V9

Question 1

what is regression analysis

Answer 1

• statistical process for estimating the relationships among variables
• many techniques for analysing several variable models
• focus is on the relationship between a dependent variable and one or more independent variables

Question 2

regression model

Answer 2

regression model variables

• unknown parameters, denoted as ß (effect size)
• independent variables (x)
• dependent variable (y)
• > independent causes / influenced dependent

Formula Y = f(X, ß)

ß is a vector
N rows of X, k column of x

Constants
N = tuber of independent measurements (more like siblings, not totally independent)
k = number of unknown parameters

Question 3

assumptions of the regression model

Answer 3

• sample is representative of the population for the interference prediction
• error is a random variable with a mean of zero conditional on the explanatory variables
• independent variables are measured with no error
• predictors are linearly independent, i.e. it is not possible to express any predictor as a linear combination of the others
• errors are uncorrelated
• variance of the data should be approximately equal across the range of your predicted values. if not, log-transform or other methods might instead be used

Question 4

how to make statistical predictions about the unknown parameters

Answer 4

when N&raquo_space; k , and the measurement errors are normally distributed then the excess of information contained in (N-k) measurements is used

Question 5

how to code linear model

Answer 5

with lm function

data(air quality)
lm.temp Ozone dependent, Temp independent

Question 6

how to plot a regression line

Answer 6

plot(airquality$Ozone ~airquality$Temp, pch=19, col=”blue”)

a

Question 7

confidence interval

Answer 7

• calculate the margin of error
• in standard regression we can use the t-statistic
• > standard error (from the lm summary)
• > critical value - probability boundary (alpha) 0.05 , 2 sided = 0.975 - degrees of freedom (N-2)
• > critique

Question 8

how to calculate Margin of error

Answer 8

Critical value x standard error

Question 9

code confidence interval and margin of error

Answer 9

lm.sum

Question 10

predicted values

- predict data from min til max temperature

Answer 10

new

Question 11

plot predicted data with confidence interval

Answer 11

plot(air quality$temp, air quality&ozone, xlab=temp, lab”Ozone”)
lines(new$Temp,as.numeric(conf[,”fit”]), lty=1, lwd=2) # regression slope
lines(new$Temp,(conf[,”upr”]), lty=2, col=”blue”) # upper limit
lines(new$Temp,(conf[,”lwr”]), lty=2, col=”blue”) #lower limit

-> can also use function visage (get from library)

Question 12

residuals

Answer 12

• describe how well the regression line fit the data (goodness of fit)
• we aim to - minimise sum of squares of the residuals, if we do - maximum likelihood model

Question 13

how to visualise residuals (code)

Answer 13

• easiest on clean data, not containing NAs

airqual. clean

Question 14

multiple linear regression

Answer 14

• every factor has a different ß

Y = a + bx1 + bx2

• modeling ozone as a function of Temp and Wind

Ozone = a + b1Temp + b2Wind

-> two factors are often better, can be checked in R^2. the higher the more amount of values explained by model

Question 15

plot observed versus estimated data points

Answer 15

first make predicted dataset

plot(airquality$Temp, airquality$Ozone) # plot observed data
plot(airquality$Temp, Ozone.est, pch=2, col”red)
legend(“top left”,legend = c(“observed”, “predicted”), pch=c(1,2), bty=”n”)

Question 16

quadratic regression + coding

Answer 16

• linearity of coefficients means: a change in ONE of the independent variables always yiels a corresponding change in the response variable
• this means the following functions are also linear

Y= a+b1(X^2)
Y=a + e^b1*X

lm(Ozone ~ Temp + I(Temp^2), data = air quality)

• around ^always needs to be I()

Question 17

how to plot quadratic regression coeffiction

Answer 17

plot(Ozone ~ Temp, data = air quality)
curve(a - b*I(X^2), add = TRUE)
- take a and b from lm summary

Question 18

how to see which model is the best ?

Answer 18

• mode selection is the task of selecting a statistical model from a set of candidate models
• AIC
• relative comparison of models
• not a test, just a guideline
• air rewards goodness of fit( as assessed by the likelihood function), bit it also includes a penalty that is an increasing function of the number of estimated parameters

Question 19

AIC

Answer 19

AIC = 2k - 2ln(L)

• L is the maximum value of the likelihood function for the model (goodness of fit)
• k is the number of estimated parameters in the model
• want the lowest AIC value
• AIC()