Introduction to Regression Analysis and R

Multiple regression in R on Automobile data to predict
Gasoline Mileage
Rachana T. Bhatia - Rutgers University

 Basics of Regression Analysis
 Addressing Model Deviation of regression models
 Model selection criterion
 Types of regression Model
 Introduction to R
 Multiple regression (including polynomial regression) on Car Data

 First step to learn predictive modelling
 Statistical technique for investigating and modeling the relationship between
variables
 Equation of straight line 𝒚 = 𝜷 𝟎 + 𝜷 𝟏 𝒙 + 𝜺
 𝜺 is a random variable that accounts for the failure of the model to fit the data
 𝒙 explanatory variable & 𝑦 response variable
 Regression does not necessarily imply causality

Linear Regression Analysis 5th edition Montgomery, Peck & ViningRachana T. Bhatia - Rutgers University

 Least squares estimation- minimize the sum of squares of the
differences between the observed response, yi, and the straight line
 Fitted values & Residuals
 Hypothesis Testing for the value of slope and intercept - T-tests
 Significant relation between the variables- reject Null Hypothesis
 Alternative approach : p-value
 Confidence Interval – associated with randomness of the data
 Prediction Interval – associated with the random variable yet to be
observed.

 Linearity
 Homoscedasticity
 Errors normally distributed (for inferential purposes)
 Independent
 Constant variance
 There is a probability distribution for y at each value of x
with mean: E 𝑌 𝑥 = β0 + β1 𝑥
Variance: Var 𝑌 𝑥 = σ2

 Looking at the scatter plot
 Q-Q plot – Quantiles of the residuals vs normal distribution
 Residual plot – Residuals Vs Explanatory variable

 Correctable non-linearity (simple and monotone )
 Non-Correctable linearity

 Define a new variable u as 𝑢 = 𝑒 𝑥

Some common transformations are:
v = ln(y)
v = p √y where p > 1 v = 1/y p where p > 0

 How well a statistical model fits observed data
 How much of the total variation in Y is described by the variation in the
explanatory variables
 square of the sample correlation of the response variable and the explanatory
variable
 Lies between -∞ to 1
 Adjusted R-squared- adjusted for the number of coefficients in the model relative
to the sample size in order to correct it for bias

 Mean Square Error
 Coefficient of Determination - R2
 Adjusted R2
 AIC (Akaike’s Information Criterion) - smaller values are better
 BIC (Bayesian Information Criterion) - smaller values are better

 LEVERAGE – ‘standardized’ measure the distance of the ith observation abscissa from
the mean of the explanatory variables
 DFBETAS - standardized measures how much estimation of βj is influenced by the ith
observation.
 DFFITS - standardized measures how much estimation of ith fitted value is influenced
by the ith observation
 COOK’S Distance -standardized measure of the distance between the fitted values
obtained using the whole sample and the fitted values obtained after removing the jth
observation

 Simple linear Model
 Polynomial regression – relationship is not linear
 Multiple linear Model – more than one explanatory variables- Categorical Data
 Robust regression (Least Absolute Deviations, Huber/ Bisquare function) - Data
contaminated with outliers
 Logistic regression – Response variable Binary (Logit and Probit link function)
 Ridge Regression – High multicollinearity
 Step wise regression – High dimensions (Forward selection/ backward elimination)

 A power tool for statistics and data modeling
 R is free
 R is a language
 Graphics and data visualization
 A flexible statistical analysis toolkit
 R Studio - an Integrated Development Environment (IDE) for the R programming
language.

 Setting the working directory
 Installing packages, updating and loading the packages
 Importing and Converting Data
 Creating vectors, data frames
 Connection to the outside world(file, gzfile,bzfile, url)
 Atomic classes of vectors : integer • numeric • character • complex • logical

 Data Frames (tabular data)-stores different class of
objects{read.table/read.csv/data.frame)
 Analogous code for writing the data
 Foreign package (read.xport, read.spss )
 Reading larger data sets (Specifying the column classes)
 Inspect objects/dataframes
 Missing Values (Na / NaN)

 Exploratory Analysis
 Subsetting (using [], [[]],$ )
 which.max/ which.min
 Handling missing values (complete.cases(), is.na…, na.rm = T)
 Splitting
 Apply , sapply, tapply, mapply
 Descriptive Analysis
 Summary()
 Str()
 Sd(), var(), median() , quantile(), hist()
 By(), table()
 Statistical test- t test , chi square test

 Variation in gasoline mileage among makes and models of automobiles is
influenced substantially by the size of the vehicle and its engine.
 Downloaded from http://lib.stat.cmu.edu/DASL/Datafiles/carmpgdat.html
 Variable Names:
 VOL: Cubic feet of cab space
 HP: Engine horsepower
 MPG: Average miles per gallon (Response Variable)
 SP: Top speed (mph)
 WT: Vehicle weight (100 lb)

 Prof. Andrew Magyar - Stat 563 - Introduction to Linear Regression_Course
Material
 Linear Regression Analysis 5th edition Montgomery, Peck & Vining
 http://www.ats.ucla.edu/stat/stata/dae/rreg.htm
 https://www.coursera.org/learn/r-programming/home/welcome

Introduction to Regression Analysis and R

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