9. Assumptions and Diagnostics

Question 1

What are the different linear model assumptions?

Answer 1

Linearity
Independence of Errors
Normality of Errors
Equal Variance

Question 2

What happens if assumptions are violated?

Answer 2

Model can’t be accurate

Question 3

Why is it useful visualise assumptions?

Answer 3

Easier to see nature and magnitude of assumption violation

Question 4

What are some drawbacks of using statistical methods of assessing assumptions?

Answer 4

Suggests assumptions are violated when they are actually small

This is due to statistical power and give no info on actual problem

Question 5

What is linearity?

Answer 5

Assumes X & Y are linear

Question 6

What happens if we estimate a linear relation when there isn’t one present?

Answer 6

Can result in underestimating that relation

Question 7

How is linearity investigated?

Answer 7

Investigated via scatterplots with loess lines (single variance)

or

Component residual’s plots (when we have multiple predictors) - Closer to black line = More linear

Question 8

How do we test for non-linearity?

Answer 8

Need to know relations between each predictor and outcome are linear by controlling other predictors

Partial residuals for x value

ei + BjXij (partial linear relation xj + y)

Question 9

What are normally distributed errors?

Answer 9

Assumes error are normally distributed about each predicted value

Question 10

How are normally distributed errors investigated?

Answer 10

Investigated via QQ plots (Quantile comparisons plots)

• Plot standardized residuals from model against theoretical expected values
• If normally distributed = Should fall neatly on diagonal plot
• If non-normally distributed = Impact shape
• Can also use histograms to see distribution

Question 11

What is equal variance (Homoscedasticity) ?

Answer 11

Assumes variance is constant across values of the predictors x1, … xk + across values of the fitted values y

Question 12

What is it called when homoscedasticity is violated?

Answer 12

Heteroscedasticity

Question 13

How is equal variance investigated?

Answer 13

Using residual plot

When comparing residual values vs predicted values = Should be the same difference above the line and below the line

Categorical predictors = Should show similar spread
Continuous predictors = Should be dots that follow the line closely

Question 14

What is independence of errors?

Answer 14

Assumes errors are not correlated with one another

Question 15

How do we test independence of errors?

Answer 15

Difficult to test, unless we know the potential source of correlation between cases

Errors are not correlated in between persons design

Can use a variant of linear model to account for independence

Question 16

What are linear model diagnostics and what the three features?

Answer 16

Explore individual cases in context of model

Model outliers, High Leverage, High Influence

Question 17

What are model outliers?

Answer 17

Cases that have unusual outcome values given their predictor values

(Show a large difference between predicted and observed)

Question 18

What are outliers?

Answer 18

Large residuals = May have a strong influence on the model

Question 19

How do you determine an outlier?

Answer 19

Size of outlier
Unstandardized individuals (same units as DV)
yi - yi (estimated)

Fine for comparison across cases in lm but difficult to compare across DVs with different units

Question 20

What is the difference between standardised and studentized residuals?

Answer 20

Standardized residuals:

• Unstandardized/Estimate SD (convert to z-score)
• Calculations can include outliers when using whole data

Studentized residuals:

• Standardized residuals w/o extreme case values of > +2 or < -2 indicate outlyingness

Question 21

What are high leverage cases?

Answer 21

Have unusual predictor value or combination of predictor values (e.g. x for away from x bar (mean))

Question 22

How do you find high leverage cases?

Answer 22

Hat values used to assess as it’s the difference between value of x and mean value of x

Mean of hat value x 2 = High leverage

Question 23

What are high influence cases?

Answer 23

Cases having a large impact on estimation of model

Have a strong effect on coefficients - e.g. if deleted a case, coefficient would change

Question 24

How do we investigate high influence cases?

Answer 24

Degree of change = one way to judge magnitude of influence

Can also consider influence via cook’s distance and DF beta

Question 25

What is cook’s distance?

Answer 25

The average distance y hat values will move if given case removed

Different cut-off suggestions

Di > 1
Dj > 4/n-k-1

Question 26

What are the three different ways you can look at Cook’s Distance in more detail (name only) ?

Answer 26

DFFit
Dfbeta
DFbeats

Question 27

What is DFFit?

Answer 27

Difference between predicted outcome (Y) of case vs without case included

Question 28

What is Dfbeta?

Answer 28

Difference between value of coefficient if case included vs not included

This is how it varies for DFFit - focuses on coefficient

Question 29

What is Dfbeats?

Answer 29

Standardised version of Dfbeta

Question 30

How do we examine influence of SE?

Answer 30

SE can impact inferences

Measure via COVRATIO

<1 = Precision is decreased by a case (SE increases)
>1 = Precision increased by case (SE decreases)

COVRATIO > 1 -/+ [3(k+1)/n) = have strong influence on SE

Question 31

What is multi-collinearity?

Answer 31

Correlation between factors

Large correlation between predictors = Increased SE so don’t want predictors to be too correlated

Question 32

What do you do if multi-collinearity occurs?

Answer 32

If it happens = Combine two predictors into single composite or drop IV that is statistically redundant

Question 33

How do you test for multicollinearity?

Answer 33

Variation inflation factor (VIF) - Measures how much SE(beta) is increased by predictor correlations

• VIF quantities are increased by predictor inter-correlations
• VIFs > 10 = Issue - want it close to 1
• Always consider influence before deleting variable

Question 34

What are sensitivity analyses?

Answer 34

Checking if you get similar results, irrespective of methodological decisions

Do coefficients change if including certain case

Question 35

What if the results from sensitivity analysis are similar?

Answer 35

Increased confidence results x based on methodology but strength

Question 36

If a case has a high COVRATIO value but a low dfbeta, what is the most likely reason?

Answer 36

It has an extreme value on x but is not a regression outlier