The analysis requires the dplyr, ggplot2, and mosaic packages. See
the Appendix for code.
The mtcars data set was transformed for easier analysis. New columns
were created in order to more easily identify the engine and
transmission variables. Also, these new variables engine and
transmission were coerced to factor variables. See Appendix for
code.
This project seeks to analyze and identify any association between fuel
efficiency and transmission. These traits were analyzed directly and by
adjusting for other factors. The data came from the mtcars data set
native to R and all analysis was performed within R version 4.4.1.
Nested ANOVA model fit testing concluded no difference between fuel
efficiency, mpg, based on transmission. There were observed
differences in average mpg based on transmission. However, these
differences did not appear significant when adjusted for other car
traits.
The 95% confidence interval for the automatic transmission coefficient was 28.6 mpg to 39.4 mpg, while the interval for the manual transmission coefficient was 32.5 mpg to 39.6 mpg. Based on these confidence intervals the fuel efficiency of automatic transmission cars could be up to 11 mpg better than or up to 6.9 mpg worse than the efficiency of manual transmission cars.Coefficients are interpreted in more detail in the Regression Analysis section.
A caveat of this analysis is that the chosen model arguably violates some conditions for linear regression models. In particular in relation to residuals. This is discussed later in the Regression Analysis section.
Two plots mpg vs transmission are included below. These plots shows
that manual cars have a higher median mpg and a larger spread than
automatics. The plots also indicate that mpg increases from automatic
cars to manual cars. See the Appendix for additional plots and
statistics. These plots provide a baseline for model selection.
Initially a regression of mpg ~ transmission was conducted. This
naive model showed significant differences in the coefficients for
automatic (17.15 mpg) and manual transmission (24.39 mpg), but
the residual plots violated the model assumptions. Next a regression was
run with mpg against all other variables; however, the full model
found no statistically significant predictors. Due to space constraints
further details for these models have been excluded.
Since weight (wt) has a known causal influence on mpg, it was
included in the model. The correlation matrix for each predictor was
consulted in choosing variables for a nested ANOVA model fit. I
elected not to use available automated stepwise regression after
encountering these
notes.
The notes question the diagnostics and inference that result from
automated model selection. Ultimately, the nested ANOVA procedure
compared 8 models.
Using nested ANOVA model testing, the following predictors were selected
for the final model: horsepower, transmission, and weight. The
nested ANOVA code is located in the Appendix. The nested process
also found qsec (quarter mile time) to be significant. However,
its inclusion caused the automatic transmission coefficient to become
insignificant. Therefore qsec was excluded from the model. The chosen
model is labeled fit_hp.
Note that many models beyond those mentioned were tested. Due to the large number of significance tests performed, models that cleared especially low p-values were prioritized to limit Type I errors for the related t-tests and ANOVA tests. However, no explicit calculations were made to identify upper bounds for p-values for multiple hypothesis testing.
Below the nested ANOVA model of mpg ~ wt + transmission + hp, named
fit_hp, is analyzed.
The results below give the coefficients, p-values, and 95% confidence
intervals for automatic and manual transmission from the nested
anova model fit_hp.
## Estimate Pr(>|t|)
## transmissionautomatic 34.00288 2.824030e-13
## transmissionmanual 36.08659 1.477711e-18
## 2.5 % 97.5 %
## transmissionautomatic 28.58963 39.41612
## transmissionmanual 32.52986 39.64331
The coefficients are significant at the 0.1% level, which provides some
protection against Type I error. Since the models were computed
without an intercept the coefficients indicate the average miles per
gallon of an automatic transmission or manual transmission car within
each model. For example, an automatic car from the fit_hp model
would be expected to have between 28.6 mpg and 39.4 mpg of fuel
efficiency. These numbers represent the average number of additional
miles driven for one additional gallon of fuel.
The plot of residuals vs the fitted values arguably shows a pattern, but
looks reasonably scattered. The residuals appear larger for extreme
fitted values suggesting heteroscedasticity, but there is little data in
these regions. The QQ plot is reasonably linear, though the residuals
show potential influential points. This could be explored further using
the influence.measures() function, but due to space constraints it is is
not explored. See the Appendix for the residual plots of the
fit_hp model and code for influence.measures().
Additional figures and code are included below.
# code for data preprocessing
mtcars$engine <- NULL # create more informative names for these vars
mtcars[mtcars$vs == 0,'engine'] <- 'V' # engine label
mtcars[mtcars$vs == 1,'engine'] <- 'straight' # engine label
mtcars$transmission <- NULL # create more informative names for these vars
mtcars[mtcars$am == 0,'transmission'] <- 'automatic' # transmission label
mtcars[mtcars$am == 1,'transmission'] <- 'manual' # transmission label
mtcars$engine <- as.factor(mtcars$engine)
mtcars$transmission <- as.factor(mtcars$transmission)
# install packages required for data processing and analysis
install.packages('dplyr', repos="http://cran.us.r-project.org"); install.packages('ggplot2', repos="http://cran.us.r-project.org"); install.packages('mosaic', repos="http://cran.us.r-project.org")
library(dplyr); library(ggplot2); library(mosaic)
# linear models for sequential ANOVA TEST
wt_only <- lm(mpg ~ wt -1, data = mtcars) # weight only model
transm_wt <- lm(mpg ~ transmission + wt -1, data = mtcars) # transmission included in nesting
fit_hp <- lm(mpg ~ wt + transmission + hp - 1, data = mtcars) # horsepower included in nesting
fit_qsec <- lm(mpg ~ wt + transmission + hp + qsec - 1, data = mtcars) # 1/4 time included in nesting
fit_vs <- lm(mpg ~ wt + transmission + hp + qsec + vs - 1, data = mtcars) # engine included in nesting
fit_carb <- lm(mpg ~ wt + transmission + hp + qsec + vs + carb - 1, data = mtcars) # carburetor included in nested fit
fit_drat <- lm(mpg ~ wt + transmission + hp + qsec + vs + carb + drat - 1, data = mtcars) # axel ratio included in nesting
fit_gear <- lm(mpg ~ wt + transmission + hp + qsec + vs + carb + drat + gear - 1, data = mtcars) #gears included in nesting
# nested ANOVA
anova(wt_only, transm_wt, fit_hp,fit_qsec,fit_vs,fit_carb,fit_drat, fit_gear) # step wise ANOVA
## Analysis of Variance Table
##
## Model 1: mpg ~ wt - 1
## Model 2: mpg ~ transmission + wt - 1
## Model 3: mpg ~ wt + transmission + hp - 1
## Model 4: mpg ~ wt + transmission + hp + qsec - 1
## Model 5: mpg ~ wt + transmission + hp + qsec + vs - 1
## Model 6: mpg ~ wt + transmission + hp + qsec + vs + carb - 1
## Model 7: mpg ~ wt + transmission + hp + qsec + vs + carb + drat - 1
## Model 8: mpg ~ wt + transmission + hp + qsec + vs + carb + drat + gear -
## 1
## Res.Df RSS Df Sum of Sq F Pr(>F)
## 1 31 3936.6
## 2 29 278.3 2 3658.3 277.7379 < 2.2e-16 ***
## 3 28 180.3 1 98.0 14.8847 0.0007998 ***
## 4 27 160.1 1 20.2 3.0709 0.0930281 .
## 5 26 159.8 1 0.2 0.0377 0.8476721
## 6 25 156.4 1 3.4 0.5204 0.4779544
## 7 24 153.2 1 3.2 0.4833 0.4938789
## 8 23 151.5 1 1.7 0.2630 0.6129269
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Note that a variety of other variables were tested in a nested model
fit, including an interaction term between wt and transmission.
However, these model resulted in insignificance of automatic
transmission or violated model assumptions (residual plot issues).
Reisdual plot of fit_hp model discussed in the Regression Analysis
section.
# resized due to page limit
plot(fit_hp, main = 'fit_hp')
# additional plots and statistics (#box plot not printed for space reasons)
#ggplot(mtcars, aes(x = mpg)) + geom_boxplot() + ggtitle('Fuel Efficency') +
xlab('Miles per gallon (mpg)') + theme(plot.title = element_text(hjust = 0.5,))
## NULL
# code for summary stats
mpg_stats <- rbind(favstats(mtcars$mpg), favstats(mtcars[mtcars$transmission == 'automatic','mpg']),
favstats(mtcars[mtcars$transmission == 'manual','mpg']))
row.names(mpg_stats) <- c('total_mpg','automatic_mpg','manual_mpg') # assign row names
mpg_stats
## min Q1 median Q3 max mean sd n missing
## total_mpg 10.4 15.425 19.2 22.8 33.9 20.09062 6.026948 32 0
## automatic_mpg 10.4 14.950 17.3 19.2 24.4 17.14737 3.833966 19 0
## manual_mpg 15.0 21.000 22.8 30.4 33.9 24.39231 6.166504 13 0
# The influence statistics for the fit_hp model are not printed due to space constraints for the assignment.
# However, note that dfbetas for transmission coefficients are small in comparison to the confidence intervals
# influence.measures(fit_hp)
# code for plots shown outside appendix
## box plot mpg vs transmission
ggplot(mtcars, aes(x = mpg, fill = transmission, y = transmission)) + geom_boxplot()
## scatter plot mpg vs transmission
ggplot(mtcars, aes(x = transmission, y = mpg, color = transmission)) + geom_point()