Linear models - chapter 7 (residuals and assumptions)

what does it mean if a estimator is resistance

and give a example

not much change when small proportion of data are dramatically altered

models ability to provide reliable parameter estimates even when the data contain outliers or influential observations

median

What does robustness mean in a linear model

the models ability to provide reliable parameter estimation even when assumptions of the model are violated.

what are the rules for resistance and robustness in linear models

• Independence assumption violated

• constant variance assumption violated

• normality (when does it not hold)

• Independence ~ Different analysis

• constant variance ~ Transform data or different analysis that allows for unequal variance

• normality ~ model not robust if the are long tailed or high skewed distributions.

name the 4 properties of residuals if the model fits well

1. mean = 0 don’t need constant variance

2. normal dis

3. indep of fitted values and covariates

4. indep across i

what are Residuals good at checking

Normality

True for errors true for them

what is the equation for ordinary residuals

e^i = yi - y^i

equation for internally Studentized residuals

ri =

E^i / sigma sqrt(1-hii)

where E^i =residuals

name the 4 properties of Studentised residuals if the model fits well

1. the s.r have mean = 0 sigma = 1

2. s.r are symmetrically distributed

3. s.r independent of fitted values and covariates

4. s.r independent across i

what are s.r good at checking

constant variance assumption

if errors have cv then so will they.

Equation for Externally studentised residuals

ti=

= yi - y^i / sqrt(var(yi - y^i))

= E^i ( n-p-1 / n-p-ri² )^1/2

where y^i = Xi^TB^i

name 4 properties of Externally Studentised residuals if the model fits well

1. mean=0 sigma =1

2. e.s.r are ti ~ tn-p-1

3. e.s.r independent of fitted values and covariates

4. e.s.r independent across i

What is a plot of theoretic quantiles against sample quantiles called and what does it show you

1. QQ plot

2. used to check normality

3. heavy tails and points departing for horizontal lines means no normality.

what plot can be used to check for constant variance

what should it show you for assumption to hold

Residuals versus fitted values plot

points form a band of roughly constant width around the horizontal zero line

what do we use to check assumptions

diagnostic plots

if we create a PI or CI for x=1 other than the usual assumptions what else do we check

That the value x=1 is in the range of values observed in the data (interpolation NOT extrapolation).

how do you set up a box plot to check assumptions

set of box plots on the same plot with one for each group

How does a box plot check for normality

each box plot is symmetrical (median line in the middle with box symmetrically shaped)

how does a box plot check for constant variance

width of the box plots are roughly the same.

name some assumption plots

• Res vs fitted (x)

• QQ plot

• leverage plot ( Standardised residuals vs leverage (x))

What assumption does a QQ plot check

what are the axis

when is this assumption of concern

• normality

• theoretical quantities against sample quantities

• diversions in the tails of the dis may cause the assumption to be questioned (don’t follow a straight line)

What assumptions does res vs fitted check

• mean 0 → scattered around 0

• constant variance → distribution of residuals is constant throughout

when does a fitted vs residual fail variance assumption, what does this mean

appear to have a pattern throughout

explanatory variables don’t capture the deterministic components

what does a leverage plot show you

points of influence, if they’re leverage is high.