assumptions and dignosics

Question 1

assumptions of linear model (LINE)

Answer 1

L = linearity
I = independence of errors
N = normality of errors
E = equal variance (homoscedecity)

Question 2

linearity

Answer 2

assumption = the relationship between x and y is linear

investigated with:
- scatterplots with loess lines (for single variables)
- component-residual plots (multiple predictors)

Question 3

component-residual plots

Answer 3

used for testing linearity between each predictor and the outcome, controlling for other predictors (also called partial residual plots)
have x values on the x axis and partial residuals on the y axis.
partial residuals for each x variable are:
residual from lm including all predictors + partial (linear) relation between x and y

Question 4

Normally distributed errors

Answer 4

assumption = the errors are normally distributed around each predicted value

investigated with:
- QQ-plots
- histograms (plot the frequency distribution of the residuals)

Question 5

QQ-plots

Answer 5

quantile comparison plots
used to test the normality of errors
- plots the standardised residuals from the model against their theoretical values. if the points are normally distributed they should fall neatly on the diagonal

Question 6

equal variance (homoscedacity)

Answer 6

assumption = the variance is constant across values of the predictors (x) and across values of the fitted values (^y)
- heteroscedacity refers to the violation of this assumption

investigated with:
- plot the residual values against predicted values -

categorical predictors should show a similar spread of residuals across their fitted values while plots for continuous should look like a random array of dots

Question 7

independence of errors

Answer 7

assumption = the errors are NOT correlated with one another

difficult to test unless we know the potential source of correlation between the cases - for now, we evaluate this based on study design

Question 8

diagnostics of linear models

Answer 8

• model outliers
• high leverage case
• high influence cases

Question 9

model outliers

Answer 9

cases for which there is a large discrepancy between their predicted value (^y) and their observed value (y)
- demonstrate large residuals so judged on a case-by-case basis

Question 10

high leverage cases

Answer 10

cases with an unusual value of the predictor (x)

• these have potential to influence our predicted β values
• assessed with hat values

Question 11

high influence cases

Answer 11

cases that have a large impact on the estimation of the model

• can have strong influence on our β coefficients and therefore can influence our model results and conclusions
• assessed with cook’s distance and dfbeta

Question 12

studentised residuals

Answer 12

provide a version of standardised residuals excluding the outlying cases - values >+2 and <-2 indicate outlyingness

Question 13

hat values (hi)

Answer 13

used to assess leverage of linear models

equation:
hi = 1/n + ( ( x - mean of x)^2 / sum of (x - mean of x)^2 squared)

the mean of hat is:
h = (k +1)/n

general rule = values more than 2*mean of hat are considered high leverage

Question 14

Cook’s distance

Answer 14

cooks distance refers to the average distance the ^y values will move if a given case is removed - determines if cases are high influence

Di = ( (studentised residual i)^2 / (k+1) ) * (hi / 1-hi)
Di = outlyingeness * leverage

there are many different suggestions for cut offs but a common one is Di > 4/n-k-1

Question 15

DFfit

Answer 15

difference between the predicted outcome values for a case, with vs without the inclusion of a different case

Question 16

DFbeta

Answer 16

the difference between the values for a coefficient, with and without a different case included

Question 17

DFbetas

Answer 17

standardised version of DFbeta - obtained by dividing an estimate of the standard error of the regression coefficient with the case removed

Question 18

COVRATIO

Answer 18

influential case’s influence on SE is measured using the COVRATIO statistic
- a covratio value <1 shows that precision is decreased by a case (SE is increased)

cases with COVRATIO > 1 + [3(k+1)/n] or < 1 - [3(k+1)/n] can be considered strong influence on standard errors

Question 19

Multi-collinearity

Answer 19

refers to the correlation between predictors. when there are large correlations between predictors the SE are increased

neither an assumption nor a diagnostic

Question 20

Variance inflation factor

Answer 20

VIF quantifies the extent to which standard errors are increased by predictor inter-correlations
we want VIF values to be close to 1, anything over 10 indicates a problem

Question 21

dealing with unusual cases

Answer 21

• is there a data entry error? can it be corrected? if not it should be deleted
• is the data legitimate but extreme? consider ways to reduce inflation
• are there unusual values as a result of skewness

Question 22

sensitivity analysis

Answer 22

refers to the idea of checking whether you get similar results irrespective of the methodological decisions you make