Regression_Model_Project Md Ahmed June 13th, 2017

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1 Regression_Model_Project Md Ahmed June 13th, 2017 Executive Summary Motor Trend is a magazine about the automobile industry. It is interested in exploring the relationship between a set of variables and miles per gallon (MPG) (outcome), particularly: Is an automatic or manual transmission better for MPG Quantify the MPG difference between automatic and manual transmissions Project progression: This is a linear regression model project. In searching answer to these questions;i will use major statistical analyses processes to verify, quantify and justify my model selection. In these steps, I will offer some statistical inference and immediate conclusion as if my model is a good fit. At the onset, I did some very basic exploratory data analysis(eda) meaning little slicing and dicing the mtcars dataset. Data manipulation is designed to get the am variable factored into two levels(auto, Manual), as per project instruction. In my regression model summary, I did try to analyze the summary-result as detail as possible to justify that Manual-transmission definitely hold upper mileage benefit.i did a simpler residual analysis to verify my model efficiency. In addition,i did some multivariable model analysis with variable adjustment and interaction to validate that none of the other models offer better mileage gain than my(fit01)model. Finally, I did use anova function to prove that my, lm(mpg ~ am) model is the right answer choice for the project questions. Project Question criteria and report writing instruction Load the mtcars data set and implement some exploratory data analysis. Design a regression model and execute some detail statistical analysis. Our linear model analysis should adhere to these instucted criteria: Interpreting the coefficients and slopes correctly. Doing some basic relevant exploratory data analyses. Fitting some multivariable linear models and evaluate reasoning for model selection. portraying a residual plot and with some diagnostics analysis. quantifying uncertainty in their(models) inferencial conclusions and/or perform an inference correctly. answering the questions of interest or detail why the question(s) is (are) not answerable? Your report should: Include an executive summary about project design progression. Written in a PDF printout format and compiled (using knitr) with a R markdown document. Concise and roughly the equivalent of 2 pages or less for the main text. Supporting figures in an appendix can be included up to 5 total pages. 1

2 1. EDA: Exploratory Data Analysis # loading 'mtcars' data set data(mtcars) # a brief data display head(mtcars) mpg cyl disp hp drat wt qsec vs am gear carb Mazda RX Mazda RX4 Wag Datsun Hornet 4 Drive Hornet Sportabout Valiant # displaying 'mtcars' data summary summary(mtcars) mpg cyl disp hp Min. :10.40 Min. :4.000 Min. : 71.1 Min. : st Qu.: st Qu.: st Qu.: st Qu.: 96.5 Median :19.20 Median :6.000 Median :196.3 Median :123.0 Mean :20.09 Mean :6.188 Mean :230.7 Mean : rd Qu.: rd Qu.: rd Qu.: rd Qu.:180.0 Max. :33.90 Max. :8.000 Max. :472.0 Max. :335.0 drat wt qsec vs Min. :2.760 Min. :1.513 Min. :14.50 Min. : st Qu.: st Qu.: st Qu.: st Qu.: Median :3.695 Median :3.325 Median :17.71 Median : Mean :3.597 Mean :3.217 Mean :17.85 Mean : rd Qu.: rd Qu.: rd Qu.: rd Qu.: Max. :4.930 Max. :5.424 Max. :22.90 Max. : am gear carb Min. : Min. :3.000 Min. : st Qu.: st Qu.: st Qu.:2.000 Median : Median :4.000 Median :2.000 Mean : Mean :3.688 Mean : rd Qu.: rd Qu.: rd Qu.:4.000 Max. : Max. :5.000 Max. :8.000 # data dimension dim(mtcars) [1] # summarizing 'mpg' values based on list-factor(auto/manual) transmission only by(mtcars$mpg, INDICES = list(mtcars$am), summary) : 0 Min. 1st Qu. Median Mean 3rd Qu. Max : 1 Min. 1st Qu. Median Mean 3rd Qu. Max

3 2. DM: Data manipulation with t-test library(dplyr) Warning: package 'dplyr' was built under R version Attaching package: 'dplyr' The following objects are masked from 'package:stats': filter, lag The following objects are masked from 'package:base': intersect, setdiff, setequal, union # factoring 'am' variable elements of 'mtcars' datasets summary(mtcars$am <- factor(mtcars$am)) # creating new levels with factored 'am-variable' data elements levels(mtcars$am) <- c("auto", "Manual") # quick view of the new 'level-set' head(mtcars) mpg cyl disp hp drat wt qsec vs am gear carb Mazda RX Manual 4 4 Mazda RX4 Wag Manual 4 4 Datsun Manual 4 1 Hornet 4 Drive Auto 3 1 Hornet Sportabout Auto 3 2 Valiant Auto 3 1 # separating 'auto' levels only into a new 'level-set' Auto_Data <- mtcars[mtcars$am == "Auto",] # separating only 'manual' levels into new 'level-set' Manual_Data Manual_Data <- mtcars[mtcars$am == "Manual",] # separating 'mpg' mean by 'Auto' and 'Manual' level summarise(group_by(mtcars, am), mn = mean(mpg)) # A tibble: 2 x 2 am mn <fctr> <dbl> 1 Auto Manual # doing t-test for verifying level-mpg-mean values t.test(auto_data$mpg, Manual_Data$mpg) Welch Two Sample t-test 3

4 data: Auto_Data$mpg and Manual_Data$mpg t = , df = , p-value = alternative hypothesis: true difference in means is not equal to 0 95 percent confidence interval: sample estimates: mean of x mean of y We are doing t-test only with mpg-mean data related to auto & manual levels to examine that these mean values are truly representattive of their group and carries a level of statistical significance. t-test analysis: Our 95% confidence interval( , to ) range does not contain zero,it is all negative values. p-value = is close to zero, which is ( < 0.05 ) at 0.05 level a statistically significant one. We can reject the assumed null hypothesis[ auto_mean == manual_mean ] at 0.05 level. Also ( Auto_mean = < Manual_mean = ), indicates the direction of the factored-element mean is significant and truly representative. 3. Regression analysis with linear model My regression model will try to Substantiate project question: Is an automatic or manual transmission better for MPG # designing first linear-model with new level with summary fit01 <- lm(mpg ~ am, mtcars) summary(fit01) Call: lm(formula = mpg ~ am, data = mtcars) Residuals: Min 1Q Median 3Q Max Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) e-15 ammanual Signif. codes: 0 '' '' 0.01 '' 0.05 '.' 0.1 ' ' 1 Residual standard error: on 30 degrees of freedom Multiple R-squared: , Adjusted R-squared: F-statistic: on 1 and 30 DF, p-value:

5 Linear model summary-result analysis: The intercept is the mean mileage for Automatic Transmission. The estimated mean for Manual Transmission is intercept plus the slope ( ) = Coefficient - Estimate: The intercept with this model is essentially the expected value of mileage attained from a car with Auto transmission, while the slope is the Manual transmission. amauto, the [Auto Transmission] cars in average attains MPG. The coefficients slope[ ammanual ], indicates mileage increases by MPG. So we can surmise that Manual transmission has better MPG than auto-transmission. Coefficient - t value: We can see that out t-static values are and = [ ] both are relatively far away from zero and large relative to corresponding standard error values, which indicates we could reject the null hypothesis meaning [ auto!= manual ]. Coefficient - Pr(> t ): Since our p-values for the intercept[1.13e-15] and slope[ ] indicates that ammanual has smaller p-value than amauto. We can infer that ammanual has higher level of p-value significance than amauto. Residual - Standar Error: Residual standar error measure the quality of a linear regression fit. The Residaul standard error is the average amount that the response(mileage) will deviate from the true regression line. In this model, the actual mileage varies between two transmission can deviate from the true regression line by approximately miles on average. In other words, given that the mean mileage for amauto are mile and that the Residual standard Error is R-squared, Adjusted R-squared: The R-squared static provides of how well the model is fitting the actual data. In our calculation multiple Rˆ2 is or rougly 35% of the variance found in the response variable(mpg) can be explained by the predictor variable am(auto/manual). Adjusted Rˆ2 is In both cases we see Rˆ2 values in range 0 < (0.3598, ) < 1 supports a good correlation between these two variables. This indicates a good linear model fit. Calculating confidence interval for the Intercept-Slope of this model: # Confidence Interval of this model [ fit01 ] with 'amauto' coefficients sumcoef <- summary(fit01)$coefficients sumcoef[1, 1] + c(-1, 1) qt(0.975, df = fit01$df ) sumcoef[1, 2] [1] # Now let's do the confidence interval of 'ammanual' slope coefficients (sumcoef[2,1] + c(-1, 1) qt(0.975, df = fit01$df) sumcoef[2, 2]) [1]

6 Analysis: So we can interpret these interval with 95% confidence that as we switch transmission from auto to manual average mileage increases to mile. Inference: we can say that manual transmission definitely produces better gas-mileage than automatic one. Residual analysis for model selection # resid function returns residuals of the linear model(fit01). residual <- resid(fit01) # a visual of the estimated residuals with model 'fit01' summary(residual) Min. 1st Qu. Median Mean 3rd Qu. Max Analysis: We can see very clearly that all the negative values, residuals ( ) = nearly equates ( ) = We know residuals must sum to.. 0, apparently ( ) = is almost close to 0. A good measurment of accurate model fit. # Plotting Residual vs fitted value par(mfrow = c(1,2)) plot(residual, pch='', xlab = "Fitted values", ylab = "Residuals") abline(0,0) # Normality of residuals(errors) qqnorm(residual, pch='') qqline(residual) 6

7 Normal Q Q Plot Residuals Sample Quantiles Fitted values Theoretical Quantiles Figure 1: Residual plot Residual vs. fitted: Residual points are in a pattern and symmetrically distributed on and below the 0-line. Residaul Q-Q plot: It is obvious that our model(fit01) residual(error) values roughly falling on a line in a normal QQ plot. These distribution verifies our model(fit01) design with potential effectiveness. plots of the regression model # drawing plots library(ggplot2) Warning: package 'ggplot2' was built under R version # dotted plot ggplot(mtcars, aes(x = factor(am), y = mpg, color=factor(am), shape = factor(am))) + geom_point(size = 3 7

8 35 mileage distribution by 'transmission and cylinder' 30 Mileage factor(am) Auto Manual Auto Manual Auto and Manual Transmission Figure 2: Dot plot: lm( mpg ~ am) It is obvious that Manual transmission getting incremental mileage. 4. Multivariable analysis with nested model testing We know omitting variables from regressors may results in bias in the coefficients of interest ( unless the regressors are uncorrelated with the omitted ones). So to avoid bias, I have decided to do a very generalized mpg measurements in connection to all the relevant regressor variables of mtcars dataset regardless of correlations. # mpg vs. all relevant regressors variables into a new linear model fit02 <- lm(mpg ~., data = mtcars) summary(fit02) Call: lm(formula = mpg ~., data = mtcars) Residuals: Min 1Q Median 3Q Max Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) cyl

9 disp hp drat wt qsec vs ammanual gear carb Signif. codes: 0 '' '' 0.01 '' 0.05 '.' 0.1 ' ' 1 Residual standard error: 2.65 on 21 degrees of freedom Multiple R-squared: 0.869, Adjusted R-squared: F-statistic: on 10 and 21 DF, p-value: 3.793e-07 Nested models: Inspecting into fit02 model Coefficients-Estimate we can surmise that variables( cyl, hp, wt, carb ) are losing more mileage than any other variables. Mileage regression is in negative territory with these variables. The rest of the variables are are gaining somewhat but trivial mileage. # Obtaining residual plot of fitted model ( fit02 ) par(mfrow = c(2,2)) residual02 <- resid(fit02) plot(fit02) Residuals Residuals vs Fitted Chrysler Imperial Fiat 128 Toyota Corolla Standardized residuals Ford Pantera L Normal Q Q Chrysler Imperial Fiat Fitted values Theoretical Quantiles Standardized residuals Chrysler Imperial Scale Location Ford Pantera Fiat L Standardized residuals Residuals vs Leverage Chrysler Imperial Cook's distance Merc Ford Pantera L Fitted values Leverage Figure 4: Detail Residual plot model-fit02 Analysis: The Residual vs Fitted plot is not exactly a smooth residual distribution. We can see from 9

10 our model(fit02), Residuals: ( ) = is not sum to zero. Our Normal QQ plot visually shows a residual normality. The scale-location plot shows some sort of linear distribution of residuals. Finally, our Residuals vs Leverage plot shows no large outlying data point holding any significant leverage. Adjustment and interaction between multiple variables with am : so I will experiment with some nested linear model with these variables adding with factored am regressor. This also called adjustment and interaction, by adding more regressor into the linear model to investigate the role of a third/fourth variable on to the relationship with outcome variable mpg. These added variable can distort, or confound the linear relationhsip between (outcome-regressor) and offer a renewed perspective about possible variable influence. # variable adjustment with possible relationship with 'cyl' fit03 <- lm(mpg ~ am + cyl, data = mtcars) summary(fit03)$coef Estimate Std. Error t value Pr(> t ) (Intercept) e-14 ammanual e-02 cyl e-07 We can see from this model that t-value = relatively away from 0, which indicates that there is a minimal relationship between mpg - (am + cyl) model. # variable adjustment with possible relationship with 'hp' interactive fit04 <- lm(mpg ~ am + cyl + hp + cyl hp, data = mtcars) summary(fit04)$coef Estimate Std. Error t value Pr(> t ) (Intercept) e-08 ammanual e-03 cyl e-03 hp e-03 cyl:hp e-02 This model (fit04) t-value correlated with [ cyl, hp = -3.08, ] is not far away from 0. We can say there exist a fainted relationship with between mileage change and (cyl + hp) predictor variable. # variable adjustment with possible relationship with 'wt' fit05 <- lm(mpg ~ am + cyl + hp + wt, data = mtcars) summary(fit05)$coef Estimate Std. Error t value Pr(> t ) (Intercept) e-12 ammanual e-01 cyl e-01 hp e-02 wt e-03 We added another variable wt into this linear model. Estimated corresponding slope, t-values all stayed in negative territory without being far away from 0. So we can infer that mileage relations will not be significant with this new variable adjustment. # On this model(fit06) we added predictor variable 'qsec' with the long list fit06 <- lm(mpg ~ am + cyl + hp + wt + qsec + wt qsec, data = mtcars) summary(fit06)$coef Estimate Std. Error t value Pr(> t ) (Intercept)

11 ammanual cyl hp wt qsec wt:qsec None of these interaction offers any new height of observation in t-values far from standard error. The only difference now p-value is significantly out of range towards accepting null values. An example of simpsons paradox. Inference: So, we can effectively assume that adding multiple variables into the linear-model wouldn t make any difference in pursuasion of mileage gain/loss on mileage coefficient-slope. 5. ANOVA - test for multiple-model statistical significance We know ANOVA test is useful for comparing two or more model for statistical significance. It is conceptually similar to multiple two-sample t-test. anova(fit01, fit03, fit04, fit05, fit06) Analysis of Variance Table Model 1: mpg ~ am Model 2: mpg ~ am + cyl Model 3: mpg ~ am + cyl + hp + cyl hp Model 4: mpg ~ am + cyl + hp + wt Model 5: mpg ~ am + cyl + hp + wt + qsec + wt qsec Res.Df RSS Df Sum of Sq F Pr(>F) e Signif. codes: 0 '' '' 0.01 '' 0.05 '.' 0.1 ' ' 1 Analysis: By analysing all four nested models with anova function, we are witnessing, Model-2 has second highest level( 0.001) of significance. There are no obvious mileage gain even with newer adjusted nested models. The only second most significant model is fit03 = am + cyl with multivariable combination.let s have a box-plot visual with this adjustment Model-2( fit03 ). boxplot(mpg ~ factor(am)+cyl, data=mtcars, col=c("salmon","dodgerblue2"), xlab="transmission-cylinder", 11

12 mileage variation with (am+cyl) Mileage Auto.4 Manual.4 Auto.6 Manual.6 Auto.8 Manual.8 Transmission Cylinder Figure 4: Box plot of ( mpg ~ am + cyl) Manual transmission still carries higher mileage with 4-cylinder combination Conclusion: All throughout these statistical verification processes, it is obvious that ammanual transmission holds a significant mileage gain in comparison to amauto cars. Our t-test, confidence interval and residual analysis offer a clear mileage preference for manual-transmission cars. Even multivariable analysis with variable adjustment and interaction forcefully confirms that A manual transmission car is better for MPG, rather than an automatic one. 12