, "+ I(", var, "^2) + I(", var, "^3)"))
nonlinear_models[[var]] <- summary(lm(formula, data = Boston))
}
nonlinear_models## $zn
##
## Call:
## lm(formula = formula, data = Boston)
##
## Residuals:
## Min 1Q Median 3Q Max
## -4.821 -4.614 -1.294 0.473 84.130
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 4.846e+00 4.330e-01 11.192 < 2e-16 ***
## zn -3.322e-01 1.098e-01 -3.025 0.00261 **
## I(zn^2) 6.483e-03 3.861e-03 1.679 0.09375 .
## I(zn^3) -3.776e-05 3.139e-05 -1.203 0.22954
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 8.372 on 502 degrees of freedom
## Multiple R-squared: 0.05824, Adjusted R-squared: 0.05261
## F-statistic: 10.35 on 3 and 502 DF, p-value: 1.281e-06
##
##
## $indus
##
## Call:
## lm(formula = formula, data = Boston)
##
## Residuals:
## Min 1Q Median 3Q Max
## -8.278 -2.514 0.054 0.764 79.713
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 3.6625683 1.5739833 2.327 0.0204 *
## indus -1.9652129 0.4819901 -4.077 5.30e-05 ***
## I(indus^2) 0.2519373 0.0393221 6.407 3.42e-10 ***
## I(indus^3) -0.0069760 0.0009567 -7.292 1.20e-12 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 7.423 on 502 degrees of freedom
## Multiple R-squared: 0.2597, Adjusted R-squared: 0.2552
## F-statistic: 58.69 on 3 and 502 DF, p-value: < 2.2e-16
##
##
## $chas
##
## Call:
## lm(formula = formula, data = Boston)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.738 -3.661 -3.435 0.018 85.232
##
## Coefficients: (2 not defined because of singularities)
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 3.7444 0.3961 9.453 <2e-16 ***
## chas -1.8928 1.5061 -1.257 0.209
## I(chas^2) NA NA NA NA
## I(chas^3) NA NA NA NA
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 8.597 on 504 degrees of freedom
## Multiple R-squared: 0.003124, Adjusted R-squared: 0.001146
## F-statistic: 1.579 on 1 and 504 DF, p-value: 0.2094
##
##
## $nox
##
## Call:
## lm(formula = formula, data = Boston)
##
## Residuals:
## Min 1Q Median 3Q Max
## -9.110 -2.068 -0.255 0.739 78.302
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 233.09 33.64 6.928 1.31e-11 ***
## nox -1279.37 170.40 -7.508 2.76e-13 ***
## I(nox^2) 2248.54 279.90 8.033 6.81e-15 ***
## I(nox^3) -1245.70 149.28 -8.345 6.96e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 7.234 on 502 degrees of freedom
## Multiple R-squared: 0.297, Adjusted R-squared: 0.2928
## F-statistic: 70.69 on 3 and 502 DF, p-value: < 2.2e-16
##
##
## $rm
##
## Call:
## lm(formula = formula, data = Boston)
##
## Residuals:
## Min 1Q Median 3Q Max
## -18.485 -3.468 -2.221 -0.015 87.219
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 112.6246 64.5172 1.746 0.0815 .
## rm -39.1501 31.3115 -1.250 0.2118
## I(rm^2) 4.5509 5.0099 0.908 0.3641
## I(rm^3) -0.1745 0.2637 -0.662 0.5086
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 8.33 on 502 degrees of freedom
## Multiple R-squared: 0.06779, Adjusted R-squared: 0.06222
## F-statistic: 12.17 on 3 and 502 DF, p-value: 1.067e-07
##
##
## $age
##
## Call:
## lm(formula = formula, data = Boston)
##
## Residuals:
## Min 1Q Median 3Q Max
## -9.762 -2.673 -0.516 0.019 82.842
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -2.549e+00 2.769e+00 -0.920 0.35780
## age 2.737e-01 1.864e-01 1.468 0.14266
## I(age^2) -7.230e-03 3.637e-03 -1.988 0.04738 *
## I(age^3) 5.745e-05 2.109e-05 2.724 0.00668 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 7.84 on 502 degrees of freedom
## Multiple R-squared: 0.1742, Adjusted R-squared: 0.1693
## F-statistic: 35.31 on 3 and 502 DF, p-value: < 2.2e-16
##
##
## $dis
##
## Call:
## lm(formula = formula, data = Boston)
##
## Residuals:
## Min 1Q Median 3Q Max
## -10.757 -2.588 0.031 1.267 76.378
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 30.0476 2.4459 12.285 < 2e-16 ***
## dis -15.5543 1.7360 -8.960 < 2e-16 ***
## I(dis^2) 2.4521 0.3464 7.078 4.94e-12 ***
## I(dis^3) -0.1186 0.0204 -5.814 1.09e-08 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 7.331 on 502 degrees of freedom
## Multiple R-squared: 0.2778, Adjusted R-squared: 0.2735
## F-statistic: 64.37 on 3 and 502 DF, p-value: < 2.2e-16
##
##
## $rad
##
## Call:
## lm(formula = formula, data = Boston)
##
## Residuals:
## Min 1Q Median 3Q Max
## -10.381 -0.412 -0.269 0.179 76.217
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -0.605545 2.050108 -0.295 0.768
## rad 0.512736 1.043597 0.491 0.623
## I(rad^2) -0.075177 0.148543 -0.506 0.613
## I(rad^3) 0.003209 0.004564 0.703 0.482
##
## Residual standard error: 6.682 on 502 degrees of freedom
## Multiple R-squared: 0.4, Adjusted R-squared: 0.3965
## F-statistic: 111.6 on 3 and 502 DF, p-value: < 2.2e-16
##
##
## $tax
##
## Call:
## lm(formula = formula, data = Boston)
##
## Residuals:
## Min 1Q Median 3Q Max
## -13.273 -1.389 0.046 0.536 76.950
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.918e+01 1.180e+01 1.626 0.105
## tax -1.533e-01 9.568e-02 -1.602 0.110
## I(tax^2) 3.608e-04 2.425e-04 1.488 0.137
## I(tax^3) -2.204e-07 1.889e-07 -1.167 0.244
##
## Residual standard error: 6.854 on 502 degrees of freedom
## Multiple R-squared: 0.3689, Adjusted R-squared: 0.3651
## F-statistic: 97.8 on 3 and 502 DF, p-value: < 2.2e-16
##
##
## $ptratio
##
## Call:
## lm(formula = formula, data = Boston)
##
## Residuals:
## Min 1Q Median 3Q Max
## -6.833 -4.146 -1.655 1.408 82.697
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 477.18405 156.79498 3.043 0.00246 **
## ptratio -82.36054 27.64394 -2.979 0.00303 **
## I(ptratio^2) 4.63535 1.60832 2.882 0.00412 **
## I(ptratio^3) -0.08476 0.03090 -2.743 0.00630 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 8.122 on 502 degrees of freedom
## Multiple R-squared: 0.1138, Adjusted R-squared: 0.1085
## F-statistic: 21.48 on 3 and 502 DF, p-value: 4.171e-13
##
##
## $lstat
##
## Call:
## lm(formula = formula, data = Boston)
##
## Residuals:
## Min 1Q Median 3Q Max
## -15.234 -2.151 -0.486 0.066 83.353
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.2009656 2.0286452 0.592 0.5541
## lstat -0.4490656 0.4648911 -0.966 0.3345
## I(lstat^2) 0.0557794 0.0301156 1.852 0.0646 .
## I(lstat^3) -0.0008574 0.0005652 -1.517 0.1299
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 7.629 on 502 degrees of freedom
## Multiple R-squared: 0.2179, Adjusted R-squared: 0.2133
## F-statistic: 46.63 on 3 and 502 DF, p-value: < 2.2e-16
##
##
## $medv
##
## Call:
## lm(formula = formula, data = Boston)
##
## Residuals:
## Min 1Q Median 3Q Max
## -24.427 -1.976 -0.437 0.439 73.655
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 53.1655381 3.3563105 15.840 < 2e-16 ***
## medv -5.0948305 0.4338321 -11.744 < 2e-16 ***
## I(medv^2) 0.1554965 0.0171904 9.046 < 2e-16 ***
## I(medv^3) -0.0014901 0.0002038 -7.312 1.05e-12 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 6.569 on 502 degrees of freedom
## Multiple R-squared: 0.4202, Adjusted R-squared: 0.4167
## F-statistic: 121.3 on 3 and 502 DF, p-value: < 2.2e-16Including polynomial terms allows us to detect non-linear relationships.
If higher-order terms (squared or cubic) are significant, it indicates that the relationship is not purely linear.
This means: - Some predictors influence crime rate in a curved or
complex manner
- Linear models alone may not fully capture these patterns
This highlights the importance of exploring non-linear models for better accuracy.
Visualization – Example Non-linear Relationship
ggplot(Boston, aes(lstat, crim)) +
geom_point(alpha = 0.3) +
geom_smooth(method = "loess") +
geom_smooth(method = "lm", linetype = "dashed", color = "red")
Chapter 4
(a) Data Exploration
We begin by exploring the Weekly dataset to understand the variables and identify potential relationships. This step includes both numerical summaries and graphical analysis.
library(ISLR2)
# Load dataset
Weekly <- Weekly
# Structure and summary
str(Weekly)## 'data.frame': 1089 obs. of 9 variables:
## $ Year : num 1990 1990 1990 1990 1990 1990 1990 1990 1990 1990 ...
## $ Lag1 : num 0.816 -0.27 -2.576 3.514 0.712 ...
## $ Lag2 : num 1.572 0.816 -0.27 -2.576 3.514 ...
## $ Lag3 : num -3.936 1.572 0.816 -0.27 -2.576 ...
## $ Lag4 : num -0.229 -3.936 1.572 0.816 -0.27 ...
## $ Lag5 : num -3.484 -0.229 -3.936 1.572 0.816 ...
## $ Volume : num 0.155 0.149 0.16 0.162 0.154 ...
## $ Today : num -0.27 -2.576 3.514 0.712 1.178 ...
## $ Direction: Factor w/ 2 levels "Down","Up": 1 1 2 2 2 1 2 2 2 1 ...summary(Weekly)## Year Lag1 Lag2 Lag3
## Min. :1990 Min. :-18.1950 Min. :-18.1950 Min. :-18.1950
## 1st Qu.:1995 1st Qu.: -1.1540 1st Qu.: -1.1540 1st Qu.: -1.1580
## Median :2000 Median : 0.2410 Median : 0.2410 Median : 0.2410
## Mean :2000 Mean : 0.1506 Mean : 0.1511 Mean : 0.1472
## 3rd Qu.:2005 3rd Qu.: 1.4050 3rd Qu.: 1.4090 3rd Qu.: 1.4090
## Max. :2010 Max. : 12.0260 Max. : 12.0260 Max. : 12.0260
## Lag4 Lag5 Volume Today
## Min. :-18.1950 Min. :-18.1950 Min. :0.08747 Min. :-18.1950
## 1st Qu.: -1.1580 1st Qu.: -1.1660 1st Qu.:0.33202 1st Qu.: -1.1540
## Median : 0.2380 Median : 0.2340 Median :1.00268 Median : 0.2410
## Mean : 0.1458 Mean : 0.1399 Mean :1.57462 Mean : 0.1499
## 3rd Qu.: 1.4090 3rd Qu.: 1.4050 3rd Qu.:2.05373 3rd Qu.: 1.4050
## Max. : 12.0260 Max. : 12.0260 Max. :9.32821 Max. : 12.0260
## Direction
## Down:484
## Up :605
##
##
##
## # Correlation matrix (excluding Direction)
cor(Weekly[, -9])## Year Lag1 Lag2 Lag3 Lag4
## Year 1.00000000 -0.032289274 -0.03339001 -0.03000649 -0.031127923
## Lag1 -0.03228927 1.000000000 -0.07485305 0.05863568 -0.071273876
## Lag2 -0.03339001 -0.074853051 1.00000000 -0.07572091 0.058381535
## Lag3 -0.03000649 0.058635682 -0.07572091 1.00000000 -0.075395865
## Lag4 -0.03112792 -0.071273876 0.05838153 -0.07539587 1.000000000
## Lag5 -0.03051910 -0.008183096 -0.07249948 0.06065717 -0.075675027
## Volume 0.84194162 -0.064951313 -0.08551314 -0.06928771 -0.061074617
## Today -0.03245989 -0.075031842 0.05916672 -0.07124364 -0.007825873
## Lag5 Volume Today
## Year -0.030519101 0.84194162 -0.032459894
## Lag1 -0.008183096 -0.06495131 -0.075031842
## Lag2 -0.072499482 -0.08551314 0.059166717
## Lag3 0.060657175 -0.06928771 -0.071243639
## Lag4 -0.075675027 -0.06107462 -0.007825873
## Lag5 1.000000000 -0.05851741 0.011012698
## Volume -0.058517414 1.00000000 -0.033077783
## Today 0.011012698 -0.03307778 1.000000000From the dataset overview:
- The dataset contains lag variables (Lag1 to Lag5), Volume, Year, and
Direction
- The response variable
Directionis categorical (Up/Down)
From the correlation matrix:
- Most lag variables show weak correlations with each other
- There is no strong linear relationship between predictors
This suggests that predicting market direction may be challenging.
(b) Logistic Regression Model
We now fit a logistic regression model using all predictors to classify the market direction (Up or Down).
model_logistic <- glm(Direction ~ Lag1 + Lag2 + Lag3 + Lag4 + Lag5 + Volume,
data = Weekly,
family = binomial)
summary(model_logistic)##
## Call:
## glm(formula = Direction ~ Lag1 + Lag2 + Lag3 + Lag4 + Lag5 +
## Volume, family = binomial, data = Weekly)
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) 0.26686 0.08593 3.106 0.0019 **
## Lag1 -0.04127 0.02641 -1.563 0.1181
## Lag2 0.05844 0.02686 2.175 0.0296 *
## Lag3 -0.01606 0.02666 -0.602 0.5469
## Lag4 -0.02779 0.02646 -1.050 0.2937
## Lag5 -0.01447 0.02638 -0.549 0.5833
## Volume -0.02274 0.03690 -0.616 0.5377
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 1496.2 on 1088 degrees of freedom
## Residual deviance: 1486.4 on 1082 degrees of freedom
## AIC: 1500.4
##
## Number of Fisher Scoring iterations: 4From the model summary:
- Some lag variables may appear statistically significant
- However, most predictors have relatively high p-values
This indicates that individual predictors do not strongly influence market direction.
The coefficients represent changes in log-odds of the market going Up.
Visualization – Probability Plot
Weekly$prob <- predict(model_logistic, type = "response")
ggplot(Weekly, aes(Lag2, prob, color = Direction)) +
geom_point(alpha = 0.4) +
geom_smooth(method = "loess")
(c) Confusion Matrix & Accuracy
To evaluate the model, we compute predictions and compare them with actual values using a confusion matrix.
# Predicted probabilities
prob <- predict(model_logistic, type = "response")
# Convert to class
pred <- ifelse(prob > 0.5, "Up", "Down")
# Confusion matrix
table(Predicted = pred, Actual = Weekly$Direction)## Actual
## Predicted Down Up
## Down 54 48
## Up 430 557# Accuracy
mean(pred == Weekly$Direction)## [1] 0.5610652The confusion matrix shows the number of correct and incorrect predictions.
The accuracy represents the proportion of correctly classified observations.
Typically: - The model predicts “Up” more often
- Accuracy is only slightly better than random guessing
This indicates that the model has limited predictive power.
Visualization – Confusion Matrix Heatmap
cm <- as.data.frame(table(Predicted = pred, Actual = Weekly$Direction))
ggplot(cm, aes(Actual, Predicted, fill = Freq)) +
geom_tile() +
geom_text(aes(label = Freq), color = "white")
(d) Train-Test Split (1990–2008 vs 2009–2010)
To better evaluate model performance, we split the data into training and testing sets.
- Training set: 1990–2008
- Test set: 2009–2010
# Training data
train <- Weekly$Year <= 2008
# Fit model using only Lag2 (commonly significant)
model_train <- glm(Direction ~ Lag2,
data = Weekly,
family = binomial,
subset = train)
# Predict on test data
prob_test <- predict(model_train, newdata = Weekly[!train, ], type = "response")
pred_test <- ifelse(prob_test > 0.5, "Up", "Down")
# Confusion matrix
table(Predicted = pred_test, Actual = Weekly$Direction[!train])## Actual
## Predicted Down Up
## Down 9 5
## Up 34 56# Accuracy
mean(pred_test == Weekly$Direction[!train])## [1] 0.625Using a train-test split provides a more realistic evaluation.
Results typically show: - Moderate accuracy
- Limited ability to generalize to unseen data
This confirms that the dataset does not contain strong predictive signals.
(e) Alternative Methods (Optional Improvement)
We can explore alternative classification approaches such as Linear Discriminant Analysis (LDA) and Quadratic Discriminant Analysis (QDA).
library(MASS)
# LDA
lda_model <- lda(Direction ~ Lag2, data = Weekly, subset = train)
lda_pred <- predict(lda_model, Weekly[!train, ])$class
mean(lda_pred == Weekly$Direction[!train])## [1] 0.625# QDA
qda_model <- qda(Direction ~ Lag2, data = Weekly, subset = train)
qda_pred <- predict(qda_model, Weekly[!train, ])$class
mean(qda_pred == Weekly$Direction[!train])## [1] 0.5865385Alternative models provide a comparison:
- LDA often performs similarly to logistic regression
- QDA may perform worse due to its flexibility and sensitivity to small datasets
Overall: - None of the models achieve high accuracy
- This indicates weak predictive structure in the data
This highlights the difficulty of predicting financial market direction.
Conclusion
Based on the analysis conducted across all chapters, several key findings can be summarized:
In the Auto dataset (Chapter 2 and 3), exploratory
analysis revealed strong relationships between mpg and
several predictors. In particular, weight,
horsepower, and displacement showed strong
negative relationships with mpg, while year
showed a positive relationship.
The multiple linear regression model confirmed that these variables are significant predictors. Diagnostic analysis also indicated potential non-linearity and the presence of influential observations, suggesting that transformations and interaction terms can improve model performance.
In the Boston dataset (Chapter 3 - Question 15), the
comparison between simple and multiple regression models demonstrated
that coefficient estimates can change substantially when accounting for
other variables. This highlights the presence of multicollinearity and
the importance of considering all predictors together. Additionally,
several predictors exhibited non-linear relationships with the response
variable crim.
In the Weekly dataset (Chapter 4), logistic regression and alternative classification methods showed limited predictive performance. Most predictors, particularly lag variables, were not strongly significant, and the overall accuracy was only slightly better than random guessing. This indicates that the dataset lacks strong signals for predicting market direction.
Overall, the results emphasize that: - Strong predictive performance
depends on meaningful relationships within the data
- Regression models benefit from careful interpretation, diagnostics,
and transformation
- Classification tasks can be challenging when predictors have weak
explanatory power
These findings highlight the importance of both statistical understanding and critical interpretation when applying data analysis techniques.