Lisbon, 1721 July 2006

1
Robustness and Multilevel Models
Silvia Salini, Pier Alda Ferrari Department of
Economics, Business and Statistics University of
Milan silvia.salini_at_unimi.it pieralda.ferrari_at_unim
i.it
Nadia Solaro Department of Statistics University
of Milan - Bicocca nadia.solaro_at_unimib.it

• Lisbon, 17-21 July 2006

2
Outline

• Introduction to Linear Multilevel Models
• Robustness in multilevel models
• Multilevel models for robustness Statistical
  Calibration
• Robustness issues in statistical calibration
• Multilevel Calibration Estimator
• Simulation Study
• Group Diagnostics
• Application Concrete Blocks data
• Conclusion

3
Introduction (1)

• Multilevel models a modelling approach for the
  analysis of data organized in hierarchical
  structures, or more generally, in multilevel
  structures
• (Goldstein (1995), Snijders and Bosker (1999))
• Parameter estimation procedures account for
  correlated data within groups
• Between-groups variance parameters involved in
  these models

4
Introduction (2)

• Connection robustness and multilevel models
  according to a dual direction of analysis
• 1. Robustness in multilevel models
• In the current acceptation Robustness with the
  sense of Resistance to the presence of either
  outliers or misspecifications of model error
  distributions
• Are maximum likelihood estimators for model
  parameters sensitive to misspecifications of
  random-effects distributions?

5
Introduction (3)

• Connection robustness and multilevel models
  according to a dual direction of analysis
• 2. Multilevel models for robustness
• Multilevel modelling referred to as a robust
  methodology when data are grouped
• In the statistical calibration framework does the
  multilevel model methodology offer a valid tool
  when data are grouped and the classical estimator
  fails the expected performance?

6
Linear multilevel models for two-level nesting
structures

• Model formulation
• nj units within group (level-1 units)
• J groups (level-2 units)
• Laird and Wares matrix formulation
  (Biometrics,1982)

Assumptions
7
Parameter Estimation
Unknown parameters fixed effects (?), level-one
and level-two variance components (?2 and ?hm)

• Main estimation methods
• Extension of Generalized Least Squares
• IGLS/RIGLS (Goldstein, 1995)
• Maximum likelihood approach ML/REML
• (e.g. Pinheiro and Bates, 2000)

• Gaussian Multilevel Model

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1. Robustness in multilevel models

• In a series of papers Solaro and Ferrari deal
  with the problem of maximum likelihood estimator
  performance when the random-effects distribution
  is misspecified
• The considered misspecifications concern kurtosis
  departures, which are described by the family of
  Multivariate Exponential Power distributions
• The main problems arise in the estimation of the
  standard errors of level-2 variance component
  estimators
• Solaro and Ferrari, Stat. Meth. Applic., 2006
• Solaro and Ferrari, in Zani S., Cerioli A.,
  Riani M., Vichi M. (eds.), Data Analysis,
  Classification and Forward Search. Springer,
  Berlin (in press)

9
2. Multilevel models for robustnessStatistical
Calibration
Calibration experiment
X(X1, X2,.Xp) standard method
Y1(Y11, Y12,.Y1q) test method
Prediction experiment
Y2(Y21, Y22,.Y2q)
?
10
Statistical Calibration
p q
MLE Classical Estimator
Classical Estimator is asymptotically correct,
some problems occurs in determining confidence
regions (Zappa and Salini, 2005)
11
Robustness issues in statistical calibration

• Robust regression methods
• (Cheng and Van Ness, 1997)
• Classic estimators on data with the outliers
  rejected
• (Walczak, 1995)
• Forward Search in Multivariate Calibration
• (Riani and Salini, 2005 Salini 2006)

12
Forward Search in Multivariate Calibration
(Salini, 2006)

• Data 1 there are not outliers but only groups,
  robust and classical estimators perform in the
  same way
• Data 2 outliers are present, robust estimators
  fit better than the classical one

.
13
Random parameters in statistical calibration
models

• Calibration with Randomly Changing Standard
  Curves (Vecchia and Chapman, 1989)
• Calibration with Random slopes (Oman, 1998)
• Prediction and inverse estimation in
  repeated-measures models (Liski and Nummi, 1995)
• Likelihood Methods for Controlled Calibration
  (Bellio, 2003)

14
Multilevel Calibration Estimator

• Three different approaches (pq1)
• Unique calibration model (ignoring the grouping
  structure)
• Within-Group calibration estimator

15
Predictions for values of Uj are given
by (BLUPs of the uj)
Multilevel Calibration Estimator

• Multilevel Calibration model
• Random intercept
• - calibr. step
• - pred. step
• Random intercept and random slope
• - calibr. step
• - pred. step

16
Simulation Study
17
Application Concrete Blocks data

• Building Industry
• The experiment is performed in order to determine
  the strength of concrete blocks
• standard method destructive, test method
  sclerometer
• N 1200, J 24
• Groups differs by concentration of cement,
  temperature,
• number of days from building
• training set (80 of data), test set (20 of
  data)

18
Application Concrete Blocks data
19
Concrete Blocks data Forward Search Group
diagnostics Forward Bivariate Ellipsoid Plot
20
Concrete Blocks data Multilevel model Level-2
residual diagnostics
Standardized level-2 residuals by Normal scores
(random intercept model), REML method
Likelihood ratio test ? 381.457, p 0.001
21
Concrete Blocks data Multilevel model Level-2
residual diagnostics
95 -Confidence Intervals (random intercept
model), REML method
22
Concrete Blocks data Multilevel model Level-2
residuals pairwise comparisons
95-Comparative Confidence Intervals (random
intercept model), REML method
23
Application Concrete Blocks data
MSE in the test set. Runs consist in different
split of training and test set
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Conclusion and Prospects

• Group diagnostics in case of Normal Mixture by
  using Random Forward Search approach (Riani et
  al., 2006)
• Developing the Multilevel model diagnostic tools
  for the prediction step
• Extension to more complex multilevel models of
  the calibration methodology, especially in the
  multivariate calibration framework
• Integrating the Forward Search and the multilevel
  model diagnostic tools to recognize grouping
  structures in data when there arent natural
  groups or when some of them are redundant

25
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26
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