Lecture 4. Checking Model Adequacy

29 Pages·2005·0.11 MB·English

Statistics 514: Checking Model Adequacy
Lecture 4. Checking Model Adequacy
Montgomery: 3-4, 15-1.1
Fall, 2005
Page 1
Statistics 514: Checking Model Adequacy
Model Checking and Diagnostics
(cid:15)
Model Assumptions
1 Model is correct
2 Independent observations
3 Errors normally distributed
4 Constant variance
− −
yij = (y:: + (yi: y::)) + (yij yi:)
yij = y^ij + (cid:15)^ij
observed = predicted + residual
(cid:15)
Note that the predicted response at treatment i is y^ij = y(cid:22)i:
(cid:15)
Diagnostics use predicted responses and residuals.
Fall, 2005
Page 2
Statistics 514: Checking Model Adequacy
Diagnostics
(cid:15)
Normality
– Histogram of residuals
– Normal probability plot / QQ plot
– Shapiro-Wilk Test
(cid:15)
Constant Variance
– Plot (cid:15)^ij vs y^ij (residual plot)
– Bartlett’s or Levene’s Test
(cid:15)
Independence
– Plot (cid:15)^ij vs time/space
– Plot (cid:15)^ij vs variable of interest
(cid:15)
Outliers
Fall, 2005
Page 3
Statistics 514: Checking Model Adequacy
Constant Variance
(cid:15) 2
In some experiments, error variance ((cid:27) ) depends on the mean response
i
E(y ) = (cid:22) = (cid:22) + (cid:28) :
ij i i
So the constant variance assumption is violated.
(cid:15)
Size of error (residual) depends on mean response (predicted value)
(cid:15)
Residual plot
– Plot (cid:15)^ij vs y^ij
– Is the range constant for different levels of y^ij
(cid:15)
More formal tests:
– Bartlett’s Test
– Modified Levene’s Test.
Fall, 2005
Page 4
Statistics 514: Checking Model Adequacy
Bartlett’s Test
(cid:15) 2 2 2
H : (cid:27) = (cid:27) = : : : = (cid:27)
0 1 2 a
(cid:15) 2 q
Test statistic: (cid:31) = 2:3026
0 c
where
P
− 2 − a − 2
q = (N a)log10Sp i=1(ni 1)log10Si
P
1 a − −1 − − −1
c = 1 + ( (n 1) (N a) )
i
3(a−1) i=1
2 2
and S is the sample variance of the ith population and S is the pooled
i p
sample variance.
(cid:15) 2 2
Decision Rule: reject H0 when (cid:31)0 > (cid:31)(cid:11);a−1.
Remark: sensitive to normality assumption.
Fall, 2005
Page 5
Statistics 514: Checking Model Adequacy
Modified Levene’s Test
(cid:15)
For each fixed i, calculate the median mi of yi1; yi2; : : : ; yin .
i
(cid:15)
Compute the absolute deviation of observation from sample median:[
j − j
d = y m
ij ij i
for i = 1; 2; : : : ; a and j = 1; 2; : : : ; ni,
(cid:15)
Apply ANOVA to the deviations: dij
(cid:15) 2 2
Use the usual ANOVA F-statistic for testing H0 : (cid:27)1 = : : : = (cid:27)a.
Fall, 2005
Page 6
Statistics 514: Checking Model Adequacy
options ls=80 ps=65;
title1 ’Diagnostics Example’;
data one;
infile ’c:\saswork\data\tensile.dat’;
input percent strength time;
proc glm data=one;
class percent;
model strength=percent;
means percent / hovtest=bartlett hovtest=levene;
output out=diag p=pred r=res;
proc sort; by pred;
symbol1 v=circle i=sm50; title1 ’Residual Plot’;
proc gplot; plot res*pred/frame; run;
proc univariate data=diag normal noprint;
var res; qqplot res / normal (L=1 mu=est sigma=est);
histogram res / normal; run;
Fall, 2005
Page 7
Statistics 514: Checking Model Adequacy
run;
proc sort; by time;
symbol1 v=circle i=sm75;
title1 ’Plot of residuals vs time’;
proc gplot; plot res*time / vref=0 vaxis=-6 to 6 by 1;
run;
symbol1 v=circle i=sm50;
title1 ’Plot of residuals vs time’;
proc gplot; plot res*time / vref=0 vaxis=-6 to 6 by 1;
run;
Fall, 2005
Page 8
Statistics 514: Checking Model Adequacy
Diagnostics Example
Sum of
Source DF Squares Mean Square F Value Pr > F
Model 4 475.7600000 118.9400000 14.76 <.0001
Error 20 161.2000000 8.0600000
Corrected Total 24 636.9600000
Levene’s Test for Homogeneity of strength Variance
ANOVA of Squared Deviations from Group Means
Sum of Mean
Source DF Squares Square F Value Pr > F
percent 4 91.6224 22.9056 0.45 0.7704
Error 20 1015.4 50.7720
Bartlett’s Test for Homogeneity of strength Variance
Source DF Chi-Square Pr > ChiSq
percent 4 0.9331 0.9198
Fall, 2005
Page 9
Statistics 514: Checking Model Adequacy
Non-constant Variance: Impact and Remedy
(cid:15)
Does not affect F-test dramatically when experiment is balanced
(cid:15)
Why concern?
– Comparison of treatments depends on MSE
– Incorrect intervals and comparison results
(cid:15)
Variance-Stabilizing Transformations
– Common transformations
p p p
x, log(x), 1=x, arcsin( x), and 1= x
– Box-Cox transformations
(cid:12)
1. approximate the relationship (cid:27)i = (cid:18)(cid:22)i , then the transformation is
1−(cid:12)
X
2. use maximum likelihood principle
(cid:3)
Distribution often more “normal” after transformation
Fall, 2005
Page 10