Statistics & Probability Letters 18 (1993) 415-420

North-Holland

2 December 1993

A note on constructive procedure for unbiased controlled rounding

Neeraj Tiwari S.D. (P. C.) College, Muzaffamagar, U.P., India

A.K. Nigam Lucknow Uniuersi@, Lucknow, India

Received August 1992

Revised March 1993

Abstract: The technique of controlled rounding is to replace each entry of an array by an adjacent integer multiple of the rounding

base, preserving the additive structure of the array. The controlled rounding is said to be zero-restricted, if it also satisfies the

condition that the values which are already the integer multiples of the base, remain unchanged. The main purpose of controlled

rounding is to ensure confidentiality of aggregate statistics. Notable amongst the various authors, who have thoroughly discussed

the problem of controlled rounding, are Nargundkar and Saveland (19721, Dalenius (1981), Ernst (1981), Cox and Ernst (1982),

Causey, Cox and Ernst (1985) and Cox (1987). In a recent work on controlled rounding, Cox (1987) has developed a procedure for

unbiased controlled rounding in two dimensions.

The purpose of this note is to improve upon the procedure of Cox (1987), so that it terminates in fewer steps. The example in the case of two-way statistical table, solved by Cox (1987), has been reconsidered to highlight the utility of the proposed procedure.

Keywords: Controlled rounding; controlled selection; zero-restrictedness; tabular array

1. Introduction

The rounding of a tabular array, A, is a method to replace each entry of A by an adjacent multiple of the rounding base b, that is, an entry ‘u’ is rounded to either b[a/b] or b([a/b] + 11, where [a] denotes the integer part of ‘a’. The rounding is said to be controlled, if the rounded array also satisfies the additivity condition along the row, column and grand totals. Zero-restrictedness is ensured through the condition that the integer multiple of ‘b’ remains unchanged.

Cox and Ernst (1982) have used the transportation theory in linear programming to solve the two-dimensional controlled rounding problem. Causey, Cox and Ernst (1985) have further investigated the method of Cox and Ernst and have used it to solve the two-dimensional controlled selection problem, originated by Goodman and Kish (1950). Cox (1987) further discussed the problem of controlled rounding and provided a constructive procedure for unbiased controlled rounding to solve the two-di- mensional controlled rounding problem. The procedure given by Cox (1987) is efficient and can easily be implemented, even by a person not having sufficient knowledge of higher mathematics and programming. Throughout, we follow the notations and definitions of Cox (1987).

Correspondence fo: Neeraj Tiwari, Nandan Villa, Adarsh Colony, Pachenda Road, Muzaffarnagar 251001, U.P., India

0167.7152/93/$06.00 0 1993 - Elsevier Science Publishers B.V. All rights reserved 415

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Let

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be a two-way table, whose internal entries (aij’s) are all real and a,,,+ i, a,, l,j and a,, i++ i are row, column and grand totals, respectively. We are interested to find the unbiased zero-restricted controlled rounding l?(A) of A, whose internal entries denoted by R(a) and totals are rounded values of internal entries and totals of A, respectively.

2. The procedure

Without altering the underlying concepts and definitions of Cox (1987), we slightly modulate procedure, so that it terminates in fewer steps.

Assuming that aij lies between 0 and 1 (that is, 0 G aij G l), A can be re-written in a larger equivalent two-way table C, whose internal entries (cij’s) are given by

the

the

but

I taij),xn

(Cii)h+l)x(n+l)= (1 -f(a,+,,j))lxn (l -f(ai,n+l))mXl

(f(am+l,n+l)) ’ 1x1 I where f(aij) denotes the fractional part of aij and 0 G cij G 1.

Adjoining to C its total entries, C can alternatively be re-written as an additive two-way table:

i

([1+%l+11),.1

i ILL 1+-a m+l,jl-[“m+l,n+ll

j=l ) 1 1X1

([ l+a m+l,jl)lxn LX, 1x1

According to Step 2 under Section 3 of Cox (1987) each of the row-column (or column-row) path of fractions, P, contains at least four entries and P gets terminated whenever it returns to the row or column, with which the cycle started. This appears to be arbitrary and does not lead to decide about the number of entries P should contain at the cycle L of each iteration. A distinct improvement to this procedure is proposed under the following steps:

Step 1. If C contains no fractions, then construct the zero-restricted controlled rounding of A, as in Cox (1987).

Step 2. Choose any fraction cij in C. Beginning with the entry (i, j), construct the alternating row-column (or column-row) path of fractions, P, of maximum length by allowing the path P to contain as much entries as it could, so that it terminates the moment it returns to the row or column of the starting entry. Clearly, as in Cox (1987), P contains at least four entries and at some stage, P either becomes a cycle containing (i, j) or a cycle, beginning at the first intersection of this row or column with P and ending at P’s terminal entry, is formed. Let L denote the first cycle formed from P.

The remaining steps are the same as those in Cox (1987) and, therefore, the procedure ensures unbiasedness. It is clear that, like Cox’s procedure, the proposed procedure also converges in fewer iterations than the number of fractions in the original C. This is so because at each iteration, at least one

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fraction in C is transformed to an integer, whereas the integers remain unchanged. The proposed procedure may even converge in fewer iterations than required in Cox’s procedure because a path of maximum length handles much more fractions at each iteration, than the path suggested by Cox. No mathematical proof can be given for establishing the superiority of the proposed procedure in terms of faster convergence. However, an attempt has been made in Section 3 to demonstrate this empirically.

3. An example: A comparison

We consider the example solved by Cox (1987) in Section 5 of his article, which was borrowed from Cochran (1977), to demonstrate the superiority of the proposed procedure over that of Cox’s procedure.

The following is the two-dimensional base 1 controlled rounding problem discussed by Cox (1987):

~~If~~I~ (3.1)

Eliminating the integral part of the internal entries, the corresponding table C is

0.91 0.27 0.03 0.55 0.24 0.61 0.49 0.79 0.42 0.69 0.36 0.55 0.30 0.49 0.30 0.24 0.18 0.36 0.36 0.86 0.18 0.12 0.30 0.49 0.91 0.70 0.39 0.22 0.69 0

(3.2)

3 2 2 3 3 113

In the following steps, we do not reproduce the total entries, as they remain fixed.

Iteration 1. L: (1, 11, (1, 21, (2, 21, (2, 31, (3, 3),(3, 4),(4, 41, (4, 51, (5, 51, (5, 4),(6, 4),(6, 3),(5, 31, (5, 2), (4, 2), (4, 1). d_= 0.14, d+= 0.09, p+= 0.6087, p-= 0.3913. Selecting d, and transforming C, we get,

Iteration 2. L: 0, 21, 0, 31, (2, 31, (2, 4), (3, 41, (3, 51, (4, 51, (4, 41, (5, 41, (5, 31, (6, 31, (6, 11, (5, 11, (5, 2). d_= 0.13, d+= 0.03, 0.8125, p+= p_= 0.1875. Selecting d, and transforming C, we get,

1 0.21 0 0.55 0.24 0.61 0.58

Ccij) = I 0.36 0.55 0.73 0.39 0.69 0.39 0.43 0.27

0.15 0.27 0.36 0.42 0.80 0.21 0 0.36 0.43 1 0.67 0.39 0.16 0.78 0

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Iteration 3. L: (1, 21, (1, 41, (2, 41, (2, 51, (3, 51, (3, 4), (4, 4), (4, 31, (5, 31, (5, 11, (6, 0, (6, 2). d_= 0.21, d+= 0.21, p+ = 0.5000, p_= 0.50000. Selecting d, and transforming C, we get,

(‘ii> =

1 0.42 0 0.34 0.24 0.61 0.58 0.73 0.60 0.48 0.36 0.55 0.39 0.22 0.48 0.15 0.27 0.15 0.63 0.80 0 0 0.57 0.43 1 0.88 0.18 0.16 0.78 0

For the sake of brevity, the matrix portion of C for the next 10 iterations has not been produced, however, other relevant details about the different iterations are given for completeness and clarity.

Iteration 4. L: (1, 21, (1, 4), (2, 41, (2, 31, (3, 31, (3, 11, (4, 11, (4, 3), (5, 31, (5, 4), (6, 41, (6, 11, (2, 11, (2, 2). d_= 0.12, d+= 0.15, p+= 0.4444, p_= 0.5556. Select d_.

Iteration 5. L: (1, 21, (1, 41, (2, 41, (2, 51, (3, 51, (3, 4), (4, 4), (4, 31, (5, 31, (5, 41, (6, 4), (6, 2). d_= 0.30, d+= 0.18, p+= 0.6250, p_= 0.3750. Select d_.

Iteration 6. L: (1, 41, (1, 5), (2, 51, (2, 3), (3, 31, (3, 2), (4, 21, (4, 0, (2, 0, (2, 21, (6, 21, (6, 31, (5, 3), (5, 4). d_= 0.15, d,= 0.03, p+= 0.8333, p_= 0.1667. Select d_.

Iteration 7. L: (1, 41, (1, 5>, (2, 51, (2, 21, (3, 2), (3, 0, (4, 11, (4, 2), (6, 2), (6, 31, (4, 3), (4, 4). d_= 0.15, d+= 0.12, p+= 0.5556, p_= 0.4444. Select d,.

Iteration 8. L: (1, 41, (1, 51, (2, 5), (2, 2), (3, 2), (3, 11, (4, 11, (4, 31, (6, 3), (6, 4). d_= 0.19, d+= 0.18, p+= 0.5135, p_= 0.4865. Select d,.

Iteration 9. L: (1, 4), (1, 5), (2, 5), (2, 2), (6, 2), (6, 31, (4, 3), (4, l), (3, 11, (3, 5), (4, 51, (4, 4). d_= 0.18, d+=0.07, p+=O.7200, p-=0.2800. Select d,.

Iteration 10. L: (1, 4), (1, 5), (3, 5), (3, 31, (6, 3), (6, 21, (2, 21, (2, 11, (4, l>, (4, 4). d_= 0.11, d+= 0.02, p+= 0.8462, p_= 0.1538. Select d,.

Iteration 11. L: (2, l), (2, 2), (6, 21, (6, 3), (4, 31, (4, 4), (3, 41, (3, 51, (4, 5), (4, 1). d_ = 0.32, d+= 0.12, p + = 0.7273, p _ = 0.2727. Select d _.

Iteration 12. L: (2, 2), (2, 41, (3, 41, (3, 3), (4, 3), (4, 51, (3, 51, (3, l), (4, 11, (4, 4), (6, 4), (6, 2). d_= 0.18, d+= 0.10, p+= 0.6429, p_= 0.3571. Select d,.

Iteration 13. L: (2, 21, (2, 41, (3, 4), (3, 11, (4, 11, (4, 31, (6, 31, (6, 2). d_= 0.30, d, = 0.08, = 0.7895, p+ p_ = 0.2105. Select d,.

Iteration 14. L: (3, l), (3, 51, (4, 51, (4, 4), (6, 41, (6, 31, (4, 31, (4, 1). d_= 0.07, d+= 0.34, 0.1707, p+= p_= 0.8293. Selecting d_ and transforming C, we get,

100 1 0 011 0 1

(‘ij> =

000 1 1 1 0 0.79 0.21 0

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Iteration 15. L: (3, 41, (3, 5), (4, 5), (4, 3), (6, 31, (6, 4). &=0.38, d+= 0.21, p+=O.6441, p-=0.3559. Selecting d, and transforming C, we get,

(‘ij) =

-1001 0 0110 1 0 1 0 0.59 0.41 1 0 0 0.41 0.59 0001 1

-1010 0

Iteration 16. L: (3, 41, (3, 51, (4, 51, (4, 4). d_= 0.59, d+= 0.41, p+= 0.5900, p_= 0.4100. Selecting d, and transforming C, we get,

(‘ii) =

1 0 0 1 0 0 1 1 0 1 0 1 0 1 0 1 0 0 0 1 0 0 0 1 1 1 0 1 0 0

Transforming back to A and reproducing the integer part, unbiased zero-restricted controlled rounding of (3.1) as follows:

1111 4 0110 2 0101 2 1000 1 0001 1

2 3 2 3110

It is, therefore, noticed that by choosing the path of maximum

as compared to 19 needed by Cox (1987) to solve the same problem.

length, only 16 iterations are required two-dimensional controlled rounding

On comparison of the proposed procedure with that of Cox, it is seen that the two entries have been rounded here as against one, after first iteration; four entries have been rounded against two, after second iteration; and so on.

which was left before, we obtain an

The example given in (3.1) was also solved in two more ways by changing the path P at different

iterations, while keeping it of maximum length. In each case, the procedure terminates again in 16 iterations. The details of the two solutions can be obtained from the authors.

Acknowledgement

The authors are grateful to the referee and the editor for their valuable suggestions and constructive comments which led to considerable improvement in presentation of this work.

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References

STATISTICS & PROBABILITY LETTERS 2 December 1993

Causey, B.D., L.H. Cox and L.R. Ernst (19851, Applications of

transportation theory to statistical problems, J. Amer. Statist. Assoc. SO, 903-909.

Cochran, W.G. (1977), Sampling Techniques (Wiley, New York,

3rd ed.).

Cox, L.H. and L.R. Ernst (19821, Controlled rounding, IN-

FOR 20, 423432. Cox, L.H. (1987), A constructive procedure for unbiased con-

trolled rounding, J. Amer. Statist. Assoc. 82, 520-524.

Dalenius, T. (1981), A simple procedure for controlled round-

ing, Statist. Tidskr. Statist. Sweden 3, 202-208. Ernst, L.R. (1981), A constructive solution for two-dimen-

sional controlled selection problems, in: Proc. of the Sur- uey Res. Methods Sect. (Amer. Statist. Assoc., Washington,

DC) pp. 61-64.

Fellegi, I.P. (1975), Controlled random rounding, Survey Methodology 1, 123-135.

Goodman, R. and L. Kish (19501, Controlled selection - A

technique in probability sampling, J. Amer. Statist. Assoc. 45, 350-372.

Nargundkar, M.S. and W. Saveland (1972), Random rounding

to prevent statistical disclosures, in: Proc. of the Sot. Statist. Sect. (Amer. Statist. Assoc., Washington, DC) pp.

382-385.

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