Nonlinear Mixed Model Methods and Prediction Procedures Demonstrated on a Volume-Age Model

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1 Nonlnear Mxed Model Methods and Predcton Procedures Demonstrated on a Volume-Age Model Prepared by: Shongmng Huang, Shawn X. Meng, Yuqng Yang August, 3

2 Techncal Report Pub. No.: T/6(3) ISBN No (Prnted Edton) ISBN No (On-lne Edton) For addtonal copes of ths publcaton, please contact: Forest Management Branch Alberta Mnstry of Envronment and Sustanable Resource Development 8 th Floor, Great West Lfe Buldng 99 8 Street Edmonton, Alberta, Canada T5K M4 Man recepton: Tel: (78) Fax: (78) Ths publcaton wll be made avalable on the Alberta Government webste. Please check the latest nformaton on the Alberta Government webste at: Copyrght 3 Forestry and Emergency Response Dvson, Alberta Envronment and Sustanable Resource Development. All rghts reserved. Edmonton, Alberta, Canada August 3

3 Executve Summary and Acknowledgement Ths study descrbes the nonlnear mxed-effects modelng technque appled to a volume-age model frequently used n Alberta. It demonstrates the procedures for usng a ftted mxed model to make subject-specfc predctons on data not used n model fttng. Generalzed and step-by-step computer programs assocated wth the predctons are provded to facltate the computatons. Crtera for obtanng the most reasonable predctons under dfferent crcumstances are dscussed. Karl Peck, Daryl Prce, Darren Atkn and Dave Morgan revewed an early draft and provded many constructve comments and suggestons. Glenn Buckmaster at West Fraser s Hnton Wood Products provded some addtonal data for model testng. Ther help s much apprecated.

4 Table of Contents Executve Summary and Acknowledgement.... Introducton.... Data and Models.... Data.... Models Nonlnear Mxed Model Methods Basc Formulaton of Nonlnear Mxed Models The Frst-Order Method The Frst-Order Condtonal Expectaton Method Model Predctons Predcton from the Frst-Order Method Predcton from the Frst-Order Condtonal Expectaton Method Goodness-of-Ft Statstcs Choosng the Best Predcton Addtonal Notes Adjustng the Predctons Populaton Average Predctons References Appendces... 8 Appendx. Generalzed Program for the Frst-Order Method... 9 Appendx. Step-by-Step Program for the Frst-Order Method... 3 Appendx 3. Generalzed Program for the Frst-Order Condtonal Expectaton Method Appendx 4. Step-by-Step Program for the Frst-Order Condtonal Expectaton Method Appendx 5. Metrc Converson Chart... 4 Page v

5 . Introducton Regresson models can generally be classfed as populaton-based models and subject-specfc models. Tradtonal regresson models estmated from the least squares method are typcally populaton-based models. They predct the populaton averages, and as such, they are also referred to as populaton-average models, populaton models, or base models (when contrasted to subject-specfc models or mxed models). One common problem wth the populaton-based models s that, due to the ntrnsc varaton and the polymorphc nature of bologcal growth, the trends exhbted by the data from ndvdual subjects wthn a populaton may not always follow the trend exhbted by the populaton averages. Ths s llustrated n Fgure, where the data from ndvdual subjects show dfferng trends than that of the populaton averages. Because of the dfferng trends, t s qute possble that a populaton-based model may ft or predct the data well on average for the entre populaton, but t could perform poorly for the ndvdual subjects wthn the populaton. Sometme populaton averages could be meanngless at a subject-specfc level. Fgure. An llustraton of populaton-based (sold lne) and subject-specfc (dashed-lnes) models, where,, 3 and 4 represent four subjects n the populaton. The populaton-based model s obtaned from all data combned. Subject-specfc models, on the other hand, descrbe the mean responses of ndvdual subjects wthn a populaton. They can account for the dosyncrases of ndvdual subjects wthn a populaton (e.g., Fgure ). Subject-specfc models are often developed from the mxed-effects modelng technque. Thus, they are often referred to as mxed-effects models, or mxed models. Snce the mxed model developed n ths study s nonlnear, the term nonlnear mxed model (NMM) s used throughout ths study. The man objectve of ths study s to demonstrate the NMM technque based on a volume-age model frequently used n Alberta. The emphass of ths study s to show how to use a ftted mxed model to make subject-specfc predctons on data not used n model fttng. For the volume-age model, each subject s a sample plot (e.g., a permanent sample plot or a temporary sample plot). The populaton s an amalgamaton of the sample plots. To facltate the computaton, generalzed and step-by-step computer programs assocated wth the predctons from specfc NMM methods are provded n Appendces. A summary of the NMM methods s also provded pror to usng them to make predctons.

6 . Data and Models. Data Black spruce (pcea marana (Mll.) B.S.P.) volume-age data from 8 permanent sample plots (PSPs) wth varous black spruce compostons were used n ths study. Among the 8 plots, 3 from the lower foothlls natural subregon were used as model fttng data. The rest (79 plots) were used as model applcaton data. Summary statstcs for the model fttng and model applcaton data are lsted n Table. Throughout ths study, volume refers to black spruce total volume (m 3 /ha). Age refers to black spruce breast heght age (years). Table. Summary statstcs for model fttng and model applcaton data. Data Varable N m Mean Mn Max SD Model fttng Volume (m 3 /ha) Age (years) Model applcaton Volume (m 3 /ha) Age (years) Note: volume and age are total volume and breast heght age, N s the total number of observatons (measurements from PSPs), m s the number of plots (subjects), mn s mnmum, max s maxmum, and SD s standard devaton. Fgure shows the model fttng and model applcaton data. It can be seen that there are cross-overs among some of the volume-age trajectores from dfferent plots/subjects n the data.. Models Fgure. Volume-age trajectores for model fttng and model applcaton data. Summary statstcs for the data are lsted n Table. Each trajectory represents repeated measures from one plot. Comparson of alternatve model forms suggests that the followng base model s approprate for descrbng the volume-age relatonshp for black spruce n Alberta: b [] Vol bage exp(-bage) where Vol s total volume (m 3 /ha), age s breast heght age (years), b and b are model parameters (also called fxed parameters) applcable to the entre populaton, and exp denotes the exponental functon.

7 The mxed model derved from the base model takes the followng form: (b + u) [] Vol (b + u)age exp(-(b + u)age) where b and b are fxed parameters applcable to every plot n the populaton, and u and u are random parameters used to account for unque characterstcs of each plot n the populaton. Parameter estmates for the base and mxed models are lsted n Table. Summary goodness-of-ft statstcs assocated wth the estmates are also lsted n Table. The parameter estmates for the base model [] were obtaned from the ordnary nonlnear least squares (NLS) method. The parameter estmates for the mxed model [] were obtaned from the frst-order (FO) method and frst-order condtonal expectaton (FOCE) method of the NMM technque. Both methods of the NMM technque are detaled n Secton 3. Table. Parameter estmates and goodness-of-ft (GOF) statstcs obtaned on the model fttng data. Parameter Model and method GOF Model and method [] NLS [] FO [] FOCE measure [] NLS [] FO [] FOCE b e b e % σ u SD σ u u MAD σ u.49. MSE σ R AIC CC BIC MPE N MAPE m e δ Note: N s the total number of observatons, m s the number of plots, random parameters, and 3 σ u, σ u and σ u u are varances and covarance for the σ s the resdual varance. The goodness-of-ft (GOF) measures are defned n Table 3. The goodness-of-ft statstcs lsted n Table were calculated from the N observatons (N389) n the entre model fttng populaton, based on the formulas gven n Table 3. They were not calculated by averagng the goodness-of-ft statstcs from the ndvdual plots n the populaton. Hstorcally, dfferent goodness-of-ft measures have been used n dfferent studes to determne the goodnessof-ft of a ftted model. Each measure has ts pros and cons, and each usually reflects one aspect of a ftted model. Ths explans why a varety of goodness-of-ft measures lsted n Table 3 were calculated n ths study. In general, for most practcal purposes, the overall accuracy (δ) value, whch combnes the mean bas ( e ) and the varance of the resduals or predcton errors (SD ), can be consdered a good overall ndcaton of model accuracy, for the δ value s a summaton of the bas ( e ) and precson (SD ). Fgure 3 shows the spaghett plots from the FO and FOCE methods of the NMM technque. The spaghett plots dsplay the plot-specfc volume-age predctons for all 3 plots of the model fttng data across the age ranges where the ftted model s lkely to be appled. Table 4 llustrates how the predctons are obtaned to draw spaghett plots for three example plots of the model fttng data. All varables and computatons lsted n Table 4 are descrbed n more detals n Model Predctons (Secton 4) and Appendces (Secton 7).

8 Table 3. Goodness-of-ft measures used n ths study. Goodness-of-ft measure Computaton formula m n m n. Mean bas (or mean error) e ( yj yˆ j) e N j N j. Percent mean bas e % e y j 3. Standard devaton SD m n (ej e) N j m n 4. Mean absolute devaton MAD y yˆ N j j j 5. Mean square error (on model fttng data) MSE N-p m n ( y j yˆ j j ) 6. Mean square error (on model applcaton data) MSE m n ( y j yˆ j) N j 7. Coeffcent of determnaton R m n ( y j j yˆ j ) m n ( y j j y) 8. Concordance correlaton coeffcent CC m n ( y j j y) m n ( y j j m n + (ˆ y j yˆ j j ) yˆ) + N( y yˆ) m n yj yˆ j 9. Mean percent error MPE ( ) N j y m n yj yˆ j. Mean absolute percent error MAPE N j y j j. Number of absolute percent errors >% number of PEj > y ˆ j yj e, PE j ( ) N y j. Overall accuracy δ e + SD 3. Akake nformaton crteron AIC -ln( L) + P 4. Schwarz s Bayesan nformaton BIC - ln( L) + Pln( m) Grand means and total number m n y y N j j m n m yˆ yˆ j N N j n Note: y j and ŷ j are the jth observed and predcted volumes for the th plot,,,, m, j,,, n, m s the number of plots n the data, n s the number of observatons n the th plot, y and ŷ are the grand means of the observed and predcted volumes, N s the total number of observatons, L s the maxmzed value of the lkelhood functon for the estmated model, p s the number of fxed parameters, P s the total number of effectve parameters n mxed model estmaton (ncludes fxed parameters, varance-covarance components of the random parameters, plus the resdual varance component), and AIC and BIC are nformaton crtera used for mxed model on model fttng data only. 4

9 Fgure 3. Plot-specfc predctons from the FO and FOCE methods across the lkely age ranges: (a) for all 3 plots of the model fttng data; and (b) for three example plots lsted n Tables 4. Table 4. Orgnal data and calculatons for three example plots of the model fttng data. Plot Tme Age Volume Vol fx u u y_pred y_res 5 FO method m m m m m m m m FOCE method m m m m m m m m Note: m, m and m3 are three example plots. Tme s measurement tme. All other varables are detaled n Secton 4.

10 3. Nonlnear Mxed Model Methods Before demonstratng how to use the ftted models to make predctons, some relevant background materal on NMM methods s presented here. Readers who are famlar wth the NMM methods can skp the background materal and go drectly to Model Predctons (Secton 4). 3. Basc Formulaton of Nonlnear Mxed Models Usng the standard termnology for nonlnear models, a populaton-based model that descrbes the populaton averages can be wrtten as: [3] y f ( x, β) + ε where y s the dependent varable (also referred to as response varable), f denotes some nonlnear functon, x s a known matrx of covarates (also referred to as ndependent varables, regressors, predctors, or x- varables), β s a vector of model parameters applcable to the entre populaton, and ε s the error term. For a subject-specfc NMM, t can be wrtten as: [4] yj f( x j, b, u) + εj where y j s the jth observaton n the th subjects,,,, m, j,,, n, m s the number of subjects n the populaton, n s the number of observatons n the th subjects, f s a general expresson of a nonlnear functon, x j s a known vector of covarates for the jth observaton n the th subjects, b s a vector of fxed parameters common to all subjects n the populaton, u s a vector of random parameters unque for the th subject n the populaton, and ε j s a normally dstrbuted wthn-subject error term. In ths study, each plot s a subject. The measurements at dfferent tmes wthn each plot are observatons. The subject-specfc NMM [4] descrbes the mean responses of ndvdual subjects wthn a populaton. Ths s acheved through the ncluson of subject-specfc random parameters u n the model. For nstance, the subject-specfc volume-age model [] can be wrtten more explctly as: (b + u ) [5] Volj (b + u )Agej exp(-(b + u)agej) where Vol j and Age j are observed volume and age for the jth measurement n the th plot, b and b are fxed parameters common to every plot n the populaton, and u and u are random parameters unque for the th plot n the populaton. In essence, a subject-specfc model has a unque set of coeffcents for each subject n the populaton. For the subject-specfc volume-age model [5] developed from m plots, there are m unque sets of coeffcents for m plots n the populaton: b Plot : Vol bagej exp(-bagej) j b b + u, b b + u b Plot : Vol bagej exp(-bagej) b b + u, b b + u bm Plot m: Volmj bmagemj exp(-bmagemj) b b + um, bm b + j m um 6

11 Because a unque set of coeffcents s developed for each subject n the populaton, rather than assgnng the same set of coeffcents obtaned for the entre populaton to each subject n the populaton, a subject-specfc model s much more flexble and powerful than a populaton-based model. It can mmc the data trends exhbted by ndvdual subjects n the populaton more closely. A subject-specfc model typcally provdes more accurate fts and predctons on a subject-specfc level than a populaton-based model. In a more compact form, the mxed model [4] can be wrtten for subject as: [6] y f( x, b, u) + ε where,,, m, y y, y,..., y ]' s a vector of observatons for the y-varable from subject, x s a [ n known matrx of the x-varables, and ε ε, ε,..., ε ]' s a vector of wthn-subject errors. The random [ n parameters vector u and the error vector ε are typcally assumed to be uncorrelated and (multvarate) normally dstrbuted wth mean zero and varance-covarance matrces D and R, respectvely. That s: u u D Cov [ u, ε ] E Var ε ε R whch can be smplfed to u N(, D) and ε N(, R ). The varance-covarance matrx D of the random parameters s generally assumed to be the same for each and every subject n the populaton (.e., D D for,,, m). The varance-covarance matrx R of the wthn-subject errors can take many forms to represent ndependent and dentcally dstrbuted (d) errors, correlated errors, heterogeneous errors, and generalzed (correlated and heterogeneous) errors. Due to ts confusng but often nconsequental nature n predctons (Huang et al. 9a, 9b; Meng and Huang ; Huang et al. ; Yang and Huang a), and to avod a dgresson from the man purpose of ths study, R was assumed to be d. That s, R σ I, where σ s the error varance and I s an n n dentty matrx (a square matrx wth ones on the man dagonal and zeros elsewhere). The key dfference between subject-specfc mxed models and populaton-based models s the ncluson of random parameters n mxed models. Random parameters prmarly serve four purposes: ). Account for the dosyncrases of ndvdual subjects wthn a populaton; ). Account for the remnant mpacts of the x-varables already ncluded n the model ths can be mportant when the true model specfcaton s unknown; 3). Account for the mpacts of other known and unknown x-varables left-out by the model wthout actually requrng these varables to be dentfed or measured ths can be a good or a bad trat; 4). Allevate or elmnate entrely the correlaton and heteroskedastcty ssues commonly occurred n forest modelng from repeatedly measured cross-sectonal data. These and other related topcs are dscussed elsewhere (e.g., Huang et al. 9c; Meng and Huang ). Dfferent methods can be used to estmate the parameters of NMMs. They nclude frst-order lnearzaton, Laplace s approxmaton, adaptve Gaussan quadrature, mportance samplng, and Bayesan estmaton (Davdan and Gltnan 995, Vonesh and Chnchll 997, Pnhero and Bates 4). The two most commonly used methods, whch were mplemented n ths study due manly to ther computatonal smplcty n makng subject-specfc predctons on new data not used n model fttng, are the frst-order (FO) method of Beal and 7

12 8 Shener (98) and the frst-order condtonal expectaton (FOCE) method of Lndstrom and Bates (99). Both methods use a frst-order Taylor seres expanson of the mxed model [6] around a b close to b and an u close to u, to lnearze [6], wth the neglgble terms (e.g., quadratcs, cubcs and cross-products) dropped: [7] f ε u u Z b b X u b x y ) ( ) ( ),, ( where X and Z, often referred to as desgn matrces n mxed model dom, are partal dervatves of y wth respect to b and u, respectvely. The methods dffer on how the u s defned n [7]. 3. The Frst-Order Method For the FO method, u n [7] s set to ts expectaton of zero,.e., u E(u ). Therefore, [7] s reduced to: [8] f ε u Z b b X b x y ) ( ),, ( where the desgn matrces X and Z are defned by: [9] b b u b x X, ' ),, ( f b u u b x Z, ' ),, ( f Rearrangng [8] yelds: [] f ε u Z X b X b b x y ),, ( To create a lnearzed form of [], the left-hand sde of [] s defned as the pseudo-response functon y : [] ),, ( b X b x y y f + Hence, [] can be wrtten as a standard lnear mxed model: [] ε u Z b X y + + Followng the standard lnear mxed model theory (e.g., see Ftzmaurce et al. 4), the generalzed least squares estmator bˆ of the fxed parameters b and the random parameters predctor û of the u n [], can be obtaned as follows: [3] ' ' ˆ ˆ ˆ m m y V X X V X b [4] ) ˆ ( ˆ ˆ ˆ ' b X y V DZ u where Dˆ s an estmate of the varance-covarance matrx D for the random parameters, and Vˆ s the estmated (margnal) varance-covarance matrx for the pseudo-response functon y, averaged over the dstrbuton of the random parameters u (Davdan and Gltnan 995, Vonesh and Chnchll 997):

13 [5] Vˆ Var y Var Zu Var ε ZDZ ˆ ' ( ) ( ˆ ) + ( ) + Rˆ where Rˆ s an estmate of the varance-covarance matrx R for the wthn-subject error term. Substtutng the Vˆ n [5] and the pseudo-response functon y n [] nto [4], and recognzng u and b bˆ once bˆ s estmated, the predctor of random parameters û n [4] can be wrtten as: [6] ˆ ' ˆ ' ˆ uˆ DZ ( Z DZ + R ) [ y f( x, bˆ, )] where the Z matrx s defned n [9](wth b bˆ ). Equaton [6] s the random parameters predcton equaton for the FO method. Once the values of bˆ and û are avalable, the pseudo-response functon [7] yˆ Xbˆ + Zuˆ Substtutng the y n the pseudo-response functon [] by the X bˆ Z uˆ y ( x,ˆ, b ) X bˆ + f + Rearrangng for y gves the predcted y ( ŷ ) for subject for the FO method: [8] yˆ f( x, bˆ, ) + Zuˆ y can be predcted from []: y ˆ n [7] and recognzng b bˆ produce: where Z s defned n [9](wth b bˆ ). Equaton [8] s the response varable predcton equaton for the FO method. The correspondng resduals or predcton errors (e ) are calculated by: [9] e y yˆ y f( x, bˆ, ) Zuˆ Notce the term Z uˆ must be ncluded n predctng the ŷ and calculatng the e values for the FO method. 3.3 The Frst-Order Condtonal Expectaton Method For the FOCE method, [7] s frst wrtten as: [] y f( x, b, u) + Xb + Zu Xb + Zu + ε where the desgn matrces X and Z are gven by: [] X f( x, b, u) b', u b Z f( x, b, u) u ', u b Defne the left-hand sde of [] as the pseudo-response functon 9 y for the FOCE method:

14 [] y f( x, b, u) + Xb Zu y + Then [] can be wrtten as a standard lnear mxed model: [3] y Xb + Zu + ε Followng the standard lnear mxed model theory (e.g., see Ftzmaurce et al. 4), the generalzed least squares estmator bˆ of the fxed parameters b and the random parameters predctor û of the u n [3] can be obtaned as descrbed n equatons [3] and [4]. For the FOCE method, substtutng the Vˆ n [5] and the pseudo-response functon recognzng u û and b bˆ, the û predcton equaton [4] can be wrtten as: [4] ˆ ' ˆ ' ˆ u ˆ DZ( ZDZ + R) [ y f( x, bˆ, uˆ ) + Zuˆ ] where the Z matrx s defned n [](wth u û and b bˆ ). y n [] nto [4], and For the FOCE method, the random parameters to be predcted ( û ) appear on both sdes of [4]. Ths s very dfferent from that for the FO method, where û appears only on the left-hand sde of equaton [6]. There s no easy algebrac soluton for û n [4]. Instead, a three-step numercal procedure needs to be mplemented to teratvely solve for û n [4]: Step. Obtan a frst estmate, termed u ˆ,, of the random parameters. Ths s acheved by assumng the ntal û appearng on the rght-hand sde of [4] equal to zero (.e., u ˆ ): [5] ˆ ' ˆ ' ˆ uˆ DZ ( Z DZ + R ) [ y f( x, bˆ, )],,,, where the ntal desgn matrx Z, s evaluated at the assumed ntal u ˆ : [6] Z, f( x, b, u) u ' bˆ, Step. Once the u ˆ, s calculated from step, the next Z, termed Z,, s evaluated at u ˆ, : [7] Z, f( x, b, u) u ' bˆ,ˆ u, The next estmaton of û, termed u ˆ,, s obtaned based on [4] agan, wth the û and Z on the rght-hand sde of [4] replaced by u ˆ, from [5] and Z, from [7], respectvely: [8] ˆ ˆ ' ˆ ' ˆ u ( ) [ (,ˆ,ˆ ) ˆ, DZ, Z,DZ, + R y f x b u, + Z,u, ]

15 Step 3. Havng the calculated u ˆ, from step, the next Z, termed Z,, s evaluated at u ˆ, : [9] Z, f( x, b, u) u ' bˆ,ˆ u, The next estmaton of û, termed u ˆ, 3, s computed based on [4] agan, wth the û and Z on the rght-hand sde of [4] replaced by the updated u ˆ, from [8] and Z, from [9], respectvely: [3] ˆ ˆ ' ˆ ' ˆ u ( ) [ (, ˆ, ˆ ) ˆ, 3 DZ, Z,DZ, + R y f x b u, + Z,u, ] Ths process s terated k tmes untl a user-specfed convergence crteron s acheved, such as: [3] u uˆ. ˆ, k,( k ) < Once the convergence crteron s acheved, the fnal predctor of the random parameters s: [3] u ˆ uˆ, k. Intrnscally, some readers may already recognze that the three-step procedure s n fact an teraton of equatons [7] and [8], wth the ntal and end condtons gven by [5] and [3], respectvely. More detals were provded n the above descrptons for the sake of other nterested readers. Wth the known bˆ and û values, the pseudo-response functon [33] yˆ Xbˆ + Zuˆ Substtutng the y n the pseudo-response functon [] by the,[] can be wrtten as: X ˆ + ˆ ˆ, ˆ + ˆ + b Zu y f( x, b u ) Xb Zu y n [3] can be predcted: y ˆ n [33], and recognzng ˆ b bˆ and u û Rearrangng for y produces the predcted y ( ŷ ) for subject for the FOCE method: [34] y ˆ f( x, bˆ, uˆ ) Evdently, for the FOCE method, the response varable for any subject can be predcted drectly by smply substtutng the known bˆ and û values nto the mxed model wthout nvolvng the term Z u ˆ. Ths s fundamentally dfferent from the FO method, whch nvolves Z u ˆ and uses [8] to predct the response varable. The correspondng resduals or predcton errors (e ) for the FOCE method are calculated by: [35] e y yˆ y f( x, bˆ, uˆ ) whch s agan dfferent from [9] for the FO method. Table 5 summarzes the formulas explct to the FO and FOCE methods of the NMM technque.

16 Table 5. A summary of the frst-order (FO) and frst-order condtonal expectaton (FOCE) methods. Method FO Method FOCE Taylor seres expanson f( x, b, ) + X( b b ) + Zu ε y + f( x, b, u ) + X( b b ) + Z( u u ) ε y + Pseudo-response functon y y f ( x, b, ) + Xb y f( x, b, u ) + Xb Zu y + Lnearzed model y X b + Z u + ε y X b + Z u + ε Desgn matrces X f( x, b, u ) b' bˆ, f( x, b, u) Z u ' bˆ, X f( x, b, u) b' bˆ, u Z f( x, b, u) u ' bˆ, u Predctor of random parameters DZ ˆ ( Z DZ ˆ + Rˆ ' ' uˆ ) [ y f( x,ˆ, b )] ˆ ' ˆ ' ˆ u ˆ DZ( ZDZ + R) [ y f( x,ˆ,ˆ b u) + Zuˆ ] Margnal predcton from fxed parameters yˆ _ f( x,ˆ, b ) yˆ _ f( x,ˆ, b ) fx Subject-specfc predcton wth ndependent and dentcally dstrbuted error structure ( Rˆ σ I fx ) yˆ f( x, bˆ, ) + Zuˆ y ˆ f( x,ˆ,ˆ b u) Resduals or predcton errors eˆ ˆ eˆ y f( x,ˆ,ˆ b u) y f( x,ˆ, b ) Zu Subject-specfc forecast wth generalzed error structure ( Rˆ σ Ψ ) ˆ, ˆ y ˆ f x b Z uˆ (, ) + + V'Ψ e yˆ f( x,ˆ,ˆ b u) + V'Ψ e ˆ Note: y and ŷ are observed and predcted values for subject, x s a matrx of covarate(s), X and Z are partal dervatves wth respect to fxed parameters b and random parameters u, respectvely, bˆ and û are predctors of b and u, Dˆ and Rˆ are estmated varance-covarance matrces for u and ε, respectvely, ŷ, x and Z denote the varables assocated wth future observatons, V contans the correlatons between the elements of past and future errors, and Ψ s the correlaton matrx of past errors.

17 It s worthwhle to reterate that the formulas summarzed n Table 5 for the FO and FOCE methods are qute dfferent, partcularly wth regard to: a) desgn matrces; b) predctors of random parameters; c) predctons of the response varable; and d) resdual or predcton error calculatons. Mxng the formulas from the FO and FOCE methods would be mathematcally ncorrect. Indeed, due to the methodologcal and computatonal dfferences between the FO and FOCE methods, when estmatng a model and usng the estmated model to make predctons, t s very mportant to ensure that the model estmaton procedure s consstent wth the model predcton procedure. Otherwse, the results could be nconsstent or smply wrong, and the reported model fttng statstcs could lose ther ntended meanng or could even gve a false ndcaton about the model s performance when predctons are made. Extensve evaluaton on model fttng and model applcaton data showed that substantal dfferences occurred when the formulas from the FO and FOCE methods were mxed (Huang 8, Meng and Huang 9). It s questonable that a number of NMM applcatons n forestry used ˆ ' ˆ ' ˆ uˆ DZ( ZDZ + R) [ y f( x,ˆ, b )] (equaton[6]) from the FO method to predct û, then used y ˆ f( x, bˆ, uˆ ) (equaton[34]) from the FOCE method to predct ŷ and came up wth an unbelevably convncng outcome. Had the correct formulas or a dfferent data set been used n those applcatons, the outcome could have been dfferent. In practce, before a ftted mxed model can be used to make predctons on data not used n model fttng, modellers must check two tems pror to recommendng a predcton procedure: ). On the model fttng data, the random parameters predcted from the predcton procedure are equvalent to the random parameters obtaned from the model estmaton procedure; and ). On the model fttng data, the predcton errors obtaned from the predcton procedure are equvalent to the resduals obtaned from the model estmaton procedure. Falng to verfy any one of the two tems could mean that the predcton procedure s ncompatble wth the model estmaton procedure. In cases where the predctons cannot be checked on model fttng data (e.g., model users typcally can only access the parameter estmates lsted n Table, but not the full model fttng data), ether the model estmaton method should be known, or a mathematcally consstent set of formulas correspondng to a specfc method should be chosen for predctons after comparng alternatve methods. It s very mportant to ensure that model estmaton procedure s equvalent to model predcton procedure. As a general rule ths prncple of equvalence between model estmaton and model predcton procedures shall be followed whenever possble n any type of model estmaton and predcton. Due to the computatonal dffcultes n developng and mplementng an equvalent procedure n model estmaton and predcton on both model fttng and model applcaton data, other NMM methods (Laplace s approxmaton, adaptve Gaussan quadrature, mportance samplng and Bayesan estmaton) are not dscussed n ths study. Interested readers may wsh to read Yang and Huang (b) on comparng some of these methods. Table 5 also lsts the formulas for subject-specfc forecasts wth a generalzed error structure ( Rˆ σ Ψ). Snce the NMM technque often allevates or elmnates entrely the correlaton and heteroskedastcty ssues that commonly occur n forest modelng, further detals on how to address these ssues n NMMs are not presented here. Generalzed error structure s used n forecastng future observatons drectly from the past measurements of the same sequence/trajectory. It has no use n predctng current or future observatons that does not drectly rely on the past measurements of the same sequence. Interested readers may wsh to read Huang et al. (9a, 9b, ), Meng and Huang (), and Meng et al. () on how to use estmated generalzed error structures to forecast future observatons from the past measurement(s) of the same sequence. 3

18 4. Model Predctons Three plots each from the model fttng data (Table 4) and model applcaton data (Table 6) are used to demonstrate model predctons. Snce the predcton procedures on the model fttng and model applcaton data are the same, detaled computatons are llustrated here only for the model applcaton data. Plot () Tme () Table 6. Example model applcaton data and computatons based on the FO method. Age Volume Vol fx u u y_pred (3) (4) (5) (6) (7) (8) (9) () v v v v v v v v v v Note: v, v and v3 are three example plots. Tme refers to measurement tme (,, 3, etc.). Age and volume refer to black spruce breast heght age (years) and total volume (m 3 /ha). All other varables are descrbed n the man text. 4. Predcton from the Frst-Order Method To make plot-specfc predctons from the FO method, random parameters unque for each plot must be predcted frst based on equaton [6]. Usng plot v of the model applcaton data n Table 6 as an example, volume predctons (Vol fx ) based on the fxed parameters only are lsted n column 5 of Table 6. They are obtaned drectly from the mxed model [], wth the random parameters set to zero and the fxed parameters gven n Table for the FO method (b.785 and b.3396): (b + ) Vol b + ) Age exp(-(b + ) Age ) [54.56, 5.355, 49.5, , ] _ fx ( y_res () Gven σ u.64, σ.6674, u u σ u.49 and σ for the FO method (from Table ), for plot v wth fve observatons and two random parameters, the varance-covarance matrces Rˆ and Dˆ for the errors and random parameters are: R ˆ σ σ σ σ σ σ D ˆ σu u u σ uu u σ

19 The partal dervatves of [] wth respect to the two random parameters are: f( x, b, ) b [36] d_b Age ( bage) exp( bage) b f( x, b, ) b [37] d_b b exp( bage) Age ln( Age) b The dervatves (columns 6 and 7 of Table 6) consttute the desgn matrx Z for plot v: Z Therefore, the two random parameters for plot v can be predcted followng equaton [6]: DZ ˆ ( Z DZ ˆ + Rˆ ) ' ' uˆ ( Vol Vol whch produces the followng random parameter predctons for plot v: u ˆ _fx [u,u ]' [.639,.933]' Random parameter predctons for other plots of the model applcaton data are obtaned n a smlar manner. They are lsted n columns 8 and 9 of Table 6. Once the û values are known, plot-specfc volume predctons are calculated followng equaton [8], whch becomes Vol ˆ Vol Z uˆ fx +. For plot v, ths gves: _ ) V ol ˆ They are lsted n column (y_pred Vˆ ol ) of Table 6. The correspondng predcton errors (y_rese ) are lsted n column of Table 6. They are computed followng equaton [9]: e Vol Vol _fx Z uˆ The computatons for the model fttng data follow the same procedures. Example results are lsted n Table 4. Appendces and provde two programs assocated wth the FO method. One s generalzed (Appendx ). The other s step-by-step (Appendx ). Both programs apply to model fttng as well as model applcaton data. Interested readers could use ether one of them to carry out the computatons nvolved n the FO method. 5

20 Fgure 4 (left-hand sde graphs) shows the plot-specfc predctons from the FO method for all 79 plots of the model applcaton data. The predctons are made wthn the observed data range (graph FO-(a)), as well as across the potental age range where the ftted model s lkely to be appled n practce (graph FO-(b)). The orgnal data and the predctons for the three example plots lsted n Table 6 are dsplayed n graph FO-(c). Note that the predctons from the FO method could be negatve n some cases (e.g., for plot v3 wth three observatons, when the age s older than about 4 years see graph FO-(c)). Fgure 4. Plot-specfc predctons from the FO and FOCE methods for the model applcaton data: (a) wthn the observed data range, (b) across the potental age range, and (c) for the three example plots lsted n Tables 6 (FO method) and 7 (FOCE method). Dashed lnes n (c) ndcate the lne of zero. 4. Predcton from the Frst-Order Condtonal Expectaton Method For the FOCE method, the predcton for the random parameters follows the three-step procedure descrbed n [5] to [3]. Plot v of the model applcaton data lsted n Table 7 s used to demonstrate the computatons. 6

21 Plot () Table 7. Example model applcaton data and computatons based on the FOCE method. Tme Age Volume Vol fx y_pred y_res u () (3) (4) (5) (6) (7) (8) (9) () Step computaton u () v v v v v Step computaton v v v v v Step 3 computaton v v v v v Fnal teraton results v v v v v v v v v v Note: v, v and v3 are three example plots. Tme refers to measurement tme (,, 3, etc.). Age and volume refer to black spruce breast heght age (years) and total volume (m 3 /ha). All other varables are descrbed n the man text. Step. Volume predctons based on the fxed parameters only are lsted n column 5 of Table 7. They are obtaned drectly from the mxed model [] wth the random parameters set to zero and the fxed parameters gven n Table for the FOCE method (b.879 and b.34): (b + ) Vol b + ) Age exp(-(b + ) Age ) [44.63, , , , 3.36] _ fx ( The partal dervatves of [] wth respect to the random parameters evaluated at u ˆ are calculated by: 7

22 f( x, b, ) (b + ) d_b Age ( (b + ) Age) exp( (b + ) Age) b d_b f(, b, ) (b b x (b + ) + ) exp( (b + ) Age) Age ln( Age ) They are lsted n columns 6 and 7 of Table 7. The dervatves consttute the ntal desgn matrx Z, evaluated at u ˆ : Z, f( x, b, u) u ' bˆ, Gven σ u.878, σ.587, u u σ u. and σ for the FOCE method (from Table ), for plot v wth fve observatons and two random parameters, the varance-covarance matrces Rˆ and Dˆ for the errors and random parameters are: R ˆ σ σ σ σ σ σ D ˆ σu u u σ uu u σ Hence, the frst estmate u ˆ, of the random parameters ûfor the FOCE method can be obtaned followng equaton [5]: DZ ˆ DZ ˆ + Rˆ ' ' uˆ,,( Z,, ) ( Vol Vol_fx 8 ) [, û, û, ] [.665,.337] Step. Once the u, s calculated from step, a new set of the partal dervatves are evaluated at u ˆ uˆ, : ˆ f( x, b, u) (b + uˆ,) [38] d_b ( (b ˆ ) ) exp( (b ˆ Age + u, Age + u, ) Age) b f( x, b, u) (b + uˆ,) [39] d_b (b ˆ ) exp( (b ˆ + u, + u,) Age) Age ln( Age) b The partal dervatves (columns 6 and 7 of Table 7) consttute a new desgn matrx Z, evaluated at u, : ˆ

23 [4] Z, f( x, b, u) u ' bˆ,ˆ u, where the elements j and j n Z, are obtaned from equatons [38] and [39], respectvely. Volume predctons for the FOCE method from the known uˆ, and b values, V ol ˆ (, ˆ, ˆ, f x b u, ), are obtaned followng equaton [34]. For plot v, ths gves: ˆ (b + û ),,,, [4] V ol (b + û ) Age exp(-(b + û ) Age ) [37.787, 8.95, 97.75, 89.55, ] Results are lsted n column 8 of Table 7. Column 9 of Table 7 lsts the correspondng predcton errors calculated by e ˆ ˆ, y f( x, bˆ, u,) Vol Vol, (note that n Table 7, y_pred Vˆ ol and y_rese ). Havng the u ˆ, and Z, values, the next estmaton of û, termed u ˆ,, s obtaned followng equaton [8], whch gves: [4] ˆ ˆ ' ˆ ' ˆ u ( ) [ (,ˆ,ˆ ) ˆ, DZ, Z,DZ, + R y f x b u, + Z,u, ] [.437,.37] They are lsted n columns and of Table 7. Step 3. Havng the calculated u, from step, the next set of the partal dervatves are evaluated at u ˆ uˆ, : ˆ f( x, b, u) (b + uˆ,) [43] d_b ( (b ˆ ) ) exp( (b ˆ Age + u, Age + u, ) Age) b f( x, b, u) (b + uˆ,) [44] d_b (b ˆ ) exp( (b ˆ + u, + u,) Age) Age ln( Age) b They (columns 6 and 7 of Table 7) consttute the next desgn matrx Z, evaluated at u, : ˆ [45] Z, f( x, b, u) u ' bˆ,ˆ u, A new set of volume predctons from the known u ˆ, and b values, are calculated followng equaton [34] agan, whch produces (column 8 of Table 7): ˆ (b + û ),,,, [46] V ol (b + û ) Age exp(-(b + û ) Age ) [74.77, 4.57, 9.74, 9.57, 7.988] 9

24 The correspondng predcton errors ( e y f( x,ˆ,ˆ b u,) Vol Vol, ) are lsted n column 9 of Table 7., ˆ Wth the known u ˆ, and Z,, the next estmaton of û, termed u ˆ, 3, s obtaned followng [3], whch gves: [47] ˆ ˆ ' ˆ ' ˆ u ( ) [ (, ˆ, ˆ ) ˆ, 3 DZ, Z,DZ, + R y f x b u, + Z,u, ] [.56,.345] They are lsted n columns and of Table 7. The process descrbed n equatons [43] to [47] s terated k tmes, wth the pror u,( k ) n the equatons replaced by the newer û, k calculated from [47]. The process s stopped once the convergence crteron specfed n [3] s acheved. For plot v of the model applcaton data, the convergence crteron s acheved after sx teratons. The fnal û s predcted to be (columns and of Table 7): û [ û, û ] [.549,.346] The fnal volume predctons for plot v are (column 8 of Table 7): ˆ (b + û) V ol (b + û ) Age exp(-(b + û ) Age ) [7.6, 38.46, 5.3, 4.85, 3.] ˆ The assocated fnal predcton errors (column 9 of Table 7) are calculated by e,ˆ,ˆ ˆ. y f( x b u) Vol Vol The fnal results from the FOCE method for plots v and v3 of the model applcaton data are also lsted n Table 7. The fnal results from the FOCE method for plots m, m and m3 of the model fttng data are lsted n Table 4. Interested readers may wsh to verfy and duplcate the computatons. Appendces 3 and 4 provde two programs assocated wth the FOCE method. One s generalzed (Appendx 3). The other s step-by-step (Appendx 4). Both programs apply to model fttng as well as model applcaton data. Interested readers could use ether one of them to carry out the teratons requred by the FOCE method. Fgure 4 (rght-hand sde graphs) shows the plot-specfc predctons from the FOCE method for all 79 plots of the model applcaton data. The predctons are made wthn the observed data range (graph FOCE-(a)), as well as across the potental age range where the ftted model s lkely to be appled n practce (graph FOCE-(b)). The orgnal data and the predctons for the three example plots lsted n Table 7 are dsplayed n graph FOCE- (c). Note that the predctons from the FOCE method are always postve across the age range, whereas the predctons from the FO method are not. Overall, Fgure 4 suggests that the predctons from the FO and FOCE methods are smlar wthn the observed data range, but beyond the observed data range, some of the predctons can be qute dfferent. Ths explans why sometmes two methods or two models wth smlar statstcs can produce very dfferent predctons. 4.3 Goodness-of-Ft Statstcs The goodness-of-ft statstcs obtaned from the FO and FOCE methods on the model applcaton data are lsted n Table 8. For a comparson, the goodness-of-ft statstcs obtaned from the drect applcaton of model [] on the model applcaton data are also lsted n Table 8. All goodness-of-ft statstcs were calculated accordng to the formulas defned n Table 3, based on the N3 observatons n the model applcaton data. They were not averaged from the goodness-of-ft statstcs calculated by ndvdual plots n the populaton.

25 Table 8. Goodness-of-ft statstcs obtaned on the model applcaton data. Model Goodness-of-ft measure e e % SD MAD MSE R CC MPE MAPE e δ []-NLS []-FO []-FOCE Note: The goodness-of-ft measures are defned n Table 3. On the model fttng data, the predctons of the random parameters and response varables from the FO and FOCE methods, as well as the computatons of the goodness-of-ft statstcs for the predctons, follow the same procedures as demonstrated on the model applcaton data. Example predctons on the model fttng data are embedded n the programs provded n the Appendces. They are also lsted n Table 4. Interested readers may wsh to verfy and duplcate the results n the programs. The predctons for all 3 plots of the model fttng data across the potental age range where the ftted model s lkely to be appled to make predctons (.e., the spaghett plots), are shown n Fgure 3. Goodness-of-ft statstcs correspondng to the model fttng data are lsted n Table. 4.4 Choosng the Best Predcton The best predcton s chosen based on a combnaton of statstcal, graphcal, bologcal and other consderatons, on both model fttng and model applcaton data. Judgng from the goodness-of-ft statstcs on the model fttng data (Table ), and usng the overall accuracy measure δ as the example, model [] estmated from the FO method s the most accurate, wth δ3.87. On the model applcaton data (Table 8), model [] estmated from the FO method s agan the most accurate, wth δ5.8. Therefore, based on the goodness-of-ft statstcs alone, model [] estmated from the FO method provded the best predcton. From the predcton graphs (also referred to as spaghett plots) shown n Fgure 3 for the model fttng data and Fgure 4 for the model applcaton data, t can be seen that the predctons from model [] estmated from the FO method can be negatve n some cases when the age s beyond certan ranges. Ths s caused nherently by the term Z u ˆ n equaton [8] for the FO method. Ths term could also cause the nnate shape of a base model to be altered (Huang et al. 9a, Yang and Huang b). For the FOCE method, equaton [34], whch mantans the nnate shape of a base model, s used to predct the response varable y. In spte of the negatve volume predctons, the predcton graphs from the FO method (n Fgures 3 and 4) show no obvous anomaly. In real-world stuatons, negatve volume predctons could be constraned to zero to be bologcally meanngful. There are plots where the volume could become zero because of mortalty. For the FOCE method, all volume predctons are non-negatve (Fgures 3 and 4). But some do not taper-off at older ages, especally on the model applcaton data (Fgure 4). Ths may not concde well wth the expected bologcal growth. It s also more dffcult to establsh a meanngful bologcal up-lmt n ths case. Hence, based on the statstcal, graphcal and bologcal consderatons, and gven the avalable data wth ther nherent qualty, quantty, relevance (e.g., data range and dstrbuton) and lmtaton, t can be nferred that model [] estmated from the FO method appears to be the best for the volume-age relatonshp consdered n ths study. Computaton-wse, the FO method s also much smpler than the FOCE method. It does not requre teraton. All predcton equatons nvolved n the FO method can be solved algebracally.

26 5. Addtonal Notes 5. Adjustng the Predctons For most practcal purposes, the nonlnear mxed model methods can generally be consdered unbased. But the unbasedness property, and ndeed, many other propertes assocated wth the NMM methods, hold only n an asymptotcally approxmated sense. When the sample sze s not large enough, or when model specfcaton s problematc, mxed model may produce based predctons. The potental bas of a nonlnear mxed model can be removed by dfferent methods. The proportonal adjustment method s the smplest and most effectve method n many cases. Ths method s mplemented through the calculaton of a rato, called the proportonal adjustment rato PAR, from the data n plot : [48] PAR y yˆ n n n y j j n yˆ j j n y j j n yˆ j j where PAR s the proportonal adjustment rato for plot n the populaton, y j s the jth observed y-value for the th plot n the populaton and ŷ j s ts predcton from the mxed model, y s the mean of the observed values, ŷ s the mean of the predcted values, and n s the number of observatons n the th plot. Re-arrange [48] produces: n n [49] y -PAR yˆ j j j j whch mples that the summaton of the observed values for plot equals to the PAR tmes the summaton of the predcted values. Defne the adjusted predctons as: [5] yˆ j_adj PAR yˆ j Then, from [5] and [48]: [5] n PAR n yˆ j adj yˆ j adj yˆ j yˆ PAR y n j n j Hence: [5] y PAR yˆ (or y ˆ _ ) y j adj whch means that the mean of the observed values for plot equals to the mean of the proportonally adjusted predctons from the mxed model. What s really mportant about the expressons gven n [48]-[5] s that, they all suggest that the mean bas of the adjusted predctons for plot s zero. That s: n n ej_ adj ( yj yˆ j_ adj) yj PAR yˆ j j j j j [53] ej _ adj n n n n n

27 where e j _ adj s the mean bas of the adjusted predcton errors for plot n the populaton. The adjusted predctons obtaned for each plot from the plot-specfc proportonal adjustment method are guaranteed to have a zero mean bas. Snce the mean bas for each plot n the populaton s zero, the mean bas of the adjusted predctons s also guaranteed to be zero for the entre populaton. The essence of the proportonal adjustment method s to utlze the power of the mxed model method to track the trends of plot-specfc data n a populaton, whle smultaneously shftng the predctons up or down proportonally to allevate the sample sze and/or nonlnear approxmaton and asymptotc ssues. Ths allows for a nonlnear mxed model to ft any plot-specfc data as close as possble. Wth the proportonal adjustment method t s always possble to acheve a bas-free ft that closely mmcs the data. But the bas-free ft stll does not mply the best ft, because the best ft s not determned by the bas alone. More specfc examples on how to use the proportonal adjustment method to adjust plot level and populaton level predctons are demonstrated elsewhere (Huang 8, Huang et al. 3). For most practcal purposes, t s recommended that adjustng mxed model predctons shall be done only when the percent mean bas of the unadjusted predctons exceeds ±5% (.e., e% >5%). Otherwse, the gans from adjustng the predctons may not be substantal. As an example, among the 79 plots of the model applcaton data, four plots have e% values that exceed 5% from the FO method. Therefore, adjusted predctons for these plots are obtaned followng the proportonal adjustment method. They are shown n Fgure 5. Actual data and computatons assocated wth Fgure 5 are lsted n Table 9. Relevant plot-specfc goodness-of-ft statstcs for unadjusted and adjusted predctons are lsted n Table. Fgure 5. Unadjusted (sold lnes) and adjusted (dashed lnes) predctons for four plots of the model applcaton data. Actual data and computatons are lsted n Table 9. 3

28 Table 9. Example calculatons for proportonally adjustng the predctons from the FO method. Plot Tme Age Volume ŷ e e% y ŷ PAR ŷ adj Note: ŷ and e are predcted volume and predcton error, e%s percent mean bas, y and ŷ are means of observed and predcted volumes, PAR s proportonal adjustment rato, and ŷ adj and e adj are proportonally adjusted ŷ and e. Table. Plot-specfc goodness-of-ft statstcs from the FO method. Plot e e % SD MAD MSE R CC MPE MAPE e δ Unadjusted predctons N/A N/A Adjusted predctons N/A.. N/A Note: actual data for the four plots are lsted n Table 9. The goodness-of-ft measures (e, e %,SD,, δ ) are defned n Table 3 and calculated by plot. N/A denotes not applcable (due to a zero denomnator). Whle vsually some of the dfferences between the unadjusted and adjusted predctons may be hard to see n Fgure 5, the plot-specfc goodness-of-ft statstcs lsted n Table clearly ndcate that the overall accuracy (δ) of the adjusted predctons s mproved for all four plots over ther unadjusted counterparts. Ths suggests that proportonally adjustng the predctons from the FO method s benefcal for all four plots. As expected, the adjusted predctons are guaranteed to be unbased (.e., e), but n general there s no guarantee that they are always more accurate than the unadjusted predctons (Huang et al. 3). 5. Populaton Average Predctons The fxed parameters estmated for model [] from the NLS method (b.6 and b.36, Table ) are ntended to make populaton average predctons. The fxed parameters estmated for model [] from the FO method (b.785 and b.3396, Table ) and FOCE method (b.879 and b.34, Table ) are ntended to make the so-called margnal predcton durng the estmaton of a nonlnear mxed model. They are not ntended to make populaton average predctons. In fact, they generally gve based populaton average predctons. 4 e adj

29 The techncal aspect of why the fxed parameters estmated as a part of a nonlnear mxed model do not represent the populaton averages s detaled n Davdan and Gltnan (3) and Ftzmaurce et al. (4). Relevant forestry examples are provded n Huang (8) for the volume-age relatonshp and Meng et al. (9) for the heght-age relatonshp. To brefly llustrate here, the three sets of fxed parameters estmated from the NLS, FO and FOCE methods (Table ) were used to make populaton average predctons. The predctons were frst made on the model fttng data, then on the model applcaton data. Fgure 6 shows the predctons overlad on the data. Fgure 6. Populaton average predctons from the fxed parameters of NLS (sold lne), FO (short-dash) and FOCE (long-dash) methods. Relevant goodness-of-ft statstcs are lsted n Table. Alternatve Populaton Average Predctons On the model applcaton data, alternatve populaton average predctons can be obtaned from the FO and FOCE methods, by treatng the entre model applcaton data as one combned plot wth N observatons (N3). The predctons follow Secton 4. for the FO method and Secton 4. for the FOCE method (except that ths combned plot has 3 observatons). Predcton results across the age range are shown n Fgure 7. For nterested readers, the predcted random parameters are u -. and u for the FO method, and u and u -.66 for the FOCE method. Fgure 7. Populaton average predctons from the FO (left) and FOCE (rght) methods. The predctons are obtaned by treatng the model applcaton data as one combned plot wth 3 observatons. Table lsts the goodness-of-ft statstcs assocated wth dfferent types of populaton average predctons on model fttng and model applcaton data. 5

30 Modelmethod Table. Goodness-of-ft statstcs from dfferent types of populaton average predctons. Goodness-of-ft measure e e % SD MAD MSE R CC MPE MAPE e δ Model fttng data usng fxed parameters only []-NLS []-FO []-FOCE Model applcaton data usng fxed parameters only []-NLS []-FO []-FOCE Model applcaton data treatng the data as from one combned plot, usng fxed and random parameters []-FO []-FOCE Note: fxed parameters for the three methods are lsted n Table. The goodness-of-ft measures are defned n Table 3. Results n Table suggest that:. On the model fttng data, the e from the NLS method s very close to zero (.e., e % s less than halfa-percent). Ths s a consequence of the NLS method. A correctly specfed and ftted nonlnear model should produce an e that s asymptotcally approxmately equvalent to zero on the model fttng data. The e values from the FO and FOCE methods usng fxed parameters only are much larger. They can be consdered based (.e., the absolute e % values exceed half-a-percent for both FO and FOCE). Typcally, the NLS method s the most accurate method (.e., wth the smallest δ value) n makng populaton average predctons on model fttng data. It s also the preferred method to use on model applcaton data f no measurement s avalable from the model applcaton data.. On the model applcaton data usng fxed parameters only, overall the FOCE method (δ5479.4) s slghtly more accurate than the NLS method (δ549.3). Both are more accurate than the FO method (δ5946.5). In general, on model applcaton data usng fxed parameters only, the most accurate method vares dependng on the specfc data nvolved. There s no generc trend wth regard to whch method s the best. 3. When treatng the model applcaton data as one combned plot and usng the FO and FOCE methods wth fxed and random parameters, the overall accuraces from the FO and FOCE methods are vrtually dentcal (δ3938. for FO and δ for FOCE). Both are substantally better (more accurate) than those on the model applcaton data usng fxed parameters only. Hence, on model applcaton data, populaton average predctons can be obtaned by treatng the model applcaton data as one combned plot, then mplementng the FO or FOCE method. The populaton average predctons obtaned n ths manner are typcally better than those obtaned from other methods wthout adjustment. In practce, such populaton average predctons can be obtaned for any user-defned model applcaton populaton. They can also be obtaned for any sub-populaton from any specfc area. It may be worthwhle to repeat that, f no y measurement s avalable from the model applcaton data, the full FO and FOCE methods cannot be mplemented. In ths case the fxed parameters from the NLS method should be used to make populaton average predctons. If necessary, these predctons can be adjusted to ncrease the accuracy of the predctons. Ths can be acheved through the proportonal adjustment method (Huang et al. 3), or some other methods f the proportonalty does not hold (Huang ). 6

31 6. References Beal, S.L. and Shener, L.B. 98. Estmatng populaton knetcs. Crtcal Revews n Bomedcal Engneerng 8: 95-. Davdan, M., and Gltnan, D.M Nonlnear models for repeated measurement data. Chapman & Hall, New York. 36 p. Davdan, M., and Gltnan, D.M. 3. Nonlnear models for repeated measurement data: an overvew and update. Journal of Agrc., Bologcal, and Envronmental Statstcs 8: Ftzmaurce, G.M., Lard, N.M., and Ware, J.H. 4. Appled longtudnal analyss. Wley, New York. 56 p. Huang, S.. Valdatng and localzng growth and yeld models: procedures, problems and prospects. An nvted paper presented at Realty, Models and Parameter Estmaton, sponsored by IUFRO, Insttuto Superor de Gestão and Insttuto Superor de Agronoma, June -5,, Sesmbra, Portugal, 8 p. Huang, S. 8. A generalzed procedure for bas-free predctons at populaton and local levels. A dscusson paper, Forest Management Branch, Alberta Mnstry of Sustanable Resource Development, Edmonton, Alberta, Pub-Feb--8, 6 p. Huang, S., Meng, S.X., and Yang, Y. 9a. Assessng the goodness-of-ft of forest models estmated by nonlnear mxed model methods. Canadan Journal of Forest Research 39: Huang, S., Meng, S.X. and Yang, Y. 9b. Predcton mplcatons of nonlnear mxed-effects forest bometrc models estmated wth a generalzed error structure. Proceedngs of Jont Statstcal Meetngs, Secton on Statstcs and the Envronment, August 6, 9, Washngton, D.C. Amercan Statstcal Assocaton, Alexandra, Va., pp Huang, S., Wens, D.P., Yang, Y., Meng, S.X., and VanderSchaaf C.L. 9c. Assessng the mpacts of speces composton, top heght and densty on ndvdual tree heght predcton of quakng aspen n boreal mxedwoods. Forest Ecology and Management 58: Huang, S., Yang, Y. and Meng, S.X.. Developng forest models from longtudnal data: a case study assessng the need to account for correlated and/or heterogeneous error structures under a nonlnear mxed model framework. Journal of Forest Plannng 6: 3. Huang, S., Yang, Y. and Atkn, D. 3. Populaton and plot-specfc ndvdual tree heght-dameter models for major Alberta tree speces. Forest Management Branch, Alberta Envronment & Sustanable Resource Development, Tech. Rept. Pub. T/6(3), Edmonton, Alberta. 8 p. Lndstrom, M. J. and Bates, D.M. 99. Nonlnear mxed effects models for repeated measures data. Bometrcs 46: Meng, S.X., and Huang, S. 9. Improved calbraton of nonlnear mxed-effects models demonstrated on a heght growth functon. Forest Scence 55: Meng, S.X. and Huang, S.. Incorporatng correlated error structure nto mxed forest growth models: predcton and nference mplcatons. Canadan Journal of Forest Research 4: Meng, S.X., Huang, S., Yang, Y., Trncado, G., and VanderSchaaf C.L. 9. Evaluaton of populaton-averaged and subject-specfc approaches for modelng the domnant or codomnant heght of lodgepole pne trees. Canadan Journal of Forest Research 39: Meng, S.X., Huang, S., VanderSchaaf, C.L., Yang, Y. and Trncado, G.. Accountng for seral correlaton and ts mpact on forecastng ablty of a fxed- and mxed-effects basal area model: a case study. European Journal of Forest Research 3: Pnhero, J.C. and Bates, D.M. 4. Mxed effects models n S and S-PLUS. Sprnger, New York. 58 p. Vonesh, E.F. and Chnchll, V.M Lnear and nonlnear models for the analyss of repeated measurements. Marcel Dekker, New York. 56 p. Yang, Y. and Huang, S. a. Estmatng a multlevel domnant heght-age model from nested data wth generalzed errors. Forest Scence 57: 6. Yang, Y. and Huang, S. b. Comparson of dfferent methods for fttng nonlnear mxed forest models and for makng predctons. Canadan Journal of Forest Research 4:

32 7. Appendces Four appendces are provded here to demonstrate and facltate the computatons assocated wth the FO and FOCE methods of the nonlnear mxed-effects modelng technque. Appendces and apply to the FO method. Appendces 3 and 4 apply to the FOCE method. Three plots from the model fttng data (plots m, m and m3) and three plots from the model applcaton data (plots v, v and v3) are used as examples n all four Appendces. They are lsted here, where year refers to measurement tme/year, and vol and age refer to black spruce total volume (m 3 /ha) and breast heght age (years), respectvely: Plotd Year Vol Age m m m m m m m m v v v v v v v v v v

33 Appendx. Generalzed Program for the Frst-Order Method Ths generalzed program for the FO method does not requre a user to know or to specfy the number of subjects (plots) n the data beforehand. It apples to any number of subjects. The program s desgned for experenced SAS/IML users. A more ntutve program s gven n Appendx. OPTIONS LS PS45; 3 4 data comb; 5 nput plotd $ YEAR vol age; 6 cards; NOTE: DATALINES 7 to 4 (for plots m, m, m3, v, v and v3); 5 ; 6 run; 7 8 proc sort datacomb; 9 by plotd; 3 run; 3 3 data wed3; 33 set comb; 34 by plotd; 35 j+; 36 f frst.plotd then do; +; j; end; 37 run; proc ml; 4 use wed3; 4 read all var {j} nto tobs; 4 read all var { age} nto age; 43 read all var {vol} nto vol; 44 fxp{ }; 45 covar{.64,.6674,.49, }; 46 dj(,,); 47 d[,]covar[]; 48 d[,]covar[]; 49 d[,]covar[]; 5 d[,]covar[3]; 5 scovar[4]; 5 bbfxp[,]; 53 bbb`; 54 tnmax(age[,]); 55 q; 56 bx{ }; 57 nnnrow(vol); 58 uj(tn,q,); 59 mcmax(tobs); 6 start sm (tn,bx,q,u,z,b,s,d,age,vol,nn,res,uv); 6 zj(nn,q,); 6 resj(nn,,); 63 uvj(tn,q,); 64 do k to tn; 65 z[:nn,].; 66 res[:nn,].; 67 do j to nn; 68 f age[j,]k then; 69 do; 9

34 7 agemage[j,]; 7 volmvol[j,]; 7 zb agemb[](-b[]agem)exp(-b[]agem); 73 z[j,]zb; 74 zb b[]exp(-b[]agem)log(agem)agemb[]; 75 z[j,]zb; 76 revolm-b[]agemb[]exp(-b[]agem); 77 res[j]re; 78 end; 79 end; 8 rz; 8 rr[loc(r[,]^.),]#bx; 8 wres; 83 ww[loc(w[,]^.),]; 84 mmnrow(w); 85 rrsi(mm); 86 uudr`inv(rdr`+rr)w; 87 ukuu`; 88 uv[k,]uk; 89 end; 9 fnsh sm; 9 run sm (tn,bx,q,u,z,b,s,d,age,vol,nn,res,uv); 9 ubuuv; 93 bfj(tn,q,); 94 do to tn; 95 bf[,]b[]; 96 bf[,]b[]; 97 end; 98 ububu bf; 99 cnm{u,u,b,b}; create rpm from ub[colnamecnm]; append from ub; qut; 3 4 data rpm; 5 set rpm; 6 _n_; 7 run; 8 9 proc sort datawed3; by ;run; proc sort datarpm; by ;run; data allp ; 3 merge wed3 rpm ; 4 by ; 5 zb ageb(-bage)exp(-bage); 6 zb bexp(-bage)log(age)ageb; 7 vol_fxbagebexp(-bage); 8 res_fxvol-vol_fx; 9 y_pred vol_fx+uzb+uzb; Y_resvol-y_pred; run; 3 proc prnt dataallp(obs8); 4 var plotd j age vol vol_fx zb zb u u y_pred y_res; 5 run; The prnt statement produces the followng results. They are lsted n Table 4 (FO method) for the model fttng data (plots m, m and m3) and Table 6 for the model applcaton data (plots v, v and v3). 3

35 Obs plotd j age vol vol_fx zb zb UI UI y_pred y_res m m m m m m m m v v v v v v v v v v

36 Appendx. Step-by-Step Program for the Frst-Order Method Ths step-by-step program for the FO method requres a user to know the exact number of subjects (plots) n the data beforehand. Ths number s specfed n lne 5. For the example data, the exact number of subjects s sx: three (m, m and m3) from the model fttng data and three (v, v and v3) from the model applcaton data. For any data set, f the exact number of subjects s known and specfed n lne 5, ths step-by-step program also apples to any number of subjects. Some readers may fnd ths step-by-step program s relatvely easer to follow than the generalzed program, although both programs produce the same results. OPTIONS LS PS45; 3 4 data comb; 5 nput plotd $ YEAR vol age; 6 cards; NOTE: DATALINES 7 to 4 (for plots m, m, m3, v, v and v3); 5 ; 6 run; 7 8 proc sort datacomb; 9 by plotd; 3 run; 3 3 data wed3; 33 set comb; 34 by plotd; 35 j+; 36 f frst.plotd then do; +; j; end; 37 run; data wed4; 4 set wed3; by plotd; 4 b.785; b.3396; 4 zb ageb(-bage)exp(-bage); 43 zb bexp(-bage)log(age)ageb; 44 vol_fxbagebexp(-bage); 45 res_fxvol-vol_fx; 46 run; flename random 'c:\_localdata\random.txt' ; 49 proc ml; 5 fle random; 5 use wed4; 5 do k to 6; 53 read all var {zb zb} nto Z where (k); 54 read all var {res_fx} nto RES where (k); 55 read all var {j} nto MM where (k); 56 ssnrow(mm); 57 R I(ss); 58 D { , }; 59 bdz`inv(z D Z` + R)RES; 6 btrans b`; 6 u btrans[,] ; 6 u btrans[,] ; 63 put k 5. + u 5. + u 5.; 3

37 64 end; closefle random ; 67 qut ; data prandom ; 7 nfle random; 7 nput u u ; 7 run ; data allp ; 75 merge wed4 Prandom ; 76 by ; 77 zb ageb(-bage)exp(-bage); 78 zb bexp(-bage)log(age)ageb; 79 vol_fxbagebexp(-bage); 8 res_fxvol-vol_fx; 8 y_pred vol_fx+uzb+uzb; 8 Y_resvol-y_pred; 83 run; 84 proc prnt dataallp(obs8); 85 var plotd j age vol vol_fx zb zb u u y_pred y_res ; 86 run; The prnt statement produces the same results as those from Appendx. They are lsted n Table 4 (FO method) for the model fttng data (plots m, m and m3) and Table 6 for the model applcaton data (plots v, v and v3). 33

38 Appendx 3. Generalzed Program for the Frst-Order Condtonal Expectaton Method Ths generalzed program for the FOCE method does not requre a user to know or to specfy the number of subjects (plots) n the data beforehand. It apples to any number of subjects. A convergence crteron of -7 s specfed n lne 97 for the teraton, whch s equvalent to a precson of.. Readers may wsh to choose a dfferent convergence crteron n some cases (such as -6 or -5 ), but are not recommended to go above -5. OPTIONS LS PS45; 3 4 data comb; 5 nput plotd $ YEAR vol age; 6 cards; NOTE: DATALINES 7 to 4 (for plots m, m, m3, v, v and v3); 5 ; 6 run; 7 8 proc sort datacomb; 9 by plotd; 3 run; 3 3 data wed3; 33 set comb; 34 by plotd; 35 j+; 36 f frst.plotd then do; +; j; end; 37 run; proc ml; 4 use wed3; 4 read all var {j} nto tobs; 4 read all var { age} nto age; 43 read all var {vol} nto vol; 44 covar{.878,.587,., }; 45 fxp{ }; 46 dj(,,); 47 d[,]covar[]; 48 d[,]covar[]; 49 d[,]covar[]; 5 d[,]covar[3]; 5 scovar[4]; 5 bbfxp[,]; 53 bbb`; 54 tnmax(age[,]); 55 q; 56 bx{ }; 57 nnnrow(vol); 58 uj(tn,q,); 59 start sm (tn,bx,q,u,z,b,s,d,age,vol,nn,res,uv); 6 zj(nn,q,); 6 resj(nn,,); 6 uvj(tn,q,); 63 do k to tn; 64 z[:nn,].; 65 res[:nn,].; 34

39 66 do j to nn; 67 f age[j,]k then; 68 do; 69 uu[k,]; 7 uu[k,]; 7 agemage[j,]; 7 volmvol[j,]; 73 zb agem(b[]+u)(-(b[]+u)agem)exp(-(b[]+u)agem); 74 z[j,]zb; 75 zb (b[]+u)exp(-(b[]+u)agem)log(agem)agem(b[]+u); 76 z[j,]zb; 77 revolm-(b[]+u)agem(b[]+u)exp(-(b[]+u)agem)+uzb+uzb; 78 res[j]re; 79 end; 8 end; 8 rz; 8 rr[loc(r[,]^.),]#bx; 83 wres; 84 ww[loc(w[,]^.),]; 85 mmnrow(w); 86 rrsi(mm); 87 uudr`inv(rdr`+rr)w; 88 ukuu`; 89 uv[k,]uk; 9 end; 9 fnsh sm; 9 run sm (tn,bx,q,u,z,b,s,d,age,vol,nn,res,uv); 93 buuv; 94 do k to tn; 95 dff; 96 dff; 97 epse-7; 98 do ter to untl ((dff<eps) & (dff<eps)); 99 run sm (tn,bx,q,u,z,b,s,d,age,vol,nn,res,uv); dffabs(uv-u); dffmax(dff[,]); dffmax(dff[,]); 3 uuv; 4 end; 5 end; 6 ubuu; 7 bfj(tn,q,); 8 do to tn; 9 bf[,]b[]; bf[,]b[]; end; ubbu ubu bf; 3 cnm{b,b,ub,ub,b,b}; 4 create rpm from ub[colnamecnm]; 5 append from ub; 6 qut; 7 8 data rpm; 9 set rpm; _n_; run; 3 proc sort datawed3;by ;run; 4 proc sort datarpm;by ;run; 35

40 5 6 data allx ; 7 merge wed3 rpm; 8 by ; 9 vol_fx (b )age(b )exp(-(b )age); 3 Y_pred (b+ub)age(b+ub)exp(-(b+ub)age); 3 Y_res vol - y_pred; 3 zb age(b+ub)(-(b+ub)age)exp(-(b+ub)age); 33 zb (b+ub)exp(-(b+ub)age)log(age)age(b+ub); 34 proc prnt dataallx(obs8); 35 var plotd j age vol vol_fx zb zb ub ub y_pred y_res ; 36 run; The prnt statement produces the followng results. They are lsted n Table 4 (FOCE method) for the model fttng data (plots m, m and m3) and Table 7 (Fnal teraton results) for the model applcaton data (plots v, v and v3). Obs plotd j age vol vol_fx zb zb UBI UBI Y_pred Y_res m m m m m m m m v v v v v v v v v v

41 Appendx 4. Step-by-Step Program for the Frst-Order Condtonal Expectaton Method Ths step-by-step program for the FOCE method corresponds to the three-step teraton procedure descrbed n Secton 3.3 and demonstrated n Secton 4.. OPTIONS LS PS45; 3 data comb; 4 nput plotd $ YEAR vol age; 5 cards; NOTE: DATALINES 6 to 3 (for plots m, m, m3, v, v and v3); 4 ; 5 run; 6 7 proc sort datacomb; 8 by plotd; 9 run; 3 3 data wed3; 3 set comb; 33 by plotd; 34 j+; 35 f frst.plotd then do; +; j; end; 36 run; data wed4; 39 set wed3; 4 by ; 4 b.879; b.34; 4 z ageb(-bage)exp(-bage); 43 z bexp(-bage)log(age)ageb; 44 vol_fx bagebexp(-bage); 45 res_fx vol - bagebexp(-bage); 46 run; flename random 'c:\_localdata\random.txt' ; 49 proc ml; 5 fle random; 5 use wed4; 5 do k to 6; 53 read all var {z z} nto Z where (k); 54 read all var {res_fx} nto RES where (k); 55 read all var {j} nto MM where (k); 56 ssnrow(mm); 57 R I(ss); 58 D { ,.587.}; 59 bdz` INV(Z D Z` + R)RES; 6 btrans b`; 6 u btrans[,] ; 6 u btrans[,] ; 63 put k 5. + u 5. + u 5.; 64 end; 65 closefle random ; 66 qut ; data prandom ; 69 nfle random; 7 nput u u; 37

42 7 run ; 7 73 data all ; 74 merge wed4 Prandom ; 75 by ; 76 proc prnt dataall; 77 var plotd j age vol vol_fx z z u u; 78 run; Ths prnt statement produces the step one results. Table 7 (Step computaton) lsts the results for plot v (shaded area) of the model applcaton data. Obs plotd j age vol vol_fx z z u u m m m m m m m m v v v v v v v v v v data wed4; 8 set all; 8 by ; 83 b.879; b.34; 84 z age(b+u)(-(b+u)age)exp(-(b+u)age); 85 z (b+u)exp(-(b+u)age)log(age)age(b+u); 86 vol_fx bagebexp(-bage); 87 y_pred (b+u)age(b+u)exp(-(b+u)age) ; 88 y_res vol- y_pred ; 89 res_c vol -(b+u)age(b+u)exp(-(b+u)age)+zu+zu ; 9 drop u u; 9 run; 9 93 flename random 'c:\_localdata\random.txt' ; 94 proc ml; 95 fle random; 96 use wed4; 97 do k to 6; 98 read all var {z z} nto Z where (k); 99 read all var {res_c} nto RES where (k); read all var {j} nto MM where (k); ssnrow(mm); R I(ss); 3 D { ,.587.}; 4 bdz` INV(Z D Z` + R)RES; 38

43 5 btrans b`; 6 u btrans[,] ; 7 u btrans[,] ; 8 put k 5. + u 5. + u 5.; 9 end; closefle random ; qut ; 3 data prandom; 4 nfle random; 5 nput u u; 6 run ; 7 8 data all ; 9 merge wed4 Prandom; by ; proc prnt dataall(obs8); var plotd j age vol vol_fx z z y_pred y_res u u ; 3 run; Ths prnt statement produces the step two results. Table 7 (Step computaton) lsts the results for plot v (shaded area) of the model applcaton data. Obs plotd j age vol vol_fx z z y_pred y_res u u m m m m m m m m v v v v v v v v v v data wed4; 6 set all; 7 by ; 8 b.879; b.34; 9 z age(b+u)(-(b+u)age)exp(-(b+u)age); 3 z (b+u)exp(-(b+u)age)log(age)age(b+u); 3 vol_fx bagebexp(-bage); 3 y_pred (b+u)age(b+u)exp(-(b+u)age) ; 33 y_res vol- y_pred ; 34 res_c vol -(b+u)age(b+u)exp(-(b+u)age)+zu+zu ; 35 drop u u; 36 run; flename random 'c:\_localdata\random.txt' ; 39

44 39 proc ml; 4 fle random; 4 use wed4; 4 do k to 6; 43 read all var {z z} nto Z where (k); 44 read all var {res_c} nto RES where (k); 45 read all var {j} nto MM where (k); 46 ssnrow(mm); 47 R I(ss); 48 D { ,.587.}; 49 bdz` INV(Z D Z` + R)RES; 5 btrans b`; 5 u btrans[,] ; 5 u btrans[,] ; 53 put k 5. + u 5. + u 5.; 54 end; 55 closefle random ; 56 qut ; data prandom; 59 nfle random; 6 nput u u; 6 run ; 6 63 data all ; 64 merge wed4 Prandom; 65 by ; 66 proc prnt dataall(obs8); 67 var plotd j age vol vol_fx z z y_pred y_res u u ; 68 run; Ths prnt statement produces the step three results. Table 7 (Step 3 computaton) lsts the results for plot v (shaded area) of the model applcaton data. Obs plotd j age vol vol_fx z z y_pred y_res u u m m m m m m m m v v v v v v v v v v Carry out the teraton manually by repeatedly runnng the program lnes 5 to 68 for 6 tmes. Ths produces the followng results: 4

45 Obs plotd j age vol vol_fx z z y_pred y_res u u m m m m m m m m v v v v v v v v v v The results obtaned after 6 teratons are equvalent to those obtaned from Appendx 3 (except occasonal decmal places for some ntermedate computatons). Notes:. Dependng on the data and model nvolved and the number of teratons carred out, some mnuscule dfferences at certan decmal places may occur between the fnal results from Appendces 3 and 4. Ths s caused prmarly by dfferent ways of computatons (e.g., the generalzed program n Appendx 3 uses a do untl statement wth a maxmum teraton of up to ).. In most cases, stable predctons for the random parameters can be acheved n less than teratons, so 6 teratons used n the above example are more than enough for most data. However, there are cases where the number of teratons may need to be ncreased (e.g., to, or even hgher) untl the fnal predctons show no practcal mprovement. Of course, the manual teraton of repeatedly runnng lnes 5 to 68 many tmes can be programmed usng a macro (avalable to nterested readers). 3. The step-by-step program for the FOCE method also apples to any number of subjects, provded that the number of subjects s specfed n lnes 5, 97 and 4. Some practtoners may fnd the step-bystep program s relatvely easer to follow than the generalzed program n Appendx 3. The step-bystep program also allows for easer dagnostcs when convergence s not acheved or does not exst for some specfc data. The non-convergence problem of the FOCE method, whch s often caused by the specfc data nvolved and the model specfcaton used, s an area that deserves further studes. 4

46 Appendx 5. Metrc Converson Chart cm.3937 n. m ft. m.936 yards ha.475 acres m ft m ft 3 m /ha ft /acre m 3 /ha 4.93 ft 3 /acre ha m km m km.637 mles km ha km.386 mles n..54 cm ft..348 m acre.447 ha ft.99 m ft 3.83 m 3 ft /acre.96 m /ha ft 3 /acre.6997 m 3 /ha mle.693 km mle.5898 km mle ha fbm ft. ft. n. fbm.3597 m 3 Mfbm foot board measure (fbm) Mfbm.3597 m 3 townshp 6 mles 6 mles 36 mle townshp km km km townshp ha m 3 log 33 board feet lumber (provncal average converson factor) Mfbm 4.3 m 3 log (provncal average converson factor) 4

47 43

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