FITTING WEIBULL AND LOGNORMAL DISTRIBUTIONS TO MEDIUM-DENSITY FIBERBOARD FIBER AND WOOD PARTICLE LENGTH John Z. Lu† Postdoctoral Research Associate Department of Wood Science and Engineering Oregon State University Corvallis, OR 97330 Charles J. Monlezun Associate Professor Department of Experimental Statistics Qinglin Wu† Roy O. Martin Sr. Professor School of Renewable Natural Resources and Quang V. Cao Professor School of Renewable Natural Resources Louisiana State University Agricultural Center Baton Rouge, LA 70803-6202 (Received July 2005) ABSTRACT Fiber lengths were analyzed for random samples of medium-density fiberboard (MDF) fibers and wood particles taken from eleven different populations. For six of the samples, the lognormal distribution fit the data, while the Weibull distribution did not. For three of the samples, the Weibull fit the data, while the lognormal did not. For two of the samples, both the lognormal and Weibull fit the data. Conclusions were based on hypothesis tests imposing a bound of 0.05 on the probability of making a Type I error for each test. Tests were based on large sample 95% nonparametric simultaneous confidence bands for the un- derlying cumulative distribution functions of the data. Keywords: Goodness of fit tests, MDF fiber, maximum likelihood estimation, non-parametric confi- dence bands, probability plots, statistical analyses. INTRODUCTION (MDF), hardboard, and wood-polymer compos- ites (Takahashi et al. 1979; Mark and Gillis Several studies have established the effect of 1983; Eckert et al. 1997; Marklund et al. 1998; wood fiber dimension and morphology on cer- Lee et al. 2001; Myers 2002; Huber et al. 2003). tain mechanical properties (e.g., bending However, suitable statistical functions that can strength and internal bonding strength) of wood be used to accurately describe wood fiber length fiber-based products such as paper, paper board, distribution over various fiber length regimes are insulation board, medium-density fiberboard still missing. For fiber dimension determination, the gen- † Member of SWST. eral measurement methods include microscopy, Wood and Fiber Science, 39(1), 2007, pp. 82 – 94 © 2007 by the Society of Wood Science and Technology Lu et al.—FITTING MDF FIBER LENGTH WITH WEIBULL/LOGNORMAL MODELS 83 projection, screen classification (e.g., Bauer- They concluded that the Erlang model provided McNett classifier), coulter counter and other an excellent fit to this data set. particle size analyzers (e.g., Kajaani FS-200, Failure to properly characterize an underlying Galai CIS-100), and image analyzers (Mark and distribution can lead to drastic errors in both Gillis 1983; Bentley et al. 1994; Carvalho et al. analytical and simulation models (Law and Kel- 1997). Particle size analyzers provide the most ton 2000). The Weibull and lognormal families popular automated techniques for fiber length of distributions have been used with great suc- measurement in the pulp and paper industry. cess to model positively (or right) skewed ran- Compared with other automated techniques, im- dom processes when the associated random vari- age analyzers have high accuracy, fast speed, able is bounded below (right skewed refers to a and high reliability for determining fiber length, process for which a large random sample would width, surface area, and coarseness (Mark and produce a histogram with the right or upper tail Gillis 1983). Image analyzers are, perhaps, most “stretched out” more than the left or lower tail). useful when large numbers of fibers must be Most of the modeling has been empirical in na- accurately detected and measured. ture, although mechanistically the Weibull arises Rotation age, especially juvenile and adult naturally as the minimum observation for a cer- woods (Haygreen and Bowyer 1994), and tain class of distributions, while the product of a growth-accelerating treatments such as fertiliza- sufficient number of positive random variables, tion (Bannan 1967) and irrigation (Klem 1968; none of which dominates the others, is lognor- Murphey and Bowier 1975) have an effect upon mal by the Central Limit Theorem (see Bury average fiber dimensional sizes in stem. Indi- 1975; Johnson et al. 1994; Kalbfleisch and Pren- vidual wood fibers are different in shape and tice 2002). The Weibull distribution may also size for different tree species. Even in the same provide the added flexibility required for an ac- tree species and the same stem, individual fibers curate analysis when theoretical considerations may have different lengths for different loca- indicate that an exponential distribution may be tions (e.g., height, distance from pith, early and adequate (Johnson et al. 1994). The Weibull and late woods, and heart- and sapwoods). Therefore, lognormal distributions may be preferable to fiber dimensional size (e.g., length, diameter or other distributions since both families assume a width, and surface area) in wood is inherently wide variety of shapes, and analyses are trac- continuous in distribution. table for both the Weibull and lognormal. This is As early as 1972, Tasman (1972) analyzed not surprising, since the lognormal is a function fiber lengths with a Bauer-McNett classifier. He of the normal, which has been studied exten- discovered that the distribution of fiber fractions sively, while the cumulative distribution for the is approximately normal. It was also reported Weibull is available in closed form; furthermore, that fiber length distribution for papermaking the natural logarithm of each produces a location furnish is approximately lognormal (Yan 1975; and scale parameter family of distributions (see Dodson 1992; Kropholler and Sampson 2001). SAS Institute Inc. 2002; Meeker and Escobar However, the data were simply fit with the log- 1998). In addition, the Weibull and lognormal normal distribution in these publications. The are “complementary” families of distributions in details on statistical analyses are not available. the following sense. When fitting both distribu- Mark and Gillis (1983) used the Erlang family tions to the same data set by the same method, of distributions (a subset of the Gamma family) the fitted lognormal pdf invariably has a heavier to describe the lengths of fibers from a bleached right hand tail than the fitted Weibull pdf, while softwood kraft sheet. They applied chi-square the fitted Weibull pdf takes on larger values in goodness of fit tests to both the Erlang probabil- the vicinity of zero (Law and Kelton 2000; ity density function (pdf) and the Erlang cumu- Meeker and Escobar 1998). Two situations are lative distribution function (cdf). The resulting then possible. Either one or both adequately fit P-values were 0.870 and 0.947, respectively. the entire data set. If neither is satisfactory for 84 WOOD AND FIBER SCIENCE, JANUARY 2007, V. 39(1) the entire data set, then it is plausible that one of letters A through K (Table 1). From each pack- the two would fit large data values, the other age, at least 500 fibers or particles were ran- would fit small data values, and both would fit domly sampled, oven-dried at 80ºC for 24 h, and the middle values. In either situation, by fitting sealed in plastic bags. Exact sample sizes for all both the Weibull and lognormal distributions, packages appear in Table 1. Fiber moisture con- the process generating the data can be success- tent was kept between three and five percent. fully modeled across the entire spectrum of the From each plastic bag, the fibers were removed, data. placed evenly on a glass dish, and clearly fo- The MDF fibers used in this study were cre- cused under the microscope. The optical lighter ated by a thermal-mechanical grinding process. was adjusted for better picture quality. With the The lengths of wood fibers obtained from such a Spot Advance imaging software (Diagnostic In- process are bounded below by zero, since no struments, Sterling Heights, MI), the image was fiber length can be negative. Such a process will then recorded with the digital camera, and the produce extremely short fibers (having lengths length of each fiber was measured in millimeters close to zero). Most of the fibers would have using the Image-Pro Plus software (Media Cy- lengths clustering around a positive value. The bernetics, Inc., Silver Spring, MD). majority of naturally occurring long fibers would be crushed into shorter ones by the grind- PROBABILITY DISTRIBUTION MODELS ing process. Some long fibers would survive, but The histograms of all wood fiber length data relatively few when compared to fibers of sets of this study were skewed to the right. The shorter length, resulting in a long right hand tail histograms for four of the data sets are represen- for the distribution of lengths. Thus, the distri- tative of the shapes for all histograms and are bution of fiber lengths should be continuous, given in Fig. 1. Because of the (right) skewness bounded below by zero and skewed to the right. of the histograms, the Weibull and lognormal For this reason, Weibull and lognormal distribu- distributions were chosen to fit the data sets of tions were fit to each of the data sets of this this study. study. The goodness of fit was determined for A random variable X follows a Weibull dis- each. Details of all statistical analyses are pre- tribution if and only if the cdf for X is sented. 1 − exp͓−͑xր␤͒␥͔ x Ն 0 G͑x; ␥, ␤͒ = ͭ EXPERIMENTAL 0 x Ͻ 0 An imaging system was used for the determi- (1) nation of wood fiber length. This system con- where both ␥ and ␤ are positive numbers. The sisted of a Leica MZFIII microscope (Leica Mi- crosystems GmbH, Wetzlar, Germany), a CCD TABLE 1. Summary of fiber information. digital camera (Diagnostic Instruments, Sterling Number of Heights, MI), a ␷-Lux 1000 optical lighter Package Fiber Fiber-based Fiber fibers per a (Volpi MFG. USH Co., Auburn, NY), a RT designation source product type sample SP402-115 power supply (Diagnostic Instru- A 1 MDF Core 503 B 1 MDF Face 505 ments, Sterling Heights, MI), and a computer.