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Using R, build a multiple regression model for data that interests you. Include in this model at least one quadratic term, one dichotomous term, and one dichotomous vs. quantitative interaction term. Interpret all coefficients. Conduct residual analysis. Was the linear model appropriate? Why or why not?

For this discussion I am going to use mtcars dataset.

mtcars
##                      mpg cyl  disp  hp drat    wt  qsec vs am gear carb
## Mazda RX4           21.0   6 160.0 110 3.90 2.620 16.46  0  1    4    4
## Mazda RX4 Wag       21.0   6 160.0 110 3.90 2.875 17.02  0  1    4    4
## Datsun 710          22.8   4 108.0  93 3.85 2.320 18.61  1  1    4    1
## Hornet 4 Drive      21.4   6 258.0 110 3.08 3.215 19.44  1  0    3    1
## Hornet Sportabout   18.7   8 360.0 175 3.15 3.440 17.02  0  0    3    2
## Valiant             18.1   6 225.0 105 2.76 3.460 20.22  1  0    3    1
## Duster 360          14.3   8 360.0 245 3.21 3.570 15.84  0  0    3    4
## Merc 240D           24.4   4 146.7  62 3.69 3.190 20.00  1  0    4    2
## Merc 230            22.8   4 140.8  95 3.92 3.150 22.90  1  0    4    2
## Merc 280            19.2   6 167.6 123 3.92 3.440 18.30  1  0    4    4
## Merc 280C           17.8   6 167.6 123 3.92 3.440 18.90  1  0    4    4
## Merc 450SE          16.4   8 275.8 180 3.07 4.070 17.40  0  0    3    3
## Merc 450SL          17.3   8 275.8 180 3.07 3.730 17.60  0  0    3    3
## Merc 450SLC         15.2   8 275.8 180 3.07 3.780 18.00  0  0    3    3
## Cadillac Fleetwood  10.4   8 472.0 205 2.93 5.250 17.98  0  0    3    4
## Lincoln Continental 10.4   8 460.0 215 3.00 5.424 17.82  0  0    3    4
## Chrysler Imperial   14.7   8 440.0 230 3.23 5.345 17.42  0  0    3    4
## Fiat 128            32.4   4  78.7  66 4.08 2.200 19.47  1  1    4    1
## Honda Civic         30.4   4  75.7  52 4.93 1.615 18.52  1  1    4    2
## Toyota Corolla      33.9   4  71.1  65 4.22 1.835 19.90  1  1    4    1
## Toyota Corona       21.5   4 120.1  97 3.70 2.465 20.01  1  0    3    1
## Dodge Challenger    15.5   8 318.0 150 2.76 3.520 16.87  0  0    3    2
## AMC Javelin         15.2   8 304.0 150 3.15 3.435 17.30  0  0    3    2
## Camaro Z28          13.3   8 350.0 245 3.73 3.840 15.41  0  0    3    4
## Pontiac Firebird    19.2   8 400.0 175 3.08 3.845 17.05  0  0    3    2
## Fiat X1-9           27.3   4  79.0  66 4.08 1.935 18.90  1  1    4    1
## Porsche 914-2       26.0   4 120.3  91 4.43 2.140 16.70  0  1    5    2
## Lotus Europa        30.4   4  95.1 113 3.77 1.513 16.90  1  1    5    2
## Ford Pantera L      15.8   8 351.0 264 4.22 3.170 14.50  0  1    5    4
## Ferrari Dino        19.7   6 145.0 175 3.62 2.770 15.50  0  1    5    6
## Maserati Bora       15.0   8 301.0 335 3.54 3.570 14.60  0  1    5    8
## Volvo 142E          21.4   4 121.0 109 4.11 2.780 18.60  1  1    4    2

The scatter plot states that, there might be a negative linear relationship between quarter mile time and hp, disp, and wt. However these aren’t very strong relationships. There is a positive linear relationship between mpg and quarter mile time.

gather( mtcars, "VARIABLE", "VALUE", 1:6,8:11) %>%
ggplot( aes(x=VALUE, y=qsec)) + 
  geom_point() + facet_wrap(~VARIABLE, scale = "free") + 
  labs(x="", y="1/4 Mile Time", title = "Scatter Plot of 1/4 Mile Time vs. Predictors" )

1. hp, am as the dichotomous term.

2. Product of disp and vs as the dichotomous vs. quantitative interaction term.

3. wt as the quadratic term.

Box-Cox will be used to transform wt.

lmBoxCox <- lm( wt ~ qsec, data = mtcars )

bc <- boxcox(lmBoxCox, lambda = seq(-2,2))

wtLambda <- bc$x[which.max(bc$y)]

wtLambda
## [1] 0.4646465

This transformation will be rounded up to the square root transformation of 0.5.

Building a Model:

my_lm <- lm(qsec ~ hp  + vs  + sqrt(wt) + vs:cyl, data = mtcars)

summary(my_lm)
## 
## Call:
## lm(formula = qsec ~ hp + vs + sqrt(wt) + vs:cyl, data = mtcars)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -1.0521 -0.3874 -0.1936  0.3262  2.4975 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 11.188146   1.280007   8.741 2.35e-09 ***
## hp          -0.017824   0.003185  -5.596 6.17e-06 ***
## vs           4.903943   1.288518   3.806 0.000738 ***
## sqrt(wt)     4.659763   0.728179   6.399 7.44e-07 ***
## vs:cyl      -0.566642   0.247907  -2.286 0.030344 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.7485 on 27 degrees of freedom
## Multiple R-squared:  0.8472, Adjusted R-squared:  0.8246 
## F-statistic: 37.42 on 4 and 27 DF,  p-value: 1.204e-10

hp: For each unit increase in horsepower, quarter-mile time decreeases by 0.0178 seconds, holding all else constant

vs: For each unit increase in vs, quarter-mile time increases by 4.9039 seconds, holding all else constant

√wt: For each unit increase in the square root of weight, quarter-mile time increases by 4.6598 seconds, holding all else constant

vs∗cyl For each unit increase in the product of the type of engine and the number of cylinders, quarter-mile time decreases by 0.5666 seconds

The predictors are significant and the R-squared value of 0.85 is high.

Is the linear model appropriate?

Due to the higher correlation among predictors linear model is not going to be appropriate here.

plot(my_lm)

data <- dplyr::select( mtcars, hp, vs, cyl, wt ) %>% 
  mutate( vs_X_cyl = vs * cyl, wt_sqrt = sqrt(wt) ) %>%
  dplyr::select( hp, vs, vs_X_cyl, wt_sqrt )
corrplot( round( cor(data, method = "pearson", use = "complete.obs"), 2 ), 
          method="circle", 
          type="upper",
          tl.col="black")