5/13/ lecture 91 STATS 330: Lecture 9

5/13/ lecture 92 Diagnostics Aim of today’s lecture: To give you an overview of the modelling cycle, and begin the detailed discussion of diagnostic procedures

5/13/ lecture 93 The modelling cycle  We have seen that the regression model describes rather specialized forms of data Data are “planar” Scatter is uniform over the plane  We have looked at some plots that help us decide if the data is suitable for regression modelling pairs reg3d coplots

5/13/ lecture 94 Residual analysis  Another approach is to fit the model and examine the residuals  If the model is appropriate the residuals have no pattern  A pattern in the residuals usually indicates that the model is not appropriate  If this is the case we have two options Option 1: Select another form of model eg non-linear regression – see other courses Option 2: transform the data so that the regression model fits the transformed data – see next slide

5/13/ lecture 95 Modelling cycle This leads us into a modelling cycle Fit Examine residuals Transform data or change model if necessary This cycle is repeated until we are happy with the fitted model Diagramatically….

5/13/ lecture 96 Modelling cycle Examine residuals Fit model Transform/ change Choose Model Bad fit Good fit Use model Plots, theory

5/13/ lecture 97 What makes a bad fit?  Non-planar data  Outliers in the data  Errors depend on covariates (non-constant scatter)  Errors not independent  Errors not normally distributed  First two points can be serious as they affect the meaning and accuracy of the estimated coefficients.  The others affect mainly standard errors, not estimated coefficients – see subsequent lectures

5/13/ lecture 98 Detecting non-planar data For the next 2 lectures, we look at diagnosing non-planar data and choosing a transformation. We can diagnose non-planar data (nonlinearity) by fitting the model, and plotting residuals versus fitted values, residuals against explanatory variables fitting additive models In each case, a curved plot indicates non-planar data

5/13/ lecture 99 Plotting residuals versus fitted values plot(cherry.lm, which=1) which = 1 selects a plot of residuals versus fitted values

5/13/ lecture 910 Indication of a curve: so data non-planar

5/13/ lecture 911 Additive models  These are models of the form Y=g 1 (x 1 ) + g 2 (x 2 ) + … + g k (x k ) +  where g 1,…,g k are transformations  Fitted using the gam function in R  The transformations are estimated by the software  Use the function to suggest good transformations

5/13/ lecture 912 Example: cherry trees > library(mgcv) This is mgcv > cherry.gam<- gam(volume~s(height) + s(diameter),data=cherry.df) > plot(cherry.gam) Don’t forget the s if you want to estimate the transformation

5/13/ lecture 913 Example Strong suggestion that diameter be transformed

5/13/ lecture 914 Fitting polynomials  To fit model y = b 0 +b 1 x + b 2 x 2, use y ~ poly(x,2)  To fit model y = b 0 +b 1 x + b 2 x 2 + b 3 x 3, use y ~ poly(x,3) and so on

5/13/ lecture 915 Orthogonal polynomials The model fitted by y~poly(x,2) is of the form Y =  0 +  1 p 1 (x) +  2 p 2 (x) where p 1 is a polynomial of degree 1 ie of form ax + b and p 2 is a polynomial of degree 2 ie of form ax 2 + bx + c p 1, p 2 chosen to be uncorrelated (best possible estimation)

5/13/ lecture 916 Adding a quadratic term (cherry trees)  Suggestion of quadratic behaviour in diameter, so fit a quadratic model in diameter (add a term diameter^2) cherry2.lm=lm(volume~poly(diameter,2)+height,data=cherry.df) Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) poly(diameter, 2) < 2e-16 *** poly(diameter, 2) e-06 *** height *** --- Residual standard error: on 27 degrees of freedom Multiple R-Squared: , Adjusted R-squared: How to add a quadratic term R2 improved from 94.8% to 97.7%

5/13/ lecture 917 Equation of quadratic To see which quadratic is fitted, use > cherry.quad = lm(volume ~ diameter + I(diameter^2) + height, data=cherry.df) > summary(cherry.quad) Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) diameter * I(diameter^2) e-06 *** height *** Residual standard error: on 27 degrees of freedom Multiple R-Squared: , Adjusted R-squared: Surface fitted fitted is x diameter x diameter x height Same model!

Splines  An alternative to polynomials are splines – these are piecewise cubics, which join smoothly at “knots”  Give a more flexible fit to the data  Values at one point not affected by values at distant points, unlike polynomials 5/13/ lecture 918

Example with 4 knots 5/13/ lecture 919

5/13/ lecture 920 Adding a spline term (cherry trees) library(splines) cherry3.lm=lm(volume~bs(diameter,6)+height,data=cherry.df) Coefficients: (Intercept) * bs(diameter, df = 6) bs(diameter, df = 6) bs(diameter, df = 6) * bs(diameter, df = 6) e-06 *** bs(diameter, df = 6) e-07 *** bs(diameter, df = 6) e-12 *** height *** Residual standard error: 2.8 on 23 degrees of freedom Multiple R-squared: , Adjusted R-squared: How to add a spline term (3 knots) R2 improved from 94.8% to 97.78%

5/13/ lecture 921

5/13/ lecture 922 Example: Tyre abrasion data Data collected in an experiment to study the abrasion resistance of tyres Variables are Hardness: Hardness of rubber Tensile: Tensile strength of rubber Abrasion Loss: Amount of rubber worn away in a standard test (response)

5/13/ lecture 923 Tyre abrasion data > rubber.df hardness tensile abloss … 30 observations in all

5/13/ lecture 924 Tyre abrasion data > rubber.lm<-lm(abloss~hardness + tensile, data=rubber.df) > summary(rubber.lm) Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) e-14 *** hardness e-11 *** tensile e-07 *** --- Signif. codes: 0 `***' `**' 0.01 `*' 0.05 `.' 0.1 ` ' 1 Residual standard error: on 27 degrees of freedom Multiple R-Squared: , Adjusted R-squared: F-statistic: 71 on 2 and 27 DF, p-value: 1.767e-11

5/13/ lecture 925 Tyre abrasion data We will use this example to illustrate all the methods we have discussed so far to check if the data are planar, scattered about a flat regression plane i.e. Pairs Spinning plot Coplot Residual versus fitted value plot Fitting GAMS

5/13/ lecture 926 Tyre abrasion data (pairs)

Spinning 5/13/ lecture 927

5/13/ lecture 928 Tyre abrasion data (coplot)

5/13/ lecture 929 Tyre abrasion data: residuals v fitted values data(rubber.df) rubber.lm = lm(abloss~., data=rubber.df) plot(rubber.lm, which=1)

5/13/ lecture 930 Tyre abrasion data: gams library(mgcv) rubber.gam<- gam(abloss~s(tensile) + s(hardness),data=rubber.df) plot(rubber.gam)

5/13/ lecture 931 Tyre abrasion data: Conclusions Pairs plot : not very informative Spinner: Hint of a “kink” Coplot: suggestion of non-linearity, a “kink” Residuals versus fitted values: weak suggestion that regression is not planar gam plots: hardness is OK, but strong suggestion that tensile needs transforming Suggested transformation: looks a bit like a cubic or 4 th degree polynomial, so try these (or a spline). Using a spline, the R 2 is increased from 84% to 94.3%.