Some Aspects of the Sampling Distribution of the Apportionment Index and Related Inference

As customer-oriented production strategies have gained ground in the sawmill industry, proper measurement of the fit between the log demand and log output distributions has become of crucial importance. The prevailing means of measuring the outcome is the so-called Apportionment Index (AI), which is calculated from the relative proportions of the observed and required distributions. Although some statistical properties of the AI have recently been examined and alternative means of measuring the bucking outcome have been suggested, properties of the sampling distribution of the AI have not yet been widely studied. In this article we examine the asymptotic sampling distribution for the AI by assuming a multinomial distribution for the outcome. Our results are based mainly on large-sample normal approximations. Also some studies of the determination of the number of logs needed to obtain a specified level of accuracy of the AI have been carried out.


Introduction
The prevailing Nordic harvesting system is to convert tree stems into smaller logs immediately at the harvesting site.Modern cut-to-length (CTL) harvesters are equipped with high-class measuring and computing systems which make Some Aspects of the Sampling Distribution of the Apportionment Index and Related Inference ented production of well-defined products.It has become more and more important not only to supply the sawmill with a sufficient number of logs at minimum cost, but also to ensure that the raw material meets the requirements of the sawmill as regards to length, diameter and quality distribution of logs (Kivinen 2004).This in turn, has made proper assessment of the goodness of the bucking outcome of crucial importance.
In general, as listed by Kivinen et al. (2005), there are two main situations where the agreement between the distribution of logs demanded by the sawmill (demand distribution) and the actual outcome (output) distribution of logs is of particular interest.These are 1) the standard pre-harvest planning procedure where bucking simulators are commonly used to determine the best available stands for prevailing customer orders, and 2) the postharvest analysis where it may be desirable to know, for example, how various harvesters have succeeded in meeting a certain demand distribution or to determine whether there are any significant differences between various wood suppliers.A proper measure for evaluating the bucking outcome also provides information on how to adjust the bucking instructions to meet the desired log distribution.According to Vuorenpää et al. (1997) the large sawmills should monitor the progress of the work of the harvesting machines on the daily basis to assure that the requirements of the sawmill are adequately satisfied.
The most widely used measure for assessing the agreement between the log demand and the log output distributions in Scandinavia is Apportionment Index (AI) or Apportionment Degree (see Section 2.1 for mathematical formulation).The AI measures the output of the actual harvesting operation by comparing the relative proportions of the outcome and target distributions.The Apportionment Index has gained ground especially by merit of its simplicity, ready interpretability and ease of use.The measure has been criticized mainly as being too crude, since, for example, it attributes the same weight to all log classes.The price-weighted version of the AI was proposed by Kivinen et al. (2005) and Nummi et al. (2005).Penalty-based variants of the traditional measure are suggested in papers by Kirkkala et al. (2000), Weijo (2000) and Malinen and Palander (2004).Sinha et al. (2005b) generalized the measure and introduced the family of Apportionment Indices.However, the Apportionment Index and its derivatives have been used mainly as a descriptive device and very little is known of their statistical properties.While there have been attempts to understand this index (Nummi et al. 2005 andSinha et al. 2005a), a serious theoretical foundation is still lacking.For a more comprehensive introduction to the topic we refer to Koskela (2007).Kivinen et al. (2005) defined four criteria for an ideal measure of bucking outcome.According to the first two criteria an ideal measure should 1) take into account the size of the stands to be compared in terms of the number of logs (i.e., given a common log demand distribution, the results from two stands with different numbers of logs should be commensurable) and 2) make it possible to compare results based on the demand matrices of different sizes (i.e., a different number of diameter or length classes or both).These criteria address the problems often encountered by forest managers in practice when determining a suitable stand or group of stands for the given demand distribution.It was concluded by the authors that the Apportionment Index, at least in principle, takes into account the stand size expressed in terms of the number of logs.This is because the AI apply proportional quantities of logs in each diameter-length combination.The issue related to handle the potentially confounding effect caused by different matrix sizes is, however, much more complicated.Kivinen et al. (2005) also conducted simulation studies to test the Apportionment Index and three other goodness-of-fit measures against the given criteria.The study showed that stands with a small number of stems or a large number of small-sized stems were likely to match the log demand distribution more poorly than stands with a large number of stems and/or a wide diameter at breast height (DBH).This behaviour can be easily understood by noting that in stands with few trees there basically are fewer opportunities to produce logs in all log categories than in stands with many trees.In some situations we may only be interested in the proportion of "correctly" located logs in the outcome distribution with respect to the demanded log distribution without taking into account any underlying conditions such as the stand size (cf.interpretation of the AI in Section 2.2).However, there may also exist situations where one is not only interested in the magnitude of the value of the measure itself but also wants to know whether the performance of the harvesting operation is tolerable with respect to some underlying conditions.For example, for a smaller number of logs harvested a lower value of the AI may not necessarily indicate a poorly performing harvesting operation.This then raises a question of the least tolerable value of the AI.This information is important especially for forest managers.
In this paper we examine some aspects of the asymptotic sampling distribution of the Apportionment Index by assuming a multinomial distribution for the outcome (see also Ransom 2000).In Section 3 we derive the approximate first two moments, i.e. the mean and the variance, of the AI assuming multinomial distribution for the bucking outcome and using normal approximations.In Section 4 the derivated moments are utilized to construct a lower tolerance limit for the AI and simulations are then carried out to evaluate the goodness of the approximations.Some studies on the determination of the number of logs needed to attain high apportionment with a given accuracy are described in Section 5.The section illustrates the effect of certain underlying conditions such as the dimension and the form of the target distribution on the value of the AI.

Mathematical Formulation of the Apportionment Index
The output of the actual harvesting operation has been measured mainly by comparing the relative proportions of the output and target distributions.More specifically, let A common practice in Scandinavia is to evaluate the fit between the demand and actual output log distributions using the Apportionment Index first introduced in forestry by Bergstrand in the mid-1980s (e.g. Bergstrand 1989).For a fixed quality class the AI is defined as where ∑ 1 are the relative proportions of the outcome and target matrices, respectively.After some simple manipulations it can be shown that the AI can be rewritten as The maximum value of the AI is 1 (100%), which indicates a perfect match between the distributions.The minimum value of the index is min ( , ,..., )  target, i.e. to attain complete agreement between the two distributions.

Some Related Measures
The overlapping coefficient (OVL) is defined as a measure of agreement or similarity between two probability distributions or two populations represented by such distributions (Inman and Bradley 1989).In a simple univariate case the OVL is simply the fraction of the probability mass common to both distributions.In a case of two discrete probability distributions, the OVL and the AI have the same formula.Both measures are also related to the so-called Dissimilarity Index (DI) or Index of Dissimilarity, which is commonly used e.g. to summarize the closeness of fit of a model to the categorical sample data (e.g.Agresti 2002, p. 329-330).One version of the DI has gained ground especially in sociology, where it has become the most common measure of social segregation.In fact, it is easy to see that OVL = 1−DI = AI for two discrete probability distributions.
The OVL has been proposed as a generalized measure of agreement.Obviously, however, the Dissimilarity Index appeared first.One of the very first instances of the DI as a measure of segregation was that of Jahn et al. (1947), and after Duncan and Duncan's (1955) methodological paper the index started to attain popularity among sociologists.Research characterizing the properties of the sampling distributions of the segregation indices, including the Dissimilarity Index, is rare.One important exception, however, is that provided by Ransom (2000), who examines the sampling distribution of the DI.He studies the segregation by sex across different occupations and derives the asymptotic sampling distribution for the Dissimilarity Index based on the statistical model provided by the multinomial distribution.Another interesting article is that by Inman and Bradley (1991), where approximations to the mean and variance of the index are given under a multivariate hypergeometric distribution for the cell frequencies in the context of a 2 × k cross-classification table subject to fixed row and column totals.However, since the target distribution is a fixed quantity, neither of these two considerations can be directly applied to the Apportionment Index.

Mean and Variance of the Apportionment Index
Appendix A includes some preliminary results used for the mathematical derivations in this section.We assume that the observed distribution (outcome) of logs follows the multinomial distribution ( , ,..., ) ~( ; , ,..., ) where ∑ 1 is the total number of logs harvested, t i * s are the relative target values with t i * ( , ) ∈ 0 1 and The distribution assumption may not necessarily make sense if a stand is tried to harvest by a target which is unsuitable for the particular stand.It is, for example, impossible to produce logs with large SED from trees with smaller DBH.A situation of this kind immediately leads to a poor fit between the outcome and demand distribution arguing that the outcome cannot be a realization from the given multinomial distribution.However, if the pre-harvest planning procedure is performing appropriately, it feels reasonable that the selection of the harvested stand is done such that the production of the requested logs is possible.
For the relative random outcome we use the and further and Under certain conditions, the binomial distribution can be approximated by the normal distribution.A conservative rule to follow in the case of X ~ Bin(n,p) is that min(np,n(1p)) ≥ 5 (see e.g.Casella and Berger 2002, p. 104-105).In terms of target values this means that e.g. for N = 500 the cell target t i * should lie in the interval [0.01, 0.99] and for N = 1000 it should belong to [0.005, 0.995].Since in practice the total number of logs (N) is usually large, the conditions will be satisfied even for quite small target values.The normal approximation gives Recall the definition of the Apportionment Index in Eq. 1.Before deriving the moments of the AI, we define If we assume normal distribution (Eq.7), upon application of Result 1 in Appendix A, E AI ( ) becomes Hence, It can easily be seen that E(AI) increases as N increases and E(AI) → 1 as N → ∞.Thus, under the assumption of multinomial distribution, index values close to one are expected for very large N.

Applying Result 1 and Corollary 2 in Appendix
As seen in Eq. 12, the variance of the AI is a decreasing function of N and V(AI) → 0 as N → ∞, i.e. very little variation is expected for a particularly large N.In a special case of equal target proportions, i.e. t * = (1/k,...,1/k), E(AI) and V(AI) reduce to for a large number of logs, N, and for a moderate number of log categories, k.E(AI) and V(AI) are given in Eq. 10 and Eq. 12, respectively.From Eq. 14 it follows that giving the approximate 100 × (1-α)% lower tolerance limit for the AI as For the output and target matrices to be considered similar, the lower tolerance limit serves as the least acceptable value of the AI.Thus, in a case where the observed outcome matrix assigns the index value which falls below Eq. 16, we would consider the outcome to indicate disagreement with the given target matrix.Overall, the larger the observed AI, the better the agreement.We hence suggest values higher than the mean (E(AI)) to indicate a satisfactory level of agreement.It is worth noting, especially for practical implementation, that E(AI) is very easy to compute.For N = 1000 and the above target matrix, E(AI) = 0.939 and V(AI) = 0.9025•10 -4 .Hence, the 95% lower tolerance limit is 0.923.For the sample outcome matrix, however, AI = 0.962.This far exceeds the mean 0.939 and we may hence conclude that the agreement in this case is highly satisfactory.Note that both the expected value and the lower tolerance limit take into account the effect of the number of logs.This may give valuable information e.g. when results of different harvesting operations with different stand sizes and targets are compared.
Remark 1.We propose to take a closer look at the two matrices in Tables 1 and 2. As noted above, there are 13 (respective 10) log categories which indicate overload (respective underload) while 2 are "neutral".We now examine the frequency distribution of the amount of overloads and underloads in this example.Both these distributions are shown in Table 3.As expected, 38 for both frequency distributions.
The computation of the AI simply depends on the total amount of overloads (or underloads) and not on their specific frequency distribution.However, from a practical point of view, the distortion or deviation from the target matrix in terms of the overload or underload distribution will have an impact on the overall cost-effectiveness of the sample output matrix.This calls for a price-weighted index of agreement (see Kivinen et al. 2005 andNummi et al. 2005).For uniform price policy (only of theoretical interest), possibly concentration of overload or underload in a few categories would be desirable.It may, for example, be preferable to have overload in categories of larger diameter-length combinations, as large logs may possibly be converted into logs of smaller dimensions, if required.

A Simulation Study for the Tolerance Limit
A simulation study was conducted to investigate the accuracy of the confidence level (1-α) of the approximate tolerance limit (TL) given in Eq. 16.
However, we carried out this study with reference to both-sided tolerance limits mainly to understand the nature of normal approximation.The approximate 100(1−α)% tolerance limits used in this simulation study are defined as We simulated 10 000 outcome matrices from , with the given target probabilities (Table 1) and N = 50, 100, 500, 1000, 5000, 10 000, 50 000, 100 000.The index value was computed for each simulated outcome (with respect to the given target matrix) and 100(1−α)% TLs were constructed using Eq. 17.The confidence levels (CL) of the approximate TLs were evaluated by calculating the proportion of the times AI falling within the respective tolerance limits.We also report the proportions of times the AI takes a value below and above the lower and upper tolerance limits.These are denoted in Table 4 by LP and RP, respectively.In Table 5 E(AI) and V(AI) are computed for all N and the given target using Eq. 10 and Eq. 12, respectively.We also report the sample point estimates of E(AI) and V(AI) calculated from the simulations.
Tables 4 and 5 show that the approximation works reasonably well.The CL values are in accordance with the respective confidence levels and the sample point estimates match nicely with E(AI) and V(AI).However, comparison of the LP and RP values in Table 4 reveals that in most of the cases LP is slighly greater than RP, which indicates some degree of non-symmetry in the distribution of the AI.Table 5 also illustrates the performance of E(AI) and V(AI) as a function of N. It can be seen that a relatively large N (≥ 500) is needed to obtain E(AI) ≥ 0.914.With respect to the given 5 × 5 target distribution, N ≥ 368 seems to be sufficient for E(AI) > 0.90.

Determination of N to Attain High Apportionment with a Given Accuracy
For very large N, index values close to 1 are expected.In this section we illustrate the effect of certain underlying conditions such as the dimension and the form of the target distribution by determinining the size of N required to attain a high value of the index with a chosen level of confidence.In other words, we determine N, say N 0 , such that for a given ε (> 0), relative target matrix and α, which is equivalent to where AI is as defined in Eq. 8. Assuming Eq. 14 and a given relative demand distribution, N 0 is solved from z = z α , where z α is the upper α100% point of N(0,1) distribution, and E AI ( ) and V AI ( ) are as defined in Eq. 9 and Eq.11, respectively.This yields where t * * * * ( , ,..., ) is the relative target, and Note that ∆(t * ) and Γ(t * ) are proportional to V AI ( ) and E AI ( ), respectively.We investigated three illustrative examples.In the first N 0 is solved for the 5 × 5 target distribution given in Table 1.In the sequel we call this target distribution the referred target matrix.In the second case the dependence of N 0 on the dimension of the target matrix is examined by solving N 0 for a k-dimensional relative target matrix t * = (1/k,1/k,...,1/k) as k varies from 20 to 30.The third example investigates the effect of the form of the target matrix on the magnitude of N 0 .

Example 1: The referred 5 × 5 target distribution
Assume the target distribution in Table 1.Table 6 displays N 0 solved for some given values of ε and α.
The above results and Eq.21 show that the number of logs N 0 depends on both the specified accuracy ε and the specified confidence level α.Table 6 shows for α = 0.05, for example, that while reducing ε from 0.1 to 0.01, N 0 must be increased from 582 to 58186.For α = 0.01 and the same decrease in ε, the number of logs needs to be increased from 685 to 68482.For both confidence levels a more than 100 times larger N is required for ε = 0.01 than for ε = 0.1.Hence, a decrease in confidence level α needs to be followed by an increase in N to attain the specified accuracy.Similarly, a decrease in ε, which determines the accuracy, requires an increase in N to attain the specified confidence.

Example 2: Target with equal probabilities
Assume the relative demand distribution t * = (1/k,1/k,...,1/k).Table 7 displays N 0 solved for α = 0.05 and some selected values of ε as k varies from 20 to 30.The bolded row corresponds to the 5 × 5 matrix.The results with α = 0.01 are shown in Table 10 in Appendix B.
Table 7 and also Table 10 in Appendix B confirm the intuitively obvious result that higher dimensional targets require a larger N to maintain the specified accuracy and/or the specified ± ± confidence level.In other words, for a fixed N, low dimensional targets tend to give higher index values than targets of higher dimension.

Example 3: Targets with a variety of forms
The results in the above examples indicate that the form of the target distribution has a demonstrable effect on the size of N 0 required.For example, for the referred 5 × 5 target distribution and the choices ε = 0.05 and α = 0.05, N 0 = 2327 is required (Table 6).For the 5 × 5 equal probability distribution, however, N 0 = 2399 is needed (Table 7).In this example we examine the effect of the form of the target distribution.
The relative target distributions to be compared are constructed by changing the frequency pattern of the relative values of the referred 5 × 5 target given in Table 1.The idea is to cover most of the commonly observed standard unimodal frequency curves.A detailed description of the construction procedure is left to Appendix C. The frequency histogram of the relative target values of the referred demand distribution and of those constructed (Target matrices 1-7) are displayed in Fig. 1.Table 11 in Appendix C reports some summary statistics related to the targets presented.
Table 8 shows N 0 solved for the target matrices 1-7, α = 0.05 and a variety of ε (= 0.01, 0.02, 0.03, 0.04, 0.05, 0.06, 0.07, 0.08, 0.09, 0.10).The bolded row corresponds to the 5 × 5 target matrix with equal probabilities.The corresponding results for α = 0.01 are displayed in Table 12 in Appendix C. Table 8 above and Table 12 in Appendix C show that N 0 depends on the form of the given target matrix.For all pairs of ε and α examined in the example, we observe the largest N 0 for the target with equal cell probabilities.This behaviour can partly be explained by observing that for a fixed N, minimum E(AI) is attained for t * = (1/k,1/k,...,1/k).The above result is easy to prove by denoting a t

Discussion
Although the Apportionment Index is the most widely used measure for evaluating the bucking outcome in Scandinavia, relatively little is known regarding its statistical properties.In this article we have examined the asymptotic sampling distribution of the AI assuming multinomial distribution for the outcome and utilizing large sample normal approximations.The assumed distribution of the random outcome should be reasonable when the pre-harvest planning procedure is performing appropriately.Under the assumption of the multinomial outcome distribution and for the given target matrix we have then derived the approximate first two moments, i.e. the mean and the variance.A simulation study is conducted to confirm that the approximations work reasonably well.
It can be easily seen from the derived formulas that the expected value of the AI increases as the number of harvested logs increases and the expectation is almost one for very large number of logs.The derivations also show that the variance of the AI decreases towards zero as the number of logs increases, i.e. very little variation is expected for highly large number of logs.The findings are important and should, especially, be taken into account e.g. when comparisons are made between two different harvesting operations.
Since, for a given target distribution, the expected value of the Apportionment Index seems to strongly depend on the number of logs harvested, there may exist situations where not only the magnitude of the measure itself is of special interest but one also wants to know whether the performance of the harvesting operation is tolerable with respect to the number of stems harvested.This kind of a situation may arise, for example, when adjusting the bucking instructions according to the harvesting information gathered.In order to answer the question of a tolerable value of the AI we have derived a lower tolerance limit for the AI.It is then suggested that the lower tolerance limit should serve as the least acceptable value of the AI while index values higher than the mean would indicate a satisfactory level of agreement.
Studies are conducted on the determination of N (the number of harvested logs) needed to obtain a high apportionment with a given accuracy under some special kinds of distributions.Although our considerations are purely theoretical, our opinion is that they also have some useful practical implications.The conclusions that can be drawn from the studies in Section 5 are that not only the number of harvested logs, as noted earlier, affect the magnitude of the AI but also the dimension of the target matrix as well as the form of the target distribution.It is hence important to also study how the values of the AI between target matrices with different dimensions (and varying number of logs) can be made commensurable.However, more thorough investigations are needed to exclusively determine how the form and size of the target matrix are actually affecting the AI.Such knowledge would be important, for example, when interpreting the values of the AI with certain other possible measures, as discussed by Kivinen et al. (2005).
Instead of using the Apportionment Index (or the OVL) alone to evaluate the similarity of two distributions, standard statistical tests (e.g.χ 2test) can also be applied (see e.g.Nummi et al. 2005).For a large sample size the standard statistical tests often possess high power to detect small differences.Thus virtually unimportant deviations from the target may in practice lead to rejection of the null hypothesis.As discussed by Inman and Bradley (1989) regarding the OVL, the AI should also be seen as supplementary to statistical tests rather than as an alternative method.
The connection of the Apportionment Index to the OVL and some other related measures is taken up in the paper to encourage those interested in the topic to do some literature research in other fields of science such as sociology.The authors strongly believe that the work done e.g. on the Dissimilarity Index and other measures of segregation may give valuable ideas for the research of the Apportionment Index as well as for the development of new measures of apportionment.

Appendix A
In this Appendix we present some results and corollaries to be used in Section 3.
The proof is omitted.

Appendix C
Let us define the summary statistics related to a target matrix as where Mean, Var and CV are the sample mean, sample variance and the coefficient of variation calculated for the target values.We also include in the summary statistics the functions ∆(t * ) and Γ(t * ), which are defined in Eq. 22 and Eq.23 in Section 5, respectively.For the referred 5 × 5 target in Table 1 in Section 4 Mean = 0.0400, Var = 0.0002, CV = 0.3458, ∆(t * ) = 0.3610 and Γ(t * ) = 4.8078.
The relative target distributions to be compared in Example 3 in Section 5 are constructed by changing the frequency pattern of the relative target values of the referred 5 × 5 target given in Table 1.(See their summary statistics in Table 11 below.)First, all 25 relative proportions of the referred target matrix are classified into 5 intervals of equal length (= 0.011) so that (f 1 , f 2 ,…, f 5 ) = (5,2,11,5,2) and ( , ,..., )   * * * t t t 1 2 5 = (0.020, 0.031, 0.042, 0.053, 0.064), where t i * denotes the midvalue of the ith interval and f i is the corresponding frequency.For a new frequency pattern, say (f' 1 , f' 2 ,…,f' 5 ), we define the ith adjusted mid-value as , i = 1,…,5.The corresponding relative target matrix is then constructed by setting the relative target values in the first f' 1 log categories equal to t 1 ** , in the next f' 2 log categories equal to t 2 ** etc. Table 12 shows N 0 solved for the target matrices 1-7, α = 0.01 and a variety of ε (= 0.01,0.02,0.03, 0.04, 0.05, 0.06, 0.07, 0.08, 0.09, 0.10).The bolded row corresponds to the 5 × 5 target matrix with equal probabilities.
Fig. 1. a) The referred target matrix b) Target matrix 1 c) Target matrix 2 d) Target matrix 3 e) Target matrix 4 f) Target matrix 5 g) Target matrix 6 h) Target matrix 7.

Table 3 .
The frequency distribution of the amount of overloads and underloads.

Table 7 .
N 0 solved for α = 0.05 and some selected values of ε as k varies from 20 to 30.

Table 6 .
N 0 solved for the referred 5 × 5 target matrix and some given values of ε and α.