Testing

• Linear model


  What makes a model linear?
• Chris Sutton


  April 2024
• We've covered a lot of modelling ground
• An unanswered question
• Linearising a model


  1. Sometimes (not always) a non-linear model can be converted into a linear one through a transformation of the response

  2. For example, is not linear but taking natural logarithms is now linear
• Care needed on the assumption we make about the residuals
• To use the techniques we have developed in this module we need for some constant variance, not the usual
• Other variations of this model can be linearised by a log transformation
• Linearising a non-linear model


  Subtracting the constant and then taking logs
• is not linear in its parameters
• This can be linearised by inverting the response as long as we are prepared to accept the condition that
• Polynomial Regression Models

  A particular set of linear regression models that are very useful in certain circumstances
• Example, the Quadratic Regression Model

  • Which is a linear model because although not linear in x it is in the betas and
• How to think of polynomial regression

  In some ways best not to think of this as multiple linear regression but as an extension of simple linear regression because the quadratic model only has one explanatory variable that appears twice as and
• Model comparison

  1. fit fitted with the set of observation data

  2. test versus Under a simple linear regression model adequately explains the data without the additional complexity of the quadratic component
• We can go beyond quadratic to cubic and higher order polynomial models
• Therefore it is usually sensible to centre these models
• Polynomial Multiple Regression


  We can also have polynomial regression with multiple explanatory variables
• Care needed with subset deletion approaches here, good practice not to remove first order terms or whilst leaving the higher order terms that rely on them in the model