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Effects of Rounding on Data Quality

Lawrence H. Cox, Jay J. Kim,Myron Katzoff, Joe Fred Gonzalez, Jr.

U.S. National Center for Health Statistics

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Outline

I. Introduction

II. Four Rounding Rules

III. Mean and Variance

IV. Distance Measure

V. Concluding Comments

VI. References

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I. Introduction

Reasons for rounding

• Rounding noninteger values to integer values for statistical purposes;

• To enhance readability of the data;

• To protect confidentiality of records in the file;

• To keep the important digits only.

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• Purpose of this paper:

Evaluate the effects of four rounding methods on data quality and utility in two ways:

(1) bias and variance;

(2) effects on the underlying distribution of the data determined by a distance measure.

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B : Base

: Quotient

: Remainder

x xx q B r

xq

xr

( ) ( )x xR x q B R r

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Types of rounding:

• Unbiased rounding: E [R (r) |r] = r

Example

E[R(3)= 0 or 10|3] = 3

• Sum-unbiased rounding: E [R (r)] = E (r)

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II. Four rounding rules

1. Conventional rounding

1.1 B even

Suppose r = 0, 1, 2, . . . ,9.

In this case B = 10.

If r (B/2), round r up to 10

else round r down to zero (0).

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1.2 B odd

Round r up to B when

r

Otherwise round down to zero.

Example:

When B = 5, round up r = 3 and 4 to 5;

Otherwise, round down.

1

2

B

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Assumptions for r and q

r follows a discrete uniform distribution;

q follows lognormal, Pareto of second kind or multinomial distribution.

Thus,

x has some mixed distribution, but the term qB dominates.

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2. Modified Conventional rounding

Same as conventional rounding, except whenrounding 5 (B/2) up or down with probability ½.

3. Zero-restricted 50/50 rounding

Except zero (0), round r up or down with probability ½.

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4. Unbiased rounding rule

Round r up with probability r/B,

and

Round r down with probability 1 - r/B

Example:

r = 1, P [R(1)=B] = 1/10P [R(1)=0] = 9/10

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III. Mean and variance

III.1 Mean and variance of unrounded number

r = 0, 1, 2, 3, . . . , B-1.

P (r) = and

E (r) =

= .

1

1

1B

r

rB

1

2

B

1

B

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=

In general when r and q are independent,

22 1

( ) ( )12

BV x B V q

2 1( )

12

BV r

( ) ( | ) ( | )V x V E x q E V x q

12 2

1

1( )

B

r

E r rB

( 1)(2 1)

6

B B

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III. 2 Conventional rounding when B is even

for unrounded number.

1( )

2

BE r

[ ] Pr[ ( ) ] 0 Pr[ ( ) 0]

2

E R r B R r B R r

B

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2

[ ]4

BV R r

2 1( ) .

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BV r for unrounded number

2 2 1( )

2E R r B

1[ ] 3 ( )

12V R r V r

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2

2( ) ( ) ,4

BV R x B V q and

2

2 1( ) ( ) .

4

BMSE R x B V q

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III.3 Conventional rounding when B is odd

Note ,

(B-1)/2 out of B elements can be rounded up.

is sum unbiased, and

for unrounded number

1[ ]

2

BE R r

2 1[ ( )] .

4

BV R r

2 1( )

12

BV r

1P ( )

2

BR r B

B

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P [R (r) = B] for modified conventional rounding.

P [R (r) = B] for 50/50 rounding: same as above.

No. of elements which can be rounded up: B-1.

All B elements. Probability of rounding up is ½.

1P ( )

2 2

BR r B

B

1 1 1P ( )

2 2 2

BR r B

B B

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P [R (r) = B] for unbiased rounding

Modified conventional rounding,

50/50 rounding

and

unbiased rounding

have the same mean, variance and MSE

as the conventional rounding with odd B.

1

1

1 1P ( )

2

B

r

r BR r B

B B B

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IV. Distance measure

Assume that when x = 0, U = 0.

Define

2

2

[ ( ) ]

[ ( ) ]x x

R x xU

x

R r r

x

1,

0, .x

x

if r is rounded up

otherwise

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Reexpressing the numerator of U, we have

With conventional rounding with B=10,

Then we have

2( )x xB r

21

0

( )( | ) | ( ).

x

x

x xx

B rE U x x P

x

1 5, 6, . . , 9x xwith r

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Expected value of U

We define

2

( ) ( | ) |

( )| ( ) | ( ) ( ).

x x x

q r

x xx x x

q r

E U E E E U x q

B rx P q P r P q

x

21

10

( )( | ) | | ( ) | ( ).

x x

x xr x x

r x x

B rU E E U x q x P q P r

q B r

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IV.1 Conventional rounding with B even

which can be expressed as

2 2/ 2 1 1

1 / 21

( ) 1

x x

B Bx x

x x x xr r B

r B rU

q B r q B r B

2/ 2 1 1

11 / 2

( ) 1

x x

B Bx

xr r B x

B rU r

r B

2 2/ 2 1 1

1 / 2

( ) 1

x x

B Bx x

x x x xr r B

r B r

q B r q B r B

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Sum of 1/r terms.

Recall the harmonic series:

The upper and lower bounds for harmonic series

2/ 2 1 1

11 / 2

( ) 1[ ]

x x

B Bx

xr r xB

B rU r

r B

2 2/ 2 1 1

1 / 2

( ) 1[ ]

x x

B Bx x

x xr r B

r B r

q B q B B

ln( 1) 1 ln( )nn H n

1 1 11 2 3 . .nH n

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The upper bound for the first term of is

The second term of is

Note that the second term of E(U) is

1ln

2

2( 1)

2

BB

B

B

1U

/ 2 1 12 2

21 / 2

1( )

1x

x x

B B

q x xr r Bx

E r B rBq

2 2

12

B

B

1U

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IV.2 Modified conventional rounding with even B

This has the same E(U) as conventional rounding.

IV.3 50/50 rounding

The first term of is

The second term of is

1U

1ln ( 1)

2 2

BB

22 3 1

6

B B

B

1U

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IV.4 Unbiased rounding

The first term of is

The second term of is

1U

1

2

B

2 1

6

B

B

1U

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IV.5 Comparison of four rounding rules

Conventional or

Mod. Conven.

50/50 Unbiased

Term

1

Term

2

2( 1) 1ln

2 2

B BB

B

2 2 1

12

BE

B q

1ln( 1)

2 2

BB

22 3 1 1

6

B BE

B q

1

2

B

2 1 1

6

BE

B q

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Comparison of four rounding rulesB = 10

Conventional or

Mod. Conven.

50/50 Unbiased

Term

1 2.61 11.49 (4.4) 4.5 (1.7)

Term

2 .85 2.85 1.651

Eq

1E

q

1E

q

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Comparison of four rounding rulesB = 1,000

Conventional or

Mod. Conven.

50/50 Unbiased

Term

1 194 3,454 (18) 500 (2.6)

Term

2 83 323 1661

Eq

1E

q

1E

q

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IV.6 E(1/q) for log-normal distribution

Suppose

and

Then, x has a lognormal distribution, i.e.,

2( , )y N

2

12

ln12( | , ) 2 , 0

x

f x x e x

yx e

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Let

Then

which is equivalent to

2

1( | , )c f x dx

21

1 1 1| 1, , ( )E q f q dq

q c q

212

11e

c

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IV.6 E(1/q) for Pareto distribution of the 2nd kind

The Pareto distribution of the second kind is

In the above k = min(q).

Let

1( ) , 0, 0

a

a

akf q a q k

q

1( | 0, 0)c f x a q k dx

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IV.7 Upper bound for E(1/q) for multinomial distribution

The multinomial distribution has the form

= 0,1,2,

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( 1)

a kE

q c a

1 2 1 21

1

!( , , . . | , , . . )

!

i

kq

k k iki

ii

nf q q q p p p p

q

iq

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In binomial distribution, we let

When is truncated at 1 from below, we have

Note that

for all i.

1s p

1 1 3

1 ( 1)( 2)i i i i

E E Eq q q q

( )if q

1( | 1, , )

1in ni

i i ni

nf q q n p p s

ns

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In general,

Using the above relationship, we have the following

does not generate any term having or .

Hence,

1 23

1|E q q

q

1 2 3 3 1 2 2 1 1( ) ( | ) ( | ) ( )P q q q P q q q P q q P q

1 2 11 2 3 1 2 3

1 1 1 1| |E E E E q q q

q q q q q q

1q 2q

1 2 11 2 3 3 2 1

1 1 1 1| |E E q q E q E

q q q q q q

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The upper bound of the expected value is

Let be the size of the category j andjn

1

1

1

1

2

211 2

5( 1)( 2) 2( 2) 6

2( 1)( 2) (1 )

1

i

jj

i

jj

n

i i i

n

i i

k

ik

n n p s n p

n n p s

Eq q q

1 21 1 11 2 1 1 1

2 2 2

1 1 3

1 ( 1)( 2)

1 3 1 3

1 ( 1)( 2) 1 ( 1)( 2)

kq q qk

k k k

Eq q q q q q

q q q q q q

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V. Concluding comments

Various methods of rounding and in some applications various choices for rounding base B are available.

The question becomes: which method and/or base is expected to perform best in terms of data quality and preserving distributional properties of original data and, quantitatively, what is the expected distortion due to rounding?

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The expected value of U, the distance measure, is intractable, so we derived its upper bound.

The expected value of 1/q is also intractable for a multinomial distribution. So we derived an upper bound. There should be room for improvement.

This paper provides a preliminary analysis toward answering these questions.

In summary,

• In terms of bias, unbiased rounding is optimal.

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• In terms of the distance measure, conventional or modified conventional rounding performs best.

• In terms of protecting confidentiality, 50/50 rounding rule is best.

VI. References

Grab, E.L & Savage, I.R. (1954), Tables of the Expected Value of 1/X for Positive Bernoulli and Poisson Variables, Journal of the American Statistical Association 49, 169-177.

N.L. Johnson & S. Kotz (1969). Distributions in Statistics, Discrete Distributions, Boston: Houghton Mifflin Company.

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N.L. Johnson & S. Kotz (1970). Distributions in Statistics, Continuous Univariate Distributions-1, New York: John Wiley and Sons, Inc.

Kim, Jay J., Cox, L.H., Gonzalez, J.F. & Katzoff, M.J. (2004), Effects of Rounding Continuous Data Using Rounding Rules, Proceedings of the American Statistical Association, Survey Research Methods Section, Alexandria, VA, 3803-3807 (available on CD).

Vasek Chvatal. Harmonic Numbers, Natural Logarithm and the Euler-Mascheroni Constant. See www.cs.rutgers.edu/~chvatal/notes/harmonic.html