![31
dataset = read.csv("SLR.csv", header=T,
colClasses = c("numeric", "numeric", "numeric"))
head(dataset,5)
#/////Simple Regression/////
simple.fit = lm(Sales~Spend,data=dataset)
summary(simple.fit)
#Loading the necessary libraries
library(lmtest) #dwtest
library(fBasics) #JarqueBeraTest
#Testing normal distribution and independence assumptions
jarqueberaTest(simple.fit$resid) #Test residuals for normality
#Null Hypothesis: Skewness and Kurtosis are equal to zero
dwtest(simple.fit) #Test for independence of residuals
#Null Hypothesis: Errors are serially UNcorrelated
#Simple Regression Residual Plots
layout(matrix(c(1,1,2,3),2,2,byrow=T))
#Spend x Residuals Plot
plot(simple.fit$resid~dataset$Spend[order(dataset$Spend)],
main="Spend x Residualsnfor Simple Regression",
xlab="Marketing Spend", ylab="Residuals")
abline(h=0,lty=2)
#Histogram of Residuals
hist(simple.fit$resid, main="Histogram of Residuals",
ylab="Residuals")
#Q-Q Plot
qqnorm(simple.fit$resid)
qqline(simple.fit$resid)
R Code for SLR](https://image.slidesharecdn.com/regression-190807101910/85/Regression-analysis-in-R-31-320.jpg)
![R Code for MLR
#loading of the data
data("mtcars")
#viewing the data
mtcars
head(mtcars)
names(mtcars)
#attach command is used in R so that we need not call the data
everytime
attach(mtcars)
#checking the realtionship between the variables
plot(mpg,cyl)
plot(mpg,disp)
plot(mpg,hp)
plot(mpg,drat)
plot(mpg,wt)
plot(mpg,qsec)
plot(mpg,vs)
47
plot(mpg,am)
plot(mpg,gear)
plot(mpg,carb)
#creating simple linear regression
#creating the multiple linear model
model <- lm(mpg~cyl+disp+hp+drat+wt+qsec+vs+am+gear+carb)
model
#checking the summary of the model
summary(model)
#various parameters to check the fitness of the model
#mean square error
sqrt(sum((model$residuals)^2)/21)
summary(model)
#hypothesis testing t-test
#test statistic is just the point estimate of the slope of the model divided by the standard error
of that coefficient/slope value
#example 1
tstat <- coef(summary(model))[3,1]/coef(summary(model))[3,2]
tstat
2*pt(tstat, 21, lower.tail=FALSE)
#example2
tstat2 <- coef(summary(model))[1,1]/coef(summary(model))[1,2]
tstat2
2*pt(tstat2, 21, lower.tail=FALSE)
summary(model)](https://image.slidesharecdn.com/regression-190807101910/85/Regression-analysis-in-R-47-320.jpg)
![49
#using the model
newdata <- data.frame(wt=2.92, qsec=20.1, am=1)
predy <- predict(model3, newdata, interval="predict")
predy
confy <- predict(model3, newdata, interval="confidence")
confy
confy %*% c(0, -1, 1)
predy %*% c(0, -1, 1)
confy[1] == predy[1]
#sample prediciton
mtcars[20, ]
pred<-
coef(summary(model3))[1,1]+coef(summary(model3))[2,1]*19.9+coef(summary(model3))[3,1]*
1.835+coef(summary(model3))[4,1]*1+coef(summary(model3))[5,1]*1.835*1
pred
33.9-31.0523](https://image.slidesharecdn.com/regression-190807101910/85/Regression-analysis-in-R-49-320.jpg)
Regression analysis in R
- 1. REGRESSION ANALYSISPresented by – Alichy Sowmya Parth Prajapati Vikrant Ratnakar Department of Pharmacoinformatics , NIPER S.A.S. Nagar
- 3. “◉ Linear Regression is a supervised modeling technique for continuous data that generates a response based on the set of input features. ◉ It is used for explaining the linear relationship between a single variable Y, called the response (output or dependent variable), and one or more predictor (input, independent or explanatory variables). ◉ It’s a simple regression problem if only a single variable X is considered, otherwise it takes the form of a multiple regression problem, that is if more than one predictor is used in the model. 3
- 4. ◉ Statistical Modelling is the process of obtaining a statistical model which adequately describes the relationships between the variables involved ◉ The model: takes the form of a prediction equation - the values of a dependent variable (DV) are predicted by a set of independent variables (IV) ◉ Simplest model: simple linear regression 4
- 6. Simple Linear Regression ◉ SLR is investigating the linear relation between two variables Y (DV) and X (IV or explanatory variable) ◉ “Linear”: used because the population mean of Y is represented as a linear or straight-line function of X ◉ “Simple”: refers to the fact that there is only one independent variable ◉ Examples: • air quality and lung function • medication dose and outcome of blood test 6
- 7. Explore the relationship Between Two Continuous Variables ◉ Step 1: Scatterplot Shape of scatterplot gives form of relation • linear • quadratic • more complex ◉ Step 2: Correlation coefficient Strength of linear relation given by correlation coefficient • r: ranges from –1 to +1 • –1 : perfect negative linear relationship • +1 : perfect positive linear relationship. • 0 : no linear relationship. 7
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- 9. ◉ Step 3: Simple linear regression This is the population line. • Y = dependent or response variable. Must be continuous. • X = independent / predictor / explanatory variable or covariate • α = population regression parameter / intercept: point where the line crosses the vertical axis • β = population regression parameter / slope: the change in the mean value of Y for each increase of one unit in the value of X • e = model error term e (residual) = deviations between predicted values of Y and the actual values of Y Assumed normally distributed with mean 0 and standard deviation 9
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- 11. Objective of SLR ◉ Objective: to predict or estimate the value of DV Y corresponding to a given value of IV X, thru the estimated regression line ◉ Sample: the observed values are Xi and Yi, I =1,2,…n ◉ Build up: an estimated regression line using the sample. The regression line from the observed data is an estimate of the relationship between X and Y in the population 11
- 12. Estimated Regression Equation ◉ a = regression coefficient = the estimate of the parameter α = the intercept of the estimated regression line = the value of Y where the line crosses the Y axis ◉ b = regression coefficient = the estimate of the parameter β = the slope of estimated regression line = the change in the mean value of Y for each one unit increase in the value of X 12
- 13. Residual ◉ For any subject i, i = 1,2,3, …, n ◉ The original observed values are Xi and Yi ◉ For any given Xi , the ‘Y’ value given by the line is called the predicted value and denoted by ◉ The residual ei is the difference between the predicted value and the observed value 13
- 14. Least Squares Estimation ◉ Least squares estimation is the method of estimating the equation / fitting the model to the data in an optimal way ◉ The sum of squares of the vertical distances of the observations from the line are minimized ◉ Least squares estimation minimizes 14
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- 16. Is X a significant predictor of Y ◉ The association between X and Y is given by the regression coefficient for the slope ◉ A zero slope means X has no “impact” on Y ◉ whereas a large value indicates large changes in Y when X changes 16
- 17. ◉ Denoted by R2 ◉ Measures the goodness ‘fit’ of the model ◉ Assesses the usefulness or predictive value of the model ◉ Is interpreted as the proportion of variability in the observed values of Y explained by the regression of Y on X ◉ E.g. R2 =71.9%, almost 72% of the variation in lung function (FEV) is explained by the regression of FEV on height ◉ R2 =SSR/SST (eg., 78.34/109.01 = 0.719 =71.9%) 17 Coefficient of Determination
- 18. R2 and b ◉ The coefficient of determination R2 describes how well the regression equation summaries the data ◉ The regression coefficient b gives the nature of the relationship between X and Y the degree of change in Y for certain changes in X ◉ Two data sets may have the same slope b but different R2 values and vise versa 18
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- 21. Assumptions 1) The observations must be independent 2) The values of the dependent variable Y should be Normally distributed (normality) 3) The variability (variance) of Y should be the same for each value of X - homoscedasticity or constant variation 4) If X is continuous, the relation between X and Y should be linear (linearity) Note • X need not be a random variable nor have a Normal distribution • In fact the assumptions need to hold for the residuals but can equivalently be tested for Y or the residuals • A transformation of Y may be required 21
- 22. Assumptions - Strategies for testing ◉ Normality • Test for Y values or for standardized residuals • using 5 measures (histogram, Normal Q-Q plot, boxplot, skewness and kurtosis statistics) ◉ Linearity • Assess from scatterplot of X vs.Y ◉ Constant variation • Plot of standardized residuals vs. X • In plot of standardized residuals vs. X the points should scatter randomly (without any pattern) and evenly (vertical spread the same) 22
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- 26. An Example: FEV (Y) and height (X) ◉ Normality of FEV: • skewness=0.867, • kurtosis -3 =1.028 26
- 27. Linearity: 27
- 28. Constant variance? 28 • Constant variance is not assumed • FEV needs a natural logarithm transformation
- 29. After transformation, reassess the assumptions for the transformed variable: ln(FEV) ◉ Normality: • skew=0.040 • kurtosis - 3= -0.433 29
- 31. 31 dataset = read.csv("SLR.csv", header=T, colClasses = c("numeric", "numeric", "numeric")) head(dataset,5) #/////Simple Regression///// simple.fit = lm(Sales~Spend,data=dataset) summary(simple.fit) #Loading the necessary libraries library(lmtest) #dwtest library(fBasics) #JarqueBeraTest #Testing normal distribution and independence assumptions jarqueberaTest(simple.fit$resid) #Test residuals for normality #Null Hypothesis: Skewness and Kurtosis are equal to zero dwtest(simple.fit) #Test for independence of residuals #Null Hypothesis: Errors are serially UNcorrelated #Simple Regression Residual Plots layout(matrix(c(1,1,2,3),2,2,byrow=T)) #Spend x Residuals Plot plot(simple.fit$resid~dataset$Spend[order(dataset$Spend)], main="Spend x Residualsnfor Simple Regression", xlab="Marketing Spend", ylab="Residuals") abline(h=0,lty=2) #Histogram of Residuals hist(simple.fit$resid, main="Histogram of Residuals", ylab="Residuals") #Q-Q Plot qqnorm(simple.fit$resid) qqline(simple.fit$resid) R Code for SLR
- 33. Multiple Regression ◉ Simple linear regression describes the linear relationship between a dependent variable Y and a single explanatory variable X ◉ Multiple regression is an extension to the case of one dependent variable and two or more explanatory variables 33
- 34. Reasons for performing Multiple regression ◉ Predictions on the basis of a number of variables will be better than those based on only one explanatory variable ◉ When testing the effect of a primary variable of interest e.g. treatment effect / exposure, one needs to account for all other extraneous influences • The need to ‘control’ or ‘adjust’ for the possible effects of ‘nuisance’ explanatory variables (known as confounders) ◉ The relationships may be complex e.g. variables may have combined or synergistic effects on the dependent variable 34
- 35. Reasons for performing Multiple regression ◉ It is almost always better to perform one comprehensive analysis including all the relevant variables than a series of two- way comparisons • Reduce chances of increasing Type I error rate beyond 5% • In multiple regression a linear model is fitted for the dependent variable, which is expressed as a linear combination of the independent variables 35
- 36. Importance of Predictors ◉ The regression coefficient bi represents the effect of that independent variable on DV Y, after controlling for all the other variables in the model ◉ The importance of each individual variable is tested by a t test or an F test as for SLR ◉ Significance of an explanatory variable is dependent on which other variables included in the regression model ◉ A confidence interval gives further information 36
- 37. Multiple regression models ◉ Multiple linear regression • predictors all continuous and linearly related to the dependent variable ◉ Analysis of covariance (ANCOVA) • both continuous and categorical predictors ◉ Analysis of variance (eg. two-way ANOVA) • predictors all categorical ◉ Polynomial regression • quadratic or higher order terms included 37
- 38. Categorical predictors ◉ Association between a continuous DV Y and a categorical IV X is assessed by comparing the mean Y values in each category of X ◉ A reference category is chosen to compare the other category/ies with ◉ The regression coefficient for a comparison represents the difference in the mean for Y for the given category vs the reference category 38
- 39. Assessing the fit of the model ◉ R2 measures usefulness or predictive value of model ◉ R2 is interpreted as the proportion of the total variability explained by the model ◉ R2 increases in value as each additional variable is added to the model ◉ adjusted R2 (preferred measure) takes into account the number of explanatory variables included in the model ◉ E.g. R2 = 0.482 Radj2 =0.462 39
- 40. ◉ Also assess fit by inspection of standardized residuals • If these follow a Normal distribution • Any value ◉ Large residual: model does not fit well for that subject ◉ Some large residuals will occur by chance, many large residuals are of concern s > 3 and < -3 are large 40
- 41. Assumptions of multiple regression ◉ The observations must be independent ◉ The relation between each continuous X and the dependent variable should be linear ◉ The values of the dependent variable Y should have a Normal distribution ◉ The variability of Y should be the same for any set of values of the explanatory variables – homoscedasticity 41
- 42. How to assess assumptions ◉ Assessing the Normality of Y (or the standardized residuals) • Obtaining scatterplots of Y (or the standardized residuals) against each continuous X primarily to assess linearity ◉ Obtaining • Levene’s test for Y (or the standardized residuals) (if categorical predictors are included in the model) to assess equal variance • a plot of the standardized residuals against each X (if continuous predictors are included in the model) primarily to assess constant variation 42
- 43. Example - assess assumptions ◉ DV: FEV1 ◉ Explanatory variables: • Height (in cm’s) • Gender (binary) • Smoking status (3categories) ◉ Normality of FEV1 (5measures) • skewness= -0.11, • kurtosis -3 = -0.80 • Assumed 43
- 44. ◉ Linearity: FEV1 vs height (scatterplot) • Assumed 44 Linearity
- 45. Constant variation: ◉ Constant variation: • standardized residuals vs height (scatterplot): no clear pattern ◉ Assumed 45
- 46. Equality Variances • Levenes’ Test (Robust): p = 0.937 >0.05, • Assumed 46 Conclusion: All the assumptions are met Note: the test could be done using standardized residuals
- 47. R Code for MLR #loading of the data data("mtcars") #viewing the data mtcars head(mtcars) names(mtcars) #attach command is used in R so that we need not call the data everytime attach(mtcars) #checking the realtionship between the variables plot(mpg,cyl) plot(mpg,disp) plot(mpg,hp) plot(mpg,drat) plot(mpg,wt) plot(mpg,qsec) plot(mpg,vs) 47 plot(mpg,am) plot(mpg,gear) plot(mpg,carb) #creating simple linear regression #creating the multiple linear model model <- lm(mpg~cyl+disp+hp+drat+wt+qsec+vs+am+gear+carb) model #checking the summary of the model summary(model) #various parameters to check the fitness of the model #mean square error sqrt(sum((model$residuals)^2)/21) summary(model) #hypothesis testing t-test #test statistic is just the point estimate of the slope of the model divided by the standard error of that coefficient/slope value #example 1 tstat <- coef(summary(model))[3,1]/coef(summary(model))[3,2] tstat 2*pt(tstat, 21, lower.tail=FALSE) #example2 tstat2 <- coef(summary(model))[1,1]/coef(summary(model))[1,2] tstat2 2*pt(tstat2, 21, lower.tail=FALSE) summary(model)
- 48. 48 #F-test summary(model) #Coefficient Confidence Intervals confint(model, level=.95) #testing the various assumptions of the model #1 checking whether the residuals are normally distributed or not #histogram resid<- model$residuals hist(resid) #quantile plot qqnorm(resid) qqline(resid) #2 checking the homoscedasticity plot(model$residuals ~ disp) abline(0,0) #residual analysis plot(model) #transformations model1 = lm(mpg ~cyl+log(disp)+log(hp)+drat+wt+qsec+vs+am+gear+carb) summary(model1) plot(model1) #reducing the model #calling of library library(MASS) #running the AIC on intial model stepAIC(model) #running the AIC on the transformed model stepAIC(model1) #constructing new models with reduced variables model2<-lm(mpg~qsec+wt+am) summary(model2) model6<-lm(mpg~log(disp)+gear+carb) summary(model6) #partial F-test nestmodel = lm(mpg ~ wt + qsec + am) anova(model,nestmodel) #Multicollinearity plot(mtcars) #checking the correlation cor(qsec, wt) cor(am, wt) cor(am, qsec) #variance inflation factor install.packages("car") library(car) vif(model2) #Polynomial Model plot(model2$residuals ~ model2$fitted.values, xlab = "Fitted Values", ylab = "Residuals") abline(0,0) quadmod = lm(mpg ~ qsec + I(qsec^2)+ wt + am) plot(quadmod$residuals ~ quadmod$fitted.values, xlab = "Fitted Values", ylab = "Residuals") abline(0,0) summary(quadmod) AIC(quadmod) #Interaction Model model3<-lm(mpg~qsec+wt*am) summary(model3) AIC(model3) resid3<- model3$residuals hist(resid3) qqnorm(resid3) qqline(resid3) plot(model3$residuals ~ disp) abline(0,0) plot(model3)
- 49. 49 #using the model newdata <- data.frame(wt=2.92, qsec=20.1, am=1) predy <- predict(model3, newdata, interval="predict") predy confy <- predict(model3, newdata, interval="confidence") confy confy %*% c(0, -1, 1) predy %*% c(0, -1, 1) confy[1] == predy[1] #sample prediciton mtcars[20, ] pred<- coef(summary(model3))[1,1]+coef(summary(model3))[2,1]*19.9+coef(summary(model3))[3,1]* 1.835+coef(summary(model3))[4,1]*1+coef(summary(model3))[5,1]*1.835*1 pred 33.9-31.0523
- 50. THANK YOU….. 50