Exercise 9

Load data

rm(list = ls()) #clean memory

data9 = read.csv("ex9data") # ta dedomena vriskontai mesa sto data frame data9

Define \(X_2=X^2\) and add it to the data.frame with the data

data9$X2 = data9$X^2

Part (i)

fit9 = lm(Y~., data9) 


names(fit9)
##  [1] "coefficients"  "residuals"     "effects"       "rank"         
##  [5] "fitted.values" "assign"        "qr"            "df.residual"  
##  [9] "xlevels"       "call"          "terms"         "model"
Yhat = fit9$fitted.values 

plot(data9$X, data9$Y)
lines(data9$X,Yhat, col=2)

Part (ii)

summary(fit9)
## 
## Call:
## lm(formula = Y ~ ., data = data9)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -3.03214 -0.58929  0.03393  1.10893  2.10000 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -1.826e+02  1.768e+01  -10.33 2.73e-06 ***
## X            8.983e+00  7.616e-01   11.80 8.91e-07 ***
## X2          -9.107e-02  7.993e-03  -11.39 1.20e-06 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.727 on 9 degrees of freedom
## Multiple R-squared:  0.9474, Adjusted R-squared:  0.9357 
## F-statistic: 81.03 on 2 and 9 DF,  p-value: 1.757e-06

Part (iii). Need to put new data in a data frame

newdata9 = data.frame(X=48, X2=48^2) 
predict(fit9,newdata9,interval="conf") 
##        fit      lwr      upr
## 1 38.78357 37.05092 40.51622

Part (iv)

predict(fit9,newdata9, interval="pred") 
##        fit      lwr      upr
## 1 38.78357 34.51055 43.05659

Part (v)

summary(fit9)
## 
## Call:
## lm(formula = Y ~ ., data = data9)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -3.03214 -0.58929  0.03393  1.10893  2.10000 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -1.826e+02  1.768e+01  -10.33 2.73e-06 ***
## X            8.983e+00  7.616e-01   11.80 8.91e-07 ***
## X2          -9.107e-02  7.993e-03  -11.39 1.20e-06 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.727 on 9 degrees of freedom
## Multiple R-squared:  0.9474, Adjusted R-squared:  0.9357 
## F-statistic: 81.03 on 2 and 9 DF,  p-value: 1.757e-06

Exercise 10

rm(list = ls())
data10 = read.csv("ex10data")

Part (i)

fit10 = lm(Y~., data10)

names(fit10)
##  [1] "coefficients"  "residuals"     "effects"       "rank"         
##  [5] "fitted.values" "assign"        "qr"            "df.residual"  
##  [9] "xlevels"       "call"          "terms"         "model"
fit10$coefficients
## (Intercept)          X1          X2 
##      37.650       4.425       4.375
b0hat = fit10$coeff[1]
b0hat
## (Intercept) 
##       37.65
b1hat = fit10$coeff[2]
b1hat
##    X1 
## 4.425
b2hat = fit10$coeff[3]
b2hat
##    X2 
## 4.375

Part (ii)

summary(fit10)
## 
## Call:
## lm(formula = Y ~ ., data = data10)
## 
## Residuals:
##    Min     1Q Median     3Q    Max 
## -4.400 -1.762  0.025  1.587  4.200 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  37.6500     2.9961  12.566 1.20e-08 ***
## X1            4.4250     0.3011  14.695 1.78e-09 ***
## X2            4.3750     0.6733   6.498 2.01e-05 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 2.693 on 13 degrees of freedom
## Multiple R-squared:  0.9521, Adjusted R-squared:  0.9447 
## F-statistic: 129.1 on 2 and 13 DF,  p-value: 2.658e-09

Part (iii)

names(fit10)
##  [1] "coefficients"  "residuals"     "effects"       "rank"         
##  [5] "fitted.values" "assign"        "qr"            "df.residual"  
##  [9] "xlevels"       "call"          "terms"         "model"
errors=fit10$res
qqnorm(errors)
qqline(errors)

plot(fit10$fitted.values, errors)

Tests for suitability of linear model

plot(data10$X1, errors) 

plot(data10$X2, errors)

For other types of residuals

library(MASS)
standard.errors = stdres(fit10) 

student.errors = studres(fit10)