Simple linear regression

Question 1

what is linear regression?

Answer 1

simple approach to supervised learning. it is used to model the relationship between several input variables (x) and a continuous response variable (y)

Question 2

assumed model?

Answer 2

y=β0+β1 X+e

Question 3

distance between observed and predicted values?

Answer 3

residual e= Yi-predicted Yi= Yi-(β0+β1Xi)

Question 4

residual sum squares (RSS)?

Answer 4

total magnitude of deviations from all squared residuals of data points (sum) (residual may be positive or negative thus square)

Question 5

to find β0 and β1, use estimation of least squares

Answer 5

first order derivatives of RSS w/ respect to β0 and β1 separately, set to 0

Question 6

predicated β1?

Answer 6

cov(x,y)/var(x)

Question 7

predicted β0?

Answer 7

mean(y) - β1*mean(x)

Question 8

what is standard error(SE) an estimator for?

Answer 8

how the estimates vary under repeated sampling

Question 9

hypothesis testing for relationship between x and y?

Answer 9

H0: β1=0 H1: β1!=0

Question 10

t-statistics(to test null hypothesis)

Answer 10

t=(β1-0)/SE(β1). n-2 degrees of freedom

Question 11

critical value and confidence interval when n is large?

Answer 11

1.96(as n increases, t-dist gets closer to normal dist) and 95%(as n increases, t-dist gets closer to normal dist)

Question 12

p-value definition?

Answer 12

probability of observing any value >= |t|

Question 13

calculate confidence interval

Answer 13

[β1+-1.96*SE(β1)]

Question 14

when to reject null hypothesis?

Answer 14

when |t| both larger than 1.96, we can reject H0 with 95% confidence

Question 15

Residual standard error means?

Answer 15

RSE measures lack of fit, if RSE=3.259, on avg, deviation of Y from regression line is 3.259 points

Question 16

R squared for?

Answer 16

measures how well regression model describes data. e.g. if R squared is 0.6119, X explains only 61.19% of subject

Question 17

how to measure RSE

Answer 17

sqrt(1/(n-2)RSS)

Question 18

measure R square?

Answer 18

1-(RSS/TSS). TSS is the total variance in response variable y. (ranges from 0 to 1)

Question 19

for 95% CI, use β1 or β0

Answer 19

β1. β0 has nothing to do with r/s between X and Y

Question 20

how to install package MASS

Answer 20

intall.package(‘MASS’)

Question 21

load MASS?

Answer 21

library(MASS)

Question 22

load data Boston in MASS?

Answer 22

data(Boston)

Question 23

documentation in data set?

Answer 23

?Boston

Question 24

number of missing values?

Answer 24

sum(is.na(Boston))

Question 25

number of duplicated values?

Answer 25

sum(duplicated(Boston))

Question 26

find outliers for both variables?

Answer 26

boxplot. stats(Boston$var1)$out

boxplot. stats(Boston$var2)$out

Question 27

reduce dataset to subset of the 2 variables?

Answer 27

name=subset(Boston, select=c(var1,var2))

Question 28

scatterplot? var1 being y and var2 being x

Answer 28

plot(var1~var2, main='Scatterplot of var1 vs var2',
xlab='var2=name',
ylab='var1=name',
pch=20
col='gray50')

Question 29

simple linear regression?

Answer 29

lmfit=lm(var1~var2,data=name)

Question 30

summary of lm?

Answer 30

summary(lmfit)

Question 31

upper and lower range of CI 95%?

Answer 31

confit(lmfit, level=0.95)

Question 32

Regression when X is binary

Create a dummy variable that equals to one if rm is above the sample median

Answer 32

mydata$dummy=ifelse(mydata$x>=median(mydata$x),1,0)

Question 33

plot scatterplot with fitted line?

Answer 33

lmfit1=lmfit(var1~mydata, data=mydata)

plot(var1~mydata$dummy, main=’scatterplot of var1 vs var2’,xlab=’var2’, ylab=’var1’,pch=20,col=’gray50’)

abline(lmfit1,lwd=2,col=’deeppink3’)