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Transcript of Measurement error in mortality models Clara Antón Fernández Robert E. Froese School of Forest...
Measurement error in mortality models
Clara Antón FernándezRobert E. Froese
School of Forest Resources and Environmental Science.Michigan Technological University
1400 Townsend Drive. Houghton, Michigan 49931
The problem
The problem
• Competition variables are measured with error
The problem
• Min 45• Max 109• Mean 70• “real” 61• Measured 65.4
The problem
• Prediction of a response versus inference for parameters– Generally, there is no need for the modeling of
measurement error to play a role in the prediction problem
– The unique situation when we need to correctly model the measurement error occurs when we develop a prediction model using data from one population but we wish to predict in another population.
The problem
• Sampling error variances change during the simulation.
• They depend on – sample plot sizes (fixed at the beginning of
the simulation but may be differ from the one used for fitting the model)
– spatial structure of the stand (tree size and spacing)
The cost
The cost
• If we ignore the changes in the error structure of the competition variables
PREDICTION
DURING MODEL FITTING
BIASED
LOSS OFPOWER
for detecting relationships among variables
The cost
TRUE
OBSERVED
Source: Carroll, R. J., D. Ruppert, L. A. Stefanski, and C. M. Crainiceanu. 2006.Measurement error in nonlinear models. Chapman and Hall/CRC.
The linear casediameter increment model
Solutions: Linear case
• Attenuation: The effect of measurement error is, generally, to bias the slope estimate towards zero.
• Stage and Wykoff (1998) proposed the Structural Based Prediction (SBP) procedure
• Results: considerable change in the magnitude of some regression coefficients and an increase in residual variance
The non-linear casemortality model
The non-linear case
• The effects of measurement error are more complex– The bias could be under or over-
estimated, even for the variables that are measured without error.
The non-linear case
• Regression Calibration
• SIMEX
• Score function methods
• Likelihood and quasilikelihood
• Bayesian methods
Computationallymore demanding
Require strongdistributional assumptions
Result in fullyconsistent estimatorsmore generally
SimpleGenerally applicable
Computationallymore intensive that RC
Once the replacement is made, essentially the same methods for ongoing analyses can be employed as if X was observed
Regression Calibration
• “Widely used, effective (and) reasonably well investigated” (Pierce and Kellerer, 2004)
• Basis: replacement of X by the regression of X on (Z,W). X variables measured with error, Z variables measured without error, W observation related with X
• Once the replacement is made, essentially the same methods for ongoing analyses can be employed as if X was observed.
Results
Data
• USDA Forest Service Region 1 Permanent Plot Program
• The set includes – regenerating stands in the Rocky Mountain
Region– control and treated (managed) stands
• 34,243 tree measurements
• 189 stands
Results and consequences
PBAL frequency distribution for western hemlock
BEFORE AFTER RC
PBAL
Fre
quen
cy
0 20 40 60 80 100
010
020
030
040
0
PBAL
Fre
quen
cy
0 20 40 60 80 1000
100
200
300
400
Results and consequencesWestern hemlock
Results and consequencesLodgepole pine
Results and consequences
• Contrary to the linear case, the effect of the sampling error in the multivariate logistic case can be under- or overestimate the effect of the variable, even for variables that are measured without errors
• Results might be influenced by the limited scope of the data
Summary
Measurement error in mortality models• Measurement error can cause
– Loss of power in the fitting phase– Bias in the prediction phase
• Regression Calibration corrects for measurement error before models are fitted or applied
• The effect of the sampling error in the multivariate logistic case can be under- or overestimate the effect of the variable, even for variables that are measured without errors