1 Effects of Rounding on Data Quality Lawrence H. Cox, Jay J. Kim, Myron Katzoff, Joe Fred Gonzalez,...
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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).
5
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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( ) .
12
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,
11
( 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