# Stratified?Formulae"?

Description

There?are?two?assignment?in?total,?you?can?find?a?Word?document?named?”Instructions”?in?attachment?for?detail

description?of?these?two?assignment.?Please?write?one?page?for?each?assignment.?

I?provide?some?links?and?context?that?could?help?you?to?do?this?assignment.?One?attachment?is?named?as

“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

Published by: INFORMS

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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

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

costeo (overhead).

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

Golden, Colorado 80401

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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