r squared and regression to the mean

Question 1

the amount of variability explained by r squared

Answer 1

the proof of variability explained is the square of the correlation coefficient
r squared is the proportion of the variability in the y-values explained by the line of best fit

Question 2

regression to the mean: another interpretation of r

Answer 2

regression is moving closer to the mean

given x, the line of best fit predicted a value of y which we call y hat

r=0 no association between x and y
r=1 there is no regression to the mean

Question 3

multiple linear regression

Answer 3

there are multiple potential explanatory variables for a response variable

Question 4

linear regression for two explanatory variables

Answer 4

by principle of least squares, line of best fit has values of a, ß1 and ß2 that minimise the sum of squared residuals

Question 5

MLR and matrices

Answer 5

do not need to be able to do by hand

Question 6

MLR in excel

Answer 6

data, data analysis, regression

plot a residual plot for each explanatory variable
randomly distributed residuals indicate a good fit
U or upside down U are bad fit

usually try to choose the simplest model (with fewest explanatory variables)
- compromise between goodness of fit (small sum of residuals) and complexity (no of parameters)

Question 7

AIC (Akaike information criterion)

Answer 7

balance complexity with goodness of fit

AIC = n ln(SSE/n) + s(p+1)

n is no of data points

SSE is the sum of squared residuals (errors)

p is number of explanatory variables

a small AIC is desired

Question 8

related types of regression

Answer 8

• MLR can fit polynomials to data by treating powers as additional explanatory variables
• MLR can consider interactions between variables - one explanatory variable changes the effect that the second explanatory variable has on the response