Regional mixed-effects height–diameter models for loblolly pine (Pinus taeda L.) plantations

Abstract

A height–diameter mixed-effects model was developed for loblolly pine (Pinus taeda L.) plantations in the southeastern US. Data were obtained from a region-wide thinning study established by the Loblolly Pine Growth and Yield Research Cooperative at Virginia Tech. The height–diameter model was based on an allometric function, which was linearized to include both fixed- and random-effects parameters. A test of regional-specific fixed-effects parameters indicated that separate equations were needed to estimate total tree heights in the Piedmont and Coastal Plain physiographic regions. The effect of sample size on the ability to estimate random-effects parameters in a new plot was analyzed. For both regions, an increase in the number of sample trees decreased the bias when the equation was applied to independent data. This investigation showed that the use of a calibrated response using one sample tree per plot makes the inclusion of additional predictor variables (e.g., stand density) unnecessary. A numerical example demonstrates the methodology used to predict random effects parameters, and thus, to estimate plot specific height–diameter relationships.

Appendices

Appendix 1. Linear mixed-effects model theory

A general linear model can be expressed as:

$$ E{\text{[}}{\mathbf{Y}}_{i} {\text{]}} = {\mathbf{X}}_{i} {\varvec{\beta}}\,, $$

(12)

where Y _i is a vector of observations from cluster i (i = 1,..,N), X _i is an (n _i  × p) regressor matrix of cluster i and β is a (p × 1) vector of regression coefficients (vector of fixed-effects parameters applicable to all N clusters). This fixed-effects model assumes the mean response of a specific set of regressor values is constant for all clusters. However, the fixed-effect model can be modified by including cluster-specific parameters (random effects) thus permitting the mean response to vary from cluster to cluster, taking the following form:

$$ E{\text{[}}{\mathbf{Y}}_{i} {\text{ $|$ }}{\mathbf{b}}_{i} {\text{]}} = {\mathbf{X}}_{i} {\text{(}}{\varvec{\beta}} \; + \;{\mathbf{b}}_{i} {\text{)}} = {\mathbf{X}}_{i} {\varvec{\beta}} \; + \; {\mathbf{X}}_{i} \,{\mathbf{b}}_{i} \,, $$

where b _i provides for cluster-specific behavior. A more general expression for this model is given by:

$$ E{\text{[}}{\mathbf{Y}}_{i} {\text{ $|$ }}{\mathbf{b}}_{i} {\text{]}} = {\mathbf{X}}_{i} {\varvec{\beta}} + {\mathbf{Z}}_{i} \,{\mathbf{b}}_{i} \,, $$

(13)

where Z _i is a (n _i  × q) regressor matrix (or design matrix) containing explanatory variables and b _i is a (q × 1) vector of random effects. The matrix Z _i can have the same regressors as in X _i or it may only contain those regressors in X _i that vary among clusters. The expectation of Eq. (13) is conditioned on the vector of random parameters, assuming that b _i is normally distributed with E[b _i ] = 0 and Var[b _i ] = D. In terms of linear models, a linear mixed-effects model can be expressed as:

$$ {\mathbf {Y}}_{i} {\text{ $|$ }}{\mathbf{b}}_{i} = {\mathbf{X}}_{i} {\varvec{\beta}} + {\mathbf {Z}}_{i} \,{\mathbf{b}}_{i} + {\varvec{\varepsilon}} _{i} $$

where ε _i is a random error assumed to be normally distributed with E[ε _i ] = 0 and Var[ε _i ] = R _i . R _i is an (n _i  × n _i ) covariance–variance matrix of cluster i. The random error is assumed to be independent of the random vector b _i with Cov[ε _i , b _i ] = 0. Then, E[Y _i ] = X _i β with a covariance matrix of Var(Y _i ) = V _i  = Z _i DZ′ _i  + R _i (e.g., Verbeke and Molenberghs 1997, p. 71). Thus, a mixed-effects model can be expressed in general form as (Laird and Ware 1982):

$$ {\mathbf{Y}}_{i} = {\mathbf{X}}_{i} {\varvec{\beta}} + {\mathbf{Z}}_{i} \,{\mathbf{b}}_{i} + {\varvec{\varepsilon}} _{i} \,, $$

(14)

where

$$ {\mathbf{Y}}_{i} \sim N({\mathbf{X}}_{i}{\varvec{\beta}}\,,\,{\mathbf{Z}}_{i}{\mathbf{DZ}}^{'}_{i} +{\mathbf{R}}_{i} ) $$

and

$$ {\left( \begin{aligned}{} & {\mathbf{b}}_{i} \\ &{\varvec{\varepsilon}} _{i} \\ \end{aligned} \right)} \sim N{\left( {{\left( \begin{aligned}{} & 0 \\ & 0 \\ \end{aligned} \right)}\;,\;{\left( {\begin{array}{*{20}c} {\mathbf{D}} & {0} \\ {0} & {\mathbf{R}}_{i} \\ \end{array} } \right)}} \right)}. $$

Estimation of the fixed vector β

An estimate of the fixed parameter vector β under model (4) can be obtained from a generalized least squares (GLS) analysis using V ⁻¹ _i as weights. Assuming all the variance and covariance parameters of D and R _i are known, e.g., which means V _i is known, results in the following estimator of $\hat{\beta }\,,$ which is the best linear unbiased estimator (BLUE):

$$ \hat{\beta } = {\left( {{\sum\limits_{i = 1}^{\text{N}} {X^{'}_{i} V^{{ - 1}}_{i} X^{{}}_{i} } }} \right)}^{{ - 1}} {\sum\limits_{i = 1}^{\text{N}} {X^{'}_{i} V^{{ - 1}}_{i} Y^{{}}_{i} } }, $$

(15)

where V ⁻¹ _i  = [Z _i DZ′ _i  + R _i ]⁻¹ and its variance–covariance matrix is:

$$ {\text{Var(}}\hat{\beta }) = {\left( {{\sum\limits_{i = 1}^{\text{N}} {X^{'}_{i} V^{{ - 1}}_{i} X^{{}}_{i} } }} \right)}^{{ - 1}} . $$

Prediction of the random vector b_i

Even though we are interested in the fixed-effects parameters, providing us with a population average curve, the main purpose for using mixed-effects models is to estimate cluster specific parameters. Knowing that b _i and Y _i are distributed jointly multivariate normal, then the conditional expectation of b _i is given by:

$$ {\text{E[}}{\mathbf{b}}_{i} |{\mathbf{Y}}_{i} {\text{]}} = {\text{E(}}{\mathbf{b}}_{i} {\text{)}} + {\text{Cov[}}{\mathbf{b}}_{i} ,{\mathbf{Y}}_{i} {\text{]}}\,{\text{Var[}}{\mathbf{Y}}_{i} {\text{]}}^{{{\text{ - 1}}}} \,{\text{(}}{\mathbf{Y}}_{i} - {\text{E[}}{\mathbf{Y}}_{i} {\text{])}}\,{\text{,}} $$

where Cov[b _i ,Y _i ] = Cov[b _i , X _i β + Z _i b _i ] = DZ′_i and E(b _i ) = 0, and thus the best linear unbiased predictor (BLUP) of b _i is given by:

$$ \hat{\mathbf{b}}_{i} = {\mathbf{D}}\;{\mathbf{Z}}^{'}_{i} \,{\mathbf{W}}_{i} {\text{(}}{\mathbf{Y}}_{i} - {\mathbf{X}}_{i} \hat{\varvec{\beta }}{\text{)}}\,{\text{,}} $$

(16)

where W _i  = V ⁻¹ _i  = [Z _i DZ′ _i  + R _i ]⁻¹ (see Rencher 2000, p. 431). As mentioned by Lappi (1991), this expression requires the inversion of a matrix with dimension equal to the number of observations. The variance–covariance matrix of the prediction errors, ${\text{Var[}}{\mathbf{b}}_{i} - {\mathbf{\ifmmode\expandafter\hat\else\expandafter\^\fi{b}}}_{i} {\text{]}}\,{\text{,}}$ is given by

$$ {\text{Var[}}{\mathbf{b}}_{i} - {\mathbf{\ifmmode\expandafter\hat\else\expandafter\^\fi{b}}}_{i} {\text{]}} = {\text{[}}{\mathbf{Z}}^{'}_{i} \,{\mathbf{R}}^{{ - 1}}_{i} \,{\mathbf{Z}}^{{\text{'}}}_{i} + {\mathbf{D}}^{{ - 1}}_{i} {\text{]}}^{{ - 1}} $$

(17)

The expressions given for $\hat{\varvec{\beta }}$ and $\hat{\mathbf{b}}_{i} \,$ in (15) and (16) assume that V _i is known, e.g., D and R _i are known. However in normal practice a consistent estimator given by $\ifmmode\expandafter\hat\else\expandafter\^\fi{\mathbf{V}}_{i} = {\mathbf{Z}}_{i} \hat{\mathbf{D}}\,{\mathbf{Z}}^{{\text{'}}}_{i} + \ifmmode\expandafter\hat\else\expandafter\^\fi{\mathbf{R}}_{i}$ must be used instead. Likelihood-based methods are used for estimating D and R _i based on the assumptions that b _i and ε _i are normally distributed (see Littell et al. 1996; Schabenberger and Pierce 2002).

Appendix 2

Fig. 4

SAS program for computing random parameters