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J OURNAL OF FOREST PRODUCTS & INDUSTRIES, 2013, 2(2), 31-39 ISSN:2325 – 4513(PRINT) ISSN 2325 - 453X (ONLINE )

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Height-Diameter Prediction Models for Some Utilitarian Natural Tree Species El Mamoun H. Osman1 , El Zein A. Idris 2 and El Mugira M. Ibrahim3 1

College of Natural Resources and Environmental Studies-Department of Forestry, University of Bahri. Khartoum- Sudan E mail: elmamounosman@yahoo.co.uk Telephone: +249911347139 2 College of Natural Resources and Environmental Studies-Department of Forestry, University of Bahri. Khartoum- Sudan E mail: elzeinadam@yahoo.com Telephone: +249911270310 3 College of Natural Resources and Environmental Studies-Department of Forestry, University of Bahri. Khartoum- Sudan E mail: elmugheira1984@yahoo.com Telephone: +249911986534 (Received: January 23, 2013; Accepted: February 07, 2013)  Abstract— This paper is a continuation of a previous paper presented by the same group in 2012. It developed and evaluated the performance of seventeen functional models that predict total tree height from diameter at breast height of five selected utilitarian natural tree species common to savannah region of the Sudan. These include Combretum micranthum, Lannea fruticosa, Sterculia setigera, Terminalia brownii, and Terminalia laxiflora. A total of 2579 tree heights and corresponding diameters at breast heights were extracted from a working plan inventory data of a natural reserved forest in the Blue Nile state. The model goodness of fit were evaluated in terms of adjusted coefficient of determination (Ra2), root mean squared error (RMSE), Akaike’s information criterion (AIC), homogeneity of the residuals and significance of the regression parameters. The results of the study indicated that the height-diameter relationship can best be described by non-linear models. Out of the evaluated models, at least 15 models were found to give reasonable results for each species with Ra2 range of 0.20 – 0.87 and RMSE of 0.8 – 4.2 m. The number of free model parameters was found to have little or no significant influence on the ranking of the model performance. The best five models for each species and their ranking in terms of goodness of fit differ from species to another with each species having different combination of models. Comparison of the range of data for the five studied species suggested that representation of various diameter-height classes had significant influence on the accuracy of prediction outside the range of the fitted data. This implies that application of the selected models is only useful at local stand level or at best in similar biological and stand structure conditions.

Index Terms— Utilitarian trees, natural forest height–diameter relationship, non-linear regression models.

I. INTRODUCTION

T

he relationship between tree diameter at breast height (dbh) and total height is a structural characteristic of a tree that describes key elements of stem form, and thus the volume of the harvestable stem. The diameter to height (d-h) relationship also affects product quality, as it influences wood structure such as lignin and cellulose content, and thus important properties of the stem (e.g., stiffness) [1]. The dbh and total height are the commonly measured variables in forest inventories as they are frequently required for both routine forest management activities and for research purposes. As such, accurate data on these two variables is quite important especially for reliable future management plans. Unlike dbh, total height is less frequently used for construction or application of forest models because measurement of dbh is more cost effective, easy and accurate than total height [2], [3]. Routine forest management usually requires updating forest growth parameters at regular intervals to refresh the future plans and decisions as affected by market and environmental variables. Investment in forest inventory operations usually weighed against the tangible benefits gained from the information obtained i.e. the direct economic return from the concerned forest. Natural forests in Sudan are generally known to be under the public ownership supervised by the local government authorities. The control over such types of forests and protective measures offered to them are generally poor due to the absence of up-to-date growth information. Any effort to ease and reduce the cost of forest inventories in such cases would be highly appreciated and could save the existence of a number of very valuable indigenous tree species in these forests. The main objective of this study is therefore to select the most


J OURNAL OF FOREST PRODUCTS & INDUSTRIES, 2013, 2(2), 31-39 ISSN:2325–4513(PRINT) ISSN 2325 - 453X (ONLINE ) 32 suitable model that could efficiently estimate tree heights without incurring unaffordable costs and time. Height-diameter relationships are used to estimate the heights of trees measured for their diameter at breast height. Such relationship describes the correlation between height and diameter of the trees in a stand on a given date and can be represented by a linear or non-linear mathematical model. In forest inventory designs diameter at breast height is measured for all trees within sample plots, while height is measured for only some selected trees, normally the dominant ones in terms of their dbh. However, for height-diameter models, more care is needed and a representative sample of accurately measured total-height is used as the response variable and dbh as the predictor variable [3]. A major difficulty in modelling the height-diameter relationship is the large number of variables influencing it and thus hindering the construction of generic models based on empirical methods such as linear and nonlinear regression [4]. Height-diameter equations can either be used for local application or they can have a more generalised use. Equations dependant on only tree diameters, without considering other stand variables, are only applicable locally to the stand where the height-diameter data were gathered. A large number of local tree height-diameter equations have been developed and/or validated and reported in the literature (e.g. [5], [6], [7], [3]). The generalized height-diameter equations, on the other hand, are normally functions of tree diameter in addition to other stand variables, and as such can be applied in similar conditions at the regional level [8]. Numerous examples of generalized height-diameter equations were tested. In addition to dbh as independent variable, reference [7] used Crown diameter, [9] used competition factor (mean neighbouring tree distance, frequency of the neighbouring tree and position of the crown), [10] used stand age, site index and altitude, [8] used dominant height, dominant under bark diameter at breast height, [11] used density and developmental effects, [12],[13] used trees/ha and basal area, [4] used tree age and genetic material, and [14] used stocking. Statistical modelling has to balance simplicity (i.e., fewer parameters in a model, lower variability in the predicted response, but with more modelling bias) against complexity (i.e., more parameters in a model, higher variability in the predicted response, but with smaller modelling bias). According to reference [15] Statistical model selection criteria have to seek a proper balance between overfitting (i.e., a model with too many parameters, more than actually needed) and underfitting (i.e., a model with too few parameters, not capturing the right signal).

highest Ra2 will have the lowest RMSE, and the Ra2 could be used as a guide [21].

II. MATERIALS AND METHODS The data for this study was extracted from an inventory carried out in El Nour Natural Forest Reserve for the purpose of updating its management plan. The forest is located in the Blue Nile State, Sudan, between longitudes 11o 48/ 19// N and 11o 53/ 30// N and latitudes 34o 28/ 47// E and 34o 32/ 35// E. The forest hosts variety of broad leaved hardwood tree species of various uses and functions for the local communities in the area. The selection of the species for the study was mainly based on their utilization preference (Table 1) for the local people around the forest [16]. For this study particular the selected tree species were Combretum micranthum, Lannea fruticosa, Sterculia setigera, Terminalia brownii and Terminalia laxiflora. The data for the selected species consists of dbh >7cm of all sampled trees and total tree heights of the dominant ones (Table 2). At the first stage, non-linear relationship between height and diameter was confirmed with a scattered plot diagram of height against dbh using polynomial equation of the second order. In a previous study, Osman et al (2012)[17] tested twenty two different non-linear models to fit the height-diameter relationship of some natural tree species in the same forest. Their results ended up with potential seventeen models that having the capability of yielding reasonable predictions. Accordingly this study has adopted the same set of models as described in Table 3. The detailed description of these models and their reference source were given by reference [2], [3] and [10]. All these models ensure that the total height equals breast height (1.3 m) when dbh = 0. The fitting of the candidate models were carried out using “DataFit-9” statistical package of the Oakadale Engineering. The evaluation of the models was based on Akaike’s information criterion (AIC), root mean squared error (RMSE), adjusted coefficient of determination (Ra2), significance of parameters estimate (P<0.05) and on numerical and graphical analyses of the residuals. Model resulting in the largest Ra2, least RMSE, and smallest values of AIC and average bias was selected as the best model for the selected tree species. The formula for AIC is (-2) times the maximized log likelihood function plus 2 times the number of free parameters, with the former term describing lack of fit and the latter penalizing the number of free parameters in the model [15] and [18]. According to reference [19] AIC can also be calculated as indicated in equation 1, while RMSE is calculated as indicated in equation 2 [20]. For the model performance, a candidate showing increased values of Ra2 and decreased values of RMSE and AIC would be regarded as the best model. When comparing regression models that use the same dependent variable and the same estimation period, the RMSE goes down as Ra2 goes up. Hence, the model with the (1) (2) Where: n = Number of data points (observations)


J OURNAL OF FOREST PRODUCTS & INDUSTRIES, 2013, 2(2), 31-39 ISSN:2325â&#x20AC;&#x201C;4513(PRINT) ISSN 2325 - 453X (ONLINE ) 33 RSS = Residual sums of squares K= Number of parameters in the model

P = Number of regression coefficients (Reg. Co)

Table1 local uses of the selected tree species

Tree species

Local uses

Combretum micranthum (CM)

fuel wood, charcoal, timber, edible seeds, bark is used for folk medicine, sapling for hut construction, basket manufacture and furniture, the inner bark fibres for binding and plaiting Lannea fruticosa (LF) Firewood and charcoal, and local buildings. Sterculia setigera (SS) Exudates a highly water-soluble gum, locally and used for cooking and binding sauces. The Bark and leaves are used for folk medicinal purposes. Bark fibre are used for mats and ropes. Terminalia brownii (TB) Fire wood, charcoal, local buildings, beams and rafters, fire wood and charcoal. The bark and fruits contain 19% tannin Terminalia laxiflora (TL) Fire wood, charcoal, railway sleepers, general construction, local furniture, joinery and turnery. Table 2 Species diameter-height characteristics Diameter at Breast Height (cm) Total Height (m) Species n Max Min Range Mean Stdev n Max Min Range Mean Stdev 261 66.2 11 53.2 37.22 13.46 162 16 7 9 11.92 1.83 CM 190 60 9.25 50.75 31.51 11.21 190 18 5 13 11.69 2.56 LF 1389 153 10.5 142.5 59.35 17.27 1389 19.7 7.7 12 12.86 1.86 SS 614 88 25 63 43.71 7.56 614 16.7 7 9.7 12.41 1.44 TB 224 82 17 65 43.12 9.09 224 17 9 8 12.62 1.53 TL Table 3 Regression models and reference source Model Formula Model Formula M1

M10

M2

M11

M3

M12

M4

M13

M5

M14

M6

M15

M7

M16

M8

M17

M9 H = total height; a, b, c = Reg. Co. and e = exponent III. RESULTS AND DISCUSSION Tables 4 - 8 list, in descending order, the fit statistics of the five best models for the five selected species, while Figures 1-7 display the corresponding height curves predicted by the same


J OURNAL OF FOREST PRODUCTS & INDUSTRIES, 2013, 2(2), 31-39 ISSN:2325â&#x20AC;&#x201C;4513(PRINT) ISSN 2325 - 453X (ONLINE ) 34 models and residual plots of some selected models. In general terms, at least twelve out of the seventeen fitted models were found to give satisfactory results with R2 range of 0.69 - 0.87. The rest of the models were either failed to give any results, or produced some negative values of height prediction, or produced a height value greater than 1.3 m at

was great variation between the values of the regression parameters of the same model across the species. In terms of prediction error, the entire range of the successful models yielded very low height prediction standard error values of less than one meter. Such results confirm the positive relationship between the dbh and the total tree height and the possibility of predicting tree height from the dbh ([2], [3], [7], [17]).

zero dbh. The successful models showed different rankings of performance with different type of tree species. Similarly, there

Code E22 E1 E11 E18 E12

Table 4 Regression parameter estimates and fit statistics for Combretum micranthum Model (n =162) Regression coefficients Ra2 RMSE S. Error R2 AIC Coefficient Value S. Error t-ratio 0.3260 0.1765 1.847 a 0.3667 0.0321 11.424 b 0.8081 0.8071 0.8047 0.8081 -66.0801 -0.0201 0.0632 -0.318 c 2.6055 0.1558 16.720 a 0.8060 0.8069 0.8057 0.8060 -65.8722 0.3935 0.0163 24.187 b 91.951 197.60 0.465 a 0.8139 0.8043 0.8019 0.8139 -63.7574 4.0674 1.5511 2.622 b 0.1781 0.1738 1.025 c 1.7735 0.1375 12.896 a 0.8150 0.8026 0.8013 0.8150 -62.3112 1.4035 0.0595 23.592 b 17.011 0.4602 36.964 a 0.8437 0.7884 0.7871 0.8437 -51.0761 20.392 1.5432 13.214 b

Prob. (t) 0.0667 0.0000 0.7507 0.0000 0.0000 0.6423 0.0096 0.3070 0.0000 0.0000 0.0000 0.0000

Table 5 Regression parameter estimates and fit statistics for Lannea fruticosa Model (n =190) Regression coefficients R2 Ra2 RMSE AIC Coefficient Value S. Error t-ratio 0.6696 0.0576 11.6293 a 0.8664 0.8657 0.9177 -28.6487 2.2154 0.0676 32.7683 b 28.467 1.4220 20.0191 a 0.8655 0.8648 0.9209 -27.3190 52.622 4.3369 12.1335 b 1.2436 0.0829 15.0092 a 0.8641 0.8634 0.9257 -25.3450 0.6189 0.0187 33.0356 b 22.459 0.6447 34.8343 a 0.8589 0.8582 0.9431 -18.2799 13.967 0.6672 20.9319 b 2.9471 0.1000 29.4636 a 0.8589 0.8582 0.9431 -18.2799 0.2110 0.0030 69.6686 b

Prob. (t) 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

Code S. Error E18

0.9177

E12

0.9209

E1

0.9257

E6

0.9431

E7

0.9431

In terms of individual models performance, Models 1, 9 and 15 (M1, M9 and M15) were found to alternate in their ranking for the position of the best three height predictors for three of the tested species. The same models were also appeared among the best five models for all species except for Terminalia brownii where completely new set of models occupied these positions. M5 was also found to be among the best five predictors for three of the five species, while M13 was found in the list of only two species and M2, M3, M4, M7, M13, M14 and M17 appeared only once in different species and positions. Species wise M1 was found to be the best predictor model for S. setigera and T. Laxiflora, M4 for T. Brownii, M15 for L. Fruticosa and M17 for C. Micranthum.

It is clear from the results that within the best five models for all the species three models (M3, M8 and M17) have three regression coefficients (a, b, c), while the rest have only two (a and b). All the parameters for the two-parameter models were statistically significant (p<0.0001) compared to those of M3 (only parameter b was statistically significant), M8 (all regression parameters were not significant) and M17 (parameter b was statistically insignificant). However, in all the cases where M3, 8 and M17 occupied a better ranking position than the two-parameter models, there was no significant difference in their values of RMSE, Ra2 or AIC compared to those of the models below them. In addition, some of the regression


J OURNAL OF FOREST PRODUCTS & INDUSTRIES, 2013, 2(2), 31-39 ISSN:2325â&#x20AC;&#x201C;4513(PRINT) ISSN 2325 - 453X (ONLINE ) 35 parameters of these three models (M3, M8 and M17) were the unnecessary overfitting. According to [20] preference found to yield large values of standard errors (26% â&#x20AC;&#x201C; 216%). should go for the model with fewer parameters, other things This result confirms what had been concluded by reference [17] being approximately equal. and will give preference to the two-parameter models to avoid Table 6 Regression parameter estimates and fit statistics for Sterculia setigera Model (n =1389) Regression coefficients Ra2 RMSE Code S. Error R2 AIC Coefficient Value S. Error t-ratio Prob. (t) 1.5778 0.0444 35.5335 0.0000 a E1 0.8424 0.7989 0.7988 0.8424 -472.3288 0.4910 0.0068 72.1326 0.0000 b 0.6854 0.0284 24.1301 0.0000 a E18 0.8618 0.7896 0.7894 0.8618 -409.3224 2.0152 0.0292 69.0618 0.0000 b 23.250 0.3454 67.3089 0.0000 a E12 0.8908 0.7752 0.7750 0.8908 -317.3078 57.301 1.7531 32.6855 0.0000 b 4.0178 0.0686 58.5918 0.0000 a E7 0.9361 0.7517 0.7516 0.9361 -179.5384 0.2224 0.0012 192.580 0.0000 b 4.0178 0.0686 58.5918 0.0000 a E13 0.9361 0.7517 0.7516 0.9361 -179.5384 0.2224 0.0012 192.580 0.0000 b Table 7 Regression parameter estimates and fit statistics for Terminalia brownie Model (n =614) Regression coefficients Ra2 RMSE Code S. Error R2 AIC Coefficient Value S. Error t-ratio Prob. (t) 41.9959 10.773 3.8983 0.0001 a 0.6855 0.4738 1.4468 0.1485 b E4 0.7900 0.7016 0.7006 0.7900 -286.4169 0.0512 0.0052 9.8796 0.0000 c 21.8351 0.3890 56.132 0.0000 a E2 0.7897 0.7014 0.7009 0.7897 -285.9932 -29.0106 0.7749 -37.438 0.0000 b 3.0835 0.0178 173.083 0.0000 a E17 0.7897 0.7014 0.7009 0.7897 -285.9932 -29.0106 0.7749 -37.438 0.0000 b 21.9971 0.3960 55.5548 0.0000 a E16 0.7897 0.7014 0.7009 0.7897 -285.9023 29.6765 0.7922 37.4595 0.0000 b 3.0984 0.0182 170.352 0.0000 a E10 0.7898 0.7013 0.7008 0.7898 -285.7785 -30.3565 0.8100 -37.479 0.0000 b Table 8 Regression parameter estimates and fit statistics for Terminalia laxiflora Model (n =224) Code S. Error

R2

Ra2

RMSE

AIC

E1

0.7934 0.7308 0.7295 0.7934 -99.7014

E18

0.8056 0.7224 0.7211 0.8056 -92.8405

E12

0.8166 0.7147 0.7135 0.8166 -86.7588

E7

0.8476 0.6927 0.6913 0.8476 -70.0826

E13

0.8476 0.6927 0.6913 0.8476 -70.0826

Regression coefficients Coefficient Value S. Error t-ratio Prob. (t) a

1.4934 0.1282

11.6447 0.0000

b

0.5396 0.0226

23.8487 0.0000

a

0.7337 0.0877

8.3632

b

2.0612 0.0894

23.0638 0.0000

a

25.110 1.3368

18.7829 0.0000

b

51.535 5.0845

10.1359 0.0000

a

3.3108 0.1615

20.5050 0.0000

b

0.2184 0.0037

58.7325 0.0000

a

3.3108 0.1615

20.5050 0.0000

b

0.2184 0.0037

58.7325 0.0000

0.0000


J OURNAL OF FOREST PRODUCTS & INDUSTRIES, 2013, 2(2), 31-39 ISSN:2325â&#x20AC;&#x201C;4513(PRINT) ISSN 2325 - 453X (ONLINE ) 36

Fig. 1. Combretum micranthum predicted heights beyond the observed data

Fig. 2. Lannea fruticosa predicted heights beyond the observed data

Fig. 3.Sterculia setigera predicted heights beyond the observed data


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Fig.4.Terminalia brownii predicted heights beyond the observed data

Fig. 5. Terminalia laxiflora predicted heights beyond the observed data

Fig 6. Residual plot of some selected models


J OURNAL OF FOREST PRODUCTS & INDUSTRIES, 2013, 2(2), 31-39 ISSN:2325–4513(PRINT) ISSN 2325 - 453X (ONLINE ) 38

Fig 7.Residual plot of some selected models Figures 1 – 5 display graphical representation of the predicted height curves given by top five successful models for each tree species. Examination of these height curves reveals that all the successful models performed quite well within the range of the observed data. Prediction outside this range produced different responses by different models and species especially at the extreme ends of the curves. However, the best sets of models were the predictors of T. brownii where almost all the models overlap as one curve. Such trend could be attributed to the quality of the sample data or to some unexplained effects that need further investigation to improve the performance of prediction for the other species. IV. CONCLUSIONS In general, the overall results of this study confirm what has been concluded by reference [17]. The height-diameter model developed in this study gave reasonably precise estimates of tree heights and could be used to predict the height of the species under consideration. Inclusion of additional prediction parameter to some models was found to yield insignificant effect on the quality of their prediction. The results of the study indicated that a successful predictor model for a given species might not be so for other species growing in the same forest under similar conditions. The study also suggested that prediction outside the range of fitting data might give misleading or unsatisfactory results. Another fact revealed by this study was that multiple criteria for examining the quality of fit would be much better to cast any doubts concerning the quality of the tested models. In General, M1, M3, M9 and M15 and M17 are recommended for the investigated species and their prediction quality could further be improved by the inclusion of other stand and growth variables. REFERENCES [1] Kroon, J., Andersson, B. and Mullin, T. J. (2008). Genetic variation in the diameter–height relationship in Scots pine (Pinus sylvestris). Canadian Journal of Forest Research, 2008, 38(6): 1493-1503, 10.1139/X07-233. [2] Moore, J. A., Zhang, L. and Stuck, D. (1996). Height-Diameter Equations for Ten Tree Species in the Inland Northwest WJAF 11(4).

[3] Sharma, R.P. (2009).Modelling height-diameter relationship for Chir pine trees. Banko Janakari, Vol. 19, No. 2. [4] Guimarães, M. A. M., Calegário,N., Carvalho, L. M. T. and Trugilho, P. F. (2009). Height-diameter models in forestry with inclusion of covariates. Cerne, Lavras, v. 15, n. 3, p. 313-321. [5] Nowak, D. J. (1990). Height-diameter relations of Maple street trees. Journal of Arboriculture 16(9): 231-235. [6] Walter Zucchini, W., Schmidt, M. and Klaus, K. V. (2001). A Model for the Diameter-Height Distribution in an Uneven-Aged Beech Forest and a Method to Assess the Fit of Such Models. Silva Fennica 35(2), 169-183. [7] Avsar M. D. (2004). The relationship between Diameter at breast height, tree height and crown diameter in Calabrain Pinus (Pinus brutia Ten) of Baskonus Mountain, Kahramanmaras, Turkey. Journal of Biological Science 4(4): 437-440, 2004. [8]

González, S. M., Cañellas, I Montero, G (2007). Generalized height-diameter and crown diameter prediction models for cork oak forests in Spain. Investigación Agraria: Sistemas y Recursos Forestales, 16(1), 76-88. ISSN: 1131-7965

[9] Dauda. T. O., Ojo, L. O. and Nokoe, S. K (2004).Unexplained relationships of height-diameter of three species in a tropical forest. Global Nest: the Int. J. Vol 6, No 3, pp 196-204. [10] Zhao, W., Mason, E.G., and Brown, J. (2006) Modelling height-diameter relationships of Pinus radiata plantations in Canterbury, New Zealand, New Zealand Journal of Forestry 51 (1): 23-27. [11] Newton, P. F. and Amponsah, I. G. (2007). Comparative evaluation of five height–diameter models developed for black spruce and jack pine stand-types in terms of goodness-of-fit, lack-of-fit and predictive ability. Forest Ecology and Management, 247(1-3), 149-166. [12] Sharma, M. and Parton, J. (2007). Height–diameter equations for boreal tree species in Ontario using a mixed-effects modelling approach. Forest Ecology and Management 249 (2007) 187–198. [13] Saunders, M. R. and Wagner, R. G. (2008). Height-diameter models with random coefficients and site variables for tree species of Central Maine. Annals of Forest Science - ANN FOR SCI , vol. 65, no. 2, pp. 203-203, 2008. [14] Vanclay, J. K. (2009). Tree diameter, height and stocking in even-aged forests. Ann. For. Sci. 66 (2009) 702. [15] Yang, H. and Bozdogan, H. (2011). Model Selection with Information Complexity in Multiple Linear Regression Modeling. Multiple Linear Regression Viewpoints, 2011, Vol. 37(2).


J OURNAL OF FOREST PRODUCTS & INDUSTRIES, 2013, 2(2), 31-39 ISSN:2325â&#x20AC;&#x201C;4513(PRINT) ISSN 2325 - 453X (ONLINE ) 39 [16] Osman E. h. and El Zain A. Idris (2012). Species Dynamics and Potential Disturbances in El Nour Natural Forest Reserve, Sudan. journal of Forest products and industries, 2012, 1(2), pp10-20, ISSN:2166. [17] Osman, E. H., Idris, E. A. And Ibrahim, E. M. (2013). Modelling Height-Diameter Relationships of Selected Economically Important Natural Forests Species. Journal of forest Products and industries, 2(1), pp34-42 ISSN: 2325 .

[19] Motulsky, H. J. and Christopoulos, A. (2003). fitting models to biological data using linear and non-linear regression. a practical guide to curve fitting. GraphPad software Inc., San Diago, CA 2 nd ed. [20] Beal, D. J. (2005). SAS Code to Select the Best Multiple Linear Regression Model for Multivariate Data Using Information Criteria. Paper SA01_05. [21]

[18] Burnham, K. P., and Anderson, D.R. (2004), "Multimodel inference: understanding AIC and BIC in Model Selection", Sociological Methods and Research, 33: 261-304.

Nau, R. F (2005). How to http://people.duke.edu/~rnau/compare.htm.

compare

models.

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