Topic 9: Remedies

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Topic 9 Remedies
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Outline

• Review diagnostics for residuals
• Discuss remedies
• Nonlinear relationship
• Nonconstant variance
• Nonnormal distribution
• Outliers

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Diagnostics for residuals

• Look at residuals to find serious violations of
  the model assumptions
• nonlinear relationship
• nonconstant variance
• nonnormal errors
• presence of outliers
• a strongly skewed distribution

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Recommendations for checking assumptions

• Plot Y vs X (is it a linear relationship?)
• Look at distribution of residuals
• Plot residuals vs X, time, or any other potential
  explanatory variable
• Use the ism in symbol statement to get
  smoothed curves

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Plots of Residuals

• Plot residuals vs
• Time (order)
• Explanatory variables
• Look for
• nonrandom patterns
• outliers (unusual observations)

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Residuals vs Order

• Pattern in plot suggests dependent errors / lack
  of indep
• Pattern usually a linear or quadratic trend
  and/or cyclical
• If you are interested read NKNW pp 104-105

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Tests for normality

• H0 data are an i.i.d. sample from a normal
  population
• Ha data are not an i.i.d. sample from a normal
  population
• NKNW (p 111) suggest a correlation test that
  requires a table look-up

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Tests for normality

• We have several choices for a significance
  testing procedure
• Proc univariate with the normal option provides
  four
• proc univariate normal
• Shapiro-Wilk is a common choice

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Test Shapiro-Wilk W
Kolmogorov-Smirnov D Cramer-von Mises
W-Sq Anderson-Darling A-Sq statistic
-----p Value------ 0.978 Pr lt W
0.8626 0.095 Pr gt D gt0.1500 0.033
Pr gt W-Sq gt0.2500 0.207 Pr gt A-Sq gt0.2500
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Other tests for model assumptions

• Durbin-Watson test for serially correlated errors
  (NKNW p 110)
• Modified Levene test for homogeneity of variance
  (NKNW p 112-114)
• Breusch-Pagan test for homogeneity of variance
  (NKNW p 115)
• For SAS commands see nknw110.sas

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Plots vs significance test

• Plots are more likely to suggest a remedy
• Significance tests results are very dependent on
  the sample size with sufficiently large samples
  we can reject most null hypotheses

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Lack of fit

• When we have repeat observations at different
  values of X, we can do a significance test for
  nonlinearity
• Browse through NKNW 3.7
• We will do details when we get to NKNW 17.9, p
  742
• Basic idea is to compare two models
• Gplot with a smooth is a better (i.e., simpler)
  approach

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Nonlinear relationships

• We can model many nonlinear relationships with
  linear models, some have several explanatory
  variables (i.e., multiple linear regression)
• Y ß0 ß1X ß2X2 ? (quadratic)
• Y ß0 ß1log(X) ?

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Nonlinear Relationships

• Sometimes can transform a nonlinear equation into
  a linear equation
• Consider Y ß0exp(ß1X) ?
• Can form linear model using log
• log(Y) log(ß0) ß1X ?
• Note that we have changed our assumption about
  the error

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Nonlinear Relationship

• We can perform a nonlinear regression analysis
• NKNW Chapter 13
• SAS PROC NLIN

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Nonconstant variance

• Sometimes we model the way in which the error
  variance changes
• may be linearly related to X
• We can then use a weighted analysis
• NKNW 10.1
• Use a weight statement in PROC REG

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Nonnormal errors

• Transformations often help
• Use a procedure that allows different
  distributions for the error term
• SAS PROC GENMOD

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GENMOD

• Possible distributions of Y
• Binomial (Y/N or percentage data)
• Poisson (Count data)
• Gamma (exponential)
• Inverse gaussian
• Negative binomial
• Multinomial
• Specify a link function for E(Y)

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Ladder of Reexpression(transformations)
1.5
p
Transformation is xp
1.0
0.5
0.0
-0.5
-1.0
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Circle of Transformations
X up, Y up
X down, Y up
Y
X
X up, Y down
X down, Y down
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Box-Cox Transformations

• Also called power transformations
• These transformations adjust for nonnormality and
  nonconstant variance
• Y Y? or Y (Y? - 1)/?
• In the second form, the limit as ? approaches
  zero is the (natural) log

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Important Special Cases

• ? 1, Y Y1, no transformation
• ? .5, Y Y1/2, square root
• ? -.5, Y Y-1/2, one over square root
• ? -1, Y Y-1 1/Y, inverse
• ? 0, Y (natural) log of Y

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Box-Cox Details

• We can estimate ? by including it as a parameter
  in a non linear model
• Y? ß0 ß1X ?
• and using the method of maximum likelihood
• Details are in NKMW p 132-133
• SAS code is in nknw132.sas

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Box-Cox Solution

• Standardized transformed Y is
• K1(Y? - 1) if ? ? 0
• K2log(Y) if ? 0
• where K2 (? Yi)1/n (the geometric mean)
• and K1 1/ (? K2 ?-1)
• Run regressions with X as explanatory variable
• estimated ? minimizes SSE

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data a1 input age plasma _at__at_ cards 0 13.44 0
12.84 0 11.91 0 20.09 0 15.60 1 10.11 1 11.38 1
10.28 1 8.96 1 8.59 2 9.83 2 9.00 2 8.65 2
7.85 2 8.88 3 7.94 3 6.01 3 5.14 3 6.90 3
6.77 4 4.86 4 5.10 4 5.67 4 5.75 4 6.23
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The first part of the program gets the geometric
mean data a2 set a1 lplasmalog(plasma)
proc univariate dataa2 noprint var lplasma
output outa3 meanmeanl
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data a4 set a2 if _n_ eq 1 then set a3
keep age yl l k2exp(meanl) do l -1.0
to 1.0 by .1 k11/(lk2(l-1))
ylk1(plasmal -1) if abs(l) lt 1E-8 then
ylk2log(plasma) output end
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proc sort dataa4 outa4 by l proc reg
dataa4 noprint outesta5 model ylage
by l data a5 set a5 n25 p2
sse(n-p)(_rmse_)2 proc print dataa5
var l sse
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Obs l sse 1 -1.0 33.9089 2
-0.9 32.7044 3 -0.8 31.7645 4
-0.7 31.0907 5 -0.6 30.6868 6
-0.5 30.5596 7 -0.4 30.7186 8
-0.3 31.1763 9 -0.2 31.9487 10
-0.1 33.0552
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symbol1 vnone ijoin proc gplot dataa5
plot ssel run
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data a1 set a1 tplasma plasma(-.5) tage
(age.5)(-.5) symbol1 vcircle ism50
proc gplot plot tplasmaage proc sort by
tage proc gplot plot tplasmatage run
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Box Cox Procedure
There is a fairly new procedure that will find
the box-cox transformation proc transreg
dataa1 model boxcox(plasma)identity(age) run
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Transformation
Information for BoxCox(plasma)
Lambda R-Square Log Like
-2.50 0.76
-17.0444 -2.00
0.80 -12.3665
-1.50 0.83 -8.1127
-1.00 0.86
-4.8523 -0.50
0.87 -3.5523 lt
0.00 0.85 -5.0754
0.50 0.82
-9.2925 1.00
0.75 -15.2625
1.50 0.67 -22.1378
2.00 0.59
-29.4720 2.50
0.50 -37.0844
lt - Best
Lambda -
Confidence Interval
- Convenient Lambda
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Background Reading

• Sections 3.4 - 3.7 describe significance tests
  for assumptions (read it if you are interested).
• Box-Cox transformation is in nknw132.sas
• Read sections 4.1, 4.2, 4.4, 4.5, and 4.6