It is a way of finding a relationship between a single, continuous variable called Dependent or Target variable and one or more other variables (continuous or not) called Independent Variables.
It’s a straight line curve. In the above figure, diagonal red line is a regression line which is also called best-fitting straight line. The distance between dots and regression line is errors. Linear regression aims at finding best fitting straight line by minimizing the sum of squared vertical distance between dots and regression line.
Variable Type
Linear regression requires the dependent variable to be continuous i.e. numeric values (no categories or groups).
Simple vs. Multiple Linear Regression
Linear regression can be simple linear regression when you have only one independent variable . Whereas Multiple linear regression will have more than one independent variable.
Regression Equation
Interpretation:
b0 is the intercept the expected mean value of dependent variable (Y) when all independent variables (Xs) are equal to 0. and b1 is the slope.
b1 represents the amount by which dependent variable (Y) changes if we change X1 by one unit keeping other variables constant.
Important Term : Residual
The difference between an observed (actual) value of the dependent variable and the value of the dependent variable predicted from the regression line.
Algorithm
Linear regression is based on least square estimation which says regression coefficients (estimates) should be chosen in such a way that it minimizes the sum of the squared distances of each observed response to its fitted value.
Minimum Sample Size
Linear regression requires 5 cases per independent variable in the analysis.
Assumptions of Linear Regression Analysis
1. Linear Relationship : Linear regression needs a linear relationship between the dependent and independent variables.
2. Normality of Residual : Linear regression requires residuals should be normally distributed.
3. Homoscedasticity : Linear regression assumes that residuals are approximately equal for all predicted dependent variable values. In other words, it means constant variance of errors.
4. No Outlier Problem
5. Multicollinearity : It means there is a high correlation between independent variables. The linear regression model MUST NOT be faced with problem of multicollinearity.
6. Independence of error terms – No Autocorrelation
It states that the errors associated with one observation are not correlated with the errors of any other observation. It is a problem when you use time series data. Suppose you have collected data from labors in eight different districts. It is likely that the labors within each district will tend to be more like one another that labors from different districts, that is, their errors are not independent.
Distribution of Linear Regression
Linear regression assumes target or dependent variable to be normally distributed. Normal Distribution is same as Gaussian distribution. It uses identity link function of gaussian family
Standardized Coefficients
The concept of standardization or standardized coefficients (aka estimates) comes into picture when predictors (aka independent variables) are expressed in different units. Suppose you have 3 independent variables – age, height and weight. The variable ‘age’ is expressed in years, height in cm, weight in kg. If we need to rank these predictors based on the unstandardized coefficient, it would not be a fair comparison as the unit of these variable is not same.
Standardized Coefficients (or Estimates) are mainly used to rank predictors (or independent or explanatory variables) as it eliminate the units of measurement of independent and dependent variables). We can rank independent variables with absolute value of standardized coefficients. The most important variable will have maximum absolute value of standardized coefficient.
Interpretation of Standardized Coefficient
A standardized coefficient value of 1.25 indicates that a change of one standard deviation in the independent variable results in a 1.25 standard deviations increase in the dependent variable.
Detailed Explanation : Standardized vs. Unstandardized Coefficient
Measures of Model Performance
1. R-squared
It measures the proportion of the variation in your dependent variable explained by all of your independent variables in the model. It assumes that every independent variable in the model helps to explain variation in the dependent variable. In reality, some variables don’t affect dependent variable and they don’t help building a good model.
In the numerator of equation above, yi-hat is the predicted value. Mean value of Y appears in denominator.
Rule :
Higher the R-squared, the better the model fits your data. In psychological surveys or studies, we generally found low R-squared values lower than 0.5. It is because we are trying to predict human behavior and it is not easy to predict humans. In these cases, if your R-squared value is low but you have statistically significant independent variables (aka predictors), you can still generate insights about how changes in the predictor values are associated with changes in the response value.
Can R-Squared be negative?
Yes, it is when horizontal line explains the data better than your model. It mostly happens when you do not include intercept. Without an intercept, the regression could do worse than the sample mean in terms of predicting the target variable. It is not only because of exclusion of intercept. It can be negative even with inclusion of intercept.
Mathematically, it is possible when error sum-of-squares from the model is larger than the total sum-of-squares from the horizontal line.
R-squared = 1 – [(Sum of Square Error)/(Total Sum of Square)]
2. Adjusted R-squared
It measures the proportion of variation explained by only those independent variables that really affect the dependent variable. It penalizes you for adding independent variable that do not affect the dependent variable.
Adjusted R-Squared is more important metrics than R-squared
Every time you add a independent variable to a model, the R-squared increases, even if the independent variable is insignificant. It never declines. Whereas Adjusted R-squared increases only when independent variable is significant and affects dependent variable.
3. RMSE ( Root Mean Square Error)
It explains how close the actual data points are to the model’s predicted values. It measures standard deviation of the residuals.
In the formula above, yi is the actual values of dependent variable and yi-hat is the predicted values, n – sample size.
Important Point
RMSE has the same unit as dependent variable. For example, dependent variable is sales which is measured in dollars. Let’s say RMSE of this sales model comes out 21. We can say it is 21 dollars.
What is good RMSE score?
There is no thumb rule regarding good or bad RMSE score. It is because it is dependent on your dependent variable. If your target variable lies between 0 to 100. RMSE of 0.5 can be considered as good but same 0.5 RMSE can be considered as a poor score if dependent variable ranges from 0 to 10. Hence there is so such good or bad RMSE by simply looking at the value.
Lower values of RMSE indicate better fit. RMSE is a good measure of how accurately the model predicts the response, and is the most important criterion for fit if the main purpose of the model is prediction.
The RMSE for your training and your test sets should be very similar if you have built a good model. If the RMSE for the test set is much higher than that of the training set, it is likely that you’ve badly over fit the data, i.e. you’ve created a model that works well in sample, but has little predictive value when tested out of sample
RMSE vs MAE
RMSE amplifies and severely punishes large errors as compared to mean absolute error (MAE).
R-Squared vs RMSE
R-squared is in proportion and has no units associated to target variable whereas RMSE has units associated to target variable. Hence, R-squared is a relative measure of fit, RMSE is an absolute measure of fit.
The code below covers the assumption testing and evaluation of model performance :
- Data Preparation
- Testing of Multicollinearity
- Treatment of Multicollinearity
- Checking for Autocorrelation
- Checking for Outliers
- Checking for Heteroscedasticity
- Testing of Normality of Residuals
- Forward, Backward and Stepwise Selection
- Calculating RMSE
- Box Cox Transformation of Dependent Variable
- Calculating R-Squared and Adj, R-squared manually
- Calculating Residual and Predicted values
- Calculating Standardized Coefficient
R Script : Linear Regression
Theory part is over. Let’s implement linear regression with R –
Load required packages
library(ggplot2)
library(car)
library(caret)
library(corrplot)
Make sure the above listed packages are already installed and loaded into R. If they are not already installed, you need to install it by using the command install.packages(“package-name”)
Read Data
We will use mtcars dataset from cars package. This data was extracted from the Motor Trend US magazine, and comprises fuel consumption and 10 aspects of automobile design and performance for 32 automobiles.
#Loading data
data(mtcars)# Looking at variables
str(mtcars)
'data.frame': 32 obs. of 11 variables: $ mpg : num 21 21 22.8 21.4 18.7 18.1 14.3 24.4 22.8 19.2 ... $ cyl : num 6 6 4 6 8 6 8 4 4 6 ... $ disp: num 160 160 108 258 360 ... $ hp : num 110 110 93 110 175 105 245 62 95 123 ... $ drat: num 3.9 3.9 3.85 3.08 3.15 2.76 3.21 3.69 3.92 3.92 ... $ wt : num 2.62 2.88 2.32 3.21 3.44 ... $ qsec: num 16.5 17 18.6 19.4 17 ... $ vs : num 0 0 1 1 0 1 0 1 1 1 ... $ am : num 1 1 1 0 0 0 0 0 0 0 ... $ gear: num 4 4 4 3 3 3 3 4 4 4 ... $ carb: num 4 4 1 1 2 1 4 2 2 4 ...
Variable description of the above variables are listed below against their respective variable names.
| Variable | Description |
|---|---|
| mpg | Miles/(US) gallon |
| cyl | Number of cylinders |
| disp | Displacement (cu.in.) |
| hp | Gross horsepower |
| drat | Rear axle ratio |
| wt | Weight (1000 lbs) |
| qsec | 1/4 mile time |
| vs | V/S |
| am | Transmission (0 = automatic, 1 = manual) |
| gear | Number of forward gears |
| carb | Number of carburetors |
Summarize Data
In this dataset, mpg is a target variable. See first 6 rows of data by using head() function.
> head(mtcars)
mpg cyl disp hp drat wt qsec vs am gear carb
Mazda RX4 21.0 6 160 110 3.90 2.620 16.46 0 1 4 4
Mazda RX4 Wag 21.0 6 160 110 3.90 2.875 17.02 0 1 4 4
Datsun 710 22.8 4 108 93 3.85 2.320 18.61 1 1 4 1
Hornet 4 Drive 21.4 6 258 110 3.08 3.215 19.44 1 0 3 1
Hornet Sportabout 18.7 8 360 175 3.15 3.440 17.02 0 0 3 2
Valiant 18.1 6 225 105 2.76 3.460 20.22 1 0 3 1
summary(mtcars)
> summary(mtcars)
mpg cyl disp hp
Min. :10.40 Min. :4.000 Min. : 71.1 Min. : 52.0
1st Qu.:15.43 1st Qu.:4.000 1st Qu.:120.8 1st 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 :146.7
3rd Qu.:22.80 3rd Qu.:8.000 3rd Qu.:326.0 3rd 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. :0.0000
1st Qu.:3.080 1st Qu.:2.581 1st Qu.:16.89 1st Qu.:0.0000
Median :3.695 Median :3.325 Median :17.71 Median :0.0000
Mean :3.597 Mean :3.217 Mean :17.85 Mean :0.4375
3rd Qu.:3.920 3rd Qu.:3.610 3rd Qu.:18.90 3rd Qu.:1.0000
Max. :4.930 Max. :5.424 Max. :22.90 Max. :1.0000
am gear carb
Min. :0.0000 Min. :3.000 Min. :1.000
1st Qu.:0.0000 1st Qu.:3.000 1st Qu.:2.000
Median :0.0000 Median :4.000 Median :2.000
Mean :0.4062 Mean :3.688 Mean :2.812
3rd Qu.:1.0000 3rd Qu.:4.000 3rd Qu.:4.000
Max. :1.0000 Max. :5.000 Max. :8.000
Data Preparation
Make sure categorical variables are stored as factors. In the program below, we are converting variables to factors.
mtcars$am = as.factor(mtcars$am)
mtcars$cyl = as.factor(mtcars$cyl)
mtcars$vs = as.factor(mtcars$vs)
mtcars$gear = as.factor(mtcars$gear)
Identifying and Correcting Collinearity
In this step, we are identifying independent variables which are highly correlated to each other. Since mpg is a dependent variable, we are removing it in the code below.
#Dropping dependent variable for calculating Multicollinearity
mtcars_a = subset(mtcars, select = -c(mpg))
#Identifying numeric variables
numericData <- mtcars_a[sapply(mtcars_a, is.numeric)]
#Calculating Correlation
descrCor <- cor(numericData)
# Print correlation matrix and look at max correlation
print(descrCor)
disp hp drat wt qsec carb
disp 1.0000000 0.7909486 -0.71021393 0.8879799 -0.43369788 0.3949769
hp 0.7909486 1.0000000 -0.44875912 0.6587479 -0.70822339 0.7498125
drat -0.7102139 -0.4487591 1.00000000 -0.7124406 0.09120476 -0.0907898
wt 0.8879799 0.6587479 -0.71244065 1.0000000 -0.17471588 0.4276059
qsec -0.4336979 -0.7082234 0.09120476 -0.1747159 1.00000000 -0.6562492
carb 0.3949769 0.7498125 -0.09078980 0.4276059 -0.65624923 1.0000000
# Visualize Correlation Matrix
corrplot(descrCor, order = "FPC", method = "color", type = "lower", tl.cex = 0.7, tl.col = rgb(0, 0, 0))
# Checking Variables that are highly correlated highlyCorrelated = findCorrelation(descrCor, cutoff=0.7) #Identifying Variable Names of Highly Correlated Variables highlyCorCol = colnames(numericData)[highlyCorrelated] #Print highly correlated attributes highlyCorCol [1] "hp" "disp" "wt" #Remove highly correlated variables and create a new dataset dat3 = mtcars[, -which(colnames(mtcars) %in% highlyCorCol)] dim(dat3) [1] 32 8
There are three variables “hp” “disp” “wt” that found to be highly correlated. We have removed them to avoid collinearity. Now, we have 7 independent variables and 1 dependent variable.
Developing Regression Model
At this step, we are building multiple linear regression model.
#Build Linear Regression Model fit = lm(mpg ~ ., data=dat3) #Check Model Performance summary(fit) #Extracting Coefficients summary(fit)$coeff anova(fit) par(mfrow=c(2,2)) plot(fit)
See the coefficients of Linear Regression Model and ANOVA table
Linear regression model tests the null hypothesis that the estimate is equal to zero. An independent variable that has a p-value less than 0.05 means we are rejecting the null hypothesis at 5% level of significance. It means the coefficient of that variable is not equal to 0. A large p-value implies variable is meaningless in order to predict target variable.
> summary(fit)
Call:
lm(formula = mpg ~ ., data = dat3)
Residuals:
Min 1Q Median 3Q Max
-5.4850 -1.3058 0.1856 1.5278 5.2439
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 16.7823 19.6148 0.856 0.401
cyl6 -1.8025 2.6085 -0.691 0.497
cyl8 -3.5873 4.0324 -0.890 0.383
drat 1.4283 2.1997 0.649 0.523
qsec 0.1003 0.7729 0.130 0.898
vs1 0.7068 2.3291 0.303 0.764
am1 3.2396 2.4702 1.311 0.203
gear4 1.3869 3.0466 0.455 0.653
gear5 2.3776 3.4334 0.692 0.496
carb -1.4836 0.6305 -2.353 0.028 *
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 3.004 on 22 degrees of freedom
Multiple R-squared: 0.8237, Adjusted R-squared: 0.7516
F-statistic: 11.42 on 9 and 22 DF, p-value: 1.991e-06
> anova(fit)
Analysis of Variance Table
Response: mpg
Df Sum Sq Mean Sq F value Pr(>F)
cyl 2 824.78 412.39 45.7033 1.464e-08 ***
drat 1 14.45 14.45 1.6017 0.21890
qsec 1 2.83 2.83 0.3137 0.58108
vs 1 1.02 1.02 0.1132 0.73969
am 1 26.35 26.35 2.9198 0.10157
gear 2 8.15 4.07 0.4513 0.64254
carb 1 49.96 49.96 5.5363 0.02798 *
Residuals 22 198.51 9.02
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
The critical plots of linear regression model are shown below –
- Residuals vs Fitted
- Normal Q-Q
- Scale Location
- Residuals vs Leverage
Calculating Model Performance Metrics
#Extracting R-squared value summary(fit)$r.squared [1] 0.8237094 #Extracting Adjusted R-squared value summary(fit)$adj.r.squared [1] 0.7515905 AIC(fit) [1] 171.2156 BIC(fit) [1] 187.3387
Higher R-Squared and Adjusted R-Squared value, better the model. Whereas, lower the AIC and BIC score, better the model.
Understanding AIC and BIC
AIC and BIC are measures of goodness of fit. They penalize complex models. In other words, it penalize the higher number of estimated parameters. It believes in a concept that a model with fewer parameters is to be preferred to one with more. In general, BIC penalizes models more for free parameters than does AIC. Both criteria depend on the maximized value of the likelihood function L for the estimated model.
AIC value roughly equals the number of parameters minus the likelihood of the overall model. Suppose you have two models, the model with the lower AIC and BIC score is better.
Variable Selection Methods
There are three variable selection methods – Forward, Backward, Stepwise.
1. Starts with a single variable, then adds variables one at a time based on AIC (‘Forward’)
2. Starts with all variables, iteratively removing those of low importance based on AIC (‘Backward’)
3. Run in both directions (‘Stepwise’)
#Stepwise Selection based on AIC
library(MASS)
step <- stepAIC(fit, direction="both")
summary(step)
#Backward Selection based on AIC
step <- stepAIC(fit, direction="backward")
summary(step)
#Forward Selection based on AIC
step <- stepAIC(fit, direction="forward")
summary(step)
#Stepwise Selection with BIC
n = dim(dat3)[1]
stepBIC = stepAIC(fit,k=log(n))
summary(stepBIC)
> summary(stepBIC)
Call:
lm(formula = mpg ~ vs + am + carb, data = dat3)
Residuals:
Min 1Q Median 3Q Max
-6.2803 -1.2308 0.4078 2.0519 4.8197
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 19.5174 1.6091 12.130 1.16e-12 ***
vs1 4.1957 1.3246 3.168 0.00370 **
am1 6.7980 1.1015 6.172 1.15e-06 ***
carb -1.4308 0.4081 -3.506 0.00155 **
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 2.962 on 28 degrees of freedom
Multiple R-squared: 0.7818, Adjusted R-squared: 0.7585
F-statistic: 33.45 on 3 and 28 DF, p-value: 2.138e-09
Look at the estimates above after performing stepwise selection based on BIC. Variables have been reduced but Adjusted R-Squared remains same (very slightly improved). AIC and BIC scores also went down which indicates a better model.
AIC(stepBIC)
BIC(stepBIC)
Calculate Standardized Coefficients
Standardized Coefficients helps to rank predictors based on absolute value of standardized estimates. Higher the value, more important the variable.
#Standardised coefficients
library(QuantPsyc)
lm.beta(stepBIC)
#R Function : Manual Calculation of Standardised coefficients
stdz.coff <- function (regmodel)
{ b <- summary(regmodel)$coef[-1,1]
sx <- sapply(regmodel$model[-1], sd)
sy <- sapply(regmodel$model[1], sd)
beta <- b * sx / sy
return(beta)
}
std.Coeff = data.frame(Standardized.Coeff = stdz.coff(stepBIC))
std.Coeff = cbind(Variable = row.names(std.Coeff), std.Coeff)
row.names(std.Coeff) = NULL
Calculating Variance Inflation Factor (VIF)
Variance inflation factor measure how much the variance of the coefficients are inflated as compared to when independent variables are not highly non-correlated. It should be less than 5.
vif(stepBIC)
Testing Other Assumptions
#Autocorrelation Test
durbinWatsonTest(stepBIC)
#Normality Of Residuals (Should be > 0.05)
res=residuals(stepBIC,type="pearson")
shapiro.test(res)
#Testing for heteroscedasticity (Should be > 0.05)
ncvTest(stepBIC)
#Outliers – Bonferonni test
outlierTest(stepBIC)
#See Residuals
resid = residuals(stepBIC)
#Relative Importance
install.packages("relaimpo")
library(relaimpo)
calc.relimp(stepBIC)
See Actual vs. Prediction
#See Predicted Value
pred = predict(stepBIC,dat3)
#See Actual vs. Predicted Value
finaldata = cbind(mtcars,pred)
print(head(subset(finaldata, select = c(mpg,pred))))
mpg pred Mazda RX4 21.0 20.59222 Mazda RX4 Wag 21.0 20.59222 Datsun 710 22.8 29.08031 Hornet 4 Drive 21.4 22.28235 Hornet Sportabout 18.7 16.65583 Valiant 18.1 22.28235
Other Useful Functions
#Calculating RMSE
rmse = sqrt(mean((dat3$mpg - pred)^2))
print(rmse)
#Calculating Rsquared manually
y = dat3[,c("mpg")]
R.squared = 1 - sum((y-pred)^2)/sum((y-mean(y))^2)
print(R.squared)
#Calculating Adj. Rsquared manually
n = dim(dat3)[1]
p = dim(summary(stepBIC)$coeff)[1] - 1
adj.r.squared = 1 - (1 - R.squared) * ((n - 1)/(n-p-1))
print(adj.r.squared)
#Box Cox Transformation
library(lmSupport)
modelBoxCox(stepBIC)
K-fold cross-validation
In the program below, we are performing 5-fold cross-validation. In 5-fold cross-validation, data is randomly split into 5 equal sized samples. Out of the 5 samples, a single sample which is random 20% of data is retained as validation data , and the remaining 80% is used as training data. This process is then repeated 5 times, with each of the 5 samples used exactly once as validation data. Later we average out the results.
library(DAAG)
kfold = cv.lm(data=dat3, stepBIC, m=5)