Chapter 5

Question 1

State the assumptions of linear-regression modelling

Answer 1

• There is a linear relationship between the outcome and predictor variables
• The variance of residuals is the same for any value of the predictor variable (this is known as homoscedasticity)
• There is independence between observations
• Residuals at any predictor value are normally distributed (normality)

Question 2

What plot can be used to check linearity and homoscedasticity

Answer 2

Residuals vs Fitted plot:
plot of fitted values vs residuals

Question 3

What plot can be used to check normality

Answer 3

Normal Q-Q plot

Question 4

Describe DV vs PRED diagnostic plots for a non-linear mixed-effects model

Answer 4

Dependent variable (DV) vs Population predictions (PRED):

DV vs PRED takes predictions based on population parameters (i.e. excluding 𝜂’s)

Question 5

Describe DV vs IPRED diagnostic plots for a non-linear mixed-effects model

Answer 5

Dependent variable (DV) vs Individual predictions (IPRED):

DV vs IPRED bases predictions on individual values for model parameters (including 𝜂’s).

DV vs IPRED takes into account random error associated with an individual.

Question 6

Describe CWRES

Answer 6

Conditionally-weighted residuals (CWRES)

Normalising residuals through a weighted step.

This is important because residuals might be influenced by the magnitude of the observation (e.g. you might find a greater range of residuals around 𝐶𝑚𝑎𝑥 compared to around 𝐶𝑚𝑖𝑛). Weighting the residuals in order to standardise them makes it easier to think about what is happening with model predictions across the full range of observations. We will use conditionally weighted residuals (CWRES).

Question 7

How is the objective function value useful in quantitative evaluation of model fit

Answer 7

Using qualitative metric of improved model fit: OFV is a measure of model fit.

Objective function can be used to compare two models.
A lower OFV indicates a superior model fit.

Question 8

Define the Akaike Information Criteria (AIC) and describe how it may be used to select a superior model

Answer 8

The Akaike Information Criteria (AIC) is a way of determining whether the magnitude of a drop in objective function is worthy of retaining the more complex model.

𝐴𝐼𝐶 = (𝑂𝐵𝑉𝑠𝑖𝑚𝑝𝑙𝑒 − 𝑂𝐵𝑉𝑐𝑜𝑚𝑝𝑙𝑒𝑥) + 2𝑘

Question 9

Describe how AIC may be used to select a superior model

Answer 9

𝐴𝐼𝐶 = (𝑂𝐵𝑉𝑠𝑖𝑚𝑝𝑙𝑒 − 𝑂𝐵𝑉𝑐𝑜𝑚𝑝𝑙𝑒𝑥) + 2𝑘

A negative 𝐴𝐼𝐶 provides (some) justification for retaining the more complex model, alongside improved diagnostic plots.

Question 10

An ideal DV vs PRED

Answer 10

A degree of even spread of points above and below the line of unity.
A line of best fit that tracks evenly along the line of unity
(Any groups of data points deviating away from this line of unity might make us suspicious that there is some underlying variability not being modelled).