回归Chapter 6

Methadone
Plot versus age
Studentised residuals
1 -2
20
-1
0
2
25
30
35
40
45
50
55
age
• Comment: There is no obvious pattern and so a linear model in age seems fine.
6.4. Leverage plots
• It can be shown that • ������ ������=1 ℎ������������ = ������ + 1 . • Thus ℎ������������ =
������+1 . ������
• Hence we use a cut-off of leverage points. • For the methadone example:
Theoretical Quantiles
6.6 Influential points
• For the multivariate case, Cook’s distance is defined in an analogous way, i.e. • ������������ =
1 (������+1) ∗ 2 ℎ������������ ������������ . 1−ℎ������������
• ������������ =
∗
������������ ������2 (1−ℎ������������ )
(6.2)
6.3 Residual plots
• We use residual plots in exactly the same way as before, except that we plot the studentized residuals against the fitted values and each explanatory variable. • We can pick up non-constant variance from the plot against the fitted values and also any possible outliers, remembering that 95% of observations should be in the range [-2, 2]. • Any curvature when plotted against an explanatory variable would suggest including a quadratic term in that variable. • For the bivariate Methadone example, we use: • >plot(fitted.values(fitbi),rstandard(fitbi),xlab="Fitted values",ylab="Studentised residuals",b=1.4)
Plot versus fitted values
Studentised residuals
1 -2 -1 0 2
5.5
6.0
6.5
7.0
• Comment: There is no obvious pattern in the plot, suggesting the constant variance assumption is OK. All the studentized residuals are in the range [-2, 2] and thus there are no obvious outliers.
• Both models should be reported in any report.
6.7 Further example
• Let us consider the ash data in females again and add in weight at 12 as a further explanatory variable. • > fitashbi=lm(ht32~ht12+wt12, data=ash30f) • > summary(fitashbi) Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 0.7014039 0.1165922 6.016 2.49e-08 *** ht12 0.6994202 0.0927759 7.539 1.56e-11 *** wt12 -0.0026917 0.0008347 -3.225 0.00167 ** Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 0.05123 on 108 degrees of freedom Multiple R-squared: 0.3808, Adjusted R-squared: 0.3693 F-statistic: 33.21 on 2 and 108 DF, p-value: 5.75e-12 • Comment: We see that weight at 12 has a negative coefficient implying that a high weight at 12 reduces height at 32.
6.2 Studentized residuals
• As with simple linear regression, the residuals have different variances dependent on the values of the explanatory variables, i.e. • ������������������ ������������ = 1 − ℎ������������ ������ 2 . • As mentioned in chapter 5, we can estimate ������ 2 by:
Chapter 6
Multivariate model validation
• We validate the multivariate model using essentially the same tools and techniques that we used for the simple linear regression model. • 6.1 Residuals and Leverage • The residual vector is: • ������ = ������ − ������. • This has variance given by: • ������������������ ������ = I − H ������ 2 . • This time we look for large ℎ������������ values(the leverages). • Usually values of ℎ������������ > 2(������ + 1)/������ are regarded as indicating points of high leverage.
Index
6.5 Normal plots
• We can check the Normality of the studentised residuals by using the same commands in R as before.
• >qqnorm(rstandard(fitbi), ylab="studentized residuals",b=1.4)
Fitted values
Plot versus Methadone
Studentised residuals
1 -2 -1 0 2
2
4
6
8
10
• Comment: There is no obvious curvature in the plot, suggesting a linear model with Methadone, but there is a suggestion that the observations at higher Methadone values are rather more variable.
• ������ 2 =
������ 2
=
������������������ (������−������−1)
=
������ ������=1
������������ −������������ 2
(������−������−1)
(6.1)
• Thus as before we standardise and use the studentized residuals defined by:
2(������+1) ������
to pick out high
• >plot(hatvalues(fitbi), ylab="leverages“,b=1.4) • >abline(h=6/17)
leverages
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
LeveragesLet us see the effect of removing point 2. >QTC1=QTC[-2] >Methadone1=Methadone[-2] >age1=age[-2] >fitbi1=lm(QTC1~Methadone1+age1) >summary(fitbi1)