Accounting For Serial Correlation Of Mixed Forest Growth Models: Prediction And Inference Implications

Comparison of a nonlinear mixed-effects basal area model fitted with and without incorporating correlated error structures

Shawn X. Meng and Shongming Huang

Biometrics Unit, Forest Management Branch, Alberta Ministry of Sustainable Resource Development, 8th Floor, 9920-108 Street, Edmonton, Alberta, Canada T5K2M4; 1-780-422-5270; [email protected]; [email protected]

Introduction

Forest growth models are typically developed using data collected from permanent sample plots or sectioned trees. A distinct characteristic of such data is they are serially correlated. If unaccounted for, the correlation would lead to biased estimates of the standard errors of model parameters and a reduction in estimation efficiency (Judge et al. 1988, Gregoire et al. 1995), even though the estimation on model parameters would remain unbiased and consistent (LeMay 1990).

To account for serial correlation, the generalized least square (GLS) and mixed-effects modeling approaches were proposed and applied in various models (Davidian and Giltinan 1995). The mixed-effects modeling approach addresses the correlation issue by incorporating the random effects parameters into models and/or directly modeling the error structure. With a model including a function to model the error structure to account for serial correlation, an improvement in the model’s fit statistics is generally resulted (Gregoire et al. 1995, Garber and Maguire 2003). Given this improvement, it is natural to assume that the model’s predictive ability could also be improved on independent data sets. This assumption tends to be supported by application of the prediction theorem detailed in Judge et al (1988). Based on this theorem, the estimated correlation of a model can be used to enhance model predictions based on known prior measurement(s) (Judge et al. 1988). Fortin et al. (2007) reported an improved predictive ability of a model fitted using the GLS when the predictions were adjusted based on estimated correlation. But this has not been properly verified under a mixed-effects model setting.

The objectives of this study were to: (1) examine if accounting for serial correlation can result in improved model predictive ability for a mixed-effects model; and (2) demonstrate if adjusting predictions based on the estimated correlation can improve the mixed model’s predictive ability.

Material and methods

Data description The data set used in this study is repeated measurements of lodgepole pine (Pinus contorta var. latifolia Engelm.) basal area collected across various ecoregions of Alberta. In this data set, lodgepole pine basal area, age and site index were measured/calculated for each plot. The data were randomly split into two parts: 245 plots for model fitting, and the rest 172 plots for model validation. The fitting and validation data of this data set are shown in Figure 1.

IUFRO. 2009. Modeling serial correlation of a mixed-effects basal area model. 1 Figure 1. Observed basal area growth trajectories of lodgepole pine for the model fitting (A) and validation data (B).

Mixed model As our main objectives of this study were to examine the impact of accounting for serial correlation on model predictive ability, we skipped the procedures to select the best mixed model, and used the following expression to model the basal area of lodgepole pine:

2 b2i BA i  (1  b1i )Agei exp[(1  b1i   3SI)Agei ]  ε i , ε i ~ N(0,R i ) (1) where BAi and Agei are vectors of basal area and breast height age of lodgepole pine for plot i;

SI is site index; 1 ,  2 and  3 are fixed-effects parameters; b1i, and b2i are random effects specific to plot i, and are assumed to follow a multivariate normal distribution with a variance- covariance structure D; ε i is a vector of the error term, assumed to follow a normal distribution with a zero expectation and a variance-covariance structure Ri.

Error structure Various structures/functions were proposed in the past and incorporated into Ri to model serial correlations (Littell et al. 2006). We examined many of them but focused only on two in this study. One is the spatial power (SP(POW)) function. The other is the Toeplitz functions with six bands (TOEP(6)). Those two error structures were compared to the iid (independently and identically distributed) error structure which assumes no additional serial correlation among repeated measurements after the random effects parameters were included in the model.

Model fitting The mixed model (1) with the iid, SP(POW) and TOEP(6) structures were fit using the %NLINMIX macro with ZERO expansion (SAS Institute Inc. 2004). The goodness-of-fit of the model applying different error structures was judged by the Akaike’s information criterion (AIC) and Schwarz’s Bayesian information criterion (BIC), which are defined in Littell et al. (2006).

Model prediction Prediction of random effects The predictive ability of the mixed model estimated with and without modeling the error structures were cross-validated on both the fitting and validation data of the data set. Prior to calculating the predictions, the random effects were predicted using the following equation: ˆ ˆ T ˆ T ˆ 1 ˆ bi  DZi (Zi DZi  R i ) (y i  f (β,0,xi )) (2)

IUFRO. 2009. Modeling serial correlation of a mixed-effects basal area model. 2 ˆ ˆ wherebi and β are the estimates of bi (a vector form of random effects parameters) and β (a ˆ vector form of fixed-effects parameters); Dˆ and R i are the estimates of D and Ri, respectively;

f (β,bi ,xi ) Zi  ; f(·) is a general form of the model (1); and x is a matrix of covariates b i  i βˆ ,0 including the age and SI for the model.

Past studies showed that the number of prior measurements used in model calibration had a big impact on predictions of random effects (e.g., Calama and Montero 2004). To examine this effect in our study, various scenarios with varied number of prior measurements were assessed for our data, two of which are reported here. In scenario 1 (R-S-1), the first observation from each plot was used to predict the random effects. Scenario 2 (R-S-2) corresponded to applying all observations from each plot for prediction of the random effects.

Once the random effects were obtained for each scenario of the model, the basal areas were predicted using eq. (3) (Davidian and Giltinan 1995): ˆ ˆ yˆ i  f (β,0,xi )  Zibi (3)

Prediction adjustment The predictions made using eq. (3) were localized predictions of the nonlinear mixed model that did not undergo the adjustment by utilizing the serial correlation estimates obtained from the model fit. To adjust the predictions, the following equation tailored for nonlinear mixed models with ZERO expansion, derived based upon the equation developed in Judge et al. (1988, p 345) was applied: ˆ ˆ T 1 ˆ ˆ y ip  f (β,0,xip )  Zip bi  V Ψ (y im  f (β,0,xim )  Zimbi ) (4) where yim and yip are vectors of prior measurements, and adjusted predictions of basal area for plot i respectively; xim and xip are matrices of covariates corresponding to yim and yip respectively; V is a matrix containing the correlations between the prior measurements and future predictions, and Ψ is the correlation matrix of the prior measurements. One or more prior measurements can be used to adjust predictions at future times. In this study, we reported only using the first observation from each plot to adjust future predictions.

Model evaluation criteria Predictive performance of the mixed effects model was compared based on the e , the mean prediction error; e %, the mean percent prediction error; and RMSE, the root mean square error.

Results

Incorporating a function to model the serial correlation resulted in obvious improvement of the model fit, as shown from the reduction in AIC and BIC values. The improvement was observed on both correlation structures assessed. Applying the SP(POW) or TOEP(6) in the model also resulted in further reduction in serial correlation, when compared to the iid structure. This reduction in serial correlation was particularly evident for the first three lags of the residuals.

IUFRO. 2009. Modeling serial correlation of a mixed-effects basal area model. 3 The cross-validation of the mixed model on both the fitting and validation data showed that the model with the inclusion of the SP(POW) or TOEP(6) function to account for the serial correlation did not outperform the models with the iid structure in terms of model predictive ability (Table 1). In fact, in most cases, especially in the cases when more prior measurements were used to generate more accurate predictions of random effects, the models with the simple iid structure were found to be better in predictions than the models with the SP(POW) or TOEP(6) structure.

Table 1. Adjusted (A) and un-adjusted (U) prediction errors obtained from different scenarios for model (1) fitted with the iid, SP(POW) and TOEP(6) structures. Fitting data Validation data Scenario e e% RMSE e e% RMSE R-S-1 iid 0.5479 1.9945 3.6512 0.2925 1.0622 3.8717 SP(POW)-U 0.7046 2.5650 5.3386 0.9669 3.5118 5.7577 SP(POW)-A 0.2386 0.8687 3.5336 0.1709 0.6206 3.7758 TOEP(6)-U 0.5339 1.9436 4.4282 0.7130 2.5897 4.8722 TOEP(6)-A 0.2926 1.0652 3.6442 0.2405 0.8733 4.0580 R-S-2 iid 0.0016 0.0057 0.9838 0.0346 0.1256 1.0191 SP(POW)-U 0.6739 2.4532 4.4666 1.0726 3.8957 4.9206 SP(POW)-A 0.1240 0.4513 2.0378 0.1976 0.7177 2.1090 TOEP(6)-U 0.3416 1.2435 3.3597 0.8112 2.9462 3.7066 TOEP(6)-A 0.1383 0.5033 2.3232 0.2741 0.9954 2.4309

Adjusting predictions based on eq. (4) resulted in consistent improvement of the model’s predictive ability as compared to those without adjustment (Table 1). The improvement was observed on both the SP(POW) and TOEP(6) structures.

Discussion

Directly modeling the error structures of the mixed model resulted in further reduction in serial correlation, agreeing to previous studies concluding that it remains necessary to further model the error structure, in addition to the inclusion of random effects into models to account for serial correlation. The application of the SP(POW) and TOEP(6) structures in this study to account for the serial correlation also resulted in drastic improvement of model fit, similar to those reported earlier (Gregoire et al. 1995, Garber and Maguire 2003). However, the improved fit observed on the mixed model did not translate into improved model predictive ability when cross-validated on both the fitting and validation data of the data set. In fact, in many cases, especially those when more prior measurements were available to generate accurate prediction of random effects, the models with the simple iid structure tend to have even better predictive ability than the models that incorporated the SP(POW) or TOEP(6) structure to account for the serial correlation (Table 1).

The outperformance of the models with the iid structure over the SP(POW) or TOEP(6) structure in these cases creates a dilemma: as successful accounting for the serial correlation resulted in

IUFRO. 2009. Modeling serial correlation of a mixed-effects basal area model. 4 reduced model predictive ability, yet ignoring the correlation yielded better predictions but could potentially invalid hypothesis testing and statistical inferences of model parameters. Therefore, depending on the top priority of what a mixed model is built for, decisions regarding if it is necessary to directly model the error structure to account for the serial correlation of the model would be made. If one’s priority is for model prediction, then it is likely that a better result would be generated when using the simple iid structure rather than entailing the complex procedure to model the serial correlation. If hypothesis testing and interval estimation about the estimated parameters need to be made, then accounting for the serial correlation becomes important.

The estimated correlation were not only used to calibrate the mixed models for obtaining local predictions, but were also used to adjust local predictions based on the prediction theorem (Judge et al. 1988). Using the modified prediction theorem tailored to nonlinear mixed models, we found that in general, the adjusted predictions had lower prediction errors than the predictions without adjustment (Table 1). This result agrees to a previous study concentrated on a GLS model (Fortin et al. 2007). However, the improvement of the adjusted predictions for our data did not seem too big to make the model with the SP(POW) or TOEP(6) structure better than the model with the iid structure. Even though in some cases minor, the adjustment can still enhance the model predictions, and is theoretically more reasonable.

Literature Cited

Calama, R., and Montero, G. 2004. Interregional nonlinear height-diameter model with random coefficients for stone pine in Spain. Can. J. For. Res. 34: 150-163. Davidian, M., and Giltinan, D.M. 1995. Nonlinear models for repeated measurement data, Chapman and Hall, New York, 359p. Fortin, M., Daigle, G., Ung, C-H., Bégin, J., and Archambault, L. 2007. A variance-covariance structure to take into account repeated measurements and heteroscedasticity in growth modeling. Eur. J. For. Res. 126: 573-585. Garber, S.M., and Maguire, D.A. 2003. Modeling stem taper of three central Oregon species using nonlinear mixed effects models and autoregressive error structure. For. Ecol. Manage. 179: 507-522. Gregoire, T.G., Schabenberger, O., and Barrett, J.P. 1995. Linear modeling of irregularly spaced, unbalanced, longitudinal data from permanent plot measurement. Can. J. For. Res. 25: 137- 156. Judge, G.G., Hill, R.C., Griffiths, W.E., Lütkepoho, H., and Lee, T.C. 1988. Introduction to the theory and practice of econometrics. John Wiley & Sons, New York, 1024p. LeMay, V.M. 1990. MSLS: a linear least squares technique for fitting a simultaneous system of equations with a generalized error structure. Can. J. For. Res. 20: 1830-1839. Littell, R.C., Milliken, G.A., Stroup, W.W., Wolfinger, R.D., and Schabenberger, O. 2006. SAS for mixed models. 2nd ed., SAS Institute Inc., Cary, NC, USA, 814p. SAS Institute Inc. 2004. SAS/STAT 9.1 User’s Guide. SAS Institute Inc., Cary, NC, 5136p.

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