Topic 22: Diagnostics and Remedies

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Topic 22 Diagnostics and Remedies
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Outline

• Diagnostics
• residual checks
• ANOVA remedial measures

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Diagnostics Overview

• We will take the diagnostics and remedial
  measures that we learned for regression and adapt
  them to the ANOVA setting
• Many things are essentially the same
• Some things require modification

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Residuals

• Predicted values are cell means,
• Residuals are the differences between the
  observed values and the cell means Yij-

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Basic plots

• Plot the data vs the factor levels (the values of
  the explanatory variables)
• Plot the residuals vs the factor levels
• Construct a normal quantile plot of the residuals

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NKNW Example

• NKNW p 712
• Compare 4 brands of rust inhibitor (X has r4
  levels)
• Response variable is a measure of the
  effectiveness of the inhibitor
• There are 10 units per brand (n10)

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Plots

• Data versus the factor
• Residuals versus the factor
• Normal quantile plot of the residuals

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Plots vs the factor
symbol1 vcircle inone proc gplot dataa2
plot (eff resid)abrand run
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Data vs the factor
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Residuals vs the factor
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QQ-plot
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Summary

• Look for
• Outliers
• Variance that depends on level
• Non-normal errors
• Plot resdiuals vs time and other variables if
  available

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Homogeneity tests

• Homogeneity of variance (homoscedasticity)
• H0 s12 s22 sr2
• H1 not all si2 are equal
• Several significance tests are available

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Homogeneity tests

• Text discusses Hartley, modified Levene
• SAS has several including Bartletts (essentially
  the likelihood ratio test) and several versions
  of Levene

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Homogeneity tests

• There is a problem with assumptions
• ANOVA is robust with respect to moderate
  deviations from normality
• ANOVA results can be sensitive to the homogeneity
  of variance assumption
• Some homogeneity tests are sensitive to the
  normality assumption

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Levenes Test

• Do ANOVA on the squared residuals
• Modified Levenes test uses absolute values of
  the residuals
• Modified Levenes test is recommended

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NKNW Example

• NKNW p 765
• Compare the strengths of 5 types of solder flux
  (X has r5 levels)
• Response variable is the pull strength, force in
  pounds required to break the joint
• There are 8 solder joints per flux (n8)

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Levenes Test
proc glm dataa1 class type model
strengthtype means type/
hovtestlevene(typeabs) run
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Output
Levene's Test ANOVA of Absolute Deviations
Source DF F Value Pr gt F type
4 3.07 0.0288 Error 35
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Means and SDs
Level strength type N Mean Std Dev 1
8 15.42 1.23 2 8 18.52 1.25 3
8 15.00 2.48 4 8 9.74 0.81 5
8 12.34 0.76
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Remedies

• Delete outliers
• Is their removal important?
• Use weights (weighted regression)
• Transformations
• Nonparametric procedures

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Weighted least squares

• We used this with regression
• Obtain model for how the sd depends on the
  explanatory variable (plotted absolute value of
  residual vs x)
• Then used weights inversely proportional to the
  estimated variance

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Weighted Least Squares

• Here we can compute the variance for each level
• Use these as weights in PROC GLM
• We will illustrate with the soldering example
  from NKNW

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Obtain the variances and weights
proc means dataa1 var strength by
type output outa2 vars2 data a2 set a2
wt1/s2
NOTE. Data set a2 has 5 cases
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Merge and then use the weights in PROC GLM
data a3 merge a1 a2 by type proc glm
dataa3 class type model
strengthtype weight wt run
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Output
Source DF F Value Pr gt F Model 4 81.05
lt.0001 Error 35 Total 39
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Transformation Guides

• When si2 is proportional to µi, use
• When si is proportional to µi, use log(y)
• When si is proportional to µi2, use 1/y
• For proportions, use arcsin( )
• arsin(sqrt(y)) in a SAS data step

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Nonparametric approach

• Based on ranks
• See NKNW section 18.7, p 777
• See the SAS procedure NPAR1WAY

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Last slide

• We finished Chapter 18 .
• We used program NKNW758.sas and NKNW768.sas.