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Example: SENIC Recall the difficulty determining the form for INFIRSK in our regression model. Last time, we settled on including one term, INFRISK^2 But, we could do an adjusted variable plot approach. How? We want to know, adjusting for all else in the model, what is the right form for INFRISK?
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Lecture 13
Diagnostics in MLRAdded variable plotsIdentifying outliersVariance Inflation Factor
BMTRY 701Biostatistical Methods II
Recall the added variable plots
These can help check for adequacy of model Is there curvature between Y and X after
adjusting for the other X’s? “Refined” residual plots They show the marginal importance of an
individual predictor Help figure out a good form for the predictor
Example: SENIC
Recall the difficulty determining the form for INFIRSK in our regression model.
Last time, we settled on including one term, INFRISK^2
But, we could do an adjusted variable plot approach.
How? We want to know, adjusting for all else in the
model, what is the right form for INFRISK?
R code
av1 <- lm(logLOS ~ AGE + XRAY + CENSUS + factor(REGION) )av2 <- lm(INFRISK ~ AGE + XRAY + CENSUS + factor(REGION) )resy <- av1$residualsresx <- av2$residuals
plot(resx, resy, pch=16)
abline(lm(resy~resx), lwd=2)
Added Variable Plot
-2 -1 0 1 2 3
-0.2
0.0
0.2
0.4
resx
resy
What does that show?
The relationship between logLOS and INFRISK if you added INFRISK to the regression
But, is that what we want to see? How about looking at residuals versus INFRISK
(before including INFRISK in the model)?
R codemlr8 <- lm(logLOS ~ AGE + XRAY + CENSUS + factor(REGION))smoother <- lowess(INFRISK, mlr8$residuals)plot(INFRISK, mlr8$residuals)lines(smoother)
2 3 4 5 6 7 8
-0.2
0.0
0.2
0.4
INFRISK
mlr8
$res
idua
ls
R code> infrisk.star <- ifelse(INFRISK>4,INFRISK-4,0)> mlr9 <- lm(logLOS ~ INFRISK + infrisk.star + AGE + XRAY + > CENSUS + factor(REGION))> summary(mlr9)
Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 1.798e+00 1.667e-01 10.790 < 2e-16 ***INFRISK 1.836e-03 1.984e-02 0.093 0.926478 infrisk.star 6.795e-02 2.810e-02 2.418 0.017360 * AGE 5.554e-03 2.535e-03 2.191 0.030708 * XRAY 1.361e-03 6.562e-04 2.073 0.040604 * CENSUS 3.718e-04 7.913e-05 4.698 8.07e-06 ***factor(REGION)2 -7.182e-02 3.051e-02 -2.354 0.020452 * factor(REGION)3 -1.030e-01 3.036e-02 -3.391 0.000984 ***factor(REGION)4 -2.068e-01 3.784e-02 -5.465 3.19e-07 ***---Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.1137 on 104 degrees of freedomMultiple R-Squared: 0.6209, Adjusted R-squared: 0.5917 F-statistic: 21.29 on 8 and 104 DF, p-value: < 2.2e-16
Residual Plots
2 3 4 5 6 7 8
-0.2
-0.1
0.0
0.1
0.2
0.3
0.4
INFRISK
mlr9
$res
idua
ls
2 3 4 5 6 7 8
-0.2
-0.1
0.0
0.1
0.2
0.3
0.4
INFRISK
mlr7
$res
idua
ls
SPLINE FOR INFRISK INFRISK2
Which is better?
Cannot compare via ANOVA because they are not nested!
But, we can compare statistics qualitatively R-squared:
• MLR7: 0.60• MLR9: 0.62
Partial R-squared:• MLR7: 0.17• MLR9: 0.19
Identifying Outliers
Harder to do in the MLR setting than in the SLR setting.
Recall two concepts that make outliers important: • Leverage is a function of the explanatory variable(s)
alone and measures the potential for a data point to affect the model parameter estimates.
• Influence is a measure of how much a data point actually does affect the estimated model.
Leverage and influence both may be defined in terms of matrices
“Hat” matrix
We must do some matrix stuff to understand this Section 6.2 is MLR in matrix terms Notation for a MLR with p predictors and data on
n patients. The data:
nY
YY
Y2
1
~
npn
p
p
XX
XXXX
X
1
221
111
1
11
~
More notation:
THE MODEL:
What are the dimensions of each?
Matrix Format for the MLR model
ne
ee
e2
1
p
1
0
eXY
“Transpose” and “Inverse”
X-transpose: X’ or XT
X-inverse: X-1
Hat matrix = H
Why is H important? It transforms Y’s to Yhat’s:
')'( 1 XXXXH
HYY ˆ
Estimating, based on fitted model
)()(2 HIMSEes
Variance-Covariance Matrix of residuals:
)1()(2iii hMSEes
Variance of ith residual:
MSEhhses ijijij )0()( 22
Covariance of ith and jth residual:
Other uses of H
YHIe )(
I = identity matrix
)()( 22 HIe
Variance-Covariance Matrix of residuals:
)1()( 22iii he
Variance of ith residual:
222 )0()( ijijij hhe
Covariance of ith and jth residual:
Property of hij’s
n
i
n
jijij hh
1 1
1
This means that each row of H sums to 1And, that each column of H sums to 1
Other use of H Identifies points of leverage
0 5 10 15
-10
010
2030
40
x
y
1 2
4
3
Using the Hat Matrix to identify outliers
Look at hii to see if a datapoint is an outlier Large values of hii imply small values of var(ei) As hii gets close to 1, var(ei) approaches 0. Note that
As hii approaches 1, yhat approaches y This gives hii the name “leverage” HIGH HAT VALUE IMPLIES POTENTIAL FOR
OUTLIER!
ji
jijiii
n
jjiji yhyhyhy
1
ˆ
R code
hat <- hatvalues(reg)plot(1:102, hat)highhat <- ifelse(hat>0.10,1,0)plot(x,y)points(x[highhat==1], y[highhat==1],
col=2, pch=16, cex=1.5)
Hat values versus index
0 20 40 60 80 100
0.02
0.06
0.10
0.14
1:102
hat
Identifying points with high hii
0 5 10 15
-10
010
2030
40
x
y
Does a high hat mean it has a large residual?
No. hii measures leverage, not influence Recall what hii is made of
• it depends ONLY on the X’s• it does not depend on the actual Y value
Look back at the plot: which of these is probably most “influential”
Standard cutoffs for “large” hii: • 2p/n• 0.5 very high, 0.2-0.5 high
Let’s look at our MLR9 Any outliers?
0 20 40 60 80 100
0.05
0.10
0.15
0.20
1:length(hat9)
hat9
Using the hat matrix in MLR
Studentized residuals Acknowledge:
• each residual has a different variance• magnitude of residual should be made relative to its
variance (or sd) Studentized residuals recognize differences in
sampling errors
Defining Studentized Residuals
From slide 15,
We then define
Comparing ei and ri• ei have different variance due to sampling variations• ri have constant variance
)1()(2iii hMSEes
)1()( ii
i
i
ii hMSE
eese
r
Deleted Residuals
Influence is more intuitively quantified by how things change when an observation is in versus out of the estimation process
Would be more useful to have residuals in the situation when the observation is removed.
Example: • if a Yi is far out then it may be very influential in the
regression and the residual will be small• but, if that case is removed before estimating and
then the residual is calculated based on the fit, the residual would be large
Deleted Residuals, di
Process:• delete ith case• fit regression with all other cases• obtain estimate of E(Yi) based on its X’s and fitted
model
)(
)(
ˆestimation from removed with valuefittedˆ
iiii
iii
YYd
YY
Deleted Residuals, di
Nice result: you don’t actually have to refit without the ith case!
where ei is the ‘plain’ residual from the ith case and hii is the hat value. Both are from the regression INCLUDING the case
For small hii: ei and di will be similar For large hii: ei and di will be different
ii
ii h
ed
1
Studentized Deleted Residuals
Recall the need to standardize, based on the knowledge of the variance
The difference between ti and ri?
)1(
)(
)( iii
i
i
ii
hMSEe
dsd
t
Another nice result
You can calculate MSE(i) without refitting the model
2/1
2)1(1
iiiii ehSSE
pnet
Testing for outliers
outlier = Y observations whose studentized deleted residuals are large (in absolute value)
ti ~ t with n-p-1 degrees of freedom
Two examples: • simulated data• mlr9