App Xro anckcs &Tecniqcs Analysis and Control of Nurse Staffing

by Robert S. Kaplan An information and reporting system based on a regression analysis of historical nurse staffing data is described. The system provides a concise monthly report from which administrators can evaluate the efficiency of scheduling procedures used by nurse supervisors to meet varying patient loads.

Efficient use of nursing personnel is an important goal for hospital administrators. The nursing service is the largest single component of hospital costs. In a 1971 survey, the direct costs of nursing represented 25-28 percent of total hospital expenses [1]. When fringe benefits and other associated overhead expenses are added to this figure, about one-third of total hospital expenses can be attributed to the cost of providing nursing services [2,3]. The predominance of nursing costs in hospitals has not gone unrecognized, and much research has been conducted to improve the efficiency and the scheduling of nursing personnel. Initial efforts focused on the various functions that nurses perform and the skill levels required to perform these tasks [3-10]. Such studies made extensive use of industrial engineering techniques, especially work sampling methods. Later work sought to determine the demand for nursing services. This was typically done by classifying patients into three or four categories (e.g., "ambulatory," "requires intensive care") that could be related to nursing time requirements [2,8,9,11-14]. Demands on nursing could then be determined from the number of patients in each category. It was also possible to recognize variations in the level of nursing services required within a 24-hour period. Knowing the demand for nursing and the tasks performed by various types of nurses, one could then vary nurse staffing levels in response to varying patient Address communications and requests for reprints to Robert S. Kaplan, Ph.D., Graduate School of Industrial Administration, Carnegie-Mellon University, Pittsburgh, PA 15213.

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demands so as to use nurses more efficiently. Cycle scheduling [2,15-17] provides long-range (several weeks or months) plans that distribute less popular shifts-evening, night, weekend-equitably among all nurses. Cycle schedules can be supplemented on a daily basis, as actual patient demands become known, by using a pool of "floating" nurses, by shifting nurses between patient units, or by overtime. Recently, a number of mathematical programming models have been proposed and tested [2,18-21] for scheduling on a daily or weekly basis. These models attempt to minimize costs while satisfying constraints on desired levels of nursing service. All these efforts can result in lower nursing costs with no reduction in the quality of care delivered when implemented properly. However, the link between lower-level or unit efficiency in scheduling and its evaluation by the hospital administrator has been mostly neglected. (Jackson and Kortge [10] and Abernathy, Baloff, and Hershey [22] discuss control and evaluation schemes but do not treat their implementation in much detail.) This is an important point because unless information on local performance is transmitted to the administrator, the motivation may be reduced for nursing supervisors and schedulers to persist in collecting detailed data and making complex decisions for the most effective use of nursing personnel. In fact, most nursing supervisors (at least at the hospital where this study was conducted) do reasonably well in balancing nursing loads over the course of a month, even without the aid of computerized models; the analysis to follow reveals evidence of variable-load scheduling despite an essentially uncontrolled environment. To the extent that mathematical models can improve this performance, supervisors will be motivated to use them. The procedure described here provides detailed feedback on past decisions to the scheduler and more concise evaluative information for alerting the administrator to those circumstances that require his attention. A regression technique is used for forecasting the number of nursing hours as a function of patient days (or some better measure of patient demand if one is available) . At the end of a reporting period an administrator can compare the predicted number of nursing hours with the actual number to determine whether the nurse scheduler succeeded in efficiently using nursing personnel. In accounting terms, this is nothing more than a flexible budgeting system [23] for nursing hours. This study was carried out solely on the basis of data currently reported on a routine basis in the study hospital. Although a more detailed examination of nursing operations in the hospital could prove useful, the effort reported here illustrates that standards and controls can be set without performing expensive and time-consuming industrial engineering studies. BACKOROUND OF THE STUDY

The hospital where this study was done is a 575-bed university teaching hospital. Because there are specialty hospitals nearby, it has no pediatrics or obstetrics facilities but otherwise provides a full range of medical services.

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Tables 1 and 2 show the data included in the nursing summary report that is prepared each month. The first page (Table 1) provides highly aggregate statistics on the nursing staff and the number of patient days. The second page (Table 2) provides information on the number of patient days in each patient unit in the hospital and the amount and type of nursing resources used in each unit. Some average statistics on nursing hours per patient day are computed, but these averages are; not compared to standards or historical trends. In this study historical data of the type shown in Tables 1 and 2 were used to estimate the normal relationship between nursing hours and patient days, both for individual and aggregated patient units. Fixed and variable components of nursing hours, as a function of patient days, were estimated using multiple regression techniques. In its simplest form, a regression with a constant and one independent variable-patient days-is a generalization of the statistic currently reported in the study hospital-average nursing hours per patient day-which, in effect, is a regression line forced through the origin. Thus one can allow for a minimum fixed complement of nurses plus additional staff as patient load increases. In addition to the expected value of the regression coefficients, the variance and covariance of the estimates are computed so that the regression model will provide confidence limits for nursing hours, given particular values of patient days. Once a satisfactory model has been estimated, the nursing hours in a particular month can be compared with the forecasted value, given the actual number of patient days, to see if the number of hours used was excessive or unusually low. Only the exceptional values need be highlighted to the administrator so that he can be relieved of the burden of examining data on 20 patient units, most of which are operating nonnally. As a final benefit, additional Table 1. Monthly Hospital Report: Summary of Distribution of Nursing Hours for May 1971 (Total Hospital Census = 16,269; Daily Average = 524.8 Patients) Care category and nurse type

Indirect care All indirect care types ................ 2304 Administratorsf .736 Supervisors .872 In-service educators .696 Direct care All direct care types .71278 Registered nurses .47230 Licensed practical nurses .7288 Nurse aides and orderlies .16759 * Does not include paid hours for attending meetings of one vacations, or holidays. f Includes directors and assistant directors in nursing.

280

average Daily hours per

Total number of hours for the month* ...........

patient

0.14 0.05 0.05 0.04

day

4.38 2.90 0.45 1.03 or more, sick leave,

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questions can be investigated, such as whether licensed practical nurses (LPNs) can be used as substitutes for registered nurses (RNs) or whether student nurses can be used as substitutes for RNs. Certain assumptions are implicit in this kind of analysis. In estimating the model from historical data, it is assumed that patient units are operating reasonably efficiently and that nurse supervisors are varying nursing loads to reflect patient demands. Fortunately, the data can be used to test this assumption. If the patient units have no significant component of variable hours worked per patient day, or if the coefficient of this component has a negative sign, the assumption must be rejected. Further, since each monthly observation is weighted equally in the statistical analysis, one must be able to assume that manpower allocation procedures are comparable for all months. Therefore, if some unusual event occurred in a patient unit one month (for example, an exceptionally high or low ratio of nursing hours to patient days, because of extraordinary circumstances), that month should be excluded from the analysis. Otherwise an extreme observation could severely distort the estimated coefficients. Such exclusion should not be done after observing the residuals from the regression, but at the beginning of the analysis, on the basis of prior knowledge that during a certain month at a given patient unit the nursing allocation departed substantially from normal procedures. If in one month there was an exceptional low level of nursing hours because of heavy absenteeism, the quality of care delivered in that month would likely be below that of other months in the sample and that month's operations would not be comparable to the rest of the sample. (In this study, no data were excluded from the analysis.) THE ANALYSIS

Data of the type shown in Tables 1 and 2 were used to estimate the variation of nursing hours in 20 patient units as a function of patient days. Initially only RN hours were considered, because they comprise about 60 percent of total nursing time and represent the highest pay category of nursing personnel. Discussion with the administrator revealed two factors that required special treatment in the regression. There had been an expansion qf nursing personnel during the second year of the data period; in addition, staffing tends to be temporarily inflated during the last three months of the fiscal year (April-June) because newly graduated nurses are hired then, while many terninating nurses wait until July to leave. To incorporate the effect of the increase in overall staffing in the second year as well as the seasonal effect, two dummy variables, J, and Sj, were included in the regression as follows: Yii = bo, + b1iX0 + b2iSj j=1, ... 24 + b.3, + ci (1) where Yi1 = nursing hours in unit i for month j Xe = patient days in unit i during month j bor = coefficient of fixed nursing hours for unit i

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b,= coefficient of variable nursing hours for unit i ij= error term for unit i in month j. The dummy variable Ij (the "jump" variable) was assigned the value 0 for j = 1, 2,..., 12 and the value 1 forj = 13,... ,24, and Sj (the seasonal variable)

was assigned the value 1 for j = 10, 11, 12, 22, 23, 24 and 0 for all other rmonths. A summary of the regression results is presented in Table 3. There are 21 regressions because unit 15 was closed and remodeled during months 13-16. When it reopened in month 17 it had almost twice the previous capacity. It was therefore treated as a new patient unit, number 16; hence there is no jump coefficient I, associated with either 15 or 16. Unit 11 was also closed and reopened later with about 50 percent more capacity. No correction for that fact is made here, and the consequence will be noted later. Seventeen of the 21 regressions had adjusted R2 values in excess of 0.45; the four exceptions were units 1 (intensive care), 2 (clinical research), 4 (renal dialysis) and 15 (which was closed after one year). The Durbin-Watson statistics [24, p. 243] indicated significant positive autocorrelation in only three of the regressions. (With n = 25 and one independent variable, the one-tailed Table 3. Regression Results (Eq. 1) for Individual Patient Units (t Ratios in Parentheses) Patient unit 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

R*

Per patient

day

ICU ..........

0.12

Clin res .......

-0.02 0.51 0.27 0.71 0.58

CCU .......... Dialysis ....... Med-surg ...... Med-surg ...... Med tchg ...... Med tchg ...... Med tchg ...... Med tchg ......

Dt 0.99 1.32 1.93 1.67 2.21 1.51

(b7)

2.30 (1.06) 0.20 (0.40) 3.71 (1.43)

RN hours Seasonal increase (b2)

408 36 -128 119 70 373 105 515

(2.21) (0.70) (2.06) (1.75) (1.63) (3.05) (1.12) (4.49) (1.98) (4.15) (2.89) (3.69) (2.92) (2.23) (1.57) (1.25) (3.68) (2.74) (2.40) (3.76) (0.24)

Permanent increase (b3) 3 (0.02)

58 225 37 252

(1.33) (4.33) (0.66) (6.90) (4.58) (5.28) (4.82) (7.78) (5.22) (5.65) (3.85) (6.20)

2.95 (2.04) 3.04 (3.42) 2.26 (3.12) 499 0.61 436 1.35 1.75 (1.80) 1.14 480 0.65 0.85 (0.49) 126 418 0.81 1.95 2.27 (5.28) 1.49 373 0.67 345 3.10 (2.88) 1.09 5.98 (2.09) 775 1461 Med-surg ...... 0.69 1.36 0.72 (0.71) 370 348 Med-surg ...... 0.52 326 1.38 1.12 (1.57) 754 Med-surg ...... 0.69 1.15 (1.08) 185 671 (9.40) 1.75 Orthoped ...... 0.80 1.67 0.54 (0.66) 105 Med-surg ...... 0.08 1.74 4.30 (3.77) 113 Med-surg ...... 0.68 307 363 (4.98) 2.34 0.20 (0.42) Med-surg ...... 0.62 1.45 0.83 (0.89) 238 451 (5.96) Med-surg ...... 0.64 0.99 239 377 (4.37) 1.49 (1.11) Orthoped ...... 0.50 642 (8.93) 325 1.72 1.05 (1.38) Med-surg ...... 0.80 65 678 (1.95) 1.31 3.55 (1.40) Med-surg ...... 0.46 * Adjusted R2. This statistic may be negative. + Durbin-Watson statistic. t Unit 15 was closed and remodeled during months 13-16 and had almost twice its previous capacity when reopened. It was thereafter treated as a new unit, no. 16. Therefore neither unit has the dummy variable Jj nor its coefficient b3.

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test at the 0.05 significance level rejects the null hypothesis if the DurbinWatson statistic is below 1.29. The test is indeterminate if the statistic is between 1.29 and 1.45. With three independent variables, the corresponding numbers are 1.12 and 1.68. These numbers were used as approximations for the situation treated here, where n = 24 for most of the patient units.) The signs of b1, the estimate of variable RN hours per patient day, were all positive, but many of the coefficients were small and insignificant. Part of this may be due to a lack of careful scheduling at those patient units. Also, in the specialty units, 1-4, staffing may be fixed because of erratic demand or permanent assignment of specially-skilled nurses, rather than responding to varying patient demands. The coefficients of both the seasonal and the jump variable, S and J, were positive in all regressions except one and were generally significant. The exception, unit 21, was not fully open until month 9. Therefore there was only one month for which both the jump and the seasonal variable were zero. This may have caused the relatively low t-values for the coefficients of these variables. Equation 1 assumes that the increase in nursing hours occurred only in the fixed component of nursing hours. Although it is traditional to handle nonstationarities by a shift in the intercept, it is possible in this instance that the increase in nursing hours was picked up in the slope coefficient, i.e., an increase in b1i, in year 2. The coefficients were therefore reestimated using the same data, as follows: Yij = bo0 + biiiXlq + b121X2V, + b2iSi + i, j=1,..., 24 (2) where X1ij (patient days, as in Eq. 1) has the value Xij for j = 1, . . ., 12 and the value 0 for j = 13, ... 24, andX2i1is assigned the value 0for j=1, ..., 12 and the value xij for j = 13, . .. , 24. The coefficients bl, and bl2 pertain to variable nursing hours in year 1 and year 2 respectively. The results are presented in Table 4. There is little to choose between Eqs. 1 and 2. The R2s, Durbin-Watson statistics, coefficients, and t-values are virtually equivalent for each patient unit in the two models, with the jump coefficient b3 in Eq. 1 replaced by an increase in the patient days coefficient b12 for the second year in Eq. 2. Both equations were retained for further study.

Analysis of Aggregate Data Many of the patient units in the hospital are functionally the same. Rather than examine staffing at each individual patient unit, it seemed possible to investigate the staffing at a more aggregate level, using groups of homogeneous patient units. Aggregation offers a number of advantages. It reduces the amount of information that an administrator needs to monitor each period and gives more leeway to the nurse scheduler. Rather than being responsible for the detailed schedule at each individual patient unit, the scheduler would only be evaluated on aggregate performance. Thus nurses could be shifted from one patient unit to another in response to circumstances not controlled for in

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Table 4. Regression Results (Eq. 2) for Individual Patient Units (t Ratios in Parentheses) Patient unit

R2*

Dt

RN hours Per patient day yr 1 (11l) y 2 (1112)

Seasonal

increase ( b2)

408 (2.21) 2.31 (1.07) 2.23 (1.00) 0.98 ICU .0 ..............12 32 (0.63) 0.28 (0.55) 1.27 -0.05 (0.09) ...........01 Clin res -122 (1.98) 4.55 (1.76) 2.00 2.69 (1.05) .............51 CCU 120 (1.74) 2.82 (1.85) 3.01 (2.06) 1.65 ...........26 Dialysis 70 (1.64) 3.26 (3.68) 2.22 2.86 (3.23) Med-surg ........ 0.71 373 (3.07) 2.65 (3.57) 1.50 2.20 (3.08) Med-surg ........ 0.58 100 (1.04) 2.18 (2.22) 1.50 (1.49) 1.38 Med tchg ........ 0.59 517 (4.05) 1.13 (0.66) 1.12 0.60 (0.34) Med tchg ........ 0.65 126 (1.88) 2.42 (5.35) 1.84 (4.03) 1.94 Med tchg ........ 0.79 350 (4.15) 3.26 (2.99) 1.49 2.85 (2.61) Med tchg ........ 0.66 773 (2.92) 6.86 (2.38) 5.75 (2.05) 1.07 Med-surg ........ 0.70 367 (3.68) 0.89 (0.88) 0.54 (0.54) 1.38 Med-surg ........ 0.53 311 (2.84) 1.38 (1.88) 1.43 0.83 (1.23) Med-surg ........ 0.70 179 (2.17) 1.78 0.87 (0.82) 1.52 (1.44) Orthoped ........ 0.80 105 (1.48) 1.67 0.54 (0.62) Med-surg ........ -0.04 t 113 (1.12) 4.30 (3.37) Med-surg ........ 0.59 1.74 t 297 (3.59) 0.44 (0.93) 0.06 (0.13) Med-surg ........ 0.62 2.31 233 (2.72) 0.98 (1.07) 1.48 0.54 (0.59) Med-surg ........ 0.65 1.71 (1.29) 237 (2.40) 1.34 (1.01) 0.97 5.O1 ....... Orthoped . 309 (3.64) 1.55 (2.06) 1.65 0.72 (0.95) Med-surg ........ 0.81 3.84 (1.52) 58 (0.20) 1.28 3.25 (1.19) Med-surg ........ 0.44 * Adjusted RW. This statistic may be negative. f Durbin-Watson statistic. t Unit 15 was closed and remodeled during months 13-16 and had almost twice its previous capacity when reopened. It was thereafter treated as a new unit, no. 16.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

the model, without necessarily affecting aggregate performance. A possible disadvantage of aggregation is that the aggregate model may not explain or predict behavior as well as all of the individual models taken together, because of the implicit assumption that each patient unit in a group operates under the same model. To investigate the effect of aggregation, we grouped three sets of selected homogeneous patient units: general medical and surgical (units 5, 6, 15, 16, 18, 20, 21); medical teaching (units 7, 8, 9, 10); and orthopedics (units 14 and 19). For each set the patient days and RN hours in each month were added together. Regression results for Eq. 2 are presented in Table 5 (p. 286); the results for Eq. 1 are not shown, but were equivalent. The results appear quite impressive. The explanatory power of the aggregate regressions exceeds that of all the component individual regressions (except for orthopedic unit 14), and the coefficients b1, for hours worked per patient day, are highly significant for medical-surgical and medical teaching units. However, as pointed out by Grunfeld and Griliches [25], it is not meaningful to compare R2 values of the individual regressions to the R2 values of the aggregate regressions. If the independent variables of the individual regressions are

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Table 5. Regression Results (Eq. 2) for Aggregated Patient- Units (t Ratios in Parentheses) RN hours

Urnit

R2*

D

yr 1

General medical and surgical .... 0.90

Medical teaching .. 0.81

Orthopedics ....... 0.72

1.53 1.64

1.24

Per patient

day

(bn)

yr 2

Seasonal

(bu2)

1.9

2.5

(3.8)

(5.8)

3.0

3.6

(3.5)

(4.1)

increase (b2) 1168

(3.0) 1068

(4.3)

1.0

1.5

416

(0.9)

(1.4)

(2.5)

* Adjusted R'. t Durbin-Watson statistic.

positively correlated, the aggregate regression will tend to have a higher R2 even though the aggregate regression does not explain aggregate behavior any better than the totality of the individual regressions. In the case treated here, there is, as one would expect, positive correlation of number of patient days among the individual patient units in a group. Therefore the increase in the aggregate R2 may be due to this synchronization effect. To control for the effect of intercorrelations among separate units, one must compute a composite value of R2 for the individual regressions and compare this with the aggregated R2 value. In this computation, residuals for each of the relevant regressions are summed together for each month and this sum is squared. These monthly squared residuals are then summed over the time period of the regression (24 months) and compared with the unexplained variance of the aggregated regressions. The details can be found in Grunfeld and Griliches [25]. Table 6 presents a summary of these computations. It shows that the aggregate regressions explain essentially the same amount of variance as the composites of the individual regressions. There is only a slight aggregation loss for medical-surgical units and a slight gain for medical teaching units. Therefore one can be relatively indifferent, on statistical grounds, to the choice between the individual or the aggregate regressions. FORECASTINO NURSINO HOURS

The regression models developed in the previous section were tested using seven new months of data generated from July 1971 to Jan. 1972. It was assumed that no further discontinuities of the type encountered at the start of fiscal year 1971 occurred during fiscal year 1972. Any nonstationarities would be revealed, however, by a consistent bias in the prediction. 286

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Table 6. Correlation Coefficients for Aggregate Regression and Composite of Individual Regressions (Eq. 2)

R2

Unit goup

Aggregate*

Composite

0.927 General medical and surgical ....... ..... 0.916 0.822 Medical teaching .............. ........ 0.838 0.758 Orthopedics ............. ............. 0.759 * Unadjusted; values are slightly higher than adjusted R' shown in Table 5.

This forecasting exercise was an experiment on paper. Nursing personnel were unaware that their actions were going to be monitored in this manner. Therefore it is reasonable to expect more extreme observations than might be seen where supervisors were aware that personnel allocations were being evaluated by a statistical model. Also, since normal staffing levels or standard operating procedures may in fact have changed, the regression models from historical data may not properly specify the current situation. Some additional methodology is needed before evaluating the forecasting ability of the regression models. There are two types of errors that arise when a regression equation is used for forecasting. One is due to the uncertainty of the regression plane itself. Since the plane is constrained to pass through the mean values of the independent variables, error due to uncertainty is smaller for values of the independent variables near their means and larger for values far from the means. The other source is the variance not explained by the model, as reflected in the error terms, c,j. If b is the vector of coefficients, Xbb is the covariance matrix of the coefficients, and if X* is a particular set of values of the independent variables for which a forecast is desired, then that part of the variance of the forecast error that arises from the uncertainty of the regression plane is ,V2=X YbbX*

(where X' is the value of the forecast) and the total variance of the forecast error is

Ft2 =

(3) where a2 is the variance of ci, [24, p. 169]. In practice, of course, Zbb and a2 are unknown and must be estimated from the data; i.e., &2 is the estimated error variance, and Zbb = &2(X'X)'1, where X is the vector of independent variables. Xbb may be more familiar if it is recognized that the square roots of the diagonal elements of Xbb yield the standard errors of the coefficients, b, from which the t-values for the coefficients are derived. Forecasts were generated using both Eqs. 1 and 2, but the predictions from the two models were virtually identical. Therefore the forecasts analyzed are

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r2

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Table 7. Forecast RN Hours and Evaluation Parameters for

Selected Individual Units (Eq. 2) Actual Unit 1 ICU

3 CCU

........

........

5 Med-surg

6 Med-surg

....

....

7 Med tchg

....

Month

patient days

1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7

273 327 305 280 309 339

1

2 3 4 5 6 8 Med tchg

.

7 i...1 2 3 4

5 6 7 9 Med tchg

....

1

2 3 4

5 6 7

288

242 111

89 106 110 121 122 105

668 600 597 608 633

538 620 1154 1137 1097 1174 1104 999 1151 694 640 638 647 650 664 674

RN hours Forecast Actual

3616 3536 3768 3800 3536 3808 3104 1720 1568 1464 1632 1662 1701

2836 2961 2910 2852 2919 2988 2765 1840

1730

1813

1336 1336

1559

1333 1432

1352 1368

1512 2544

1740

1817 1835 1885 1890

1338 1328 1364 1445 1136 1403 2427

Percent Percent

~t

error 22 16 23 25 17 22 11 -

7

-11 -24 -12 -13 -11 5

1.90 1.33

2.05 2.31 1.46 1.86 0.82 -0.88 -1.09

-2.54 -1.49

5 7 17 7 5

-1.69 -1.43 -0.59 -2.24 -0.02 0.05 0.72 -1.00 1.90 1.17 0.43

6 -18 -35 -33 2 -13

-0.50 -1.26 -2.37 -2.13 -0.17 -1.02

- 1 -10

-0.07 -0.73

-16 3

-

-17 -

-

0 0

2248 1936 1832 1720 1968 2144

2295 2017 2419

1808 1552 1472

1824 1707 1703

1680 1488

1722 1729

1592 1744 2168

1759

1781 1713

2096 1701 1992 1936 1784

1716 1684 1703 1688 1686

5

0.39

1723

-15

-0.90

845

1496 1721

-1.68

1632 1720 1888 1840 1600 1664

1987 1622 1721 1769 1784

-15

694 735 755 761 698 792

913 916 888

905 891 890 922

2382 2276 2480

1632 1859

-

-

-16

-1.09 -0.20 -1.13

-10

-0.78

2

-0.17

-

-

21

1.82

18 1 14

1.52 0.07

13

0.98

1 0 6 3 2 -12 -

-

1.15

0.07 -0.01 0.81 0.38

-0.22 -1.30

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Table 7. Continued Actual patient days

Actual

Forecast

Percent error

7

902 899 883 902 887 815 919

2208 1989 1869 1976 1912 2120 2024

1682 1672 1620 1682 1633 1398 1737

24 16 13 15 15 34 14

2.86 1.72 1.34 1.60 1.51 3.45 1.55

....

1 2 3 4 5 6 7

1309 1237 1242 1345 1198 1094 1295

4638 4349 4243 4376 4192 4402 4360

3621 3127 3162 3868 2860 2147 3525

22 28 25 12 32 51 19

1.91 2.09 1.86 0.95 2.08 2.66 1.56

13 Med-surg ....

1 2 3 4 5 6 7

848 849 812 1085 1283 1069 1433

2261 2112 1941 2104 2168 2444 2864

1901 1903 1852 2227 2499 2205 2706

16 10 5 - 6 -15 10 6

0.84 0.49 0.20 -0.41 -1.39 0.78 0.64

1

1054 1049 998 1061 966 817 1039

2232 2296 1704 1765 1720 1808 1952

2176 2168 2090 2186 2042 1815 2153

3 6 -23 -24 -19 - 0 -10

0.30 0.69 -2.09 -2.26 -1.66 -0.02 -1.10

1034 1039 978 1051 1005 967 1051

1912 1960 1840 1976 1864 1909 1837

2068 2073 2013 2085 2040 2002 2085

-

8 6 9 -6 -9 -5 -13

-0.82 -0.59 -0.90 -0.57 -0.92 -0.48 -1.29

1

804 774 736 799 797 503 741

1760 1818 1853 1904 1736 1280 1720

1746 1700 1641 1738 1735 1281 1649

1 7 11 9 0 -0 4

0.08 0.65 1.15 0.91 0.00 -0.00 0.39

1

1193 1142 1087 1204 1074 1071 1192

3048 3141 2685 2816 2816 2360 2664

2976 2780 2569 3018 2519 2508 2972

2 11 4 - 7 11 - 6 -12

0.15 0.81 0.25 -0.42 0.63 -0.31 -0.66

Unit

Month

10 Med tchg ....

1 2 3 4

5 6 11 Med-surg

14 Orthoped

....

2 3 4 5 6 7

18 Med-surg ....

20 Med-surg

....

21 Med-surg ....

Fall 1975

1 2 3 4 5 6 7 2 3 4 5 6 7 2 3 4 5 6 7

RN hours

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those generated by Eq. 2, which assumes a shift in the variable nursing hours coefficient at the start of year 2 (July 1970). Table 7 shows the forecast and evaluation for 14 of the units for each of the first seven months in fiscal year 1972 (the seven remaining units are not shown because of space limitations). The next-to-last column is the percentage error of the prediction computed as actual value minus predicted value times 100 divided by actual value. A positive error indicates that the actual RN hours exceeded the predicted value. The final column is the t-value of the prediction error, computed by dividing the forecast error (actual minus predicted value) by the standard deviation of the estimate, calculated from Eq. 3. A t-value in excess of 2.09 (based on a two-tailed test with d.f. = 20) indicates a substantial departure from the model estimated from historical data. In units 1, 10, and 11, the actual number of hours worked exceeded the predicted values in every month. For each unit there is at least one error with a t-value in excess of 2.09 as well as two or more observations with errors exceeding 20 percent. It is therefore highly probable that these units had increases in staffing levels in fiscal year 1972, so a new set of estimates needs to be derived for them for future control. The results are most suspect for unit 11, which was closed from Oct. 1970 to Feb. 1971. When it reopened, the staffing levels were more than 50 percent higher (per patient day) than previously. Therefore the model for this unit would need to be reestimated. In particular, the estimated value of 6.86 marginal nursing hours per patient day (see b12 in Table 4) seems much too high. There are also some patient units (3, 6, 14, and 18) where the model generally overestimates RN hours. However, even in these units the model is useful in highlighting substantial variations in staffing patterns from month to month. For example, in unit 6, there were 1,154 patient days and 2,544 RN hours during month 1. The model predicted 2,427 RN hours for less than a 5-percent error, and everything seems to be operating normally. But in month 4 there were 1,174 patient days (20 more than in month 1) and only 1,832 RN hours (712 fewer than in month 1). The model predicts 35 percent more patient days, and the error is highly significant, with t = -2.37. This situation deserves the attention of an administrator to determine whether the unit is operating at distressed staffing levels in month 4 (and in month 5 as well; in this month too an abnormally low number of RN hours was recorded) or whether it was operating with too many RNs in most of the previous months. A similar situation exists in patient unit 14, where the first two months were normal but the next three months indicated unusually low staffing levels. In the remaining units (not all of which are shown) the actual number of RN hours coincides reasonably well with the predictions generated by the regression model, with occasional outliers appearing; for example, months 1 and 6 for unit 5 showed a decrease of 130 patient days but an increase of 32 RN hours. Despite the discrepancies mentioned, forecasts for the units were

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accurate enough, in the main, that four of seven monthly predictions for all medical-surgical units combined were within 4 percent of the actual hours worked and six were within 9 percent. Aggregate Forecasts Forecasts were also made for the aggregated patient units-the medicalsurgical, medical teaching, and orthopedic units listed before-using the aggregate regressions of Eq. 2 reported in Table 5. (Again, Eqs. 1 and 2 gave virtually identical forecasts.) The forecasts are shown in Table 8. For the medical-surgical units, the largest category in the hospital, the model's predictions were close to the actual values with errors of 1 percent or less in three of the seven months. Particularly interesting is month 6, which shows a 10-percent drop in patient days (probably due to the Christmas vacation). Staffing dropped accordingly, to the level predicted by the model. There is a tendency toward "understaffing," with three forecasts being 8 to 9 percent high, but this seems due to factors internal to these units (months 4 and 12 had more patient days but 888 fewer RN hours). The t-values indicate that these fluctuations of less than 10 percent are not inconsistent with the historical data. Table 8. Forecast of RN Hours and Evaluation Parameters for Aggregated Patient Units (Eq. 2) Unit

General medical and surgical

.....

Medical teaching ...

Orthopedics

Fall 1 975

.......

Actual

RN

hours

Patientt daYS

Actual

Forecast

1 2 3 4 5 6 7

5638 5473 5249 5650 5391 4813 5556

12600 12367 11343 11712 11192 10773 11571

12730 12323 11772 12760 12121 10697 12528

1 2 3 4 5 6 7

3354 3149 3144 3209 3189 3067 3307

7905 7269 6762 7536 7176 7096 6928

1 2 3 4 5 6 7

2112 2105 1996 2116 1974 1790 2081

4104 4072 3269 3301 3440 3568 3672

MOnth

group

FaIl 1975

Percent error 1 0 4 9 8 1 8

-0.14 0.05 -0.48 -1.13 -1.03 0.09 -1.04

7450 6721 6703 6934 6863 6429 7283

6 8 1 8 4 9 - 5

0.79 1.00 0.11 1.09 0.57 1.21 -0.63

4026 4015 3854 4032 3821 3548 3980

2 1 - 8 -22 -11 1 - 8

0.21 0.15 -1.59 -1.99 -1.02 0.04 -0.85

-

-

291~~~~~~~~~~~~~~~

291

Kaplan

The medical teaching units also have no month in which the forecast error exceeds 10 percent or the t-value exceeds 1.25. Staffing levels appear higher than average but not dramatically so. The most interesting set of observations occurred for the orthopedic units. Initially the actual and predicted values were virtually identical. For the middle three months staffing levels were much below the expected level (for example, month 4 had four more patient days than month 1 but 803 fewer RN hours), and this deficit is clearly marked by the data exhibited in Table 8.

LPN and Student Hours Hours worked by LPNs, aides, and orderlies are reported monthly by patient unit (see Tables 1 and 2). LPN hours were analyzed in the same way as RN hours, but the results were much less conclusive. The coefficient for marginal LPN hours worked per patient day was generally small and frequently negative. Only two patient units had a significantly positive coefficient. When patient units were aggregated, the medical-surgical group had a significantly positive coefficient of 0.5 LPN hours per patient day. The other two aggregate units, medical teaching and orthopedics, also had coefficients of this magnitude, but the significance levels were much lower. To see whether LPNs are substitutes for RNs, the regressions previously described for RNs were repeated with an added variable for LPN hours. If the coefficient of this variable were negative (i.e., if LPN hours increased as RN hours decreased, holding other variables constant), one could conclude that LPNs do substitute to some extent for RNs. However, no strong effect was noticed and, in fact, the medical-surgical and medical teaching groups had positive signs for this coefficient. A similar analysis was performed for student nursing hours. It is possible that student nurses increase the load on RNs if they simply add a teaching function to other RN duties and do not contribute much to patient care. An alternative hypothesis is that student nurses can perforn a number of nursing functions satisfactorily with a minimum of supervision and hence reduce the number of RNs that need to be on duty. When a regression was performed using either Eq. 1 or Eq. 2, but adding a fourth variable representing the number of student nursing hours in the patient units each month, the sign of the coefficient of this new variable was consistently negative. The magnitude of this negative coefficient was small (usually between -0.5 and zero), but it was frequently significant, especially in the aggregate regressions. It appears that the contribution of student nurses to patient care somewhat exceeds the burden of their supervision. Finally, the analysis described previously for RN hours in the three groups of aggregated patient units was repeated using total nursing hours. Only Eq. 1 was used. The regression results, presented in Table 9, are comparable to those in Table 5, with an increase between 1.0 and 1.5 in marginal nursing hours per patient day. Equation 1 was then used to forecast total nursing hours in the seven-month 292

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Table 9. Regression Results (Eq. 1) for Aggregated Patient UnitsTotal Nursing Hours (t Ratios in Parentheses) Nursing hours (total)

R*

Unit

group

Df

General medical and surgical . 0.89

1.44

Medical teaching .0.80

2.09

Orthopedics .0.68

2.37

Per

patient day (bi) 3.6 (5.2) 4.0 (4.6) 3.0 (2.6)

Permanent

Seasonal increase

increase

&) (2

(b3 ) 2747 (3.6) 1476 (6.8) 939 (6.3)

838 (1.1) 954 (3.8) 421 (2.4)

* Adjusted R. f Durbin-Watson statistic.

test period. The output is shown in Table 10; again, the model appears to do an excellent job of aggregating and presenting the data for administrative review. All seven observations in the medical-surgical units are within an admissible range. Some observations in the medical teaching units (months 2 and 6 and Table 10. Forecast (Eq. 1) of Total Nursing Hours and Evaluation Parameters for Aggregated Patient Units Actual

Unit group

General medical and surgical .........

Medical teaching .......

Orthopedics

...........

Month

patient

1 2 3 4 5 6 7

Percent

Actual

Forecast

5638 5473 5249 5650 5391 4813 5556

21587 20488 17534 19040 18046

20009 19421 18621 20052 19128 17066 19717

7 5 - 6 - 5 - 6

1 2 3 4 5 6 7

3354 3149 3144 3209 3189 3067 3307

13121 12461 10730 11789 12309 12226

12089 11262 11242 11504 11424 10931 11900

8 10 - 5 2 1 1 3

1.78 2.16 -0.92 0.51 0.18 2.46 0.57

1 2 3 4

2112 2105 1996 2116 1974 1790 2081

7096 7056 5669 5898 5712 6232 6672

6650 6629 6300 6662 6233 5678 6556

6 6 -11 -13 - 9 9 2

1.15 1.10 -1.61 -1.96 -1.32 1.12 0.30

5 6 7

Fal17 1 975 Fall

Nursing hours (total)

17531 19317

11526

error

3 - 2

1.17 0.81 -0.84 -0.75 -0.83 0.36 -0.30

9

293

Kaplan

possibly month 1) are signaled for investigation, and the decrease in staffing in the orthopedic units during months 3-5 is again identified. DISCUSSION

In many previous studies, efforts have been undertaken to weight patient days by the proportion of patients in various patient care categories so as to obtain a more accurate index of demands on nursing personnel. If such an index is available the methods described here are still satisfactory. Instead of using patient days as the independent variable in the regression, the weighted index of patient demands on nursing would be used instead. Though such an index was not available at the hospital studied, its lack does not seem critical when monthly observations are used: there may be considerable day-to-day fluctuations in the severity of illness of patients in a patient unit, but it is likely that over a period of 30 days such fluctuations will tend to cancel out. Thus patient days should be a reasonable surrogate for patient demands on nursing, but naturally there is no way of verifying this assumption with the available data. Some more work on updating estimates to reflect changes at certain patient units would need to be performed before the system could be implemented, but the information generated by the model should be very useful to an administrator. One need only compare Table 7 (which, if complete, would be about one-third larger) or Table 8 with seven copies of Tables 1 and 2 to see which provides the most useful and convenient information to the administrator. The data as displayed in Table 7 can be used in conjunction with standard industrial quality-control procedures for evaluation. For example, using a control chart procedure [26,27] one would plot for each unit the deviation of actual from predicted hours worked each month (Mansfield and Wein [28] describe the implementation of a control chart for costs based on a regression equation.) Some heuristic criteria could then be established as to when to signal the administrator of an exceptional occurrence. Such criteria might include, for example, a deviation whose t-value exceeds 2, a deviation of more than 20 percent, or three consecutive deviations with t-values exceeding 1.0 or percentage errors exceeding 10 percent. Thus the bulk of the observations would not need to be displayed to an administrator, and he could concentrate on those units in which staffing exceeded the bounds expected during normal fluctuations. The behavioral aspect of the control procedures should not be overlooked. In this study there was no attempt to tell nursing supervisors how they should be operating or staffing their patient units. Rather, the model describes how they have been performing in the past; it only makes explicit what previously may have been an informal, somewhat unmonitored procedure. If some of the estimates are not considered satisfactory (e.g., variable RN hours per patient day less than 1.0 or 1.5 in a patient unit), new standards can be negotiated between the administrator and nurse supervisor.

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The main objective of the statistical analysis is to obtain an estimate of marginal nursing hours per patient day around the point (as measured by average patient days per month) where patient units expect to operate, and to provide a confidence limit for judging deviations from that point as excessive or normal. This estimate is different from average nursing hours per patient day unless one assumes that the fixed component of nursing hours per month is

zero. If a change in staffing level occurs, or if historical data of the type used here are not available, the figure for marginal nursing hours could be estimated or agreed upon without the aid of statistical analysis. The flexible budgeting

scheme described in this article can be used as long as the marginal nursing hours coefficient is determined and the average patient days and average nursing hours in a month are also known: the budget line can be determined if one point on the line and its slope are given. Presumably these average patient days and nursing hours are available or previously set, so that only the slope needs to be negotiated between the nursing supervisor and the hospital administrator. The kind of information provided by a report in the form of Table 7 could also be useful in initiating or evaluating requests for additional nurses. The administrator might agree to such requests, provided that the added staff were used to increase the variable staffing component rather than the fixed component, i.e., the slope would be increased but not the intercept. REFERENCES

1. Administrative Profiles. Hospitals 40:20 Jan. 16, 1972. 2. Maier-Rothe, C. and H. Wolfe. Cycle scheduling and allocation of nursing staff. Working paper, Arthur D. Little, Inc., Cambridge, MA, 1971. 3. Offensend, F. L. Design, control, and evaluation of inpatient nursing systems. Doctoral dissertation, Department of Engineering-Economic Systems, Stanford University, 1971. 4. Connor, R. J. et al. Effective use of nursing resources. Hospitals 35:30 May 1, 1961. 5. Connor, R. J. A work sampling study of variations in nursing work load. Hospitals 35:40 May 1, 1961. 6. Connor, R. J. Hospital work sampling with associated measures of production. J Ind Eng 12:105 Mar.-Apr. 1961. 7. Feyerherm, A. and W. Kirk. Effect of census variation on nursing activity pattems. Hospitals 38:62 Apr. 1964. 8. Harris, H. Nursing-staffing requirements. Hospitals 44:64 Apr. 16, 1970. 9. Saren, M. and A. Straub. Nursing service effectiveness. Hospitals 44:45 Jan. 16, 1970. 10. Jackson, R. M. and C. Kortge. Automated proficiency reports. Hospitals 40:76 Jan. 1, 1971. 11. Wolfe, H. and J. Young. Staffing the nursing unit: Controlled variable staffing. Nurs Res 14:236 Summer 1965. 12. Pardee, G. Classifying patients to predict staff requirements. Am J Nurs 8:517 Mar. 1968. 13. McCartney, R., B. McKee, and I. Cady. Nurse staffing systems. Hospitals 44:102 Nov. 16, 1970. 14. Clark, E. and W. Diggs. Quantifying patient care needs. Hospitals 45:96 Sept. 16, 1971.

Fall 1 975

FaI~~l 17

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Kaplan 15. Frances, M. A. Implementing a program of cyclical scheduling of nursing personnel. Hospitals 40:108 July 16, 1966. 16. Howell, J. Cyclical scheduling of nursing personnel. Hospitals 40:77 Jan. 16, 1966. 17. Morrish, A. and A. O'Connor. Cyclic scheduling. Hospitals 44:66 Feb. 16, 1970. 18. Wolfe, H. and J. Young. Staffing the nursing unit: The multiple assignment technique. Nurs Res 14:299 Fall 1965. 19. Warner, C. and J. Prawda. A mathematical programming model for scheduling nursing personnel in a h9spital. Manage Sci 19(B):411 Dec. 1972. 20. Hershey, J., W. Abernathy, and N. Baloff. Comparison of nurse allocation policies: A Monte Carlo model. Decis Sci 5:58 Jan. 1974. 21. Abernathy, W., N. Baloff, J. Hershey, and S. Wandel. A three-stage manpower planning and scheduling model: A service-sector example. Oper Res 21:693 May-June 1973. 22. Abernathy, W., N. Baloff, and J. Hershey. The nurse staffing problem: Issues and prospects. Sloan Manage Rev 12:87 Fall 1971. 23. Homgren, C. Cost Accounting: A Managerial Emphasis. 3d ed. Englewood Cliffs, NJ: Prentice-Hall, 1972. 24. Goldberger, A. Econometric Theory. New York: Wiley, 1964. 25. Grunfeld, Y. and Z. Griliches. Is aggregation necessarily bad? Rev Econ Stat 42:1 Feb. 1960. 26. Duncan, A. Quality Control and Industrial Statistics. 3d ed. Homewood, IL: Irwin, 1967. 27. Bowman, E. and R. Fetter. Analysis for Production and Operations Management. 3d ed. Homewood, IL: Irwin, 1967. 28. Mansfield, E. and H. Wein. A regression control chart for costs. Appl Stat 7:48 Mar. 1958.

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