# Stratified?Formulae"?

Description
There?are?two?assignment?in?total,?you?can?find?a?Word?document?named?”Instructions”?in?attachment?for?detail
“Stratified?Formulae”?would?help?you?to?do?assignment?1.?
1.?https://onlinecourses.science.psu.edu/stat100/node/18
2.?https://www.greenbook.org/marketing?research/non?response?bias
3.?http://surveytoolkit.micronutrient.org/

Assignment 1:
Provide some examples of surveys (perhaps some from your own experience) where non-response is possible. How serious is the issue of non-response in each of your examples? How might one increase the response rate? What might one do to estimate the non-response bias?
The sampling plan below was used in a statistical audit. The columns of the table list the strata indices, strata sizes, sample sizes and a description of the strata.
1 25 25 > \$ 40,000.00
2 250 50 15,000.00 – 40,000.00
3 450 38 5,001.00 – 14,999.00
4 800 32 1,000.00 – 5,000.00
5 2,000 30 <1,000.00
Totals 3,525 175
The means and standard deviations of erroneously paid dollars by strata were (\$10,045, \$623), (\$8,756, \$401), (\$4,265, \$211), (\$809, \$75) and (\$53, \$45), respectively.
1. Calculate an estimate of the average error per case in the sampling frame (population) and a 90% confidence interval for the same.
2. Calculate an estimate of the total dollars in error in the sampling frame (population) and a 90% confidence interval for the same.

Assignment 2:
Based on your calculations in the assignment 1, what restitution amount is the judge likely to set? Why?
An expert for the audited party argued that all 25 cases in strata 1 were included in the sample. This is extremely unlikely if random sampling was used. Furthermore, the sample sizes in the other strata are disproportionate to the strata sizes, so this sample cannot be representative and the estimates based on this sample should be disregarded. What do you think?
The following references may provide some context:
? Heiner, K. W., Fried, A., and Wagner, N., (1984), ?Successfully Using Statistics to Determine Damages in Fiscal Audits?, Jurimetrics, 24, No. 3, Spring.
? Karl W. Heiner, Owen Whitby, (1980) Maximizing Restitution for Erroneous Medical Payments When Auditing Samples. Interfaces 10(4):46-54. (You will find this article in attachment)
? Heiner, K. W., (1995), ?Computerized Interactive Stratification in Statistical Audits:, Mathematics with Vision, Computational Mechanics Publications, Southampton, UK, pp. 199-206.

Maximizing Restitution for Erroneous Medical Payments When Auditing Samples
Author(s): Karl W. Heiner and Owen Whitby
Source: Interfaces, Vol. 10, No. 4 (Aug., 1980), pp. 46-54
Stable URL: http://www.jstor.org/stable/25059952
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INTERFACES Copyright ? 1980, The Institute of Management Sciences
Vol. 10, No. 4, August 1980 0092-2102/80/1004/0046\$01.25
MAXIMIZING RESTITUTION FOR ERRONEOUS
MEDICAL PAYMENTS WHEN AUDITING SAMPLES
Karl W. Heiner
School of Management, Rensselaer Polytechnic Institute, Troy, New York 12181
and
Owen Whitby
SwissRe Holding (N.A.) Inc., New York, New York 10017
Abstract. In auditing of paid Medicaid claims, estimates of payments erroneously
paid to providers are frequently based on statistical samples. When such procedures are
used, hearing officers are often inclined to use the lower end of an interval estimate to
determine restitution by a provider. In this paper, the advantages of estimating from large
samples are weighed against the larger costs involved. A procedure for choosing a sample
size that would maximize net recovery is proposed.
Background
In health care in the United States, most medical expenses are paid by a third
party, usually an insurance company or the state (e.g., Medicaid). As the size of a
health care provider’s insured business becomes large, and increases in volume, the
likelihood of a wrong payment seems to increase. Providers may not be paid or they
may be paid more than once for a service. A payment may be the wrong amount for
the type of service rendered. The greater the cost for a specific service, the more
costly errors become. In some large states (e.g., Illinois, New York), costs of wrong
Medicaid payments are measured in millions of dollars (see letter following article).
Government agencies have been formed to identify causes of wrong payments,
to recoup them, and to take corrective actions. Recoupment of such Medicaid pay
ments from a specific service provider (hospital, clinic, physician, pharmacy, nurs
ing home, etc.) must be based on an audit process which reviews state and provider
accounts and client medical records. The number of paid claims per provider in an
audit period is usually quite large. In those cases where the amount of wrong pay
ment is substantial, an audit of all transactions is often not feasible. Consequently,
state Medicaid auditing units have elected to sample from provider files and estimate
the amount associated with such payments. Based on these estimates, courts and
hearing officers have granted restitution to the state. At times, hearing officers have
conservatively elected to equate restitution to the lower end of a two-sided interval
estimate of the dollars overpaid.
Sampling units are sometimes taken to be transactions. At other times, a case or
Medicaid family is the appropriate sampling unit. In either case, the auditing cost per
unit can be substantial since it involves reviewing state and provider accounting
records and the medical records of the clients involved in the transactions. The cost
of the audit must be balanced against the anticipated amount of restitution. There
fore, the question of the most efficient sample size is important.
STATISTICS?SAMPLING; HEALTH CARE
46 INTERFACES August 1980
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Maximizing Net Recovery
When the lower end of the 95% confidence interval is equated to the amount of
restitution, the cost of increasing the sample size must be weighed against the effect
of an increased sample size on the width of the confidence interval. Larger, more
costly samples yield smaller standard errors, narrower confidence intervals, a larger
expected lower limit to the interval estimate, and therefore, greater expected restitu
tion.
Using the notation in Cochran [ 1977], a 1 ? a confidence interval for the amount
of wrong payment Y is given by
zns ru p zns r??
where a sample of size n is taken from a population oiN units with dispersion
_> = ‘
N
where y = (1/7V)^ y^ z is the(l-a/2)quantile of the standard normal distribution, and
i=i
Y = Wlnytyv /=i
Assuming that audit cost is C = co + nc, where Co is the fixed cost associated
with the audit (e.g., hearing costs, computer costs, etc.) and c is the additional cost
per unit sampled, and the restitution is the lower limit of the confidence interval, then
the net amount recovered, R, equals restitution minus cost, or
R = Y-^? y/l-n/N-c0-cn. (1) yjn
Note that in Expression (1) the point estimate of wrong payments is reduced by
three terms. One is the fixed cost co. The other two terms involve n, the sample size.
As n increases, the auditing cost en increases while the reduction term used to
establish the lower end of the confidence interval decreases. When n is small, and
consequently small with respect to N, small increases in n have a dramatic effect on
decreasing the second term on the right-hand side in Equation (1), while having
relatively little impact on the last term, en. However, for moderate sample size n,
increases in n have little effect on (zNS/n1/2)(l-n/N)112, while still having the same
effect on audit costs. Only as n grows large with respect toN does the reduction due to
(zNSlnll2)(?nlN)112 again become substantial due to (l?nlN)1!2 becoming small,
small.
For 1?a=0.95, Figures 1 through 4 depict the relationship between n and the
variable reduction
zNS
VI- n/N + en.
y/n
INTERFACES August 1980 47
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In these figures, cost per audited case c is expressed in terms of “standard de
viations,” S, of wrong payments. Figures 1 through 4 display this relationship for
N = 100, 500, 1000, and 5000, respectively. The curves in these figures are of two
types; those that are monotonically decreasing (achieving a minimum atn = N), and
those that achieve a minimum at some point? less thanAf. In every instance where a
minimum is achieved ain FIGURE 1. REDUCTION FROM FULL RECOVERY DUE TO CHOICE OF
SAMPLE SIZE (N =100).
^-i-1-1-1-1-1-1 i-1-1-1
0 10 20 ?D 40 SO CO 70 80 SO 100
SAMPLE SIZE
48 INTERFACES August 1980
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FIGURE 2. REDUCTION FROM FULL RECOVERY DUE TO CHOICE OF
SAMPLE SIZE (/V=500).
0 50 100 150 200 250 300 350 400 450 500
SAMPLE SIZE
FIGURE 3. REDUCTION FROM FULL RECOVERY DUE TO CHOICE OF
SAMPLE SIZE (#=1000).
INTERFACES August 1980 49
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FIGURE 4. REDUCTION FROM FULL RECOVERY DUE TO CHOICE OF
SAMPLE SIZE (#=5000).
I i i-1 i ? i-I-‘ <
0 SOD 1000 1500 2000 2S0O 9000 3S0O 4000 4500 SOO0
SAMPLE SIZE
When the sum of the two terms depending on n is minimized at n = N, it is due
to the low cost of an audited case, and in situations like these an audit of the entire
paid claims universe is advisable. For some combinations of population size and cost
(e.g.,W=100, c = AS, or N = 500, c =.25, orN = 1000, c= A5S,orN =
5000, c = .085), there appears to be a value of n beyond which there is little difference
in net recovery. However, for larger values of c, the process of accurately determining
the n to minimize reduction of restitution and consequently to maximize net recovery
becomes important.
Setting the partial derivative of net recovery R, with respect to n equal to zero,
net recovery is maximized by solving the following equation for n, when such a
solution exists. (When no solution exists, the optimal sample size isN.)
50 INTERFACES August 1980
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Straightforward calculus verifies that if this equation has a root n ^ Nil, then that
root is the optimal sample size. If the equation has no real root, or has no root less
than or equal to Nil, then the optimal sample size is n =N. The above equation may
be solved numerically, or the equivalent quartic
c2n4 – c2Nn3 + 0.25 Z2N3S2 =0
may be solved explicitly, or the following tabular method may be used.
Calculating Optimal Sample Size
Optimal sample size is determined by referring to Table 1, where optimal
sample size is a function of a parameter y. y depends on c, the cost to audit a unit
(usually a Medicaid case or a transaction), on N, the number of units in the popula
tion (the provider’s business during an audit period) and onS2, the “variance” of the
amount erroneously paid per unit in the population. It will be necessary to estimate c
and S2 from previous experience or pilot auditing.
TABLE 1. SAMPLING FACTORS FOR MAXIMIZING NET RECOVERY
BASED ON THE LOWER END OF A CONFIDENCE INTERVAL.
y* Xa y k
.01 .0100 .21 .2290
.02 .0201 .22 .2412
.03 .0303 .23 .2535
-04 ,0406 -24 .2661
.05 .0509 .25 .2788
.06 .0613 .26 .2917
.07 .0718 .27 .3048
.08 .0823 .28 .3181
.09 .0930 .29 .3317
.10 .1037 .30 -3455
.11 .1146 .31 .3597
.12 .1255 .32 .3741
.13 .1365 .33 .3889
-14 -1477 -34 .4040
.15 .1589 .35 .4196
.16 .1703 .36 .4356
.17 .1818 .37 .4522
.18 .1934 .38 .4694
.19 .2051 .39 .4873
.20 .2170 .39685 .5000
ay = (z2S2l4Nc2y/3 and a is the sampling fraction, where z is the ( 1 -a/2) quantile of the standard normal
distribution, S2 is the “variance” of wrong payments per sampling unit, N is the size of the universe, ande is
the cost of auditing one unit.
To find the optimal sample size, first compute y:
y = (z2S2/4Nc2) l’ (2)
where z is the (l-a/2) quantile of the standard normal distribution. If
y =?r21* = .39685, then? = Nil.
If y is greater than 4″2/3 = .39685, then n = N. Otherwise, the proportion of the
population to be included in the sample is X as found in Table 1. That is, n = (X) N.
Net recovery may be estimated by using Equation (1).
INTERFACES August 1980 51
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It should be noted that this methodology only determines the cost-efficient
sample size, given that it has been decided that an audit will be conducted. In order to
decide whether to conduct the audit, an estimate of the net recovery is useful. An
estimate of net recovery must include estimates of the total overpayment Y and fixed
Example 1. An audit is planned for provider A. The audit will be conducted on
randomly selected cases and each case will be audited with respect to each aspect of
the case. From past experience, the cost per case audited is assumed to be \$45, which
includes salaries of Health Department physicians as well as auditors. For this type of
audit \$40 is a typical figure for the standard deviation of overpayments of audited
cases. There are 1,000 cases in the provider’s practice for the audit period of interest.
To determine the sample size that maximizes net recovery, the correct parameter
y is calculated from Equation (2), and the proportion of the population to be sam
pled, X, is obtained from Table 1. (Note: A more finely tabulated version of Table 1
has been used for the examples. Proper interpolation in Table 1 would yield the same
results.)
For a 95% confidence interval z is approximately 2, and
y = (z2S2lANc2yi
= [(If (40)2/4(1000) (45)2]1’3
= (6400/8,100,000) !/3
= (.0008) */?
= .092.
Consequently, from Table 1, .095 of the population should be used in the sample to
maximize net recovery, and optimal sample size is
n = (.095) (1000) = 95.
Example 2. During a particular audit period Pharmacy Z filled approximately
8600 prescriptions, for which it billed Medicaid \$42,000. From a previous audit of
this provider’s records it was estimated that the cost per transaction audited was
approximately \$1.25, since on the average 40 transactions could be audited in one
\$50 man-day. The standard deviation of the amount erroneously paid per transaction
was estimated to be \$5. To find the optimal size for a 95% confidence interval, take
y=(z2S2l4Nc2yt3
= [(4)(25)/(4)(8600)(1.25)2]1/3
= (.0019)1/3
= .123.
Therefore, from Table 1, X is .129 and the optimal sample size is . 129/V or 1109.
For this pharmacy, it is believed that approximately 60% of \$42,000 is in error.
Consequently, the estimated net recovery
R = Y-z? VI ? ~c0~cn Jn N
(8600X5) = 25,200-(2)V7Zi; Vl-.129-c0-(1.25)(1109) V?T?9
= 25,200- 2,410 -c0- 1,386
= 21,404-c0.
52 INTERFACES August 1980
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Net recovery is estimated to be \$21,404 less fixed costs.
Example 3. Clinic S processed 75,000 patients over a two-year period, for which the
state paid \$35 per visit, or \$2,625,000. It requires 25 man-days at \$50 per day to
review 100 transactions, i.e., the cost per audited transaction is \$12.50. From a pilot
audit, it is estimated that 15% of the transactions should not have been paid. Con
sequently, the standard deviation of the distribution of erroneous payments is 35
((.15)(.85))1/2or\$12.50.
To determine the optimal sample size for a 95% confidence interval, calculate y
from Equation (2):
y = [(4) (12.50)2/(4) (75,000) (12.50)2]1/3
= .024.
From Table 1, the sample size should be .024N or 1800. This sample size could be
expected to yield a net recovery of
NS I n~ R = Y-z-^r sj—c^-cn sfn~ N o
= (.15)(2,625,000) – 2 i75*00^2-50) ^1 – .024-c0 – (12.50)(1800) VI800
= 393,750 – 43,661 – c0 – 22,500
= 327,589 -c0,
or \$327,589 less fixed costs.
A similar clinic, clinic P, only processed 3,500 patients. For clinic P,
y = [(4) (12.50)2/(4) (3500) (12.50)2]1/3
= .066.
Table 1 indicates that the optimal sample size is .06SW or 238. The net recovery for
clinic P would be
R = (.15)(35)(3500) – 2 (35Q?/_2 – y/ – .068 – c0 – (12.50)(238) V238
= 18,375- 5476 – c0 -2975
= 9,924 -c0.
Conclusions
Selecting the appropriate sample size when auditing paid claims for determina
tion of restitution can result in considerable saving. The procedure outlined is being
used by the Division of Audit and Quality Control, New York State Department of
Social Services, to determine sample sizes for auditing Medicaid service providers’
paid claims. During the first year of this process, the state has recovered in excess of
1.5 million erroneously paid dollars.
INTERFACES August 1980 53
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NEW YORK STATE
DEPARTMENT OF SOCIAL SERVICES
40 NORTH PEARL STREET, ALBANY, NEW YORK 12243
BARBARA B. BLUM
June 1, 1979
Dr. R. E. D. Woolsey
Institute for Operations Research
Colorado School of Mines
Dear Professor Woolsey:
This is to document that the paper by Karl Heiner and Ohen Whitby
describes a procedure used by the New York State Department of Social
Services for choosing sample size in certain Medicaid expenditure
reviews. The procedure is used primarily for audits of hospitals,
physicians, and pharmacies in the Western Region of the State, which
includes Buffalo, Rochester, and Syracuse. The procedure is applicable
in this region because the potential number of audit targets is lower
in relationship to the number of auditors, than in New York City. For
the-New York City situations Drs. Heiner and Whitby are currently
developing a procedure for allocating auditor time to audits that would
maximize net recovery of erroneous Medicaid payments.
The procedure described in this paper has been particularly useful
in minimizing the loss of potential recovery due to undersampling and
the unnecessary expenditures of auditor time due to oversampling. This
technique is a part of an extensive audit operation which recovers
several million dollars annually in Medicaid overpayments.
Yours tjru 1 y, 1
f / John M. ?Jordan
^ PrincipaKAccountant
JMJ:dat
54 INTERFACES August 1980
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We wish to find ??
????, a stratified estimate of the total disallowance dollars in the
universe. This estimate and its standard error are based on the individual strata means, ???,
?? ,variances
?, and weights,??. The weights ?? ? ??
? , where
??
?
L
h
N Nh
1
and L represents the number of strata.
The point estimate of the mean or average disallowance in the stratified sample is:
?????? ????
?
???
??? ? ???
?
?
???
???
The stratified point estimate of the total disallowance dollars in the universe
??
???? ? ? ??????.
.
The interval estimates depend on the variance of ?????? and the variance of ??
????, which
are, respectively,
?????????? ? ?
?? ? ????? ? ??? ??
?
??
?
???
and
?????
????? ? ? ????? ? ??? ??
?
??
?
??? .
The ?1???100% confidence interval estimates for the mean disallowance and the total
disallowance are:
?????? ? ?????,??? ?/ ? ?????????, ?????? ? ?????,??? ?/ ? ?????????
and
??
???? ? ?????,??? ?/ ? ????
?????, ??
???? ? ?????,??? ?/ ? ????
?????,
respectively.

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