Estimating the effect of early discharge policy on readmission rate. An instrumental variable approach

doi:10.4236/health.2010.25075

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Vol.2, No.5, 504-510 (2010)

doi:10.4236/health.2010.25075

Health

Openly accessible at

Estimating the effect of early discharge policy on

readmission rate. An instrumental variable approach

Eugenia Amporfu

Kwame Nkrumah University of Science and Technology, Kumasi, Ghana; [email protected]

Received 23 January 2010; revised 7 February 2010; accepted 8 February 2010.

ABSTRACT*

Early discharge policy, common in the devel-

oped countries, refers to the reduction of hos-

pital length of stay as a way of reducing the cost

of care. The effect of the policy on quality of

care has received a lot of attention in the litera-

ture. Some of the earlier papers have ignored

the endogeneity of length of stay in the read-

mission equation, an approach that could lead

to inconsistent estimation. This study develops

a statistical technique for the consistent estima-

tion of the effect of the early discharge policy. An

instrument that can be used extensively across

different diagnostic groups is provided, hence

solving the difficult problem of finding an in-

strument for length of stay. The exogeneity test in

Gorgger (1990), the test for weak instruments in

Staiger and Stock (1997) as well as the Hensen

(1982) for over identification confirmed respec-

tively that length of stay is endogenous the in-

strument is strong and the valid.

Keywords: Instrument; Length of Stay;

Early Discharge; Endogeneity;

Instrumental Variable Estimation

1. INTRODUCTION

Instrumental variable estimation is a two stage estima-

tion technique that first purges endogenous regressors,

and hence exogenizes them before including them as

regressors in a regression equation for estimation. An

important reason why instrumental variable estimation is

becoming less popular is the difficulty of finding an in-

strument. This difficulty is encountered in models that

measure the impact of early discharge policy on quality

of care. Many studies have thus ignored the need for

instrumental variable estimation and hence the consis-

tency of their estimated impact of the policy is ques-

tionable. The objective of this paper is to develop a sta-

tistical technique for consistent estimation of the impact

of early discharge on quality of care regardless of the

diagnostic group used. The British Columbia (Canada)

Ministry of Health and Center for Health Services and

Policy Research linked database provided the maternity

data used for the study.

The models that measured the impact of the early

discharge policy on quality of care used readmission and

mortality rates as proxies for quality of care with length

of stay as one of the independent variables. Length of

stay, because it is correlated with an unobservable vari-

able, the severity of illness, is endogenous in the regres-

sion equation. Severity of illness is unobservable to the

researcher but observable to the patient (or doctor) and a

severely ill patient is likely to stay long in the hospital

and be readmitted. Ignoring the endogeneity of length of

stay then can lead to inconsistent estimation and hence

unreliable conclusions. Examples of papers that ignored

the endogeneity of length of stay in the readmission eq-

uation can be cited [1].

Consistent estimation of the impact of early discharge

policy on quality of care is important because the policy

is implemented extensively across North America and

Europe (e.g., Sweden, Norway in the 1990s), as a cost

containment strategy. In the United States the policy was

induced by the introduction of prospective payment [2]

and capitation [3] in the 1980s. In Canada, the policy

was introduced by provincial governments in the 1990s

(example, 1994 in British Columbia and Alberta) and

was often accompanied by home visits implemented thr-

ough public health nursing programs. In general, the

early discharge policy has raised concern about its im-

pact on quality of care and has attracted a lot of studies

in the area [2,4].

Information on the impact of the policy on quality of

care will serve as an important guide to policy makers on

the extent to which the policy is able to reduce the cost

of care. Inconsistent estimation of the impact then can

lead to misleading results and hence mislead policy

makers on the efficiency of the allocation of health care

*Data used for this study was provided by British Columbia (Canada)

Ministry of Health and Center for Health Services and Policy Research

linked database.

E. Amporfu / HEALTH 2 (2010) 504-510

505

Openly accessible at

resources. The endogeneity problem can be solved if

severity of illness can be accurately measured and in-

cluded in the regression equation. This requires clinical

information from medical records [5]. Such an approach

is however not used in the literature because it is ex-

tremely costly to gather the information [6].

Some studies such as [7] used the cost weight of pa-

tients diagnostic group, hospital entry type (i.e., whether

it was emergency or elective) and whether the diagnostic

group was a surgical or non surgical type as proxies for

severity of illness. These variables are reasonable prox-

ies for the study’s data: all elderly patients across differ-

ent diagnostic groups. Such variables however cannot

pass as proxies for all diagnostic groups especially ma-

ternity data. This is because for maternity, an emergency

entry usually implies advancement in labor but does not

necessarily imply severity of illness. Caesarean section

could occur by choice, or as a result of the woman’s pel-

vic or birth canal or for medical reasons. Thus a woman

who undergoes caesarean section is not necessarily ill.

Hence surgical delivery cannot be an indicator of sever-

ity of illness for a maternity patient and the difficulty of

finding a proxy for the severity of illness remains. Find-

ing an appropriate instrument for length of stay then

would lead to a consistent estimation of the policy’s im-

pact on quality of care and nullify the need to find a

proxy for severity of illness.

The first to alert of the endogeneity problem with

length of stay and the need to use instrumental variables

to ensure consistent estimation did not come till 2000 [8].

However, [8] used time of delivery and the method of

delivery as instruments for length of stay. While these

instruments might have produced consistent estimates,

the instruments chosen are only restricted to the data set

used, newborns, and so cannot be applied to other diag-

nostic groups. It is not surprising that more recent papers

[1] that did not use infant data ignored the endogeneity

problem. What is needed then is an instrument that is

highly correlated with length of stay but uncorrelated

with severity of illness regardless of the diagnostic group

used for the analysis of effect of the early discharge pol-

icy. Such an instrument is very important because the

early discharge policy is widely implemented for differ-

ent diagnostic groups.

One such variable in this context is a policy instru-

ment for the early discharge policy: the average number

of patients per bed in a given year for a given hospital.

To implement the early discharge policy hospitals may

have to increase the number of patients per bed that is

allocated to the diagnostic group targeted by the policy.

This could be done by either increasing the size of the

population served by the hospital through the closure of

other hospitals or by reducing the number of beds in the

hospital or both. Because some hospitals are likely to

attract the severely ill patients the average number of

patients per bed in the hospital of delivery could be cor-

related with the severity of illness. Thus for each obser-

vation (patient) I computed the average number patients

per bed whether or not it was the choice hospital. The

number of patients per bed is a better instrument than the

year of policy used in an earlier work [9] because year of

policy is only valid as an instrument if no other policy

that could affect readmission rate was implemented in

the years of and after the policy. Weak instrument test

was used to test for the strength of the instrument [10].

The paper is organized as follows. Section 2 describes

the model, its estimation and various tests performed.

Section 3 describes the data while Section 4 reports the

results and Section 5 discusses the results.

2. METHODS

2.1. The Model

The model for the analysis has two regression equations,

an OLS regression Eq.1, and a discrete time duration

hazard Eq.2:

LOSZ u







 (1)

it it

yLOSZ







 (2)

where LOS represents length of stay in days,



is a

vector of the instruments, the annual average number of

patients per bed for each hospital; Z represents the ob-

servable characteristics of the patient: age, income, me-

thod of delivery, complications, Indian status, education

and birth rate of patient’s neighborhood; ui is the unob-

served characteristic, severity of illness, of the patient

that affects length of stay. Complications is a dummy

variable indicating whether the patient’s diagnostic short

list falls between 147 and 151 which includes hemor-

rhage of pregnancy, care during pregnancy as well as

complications in labor, delivery and pueperium. Note

that complication is observable and it differs from sever-

ity of illness in that two patients may both have hemor-

rhage but differ in severity.

In (2), yit is a binary variable indicating whether or not

patient i is readmitted t days (t is from 1 to ninety) after

discharge. Thus yit = 1 if the patient is readmitted and

zero otherwise. Even though the body takes about 60

days to return to its pre-pregnancy state, when psycho-

logical adjustment is taken into account, the woman

needs at least 90 days to adjust. Thus 90 days is long

enough for any impact of the early discharge policy to be

found. The error term it



= it it





, where vit is the

severity of illness component of



and εit is identi-

cally and independently distributed. Thus it is assumed

that apart from the severity of illness the error term, it



is identically and independently distributed. The pres-

ence of severity of illness means that vit is correlated

E. Amporfu / HEALTH 2 (2010) 504-510

506

Openly accessible at

with uit which implies that LOS is endogenous and the

estimates of ,





and



are inconsistent.

The hazard equation specifies the probability of being

readmitted conditional on not being readmitted. Follow-

ing the standard data transformation for discrete duration

hazard, each patient contributed several observations to

the data depending on the number of days, after dis-

charge, it takes before she is readmitted. Thus a patient

that is readmitted 18 days after discharge contributes 18

observations to the model. The observations are trun-

cated after 90 days making those not readmitted after 90

days the censored group, each with 90 observations. The

transformed data was then estimated by logit.

2.2. Estimation and Tests

To technically confirm the endogeneity of length of stay

in the readmission equation, the exogeneity test for

logit was used to test for the exogeneity of length of

stay [11]. The test statistic, nR2, was obtained from a

regression of a vector of units on ˆ

[()

ii ]

FZ X





and ˆ

[( )

NLIV



LO]

S where ˆ



is the vector

of the estimated parameters in (2) without IV estimation

and is the vector of estimated parameters of (2)

using IV estimation by nonlinear least squares. Under

the null of exogeneity, the test statistic follows

ˆNLIV





where G is the number of instrumental variables and n is

the sample size.

To ensure the consistent estimation of the coefficients

a test for weak instruments as shown in [10] was per-

formed. It is basically a Wald test for the significance of

the instruments in (1), i.e. , the null hypothesis is δ = 0.

The test statistic is 1

ˆˆˆ ˆ

'( ')





, where ˆ



is a vector

of V restrictions, δ = 0, evaluated at ˆ



, is a KXV

matrix of the derivatives of the restrictions with respect



, and is the asymptotic Cramer-Rao lower

bound variance both evaluated at





. The test statistic

is 2



where V is the number of instruments, which is

16 in this context.

Over identification test [12] was performed to test for

the appropriateness of the instruments for length of stay.

The null hypothesis states that the instruments are not

significant in the readmission equation implying that

they are appropriate instruments for length of stay. The

test statistic here is also nR2 from the least square re-

gression of the residual of (2) on the instruments and the

exogenous variables. nR2 is also 2



where V again is

the number of instruments.

2.3. Data

The study used a four year (1993-1996) maternity data

on all deliveries from sixteen acute care hospitals in

British Columbia, Canada, provided by the British Co-

lumbia Ministry of Health and Center for Health Ser-

vices and Policy Research linked database. For the pur-

poses of this study, the data contained information on

age in years, length of stay in days, method of delivery,

complications, Indian status, local health area, dates of

admission and discharge, transfers and hospital of deliv-

ery. Since transfers involve admission and discharges

from different hospitals, a transferred patient that returns

to her original hospital can easily be mistaken for a re-

admitted patient. Thus transferred patients were removed

from the data. This reduced the sample size from 92,595

to 90,658 deliveries. Readmissions were also reduced

from 3492 to 3326. To ensure that readmissions are re-

lated to the initial hospitalization only those readmis-

sions caused by obstetric problems were included.

Information on patients’ income and educational back-

ground were not available and so patients’ neighborhood

information was used. Neighborhood characteristics of

the patients’ local health areas for education, income and

fertility rates were obtained from the website of the gov-

ernment of British Columbia. The number of maternity

beds in each hospital which was needed for the construc-

tion of the instruments was obtained from the Directory

of Canadian Hospitals. Since the data covered all deliv-

eries in each of the hospitals, the number of patients per

bed was computed by dividing the total deliveries in

each hospital in a year by the number of maternity beds

in the hospital for the year.

As shown in Table 1 the average age of the mothers

remained at about 29 over the four years. Length of stay

decreased gradually over the years. The number of ma-

ternity beds in all the hospitals decreased relative to the

number of patients who delivered hence the number of

patients per bed increased over time. There was a sig-

nificant (17.4%) reduction in the number of maternity

beds in 1994, the year of the policy, followed by smaller

(0.05% and 3%) reduction in the two years that followed.

The number of maternity beds was therefore reduced

over time to implement the early discharge policy hence

making it correlate with length of stay.

Table 1. Data summary.

1993 1994 1995 1996

Sample size23149 23325 22795 22583

Average

length of

stay

3.71 3.47 3.29 3.22

Average age29.2 29.3 29.5 29.7

Number of

maternity

beds

432 357 355 344

Patients per

bed 53.6 65.3 64.2 92.6

E. Amporfu / HEALTH 2 (2010) 504-510

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The data used had several advantages in aiding with

the consistent estimation of the parameters. First, the

omission of breastfeeding and home visit are not

likely to affect the consistency of the estimation. In ad-

dition to the severity of illness, home visit and breast-

feeding are also correlated with both readmission rate

and length of stay [8]. As noted in [13], newborns who

receive home visits have short stays. Since mothers are

unlikely to outstay their babies in the hospital, at least

during the period under study, the availability of home

care should in general, reduce length of stay for mothers.

Home visit also reduces readmissions because the home

visit nurse is likely to detect a developing infection and

treat before it develops to require readmission. Thus, the

omission of home visit can result in inconsistent estima-

tion of the readmission equation. The source of the data

used in this study however minimizes any such problem

because the early discharge policy in British Columbia

was implemented through the introduction of home visit

program funded by the Closer to Home Fund available

to all hospitals. All patients in the data then had access to

home visits and so the impact on readmission is captured

by the intercept and not the error term.

Mothers with no lactation problem have short stay and

are less likely to be readmitted. Thus the omission of

breastfeeding from the estimation equation could render

length of stay inconsistent. However, breastfeeding is

not likely to affect the consistency of the estimates in the

study because lactation problems are captured under

complications as care during puerperium. Thus the as-

sumption that without the severity of illness the error

term in (2) is identically and independently distributed is

valid.

Second, maternity data is appropriate for the study

because the early discharge policy has in general being

applied to maternity patients across North America and

Europe and so the resulting impact on readmission rates

of maternity patients have received a lot of attention in

the literature [1,3].

Lastly, studies have shown that severity adjusts well

when it is disease specific [14]. This is to ensure that the

clinical parameters in the model such as complications

have similar effect on outcome hence validating the ex-

pectation that hemorrhage, for example, increases the

probability of readmission rate.

3. RESULTS

The test statistic of the endogeneity test was 97835.29

with 16 degrees of freedom and a p-value of zero. Thus

the null of no endogeneity was rejected. This confirms

the expectation that length of stay is endogenous in the

readmission equation; hence previous studies that ig-

nored the endogeneity problem could have produced

inconsistent estimates, making their conclusions unreli-

able.

The test for weak instruments yielded a test statistic of

104458.8 with 16 degrees of freedom and a p-value of

zero, leading to the rejection of the null hypothesis of

weak instruments. As stated in [10] weak instruments

yield inconsistent estimates. The test results then con-

firm that the instruments used, number of patients per

bed, are highly correlated with length of stay and uncor-

related with the severity of illness hence they are not

weak instruments for length of stay. Finally the over

identification test yielded a test statistic of 14.67 with 16

degrees of freedom and a p-value of 0.547. Thus, the

null was not rejected implying that the instruments are

appropriate for length of stay. Having been assured of

consistent estimation of the coefficients, attention is now

turned to the results of the regression.

As shown in Table 2, all the instrumental variables

had negative signs confirming the expectation that

length of stay falls as the number of patients per bed

increases. The results also showed that length of stay

dropped gradually even after 1994, the year of the pol-

icy.

The results from the hazard estimation, in Table 2,

show that all the estimated coefficients in the IV and the

non IV estimation had the same signs. Both show that

there was no significant change in readmission rate in,

1995, the year after the policy but readmission rate in-

creased in 1994 and 1996. As expected the readmission

rate of those who lived in local health areas with high

education level and/or high income were less likely to be

readmitted conditional on not yet been readmitted. Pa-

tients with complications as well as those with Native

Indian status were more likely to be readmitted condi-

tional on not yet been readmitted, than those without

complications and/or Native Indian status. As expected,

Native Indians were more likely to be admitted than

non-native Indians. In general, the IV estimates were

less efficient than the non-IV estimates.

Since readmission rates are considered often as un-

conditional, unconditional readmission rates were com-

puted to compare the impact of the policy on readmis-

sion rate using the IV and the non-IV estimations. The

computation involves calculating, for each patient, the

survival rate, subtracting it from one and then averaging

over all the patients. The coefficient for length of stay

under the non-IV was –0.051 + 0.0008*LOS and that of

the IV was –0.115 + 0.006*LOS. The average length of

stay of 1994 in the above coefficients and unconditional

readmission rates were used to compute the marginal

effect of length of stay on readmission rate. The results

showed a marginal effect of –1.54% under the IV esti-

mation and –1.1% under the non-IV estimation, i.e., an

increase in length of stay by a day reduced readmission

rate by 1.54% under the IV estimation and 1.1% under

E. Amporfu / HEALTH 2 (2010) 504-510

508

Table 2. Regression results (numbers are estimated coefficients).

Dependent Variables

Independent

Va ri a bl e s

Length of Stay Readmission Equation

(with IV estimation)

Readmission Equation

(without IV estimation)

Number of patients per bed 1 –0.00003

Number of patients per bed 2 0.0002

Number of patients per bed 3 –0.015

Number of patients per bed 4 –0.007

Number of patients per bed 5 –0.028

Number of patients per bed 6 –0.016

Number of patients per bed 7 –0.019

Number of patients per bed 8 –0.032

Number of patients per bed 9 –0.043

Number of patients per bed 10 –0.167

Number of patients per bed 11 –0.026

Number of patients per bed 12 –0.150

Number of patients per bed 13 –0.00005

Number of patients per bed 14 –0.091

Number of patients per bed 15 –0.015

Number of patients per bed 16 –0.047

1994 –0.174 0.104 0.105

1995 –0.344 0.002 0.008

1996 –0.404 0.092 0.103

Income 0.841 –0.174 –0.175

Native Indians 0.130 0.277 0.283

Fertility 0.239 0.097 0.103

Age –0.122 –0.073 –0.092

Age*age 0.002 0.001 0.002

Education –0.897 –0.142 –0.154

Complication 0.450 0.544 0.523

Length of stay –0.115 –0.051

Length of stay * length of stay 0.031 0.0001

T –0.59 –0.059

T2 0.001 0.001

constant 5.288 –5.477 –5.431

All estimates are significant at 5% significance level.

the non-IV estimation. Thus, there is a greater marginal

impact of length of stay on readmission rate under the IV

than the non IV estimation. This result is similar to that

in [8] as well as in [6].

E. Amporfu / HEALTH 2 (2010) 504-510

509

An LM test was run to check if the coefficients of

length of stay in the IV estimation were statistically dif-

ferent from those of non-IV. This null hypothesis is φIV =

φnonIV. The test involves running the residuals of the re-

stricted regression on the derivatives of the logit with

respect to each of the coefficients in the model. The test

statistic is nR2 which follows the chi square distribution

with two degrees of freedom (number of restrictions).

The resulting test statistic was 2445.7 with a p-value of

zero, implying that the coefficients were statistically

different from each other. Hence the earlier studies that

ignored the endogeneity of length of stay could be

flawed.

The extend to which the policy contributed to the in-

crease in readmission rates was found by rerunning the

readmission regression after including interactions of

length of stay with the year dummies. The resulting es-

timated coefficients for the interactions of the year

dummies with the square of length of stay were not sig-

nificant for either the IV or the non-IV estimates so the

regressions were reran without those interactions. The

results are reported in Table 3 and they show that the

interaction coefficients for the years and length of stay

were negative for the IV results and positive but close to

zero for the non-IV results. This implies that according

to the IV results, the reduction in readmission rate from

a day’s increase in length of stay is greater after the pe-

riod of the policy. The opposite is the case under the

non-IV results.

To translate these into readmission rates, the results in

Table 3 were used to compute the contribution of the

policy to any change in readmission rate as the differ-

ence between the readmission rates with and without the

interactions. The readmission rates without the interac-

tion were the readmission rates for each year without

any contribution from the early discharge policy. The

readmission rates that included the interactions repre-

sented the readmission rates for each year including the

impact of the policy. The difference between the two

then is the change in readmission rate as a result of the

policy. To compare the impact of the policy with the

overall change in readmission rate over time, the results

in Table 2 were used to compute the change in readmis-

sion rate for each year after the policy.

The results, reported in Table 4, show that readmis-

sion rates increased over the years under both the IV and

the non-IV estimation. However, the increase was con-

sistently higher under the IV than the non-IV estimation.

The results under IV also showed that with the exception

of 1994, where other factors must have contributed to

the increase in readmission rate, any increase in read-

mission rate in the years that followed was due to the

policy and other factors must have reduced the impact of

the policy. Such information was not captured by the

non-IV results.

Table 3. Regression results from interacting length of stay with

the year dummy variables.

Readmission Rate

(with IV)

Readmission rate

(without IV)

1994* 0.128 0.095

1995* 0.111 –0.023

1996* –0.212 0.050

1994*length of stay*–0.004 0.002

1995*length of stay*–0.027 0.007

1996*length of stay*0.102 0.013

Income –0.174 –0.177

Native Indians 0.275 0.283

Fertility 0.100 0.103

Age –0.079 –0.092

Age*age 0.001 0.002

Education –0.141 –0.155

Complication 0.494 0.523

Length of stay* –0.036 0.045

Length of stay *

length of stay* 0.017 –0.0004

T –0.059 –0.059

T2 0.001 0.001

constant –5.516 –5.406

*Significant at 10% level. All other variables are significant at 5%

significance level.

Table 4. Effect of policy on readmission rates.

1994 1995 1996

IVNon-IVIV Non-IV IV Non-IV

Change in

readmission

rate (%)

2.8 2.6 2.1 0.19 2.6 2.5

Change in

readmission

rate due to

policy (%)

1.5 1.1 2.4 0.23 5.2 .101

Number of

readmissions

due to policy

53623 547 52 1174 23

As shown in Table 4, for 1994 the policy increased

readmission rate by 0.1 percentage points under the

non-IV estimation and 2.3 percentage points under the

IV estimation. Considering the number of deliveries of

23,325 in the selected hospitals in 1994, it implies, ac-

cording to the non-IV estimation about only 23 women

were readmitted as a result of the early discharge policy.

E. Amporfu / HEALTH 2 (2010) 504-510

510

The number however increased to about 536 women

under the IV estimation.

Openly accessible at

4. CONCLUSIONS

The current study has one main weakness. The IV esti-

mation, when the second stage regression is logit or pro-

bit, requires that the endogenous variable is continuous.

This implies that length of stay should be measured in

hours and not in days as was used. The requirement that

the problem variable be continuous is consistent with

probit estimation which is continuous. However, in the

current study the second stage regression is discrete and

so a discrete endogenous variable for the first stage may

not be problematic.

The study however has several strengths. First, the in-

struments, average patients per bed, proposed to ensure

consistent estimation of the impact of the early discharge

policy on readmission rate are not weak. Second the in-

struments are not significant in the readmission equation

implying that they are not one of the regressors for re-

admission. Hence they are strong and appropriate in-

struments. Third, the instruments are not restricted to

any diagnostic group and so can widely be applied to

any diagnostic group to find consistent estimation of the

impact of the early discharge policy. Finally, information

on the number of beds for a diagnostic group can easily

be found for the computation of the average number of

patients per bed.

The study has shown that the non-IV estimates are

statistically different from the IV estimates implying that

earlier studies that ignored the endogeneity of length of

stay might have produced misleading results. The non-

IV estimates underestimate the impact of the policy on

readmission rate. That could explain why some of the

studies that ignored the endogeneity problem found no

impact of the policy on readmission rate. The results

from the non-IV estimates inform policy makers that the

policy as it was implemented did not deteriorate read-

mission rate and so provided no reason for policy mak-

ers to make any adjustment. The results from the IV es-

timates imply that, since readmissions are expensive, the

policy as implement might not contain as much cost as

was expected and so further amendments such as im-

provement in home care is necessary to make the policy

more able to reduce the cost of care.

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