Measurement error in mortality models Clara Antón Fernández Robert E. Froese School of Forest Resources and Environmental Science. Michigan Technological University 1400 Townsend Drive. Houghton, Michigan 49931

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

If we ignore the changes in the error structure of the competition variables PREDICTION DURING MODEL FITTING BIASED LOSS OF POWER for detecting relationships among variables

The cost TRUE OBSERVED Source: Carroll, R. J., D. Ruppert, L. A. Stefanski, and C. M. Crainiceanu Measurement error in nonlinear models. Chapman and Hall/CRC.

The linear case diameter 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 case mortality 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 Computationally more demanding Require strong distributional assumptions Result in fully consistent estimators more generally Simple Generally applicable Computationally more 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

Results and consequences Figure 4. Western larch (top) and western hemlock (bottom) sensitivity graphs. Western hemlock

Results and consequences Lodgepole 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