The Effect of Rounding on the Probability Distribution of Regrading in the U.S. Peanut Industry

Transcription

1 The Effect of Rounding on the Probability Distribution of Regrading in the U.S. Peanut Industry Edgar F. Pebe Diaz Graduate Research Assistant Department of Agricultural Economics Oklahoma State University 555 Ag Hall Stillwater, OK Phone: (405) B. Wade Brorsen (Corresponding Author) Regents Professor Department of Agricultural Economics Oklahoma State University 56 Ag Hall Stillwater, OK Phone: (405) Kim B. Anderson Professor Department of Agricultural Economics Oklahoma State University 517 Ag Hall Stillwater, OK Phone: (405) Phil Kenkel Professor Department of Agricultural Economics University of Tennessee 314-B Morgan Hall Knoxville, TN Phone: (43) WAEA Selected Paper Vancouver, Canada 000 Revised Version H:\Brorsen\Papers\PeanutRounding.doc 10/4/03

2 Abstract This article determines the effect of rounding (pointing-off) of grade percentages to the nearest whole number on the probability distribution of regrading in the peanut industry. The probability of regrading is important because regrading is costly and rounding creates an incentive for peanut graders to use overweight samples in order to avoid regrading. A new probability distribution called the normal-jump distribution was developed to model the distribution of the discrepancy between the sample weight before and after grading. A multiple-dimension Monte Carlo integration was then used to calculate the estimated probability of regrading with and without rounding. The probability of regrading was higher with rounding than without it. When rounding was not used, the sample weight had little effect on the probability of regrading. With rounding, the probability of regrading was reduced by beginning with a larger than 500- gram sample. Thus, rounding provides an incentive to take overweight samples. A low cost way to improve peanut grading accuracy would be to round to tenths rather than whole numbers. Key words: grade percentages, normal-jump distribution, peanuts, regrading, rounding.

3 The Effect of Rounding on the Probability Distribution of Regrading in the U.S. Peanut Industry The United States produced 1.6 million tons of peanuts, the production value of which was 1 billion dollars in 1998 (USDA, 1999). Under the U.S. peanut grading system, all loads of peanuts are officially inspected and graded by the U.S. Department of Agriculture Federal/State Inspection Service (FSIS). The loads are brought by the producers or sellers to the buying points where the loads are graded by peanut contract graders. The FSIS employs about,000 graders at about 500 buying points across the producing areas to grade peanuts during harvest from August to November (Dowell, Meyer, and Konstance). Accurate pricing of peanuts depends on accurate grading. One possible source of inaccuracy in peanut grading is that some graders have been observed taking samples slightly greater than the prescribed 500g, presumably to reduce chances of regrading (Anderson). Due to time constraints and pressure during rush hours of the grading season, graders may use an overweight sample to ensure the allowable tolerance is met if some of the sample weight is lost (Dowell, Meyer, and Konstance). For example, if a 500g sample is required, graders may begin with a 501g sample. Sample weights greater than 500g result in more peanuts in the sample. With overweight samples, graders tend to overestimate the grade factors measured and thus assign peanuts a higher price than is merited (Tsai et al; Whitaker, Dickens, and Giesbretch; Whitaker et al). Regardless of the actual sample weight, graders calculate grade percentages as if the initial sample weight were exactly 500g and round off the percentages to the nearest whole number, as required by the current grading procedure (USDA, 1996). Such severe rounding no longer seems necessary. The probability of regrading is important because 3

4 regrading is costly and because, as we show, rounding creates an incentive for graders to use overweight samples to avoid regrading. Also, how often rounding causes the need to regrade is not available in any published literature. The implications of this study may be useful for government policy, when revising the Farmers Stock Peanuts Inspection Instructions Handbook by the U.S. Department of Agriculture Federal/State Inspection Service, and for formal training programs for peanut graders. To get the policy changed, the benefits of a change must be shown. The purpose of this study is to determine the effects of rounding on the probability of regrading. A designed experiment is used to generate data. Nonparametric and parametric methods are then used. The nonparametric approach uses the empirical p.d.f. Under the parametric approach, a new probability distribution function called the normal-jump distribution is developed to estimate the probability of regrading with and without rounding. The normal-jump distribution of weight discrepancy and regression equations for grade factors are used to set up the integrals to calculate the probability of regrading. The integrals are solved with Monte Carlo methods. The Regrading of Peanuts With the current peanut grading procedure, the measurement of grade factors begins with a sample weight of 500g for truckloads of 10 tons or less. For single loads of over 10 tons, a sample of 1,000g is used (Oklahoma Federal/State Inspection Service). The set of grade factors G measured with the initial sample weight S are the percentages of sound mature kernels (SMK), sound splits (SS), other kernels (OK), total damage (TD), and hulls. 4

5 Measuring grade factors involves randomness. Some sample weight is lost during the grade analysis. When measured in tenths of grams, regardless of the grader s ability, there will be a discrepancy between the weight before and after grading. This discrepancy is defined by the difference between the initial sample weight S in grams and the weight in grams of total kernels and hulls K, or the total weight of graded material. That is, W = S K. The weight discrepancy found after grading comes from (i) the sample weight lost as dust and kernels during the analysis, and (ii) infrequent human errors. The weight lost is mostly dust or dirt created when the sample is shelled. Also, small pods or kernels can fall through the sheller grate or get stuck in the grading screen. Under time pressure, graders may forget to clean the pan containing kernels from a previous analysis or might accidentally drop some kernels into the pan when grading. These kernels will show up in the next sample graded. Hence, graders sometimes have total kernels and hulls greater than the initial sample weight. Human or mental errors include errors in recording weights, calculating the percentages, and transcribing the results (Powell, Sheppard, and Dowell). Usually graders do not lose pods and kernels or make errors, but when they do, the errors may be large. To check the accuracy of the grade factors measured, graders first round the percentages in each of the five categories to the nearest whole number. If the sum of these percentages is less than 99% or greater than 101%, peanuts must be regraded. Rounding can cause the sample to have to be regraded even when the grader has made no errors. 5

6 Data and Methods Because of the lack of data on initial sample weights and thus on weight discrepancy, there was a need to develop a designed experiment. Grade factors were measured by three professional graders and two aides. They were asked to complete the measurements quickly to simulate conditions during the peak of the grading season. The experiment used eighty-three,100g samples of peanuts. Twenty-five samples were runner-type peanuts and fifty-eight samples were spanish-type peanuts. The samples included a range of quality and cleanliness characteristics. Each sample was then divided into four subsamples of approximately 500, 501.4, 503.7, and 505g in order to have four different sizes from the same sample. All grade factors were calculated in grams and percentages with and without rounding. The probability of regrading was estimated using nonparametric and parametric methods. Under the nonparametric method, the probability of regrading with and without rounding was estimated using the empirical p.d.f. That is, the probability of regrading is calculated by dividing the observed number of times the sum of kernels and hulls falls outside the % interval by the total number of observations in each size. The weakness of the nonparametric approach is that estimates are not precise. Under the parametric approach, the probability of regrading was estimated using a new probability distribution and Monte Carlo methods. Equations, specifying grade factors as a function of weight discrepancy are needed. The grade factor equations were estimated using random effects models (fixed effects models give very similar answers). For example, the equation for SMK was (1) SMK ij = β 0 + β1 SAMPLE ij + β DISCREPANCYij + υ i + ν ij 6

7 where SMK ij is the weight of sound mature kernels in grams measured from the ith truckload and jth subsample size, SAMPLE ij is the initial sample weight in grams, DISCREPANC Y ij is the weight discrepancy in grams, and i ν ij υ + is the error term of the random effects model. Parameters in equation () were estimated with PROC MIXED in SAS (SAS Institute). The Normal-Jump Distribution To estimate the probability of regrading without rounding, the probability distribution of weight discrepancy in grams Z(W ) was needed. Weight discrepancy W is assumed to combine two sources of errors. One source is associated with the weight lost as dust and kernels during grading and the other with infrequent human errors. Given these specific characteristics, a new probability distribution called the normal-jump distribution was developed to model the distribution of weight discrepancy. This distribution is a modification of the mixed diffusion-jump process that has been used to model asset prices (Steigert and Brorsen) and exchange rate movements (Akgiray and Booth (1986, 1988); Oldfield, Rogalski, and Jarrow; Tucker and Pond). The major differences are that a normal distribution is used rather than a lognormal and the model is not a continuous-time diffusion process. The normal-jump distribution combines a normally distributed process and a jump process. We assume that the process associated with weight lost as dust and kernels follows a normal distribution with mean α and variance σ N. The occurrence of infrequent human errors is described by a Poisson distribution. The size of the jump or error is normally distributed with mean µ and variance σ J. The jump intensity λ 0 7

8 indicates the probability of occurrence of a large human error. The jump process is assumed to be independent of the normal process. The distribution is skewed if µ is not zero and the direction of skewness is the same as the sign of µ. The distribution is leptokurtic if λ is greater than zero. The normal-jump distribution of weight discrepancy W is given by () Z( W; θ ) = n= 0 exp n! λ [ W ( α + nµ )] ( σ + N n n exp λ π ( σ + nσ ) N J σ J ), where θ is the vector of five parameters of the normal-jump distribution. The summand tends to zero as n increases. For estimation purposes, the summation is calculated up to n = 7. That Z ( W; θ ) integrates to one was verified using the MAPLE 5.3 mathematical program (Char et al). The function Z ( W; θ ) is nonnegative since each of the terms are nonnegative. Therefore, Z( W; θ ) is a valid p.d.f. The estimation problem is to maximize the following log-likelihood function Ln Ω with respect to the five parameters: α, σ N, λ, µ, and σ J (3) [ W ( α + nµ )] n Ω = 1 exp( λ) n k Ln ln λ exp t 1 n= 0 π ( σ + nσ ) n! ( σ + nσ N J N J = ). The estimation of this nonlinear model requires a numerical optimization algorithm. The parameters are estimated using the NL command and the LOGDEN option in SHAZAM. The numerical Hessian matrix is computed using the NUMCOV option to calculate standard errors. Likelihood odds ratios were used to compare the fit of the proposed normal-jump distribution versus normal, gamma, and Student t-distributions. For example, a test of the 8

9 hypothesis that the normal-jump distribution was superior to a pure normal process is to test that weight discrepancy is normally distributed and, therefore, there is no jump process. The null hypothesis was H : µ = σ J = λ 0. 0 = The distribution of weight discrepancy was then used to estimate the probability of regrading without rounding This probability was given by (4) Γ = 1 Z( W; θ ) Γ NR for a set of initial sample weights S, given W and θ. NR S 495 S 505 dw. To numerically evaluate the integral in (4), the MAPLE 5.3 mathematical program was used (Char et al). The probability of regrading with rounding Γ R involves a multiple integral. Since each of the five grade factors must be rounded individually, the probability distribution of each grade factor conditional on weight and sample size is used as defined in (1). The multiple integral is (5) Γ = 1 I [ j' round( G) ] 99,101] p( G; S, W ) dg Z( W; ) dw R [ θ where I []. is an indicator function that is one when the argument is inside the 99- [99,101] 101% interval and zero otherwise, j' is a 1x5 vector of ones, G is a 5x1 vector of grade factors including sound mature kernels (SMK), sound splits (SS), other kernels (OK), total damage (TD), and hulls expressed in percentages. The round function rounds each argument to the nearest whole number. The function p( G; S, W ) is the joint density functions for the grade factors estimated with (1), and Z( W; S, θ ) is the distribution of 9

10 weight discrepancy. The lower and upper limits of the integrals were 0 and 100, respectively. The multiple integral in (5) was calculated with Monte Carlo integration. Monte Carlo integration uses stochastic simulation to generate random values of weight discrepancy and grade factors conditional on the initial sample weight. The integral then was the percentage of times that regrading was necessary. Ten thousand replications were used. Empirical Results Table 1 shows that rounding introduces noise into the measurement of producer prices. The standard deviation of the difference in producer prices with and without rounding when dividing by 500g was $3.30/ton, and when dividing by the actual sample weight was $3.35/ton. Table shows the estimated parameters for the grade factor equations. Note that the effect of sample weight on grade factors was only significant for SMK and hulls. The other categories made up a small part of total weight and so their equations were estimated less accurately. Weight discrepancy was found to negatively influence SMK. This implies that most of the large errors are in measuring SMK. Table 3 reports the probability of regrading with and without rounding using the empirical p.d.f. Rounding does tend to increase the probability of regrading as sample weights increase. The weakness of the nonparametric method is that estimates are imprecise. Values of the weight discrepancy ranged from 4.0 to 47.70g, with the mean and the variance being.77 and 4.44g. Table 4 shows that all five parameter estimates of the normal-jump distribution of weight discrepancy are statistically significant. The 10

11 process associated with the sample weight lost as dust and kernels suggests that graders making no errors average losing.6g of the sample. The 95% confidence interval around these estimates, g, falls within the allowable tolerance (5g or 1% of the sample weight). The probability of losing 5g when no large error is made was only This result suggests that the range of the allowable tolerance is reasonable. The estimates of the jump process showed 7.05 large errors for every 100 samples, with the mean of this error being 5.90g. The probability of at least one large error is as calculated using the Poisson probability mass function. Figure 1 shows the normal-jump distribution of weight discrepancy given an initial sample weight of 500g. The log-likelihood function values of the normal, Gamma, and Student t- distributions were , , and For the nested test of a normal distribution, the chi-square test statistic of is considerably greater than the critical value of So, the likelihood ratio test leads to the rejection of a normal distribution. Likelihood odds ratios strongly favor the normal-jump distribution over the Gamma, and Student t-distributions. The distribution and the estimates of the grade factor equations were used to calculate the probability of regrading with and without rounding for a set of initial sample weights ranging from 495 to 505g. Table 5 reports the results. Note that when rounding was not used, sample weights above 500g had little effect on the probability of regrading. With rounding, the probability of regrading was greatly reduced when beginning with a 50g or 503g sample. Thus, rounding does provide a strong incentive to use overweight samples. Therefore, there is strong evidence that the policy of rounding increases costs 11

12 due to more frequent regrading and reduces accuracy of grades and prices obtained with the current grading procedure. With a 500g sample and no rounding, 99.6% of the samples that are regraded are ones where the grader made a mistake. With rounding, only 46.9% of the samples that are regraded are ones where the grader made an error 1. Without rounding, 80% of the samples on which a human error occurs are regraded, but with rounding only 63% of them are. Thus, regrading the wrong samples is another source of error created by rounding. The savings from reduced regrading are modest, but they likely are enough to pay for the cost of making a change. Assuming peanut trailers average 5 tons, the 1.6 million tons produced in the United States would provide 30,000 samples to be graded. If all samples were 500g, the decrease in the probability of rounding would be.0348 and so there would be 11,136 fewer regrades. At $9/grade (the rate paid to private graders) this would mean reduced costs of around $100,000/year. At a 5% discount rate the present value of an income stream of $100,000/year would be near $M. Such cost savings would not be fully observed because some graders now begin with samples greater than 500g. The incentive to take overweight samples may be the greatest concern. Biases created by taking overweight samples will not be reduced by aggregation. Graders in some grading stations never use overweight samples. This creates an inequity for producers and increases the buyer s risk. 1 The percentages without rounding are calculated analytically. The percentages with rounding are calculated using a contingency table produced from the Monte Carlo simulations. 1

13 Conclusions The proposed normal-jump distribution seems to be a good model for a process composed of many small errors and occasional large errors. The current use of rounding of grade factors to the nearest whole number introduces noise, increases costs due to more frequent regrading, causes the wrong samples to be regraded, and provides a major incentive for graders to take overweight samples. A very low cost way to improve the peanut grading procedure would be get rid of rounding. Stopping the rounding of grade factors could pay for itself since it would reduce the frequency of regrading Now that we have computers and calculators, rounding to whole numbers is certainly not necessary. 13

14 References Akgiray, Vedat, and Geoffrey Booth. Stock Price Processes with Discontinuous Time Paths: An Empirical Examination. The Financial Review 1(May 1986): Mixed Diffusion-Jump Process Modeling of Exchange Rate Movements. Review of Economics and Statistics 70(1988): Anderson, Kim B. Personal communication, November Char, Bruce W., Keith O. Geddes, Gaston H. Gonnet, Benton L. Leong, Michael B. Monagan, and Stephen M. Watt. First Leaves: A Tutorial Introduction to Maple V. New York: Springer-Verlag, Dowell, Floyd E. Sample Size Effects on Measuring Grade and Dollar Value of Farmers Stock Peanuts. Peanut Science 19(199): Dowell, Floyd E., Charles R. Meyer, and Richard P. Konstance. Identification of Feasible Grading System Improvements Using Systems Engineering. Applied Engineering in Agriculture 10(1994): Oldfield, G.S., R.J. Rogalski, and R.A. Jarrow. An Autoregressive Jump Process for Common Stock Returns. Journal of Financial Economics 5(1977): Oklahoma Federal/State Inspection Service, Agricultural Products Division, Fruit and Vegetable Section, Department of Agriculture. Standard Operating Procedures for Peanut Grading Operations. SOP-46, Oklahoma City, Oklahoma, August 14, Powell, Jr., J. H., H. T. Sheppard, and F. E. Dowell. An Automated Data Collection System for Use in Grading of Farmers Stock Peanuts. Paper presented at the 1994 International Winter Meeting sponsored by the American Society of Agricultural Engineers (ASAE), Atlanta, Georgia, December 13-16, Steigert, K.W., and B.W. Brorsen. The Distribution of Futures Prices: Diffusion-Jump vs. Generalized Beta-. Applied Economics Letters 3(1996): Tsai, Y. J., F. E. Dowell, J. I. Davidson, Jr., and R. J. Cole. Accuracy and Variability in Sampling and Grading Florunner Farmers Stock Peanuts. Oléagineux 48 (December 1993): Tucker, Alan L., and Lallon Pond. The Probability Distribution of Foreign Exchange Price Changes: Tests of Candidate Processes. The Review of Economics and Statistics 70(November 1988):

15 U.S. Department of Agriculture, Agricultural Marketing Service, Fruit and Vegetable Division, Fresh Products Branch. Farmers Stock Peanuts Inspection Instructions, Handbook Update 133, August U.S. Department of Agriculture, National Agricultural Statistics Service. Agricultural Statistics Washington, DC: United States Government Printing Office. Whitaker, Thomas B., J. W. Dickens, and F. G. Giesbretch. Variability Associated with Determining Grade Factors and Support Price of Farmers Stock Peanuts. Peanut Science 18(1991):1-16. Whitaker, Thomas B., Jeremy Wu, Floyd E. Dowell, Winston M. Hagler, Jr, and Francis G. Giesbrecht. Effects of Sample Size and Sample Acceptance Level on the Number of Aflatoxin-Contamined Farmers Stock Lots Accepted and Rejected at the Buying Point. Journal of the Association of Official Analytical Chemists 77 (1994): White, Kenneth J. SHAZAM User s Reference Manual Version 7.0. Second Printing. Vancouver, Canada: McGraw-Hill Book Company,

16 Table 1. Descriptive Statistics of Four Pricing Methods Using Paired Differences on Peanut Producer Prices Standard Description Mean Deviation Minimum Maximum Prices with Rounding/500-g Sample Prices with No Rounding/500-g Sample Paired Differences a Mean Square Error Number of Observations 314 Prices with Rounding/Actual Sample Prices with No Rounding/Actual Sample Paired Differences b Mean Square Error Number of Observations 96 Note: All prices related to observations requiring regrading have been excluded. Variability of prices is defined as the standard deviation of the paired differences. Producer prices are expressed in dollars. Total number of observations is 33. a The observed value of the t-statistic is 1.34 which is less than 1.96 (the critical value for the two-tailed test of the t-statistics with 313 degrees of freedom and a 5% significance level). Thus, we failed to reject the null hypothesis of average producer prices being the same under these two pricing methods. b The observed value of the t-statistic is

17 Table. Parameter Estimates of the Grade Factor Linear Equations with Random Effects Models Standard Parameter Estimate Error t-ratio Sound Mature Kernels (SMK) Intercept Sample Weight Weight Discrepancy Between-Groups Variance Within-Groups Variance Sound Splits (SS) Intercept Sample Weight Weight Discrepancy Between-Groups Variance Within-Groups Variance Other Kernels (OK) Intercept Sample Weight Weight Discrepancy Between-Groups Variance Within-Groups Variance Total Damage (TD) Intercept Sample Weight Weight Discrepancy Between-Groups Variance Within-Groups Variance Hulls Intercept Sample Weight Weight Discrepancy Between-Groups Variance Within-Groups Variance Note: Expected grade factors are expressed in grams and have been calculated with no rounding since weight discrepancy is a continuous variable. 17

18 Table 3. Probability of Regrading Based on the Empirical p.d.f. Method of Grade Factor Determination Size A (500.0g) Size B (501.4g) Size C (503.7g) Size D (505.0g) Rounding/500g Sample No Rounding/500g Sample Rounding/Actual Sample No Rounding/Actual Sample Note: The probability of regrading is calculated by dividing the number of times the sum of total kernels and hulls (grade factors measured) falls outside the % interval by the total number of observations in each size (83). Note that with 100 observations and five regrades, the 95% confidence interval would be

19 Table 4. Parameter Estimates of the Normal-Jump Distribution of Weight Discrepancy in Grams Standard Parameter Symbol Estimate Error t-ratio Mean of the Normal Process α Variance of the Normal Process σ N Jump Intensity λ Mean of the Poisson Jump µ Variance of the Poisson Jump σ J Note: The log of the likelihood function is The number of observations is 33. No rounding of grade percentages is done. 19

20 Table 5. Probability of Regrading in the Peanut Industry Initial Probability of Probability of Sample Regrading Regrading Weight with Rounding without Rounding Note: The initial sample weight is expressed in grams. 0

21 Figure 1. The normal-jump distribution of weight discrepancy in grams for a 500-g sample 1