Non linear regression

Question 1

Detecting non- linearity

Answer 1

Plot residual from linear regression against fitted values -> there should be no pattern in the residuals (e.g. funnell)

Normal quantile plot

Question 2

Assumptions

Answer 2

Same (except does not assume linear relationsip)

Question 3

What to do it data is not linear

Answer 3

1) Transform X or Y
- Log-transformation = most versatile = good if Y is related to X by a power
- Reciprocal 1/x
- Square root transformation √x – usually for count data
- Arcsine – for proportions
- Raise x to a power xn
- Box-Cox - makes data normal

2) Check whether other models fit (piecewise linear, polynomial, piecewise polynomical, use models that assume a different disitribution like logistic regression)

3) predict response from explanaotry

Question 4

Non- Linear regression overview

Answer 4

predicts the values of a response variable from values of an explanatory variable.

It fits a non-linear model to data (many different models) -> many options

Calculations to minimise distance between observations and predictions.

Question 5

Piece wise linear models

Answer 5

Fits different linear models to different sections of data

knots / switch points are when gradient changes (although assumes lines are linear between knots

Question 6

Polynomials (e.g. quadratic)

Answer 6

y= mx+nx^2+/-c

Can also do piecewisr polynomial models

Can use ANOVA table + partition variation by linear, quadratic and error components to determine which components explain significant proportions of the variation (eg. here is linear + quadratic)

Question 7

Different distributions

Answer 7

Til now we have assumed we want to fit models assuming a normal disitrbution -> may not be the case

Poisson disitrbution
-> Used for count data
-> It gives the probability of an event happening a certain number of times (k) within a given interval of time or space

Binomial disitrbution
-> Used for proportion data (fixed between O and 1)

Exponential disitrbution
-> Used for survival data

Question 8

Tranformations

Answer 8

• Instead of choosing a non-linear model, we could transform the data to make it linear.

Question 9

Dealing with binary data

Answer 9

Either:

• Binary data can be fitted with a logistic regression model.
• Logistic regression predicts the probability of occurrence of a binary variable as a function of a continuous numerical variable.
• Binary data can be tranformed using the logs odds ratio and a linear model can be fitted.
• Linear model can be back tranformed so that occurance data can be predicted.