4.3 Estimating the Probability and .Amount of Decayed Wood in Standing Trees
Valerie M. LeMay
Summary
Estimation of the amount of internal structural damage (decay) in standing trees is necess ary for assessing the amount of timber volume available for harvest. Determining which trees are likely to have some decay is important in deciding which trees to leave for wildlife habitat and also for selecting sound trees for specialized products.
Recent research has shown that the measurement of the amount of decayed wood area at the base of the tree substantially improves the estimate of percent decay in standin g trees. Also, because of the number of zero percents usually present in decay data, the tobit estimator may be preferred to the usual least squares estimator in some instances. Finally, research has shown that it is possible to develop objective rules for classifying standing trees as decayed or sound. Classification trees analysis is recommended as a tool for developing these rules.
4.3.1 Introduction
An estimate of the expected loss of wood due to internal structural damage {decay) caused by decay organisms is necessary for determining the amount of available stand
ing timber volume and for selecting trees for products which require sound wood, such as beams and telephone poles. Also, determining which trees are likely to have some decay may be useful in assessing which trees would be preferred for wildlife use.
Over the last 10 years, research into estimating decay volume and predicting which trees are likely to have some decay has been performed for common tree species of the provin ces of Alberta and British Columbia (B.C.). Species selected for study have been largely restricted to those for which decay occurs long before timber harvest generally occurs. These include species with high succeptibility to decay and those that are very long lived such as some of the species on the Coast of B.C.
A brief summary of some of the results of recent study is presented in this paper.
Possiblities for improvements are also suggested. More detailed results were recently published (LeMay 1993; LeMay et al. 1993; LeMay et al. (in progress)).
4.3.2 Percent Decay Estimation
Rather than estimating decay volume directly, the percent of decay volume to gross volume is usually estimated. This percent can then be used for a limited range of timber utilization specifications. Research into estimating percent decay has been focused on the problem of finding reasonably precise estimates from measures that are readily taken on standing trees ( e.g., Aho 1974; Aho and Simonski 1974; Aho 1977; Bailey and Dobie 1977; Basham 1958; Brown 1934; Filip et oJ. 1990; LeMay 1982; Morawski 1967;
Wall 1971). Tree size measures such as the diameter outside bark at 1.3 metres above ground ( dbh) and height have been used as predictor variables. Age has also been used, but may have to be estimated for trees with substantial decay at the tree base.
Indicators of decay such as the presence of conks, scars, a broken top, or a pronounced fork have been frequently considered for estimating percent decay. Measures of site such as moisture do help in providing more precise estimates of decay, but usually site measures are restricted to stratification of the land areas into broad regions because of cost. A common approach to estimating decay for application to broad lan dscapes is to produce a regression equation using diameter outside bark at breast height and external indicators to predict percent decay for a particular species and area.
Two problems have been recognized for estimating percent decay on standing trees.
1. The variability of decay measures is very high over values of the commonly used predictor variables.
2. There are a large number of zero percents which occur over wide ranges of the predictor variables.
As an example of these two problems, Figure 1 shows percent decay versus dbh for aspen (Populus tremuloides Michx). The high variability of percent decay results in low precision. The large number of zero percents increases this variability and introduces bias into coefficients estimated using least squares regression. Studies were initiated to test alternative solutions for these two problems.
Addition of a Measure of Decay at Breast Height
Data for aspen, cedar ( Thuja plicata Donn), and true fir ( Abies lasiocarpa (Hook.) Nutt.) were obtained for the B.C. Ministry of Forests, Inventory Branch in Victoria, B.C. These data represented a range of sites and tree sizes. The aspen and fir data were largely from northeastern B.C., whereas the cedar data were from the southern coast of B.C. These three species represent a large part of the timber volume in B.C.
and often have internal decay. The trees were destructively sampled to determine the decay volume.
100
p 1
e r 75 1 1
C 1 2 1 1 1 1 1 2
e 1 1 3 1 1 1 1 3 1 1
n 1 2 1 1 2 1 2 1 1 1 1 2 1 1 1 1 1 1 2 1 1 t 1 3 1 32 1 1 1 2 1 3 1 1 2 2 1 1 1 1
50 1 1 1 1 1 44432344223332 1 1 1 1 1 1 1
D 222541 23248442562 1 1 3 3 1 2 1 1 1
e 1 1 1 23366247247 1 55 1 6 1 342 1 1 1 1
C 5662 3* 3554433542 1 5 1 1 1 1 2 1 1
a 25 12794 7536246632 3 3 1 2 1 1 1 1
y 25 546 * 2 * 56 * 6446 1 372223 1 3 1 1 1 1 1 1 3 1 379435544 16756 1834 1 24 1 1 1 1 3
3 6773*43534737523444322 1 1 1 1 1 1 3 1 1 33689 * 58* 535554234 1 1 22 1 1 221 1624 3 * 6536757 * * 3344721 4 1 34223 1 1 0 2259562546 • 9 • 9 • • • 5 • • • 95 • 525442 23 1 2 2 1 1 1 1
10 20 30 40 50 60 70
D i ameter at 1 . 3 m
Figure 1: Variability in percent decay for aspen
To represent the commonly used equations, an equation using dbh above ground and classes of external indicators ( risk groups) was fit for each species (Model 1 ). Risk group was defined as no external indicators (O); one external indicator but no conks (1 ); two external indicators but no conks (2); three or more external indicators but no conks (3); and conks present (4). The external indicators that were noted were (1) scars (not recent); (2) fork or pronounced crook on the main bole of the tree; (3) frost cracks; (4) mistletoe on the trunk (relevant for fir only); (5) large rotten branches of more than 10 cm in diameter; and (6) dead or broken tops which are not recent and show signs of weathering. Alternative equations included a measure of the decay area at the base of the tree as a percent of the total area. These measures were calculated for 1.3 m (Model 2) and 0.3 m (Model 3) above ground, and were based on decay area measured on felled trees. In practice, the measurement would have to be made on increment cores, or a wood density measuring device, such as the RESISTOGRAPH could be used to detennine if decay is present at breast height (RESISTOGRAPH is manufactured and distributed under license of W.F.G. Kamm and F. Rinn, Germany).
Some measurement error would occur in using these less destructive measures, however.
The inclusion of a measure of the decay at the base of the tree substantially improved the prediction of the percent decay over the usual model (Table 1 ). This procedure could be practically used for high value trees to estimate decay. For larger tracts of timber, subsampling designs could be used by obtaining good estimates of percent decay on some trees and by using these trees to improve estimates for other trees in the area.
Table 1: Comparison of percent" decay estimation equations.
Equationa Cedar (5841b)
1 2 3
R2 c Se d
.06 .42 .44
21.2 16.5 16.2
Fir (5984) R2 Se
.24 13.4 .55 10.3 .50 10.9
Aspen (2332)
R2 Se
.15 .62 .61
17.4 11.6 11.7
a Model 1 used log10(dbh) and risk groups entered as dummy variables. Model 2 in
cluded also percent of decay area at 1.3 m above ground, whereas Model 3 included percent of decay at 0.3 m above ground.
bNumber of trees in the sample.
ccoeffi.cient of determination.
dStandard error of the estimate.
Alternative Estimation Methods
The presence of zero percent decay over a wide range of predictor variables occurs since the tree has no decay until some threshold of resistence is crossed and decay appears. The use of a tobit estimator (Tobin 1958) was tested as an alternative to the ordinary least squares regression (OLS) estimator.
For an unlimited dependent variable, under the assumptions that the error terms are normally distributed, have equal variance, and the data are independent, the expected value for the dependent variable given the set of independent variables is:
[1] [2]
[3]
E[Yil:z:i]
E[ei]
E[Yi l:z:i]
:z:i/3 + E[ei]
0 :z:i/3
For the percent decay data, however, the expected value of percent decay is:
[4] [5]
E[Yi l:z:i]
E[Yi l:z:i]
- Prob(yi = 0) x E[YilYi = 0, :z:i]
+
Prob(yi>
0) X E [Yi lYi>
0, :z:i]Prob(yi > 0) X (E [yilYi > 0, :z:i])
The first term of Equation 4 is zero, since the expected value of percent decay for trees without decay is zero. ff the decay percents are similar to the ratios of expenditure to income used by Tobin {1958) in his example, the nonzero percent data could be assumed to follow a truncated normal distribution with equal variance. The expected value for all data including the zero percents then becomes:
(6]
(7]
E(yil:z:i] - c) ( :,;:) x (:z:i{J
+
E(eilYi > O]) E(yil:z:i] - c) ( :,;:) x (xi/3+
u..\i)where c) (
z1)
is the cumulative probability from the standard normal distribution function at z:: , the probability that the percent decay is greater than zero at z:;3 ; and,\i is the inverse Mills ratio defined as:
[8] ,\ -'
"' (�)
c)
( zt)
where <j, (
=:fl)
is the probability from the standard normal probability density function atz! .
The inverse Mills ratio represents the conditional probability given that the normal distribution is truncated. The term u,\i of Equation 7 is the expected value for the error terms when Yi is greater than zero. The proof for this expected value was first presented by Tobin (1958), but can be found in many econometric references ( e.g., Judge and others 1985; Greene 1990; Kmenta 1986; and Maddala 1983). The bias in estimating percent decay using OLS with all observations is shown by the difference between Equation 7 and Equation 3. The use of OLS with only the non-zero data points ( called positive least squares or POLS), also results in biased coefficients since E[YilYi > 0, xi] = zi/3+
O'Ai rather than zi{3. The OLS and P OLS estimates are also inconsistent 1 (Greene 1990; Judge et al. 1985). However, Equation 7 approaches Equation 3 as the number of zero percents decreases, since E [ei lYi > O] approaches zero. Therefore, for datasets with a low proportion of zero percent decay points, little bias will occur if OLS is used. Conversely, the bias increases with the proportion of zero percents in the data. These biases tend to be toward estimating zero coefficients (Greene 1990); tests to determine if the true coefficients could be zero will tend to be biased toward not rejecting the null hypothesis.1Consistent estimators are estimators for which the probability that the absolute difference between the estimate and the parameter is greater than a small factor approaches zero as the sample size increases.
As an alternative estimator, Tobin {1958) suggested a combined regression and probit estimator based on a maximum likelihood equation, which has since been termed the "to bit estimator" . For datasets that have zero as the lower limit ( as with percent decay data), and are independent, homoscedastic, and distributed as the truncated normal distribution above the lower limit, the maximum likelihood function is:
[9]
where Ily; =O is the product over all sample observations with the value of the dependent variable equal to zero; and Ily; >O is the product over all sample observations with the value of the dependent variable greater than zero. Tobin proved that the estimator resulted in consistent estimates of the coefficients, and that the negative inverse of the second derivatives of the maximum likelihood function, evaluated at the maximum likelihood solution, resulted in a consistent estimate of the variance/ covariance matrix of the coefficients. The estimated coefficients were shown to be asymptotically nor
mally distributed, and a likelihood ratio approach was recommended for testing the significance of particular independent variables. In addition to providing consistent estimates of the coefficients and of their variances, the benefit of Tobin's approach is that the probability of the dependent variable being above the lower limit (or below the upper limit), the expected value if the dependent variable is above the lower limit (or below the upper limit), and the expected value for all values of the dependent variable are estimated simultaneously.
In order to evaluate the practical implications of using the tobit estimator instead of OLS, a Monte Carlo simulation to examine the differences in estimated coefficients and in estimated percent decay was performed. Again, aspen, cedar, and fir data were used. One hundred samples of 300 trees each were selected from each data set. For the aspen data set, only 93 sample sets were retained since seven of the samples did not have any risk group 2 data. For each sample, a decay estimation equation using the logarithm of dbh and indicator variables representing risk groups was fit using the regression procedure of SPSSX (SPSSX, Inc. 1985) and also using the tobit estimator of SHAZAM (White and Horsman 1986). The results indicated that although there are theoretical reasons to employ the tobit estimators, no practical gains occurred from their use. The estimated coefficients, and, more important, the differences between the observed and predicted percent decay values (Table 2) were similar for the two estimating systems. The wide variability in decay may have contributed to the result of having no improvements in using OLS. For a more precise model, the tobit estimator would possibly result in noticeably improved (less biased) estimates.
Recommendations for Estimating Percent Decay
For estimating percent decay, it is recommended that a measure of the decay at the base of the tree be used for valuable species or small tracts of timber. For less valuable species, the use of size variables and external indicators to predict tree decay is more practical, unless a subsampling approach is adopted. The use of the tobit estimator for
fitting decay prediction equations is theoretically attractive, but showed no practical gains over the OLS estimator. This may have been due to the wide variability in the decay estimation model used, however. If the software is available, the use of the tobit estimator is recommended.
Table 2: Summary Statistics for Expected Percent Decay Using OLS Versus the Tobit Estimator by Species
Mean OLS
Min Max
Cedar1
AD 2 -0.072 -3.623 AAD 3 15.705 15.409 ASD 4 376.647 369.871 Pseudo-R2 0.05 0.01 Fir1
AD -0.066 -1.737
AAD 10.296 9.691
ASD 201.057 197.477 Pseudo-R2 0.32 0.29 Aspen6
AD 0.038 -1.440
AAD 6.677 6.217
ASD 118.795 116.958 Pseudo-R2 0.14 0.06
1 Based on 100 Sample Sets
2 Average Difference
3 Average Absolute Difference
4 Average Squared Difference
1.960 16.364 392.488 0.06
1.606 10.854 210.124 0.40
1.707 7.320 129.362 0.15
Var Mean
1.169 -0.424 0.033 15.733 19.627 376.106 0.00 0.05
0.541 -0.859 0.052 10.506 5.982 199.036 0.00 0.33
0.402 -0.411 0.053 6.993 3.380 119.569 0.00 0.14
Tobit
Min Max
-3.917 1.566 15.431 16.416 369.500 393.886 0.00 0.06
-2.706 0.857 9.803 11.129 194.498 206.462 0.30 0.41
-2.195 1.570 6.343 7.868 117.070 130.607 0.06 0.15
5 Calculated in a similar manner as the Coefficient of Determination
6 Based on 93 Sample Sets
Var
1.110 0.034 19.529 0.00
0.573 0.071 6.116 0.00
0.505 0.072 4.443 0.00
4.3.3 Classification of Decayed Versus Sound Trees
As well as estimating the amount of decay, classifying standing trees as decayed or sound is of interest. Two estimating methods, discriminant analysis (SPSSX, Inc.
1986) and classification trees analysis (Brieman et al. 1984) were used. Discriminant analysis is commonly used to classify data; the classification trees analysis was de
veloped relatively recently and produces dichotomous trees based on the criteria of minimizing the number of incorrectly classed items. Three general rules were proposed for testing.
Rule 1 All trees in the learning data set were used in discriminant or classification trees analysis to obtain classification rules. The results from these analyses were used to classify trees.
Rule 2 Trees in the learning data set were separated into two groups, depending on whether or not the presence of conks was recorded. Data for which the presence of conks was recorded were classified as decayed with certainty. The remaining data were used in discriminant or classification trees analysis. The rule system for the test data set was first based on correctly classifying trees with conks as decayed. The remaining trees were classified using the results of the discriminant or classification trees analysis.
Rule 3 An additional variable, the percent decay area at breast height of 1.3 metres above ground (BHPDEC) measured on the cut surface, was used for the third rule. As with Rule 2, trees in the learning data set were initially separated into two groups. The first group included trees for which either the presence of conks was recorded, or for which BHPDEC was greater than zero. The remaining data were used in discriminant or classification trees analysis to develop classification rules. The rule system for the test data set was based on the correct classification of trees in the first group as decayed, coupled with classifying the remaining trees using the results of discriminant or classification trees analysis.
The three species used for testing the estimation of percent decay again used. The data for each species were randomly di\-;ded into model estimation (66%) and model testing (33%) data sets. For all data sets, the following variables were used in developing classification rules.
1. An index, used as the dependent variable, where "1" represented decayed trees and "O" represented sound trees.
2. Tree age (AGE), obtained by counting the number of rings on the cut surface at stump height (0.3 metres above ground) and then adding a constant to represent the number of years to reach the stump height. For trees with a substantial amount of decay (structural damage), age was estimated;
3. dbh;
4. Total tree height in metres (HT)';
5. The ratio of dbh to HT (DOH) as a measure of tree form;
6. llisk group as defined previously;
7. Crown class, reflecting the position of the tree in the canopy, was defined as dominant (1), codominant (2), intermediate (3), or suppressed (4);
8. Site class, defined as Good (1), Medium (2), Poor (3), and Fair ( 4). These were based on site index ranges and assigned to each unique stand;
9. Elevation in metres (ELEV).
Most of these variables were selected because they are easy to measure on standing trees and have been used in previous studies to estimate the amount of decay. Age was included in the study, but often only an estimate of age is possible for trees with much decay. Site variables such as moisture, temperature, elevation, or classification into ecological types are likely related to the amount of decay. However, only elevation and site class were available for these data. It was recognized that elevation alone does not account for all site differences (Klinka et al.
1989,
p.6-9)
and the site class used was only a broad site index class based on stand level classification.Class variables were represented by dummy variables for discriminant analysis, and therefore, the assumption of a multivariate normal distribution for the independent variables was not met. As a result, statistical tests to determine which variables are important to the discriminating functions were invalid. Instead, the assessment of each discriminant analysis was based on the resulting error rates, which also facilitated comparisons to the classification trees analyses. The number and percent of incorrectly classified trees ( misclassification error rate) were calculated for the model estimation and model testing data sets.
As an example for the discriminant analysis, Fischer's linear discriminating func
tions (Dillon and Goldstein
1984,
pp.369-370)
for aspen, Rule1,
were:yl
i= -474. 14 - 0. 1613
xAGE
i- 14.9374
xdbh
i+ 352.8277
xDOH
i+21 .0184
xHT
i- 0.6335
xXl
i- 5.8776
xX2
i- 19.2278
xX3
i+2.7479
xX4
i+ 414.9717
xX5
i+ 422.6598
xX6
i+ 435.6958
xX7
i-12.6988
xX8i + 5. 7045
xX9
i+ 0.06385
xELE½
y2
i-481.23 - 0. 1 1 18
xAGE
i- 15.2916
xdbh
i+ 358.6050
xDOH
i+21 .3774
xHTi + 0.8198
xXl
i- 3.6578
xX2i - 16.8565 x X3i +4.643 1
xX 4i + 412. 7362
xX5
i+ 419.6378
xX6
i+ 432.9486
xX7i -12.2240
xxs
i+ 6.3061
xX9
i+ 0.06244
xELE½
if yl
i> y2
ithen classify tree i as Saund;
if y2
i> yl
ithen classify tree i as Decayed ;
For aspen, Rule 1, the classification tree is shown in Figure 2.
The classification trees were easier to interpret than the discriminating functions.
In terms of the misclassification error rates, the two methods produced similar results (Table 3). The third rule resulted in an improved classification overall, but sound trees better classified using Rules 1 or 2. The reduction in the number of trees entering the analysis methods for Rule 3 likely resulted in this poorer classification of sound trees.
The results indicate that classication rules could derived with reasonable success.
These rules were not optimal for classifying these species, and would likely be improved by deriving particular rules for each species and region. The classification trees analysis is recommended for ease of interpretation over discriminant analysis.
AGE � 85.5 years
Yes No
ELEV � 617 m Risk
=
0 or 1Yes No Yes No
ELEV � 477 m Sound ELEV � 777 m Decayed Yes
Sound
No Decayed
Yes Decayed
No Site
=
3 (Poor)Yes No
ELEV � 1020 m Decayed
Yes No
Sound Decayed
Figure 2: Classification Tree for Aspen, Rule 1
Table 3: Number and Percent (in Brackets) of Trees Incorrectly Classified Using Two Estimating Systems and the Model Testing Data by Species
Model/
Rule Class
Discriminant Analysis
1 Sound Decayed TOTAL 2 Sound
Decayed TOTAL 3 Sound
Decayed TOTAL Classification Trees
1 Sound Decayed TOTAL 2 Sound
Decayed TOTAL 3 Sound
Decayed TOTAL
Aspen {636a) Correct Wrong
79 21
76 24
76 24
79 21
92 8
89 11
69 31
98 2
95 5
90 10
81 19
82 18
90 10
82 18
83 17
81 19
97 3
96 4
a Number of trees in the sample.
Cedar {686) Correct Wrong
69 31
8 1 19
80 20
69 31
81 19
80 20
69 31
98 2
96 4
79 21
83 17
83 17
79 21
83 17
83 17
67 33
98 2
96 4
Fir (1021) Correct Wrong
62 37
70 30
68 32
60 40
72 28
69 31
64 34
87 13
82 18
64 36
66 34
66 34
67 33
64 36
65 35
60 40
92 8
83 17
4.3.4 Conclusions
The estimation of percent decay in standing trees is difficult because of the wide variability among trees. However, the addition of a decay measurement at the base of the tree could be used for a small number of valuable trees. The use of tobit estimators is not recommended unless a computer routine to obtain the estimates is readily available.
The development of rules to classify trees as decayed seems feasible. For low error rates, the rules would have to be specific to the tree species and location of interest.
Classification trees analysis is recommended as a practical tool for developing these rules.
4.3.5 References
Aho, P.A. 1974: Defect estimation for grand fir in the Blue Mountains of Oregon and Washington. USDA Forest Service Res. Pap. PNW-175. 11 pp.
Aho, P.E. and Simonski, P. 1975: Defect estimation for white fir in the Fremont Na
tional Forest. USDA Forest Service Res. Pap. PNW-196. 9 pp.
Aho, P.E. 1977: Decay of grand fir in the Blue Mountains of Oregon and Washingon.
USDA Forest Service Res. Pap. PNW-229. 18 pp.
Bailey, G.R. and Dobie, J. 1977: Alberta poplars-tree and log quality. Western Forest Products Laboratory. Information Report.
Basham, J.T. 1958: Decay of trembling aspen. Canadian Journal of Botany, vol. 26, pp. 491-505.
Breiman, L., Friedman, J.H., Olshen, R.A. and Stone, C.J. 1984: Classification and regression trees. Wadsworth & Brooks, Monterey, California. 358 pp.
Brown, R.M. 1934: Statistical analysis for finding a simple method for estimating percentage heart rot in Minnesota aspen. Journal of Agricultural Research, vol.
40, no. 10, pp. 929-942.
Dillon, W.R. and Goldstein, M. 1984: Multivariate analysis: methods and applications.
John Wiley & Sons, Toronto, Ont. 587 pp.
Filip, G.M., Schwandt, J.W. and Hagle, S.K. 1990: Estimating decay in 40- to 90-year old grand fir stands in the Clearwater Region of Northern Idaho. USDA Forest
Service Res. Pap. PNW-RP-421.
Greene, W.H. 1990. Econometric analysis. MacMillan Publishing Company, New York.
pp 715-753.
Judge, G.C., Griffiths, W.E., Hill, R.C., Liitkepolh, H. and Lee, T. 1985: The theory and practice of econometrics. 2nd. ed. John Wiley and Sons, Toronto. pp.
779-783.
Klinka, K., Krajina, V.J., Ceska,
A.
and Scagel, A.M. 1989: Indicator plants of Coastal British Columbia. University of British Columbia Press, Vancouver, B.C., 288 pp.Kmenta, J. 1986: Elements of econometrics. 2nd. ed. MacMillan Publishing Company, New York. pp. 560-566.
LeMay, V. M. 1982: Estimating total, merchantable, and defect volumes of individual trees for four regions of Alberta. M.Sc. (Forest Science) thesis. University of Alberta, Edmonton, Alberta. 83 pp plus Appendices.
LeMay, V.M. 1993: Percent decay estimation using decayed wood area at breast or stump height. Canadian Journal of Fore�t Research, vol. 23, pp. 307-312.
LeMay, V.M., Kozak, A., and Marshall, P. 1993: Using limited dependent variable estimators for estimating percent decay. Canadian Journal of Forest Research, vol. 23, pp. 266-274.
LeMay, V.M., Tait, D.E., and vanderKamp, B.J. in progress: Classification of cedar, aspen, and fir trees as decayed versus sound. 34 pp.
Maddala, G.S. 1983: Limited-dependent and qualitative variables in econometrics.
Economics Society l\4onographs No. 3. Cambridge University Press, New York.
400 pp.
Morawski, J.R. 1967: Assessment of cull. In: Wood Measurement Conference Proceed
ings. Technical Resport No. 7. University of Toronto. pp 24 to 28.
SPSSX, Inc. 1986: SPSSX User's Guide. Edition 2. Mc-Graw Hill Book Company, Toronto. pp 602 to 619.
Tobin, J. 1958: Estimation of relationships for limited dependent variables. Econometrica, vol. 26, pp. 24-36.
Wall, R.E. 1971: Variation in decay in aspen stands as affected by their clonal growth pattern. an. J. of For. Res. Canadian Journal of Forest Research, vol. 1, pp.
141-146.
White, K.J. and Horsman, N.G. 1986: SHAZAM: The econometrics computer program, Version 5.1, User's Manual. Dept. of Economics, Univ. of B.C., Vancouver, B.C.
291 pp.
