Impact of Expanded-Duty Assistants on Cost and Productivity in Dental Care Delivery

by Joseph Lipscomb and Richard M. Scheffler Data from an experimental dental program are used to develop a linear programming model of dental care delivery that the authors use to examine the economic implications of introducing expanded-duty dental assistants (EDDAs) in three types of dental practices. The authors examine the changes in productivity and profitability that result from hiring one or more EDDAs and conclude that a dentist in solo practice can more than double his net revenue by hiring one EDDA but will not increase his productivity further by hiring additional EDDAs. Two- and three-dentist groups also can increase revenue by hiring EDDAs, but, beyond a certain point, an inverse relationship exists between the number of auxiliaries hired and net revenue generated. Use of expanded-duty dental assistants (EDDAs) has the potential to increase the efficiency and profitability of services delivered in private dental practices, and in many states EDDAs are now allowed to perform many tasks that until recently were performed only by dentists [1]. However, in most practices, only conventional dental assistants are used; in such practices the dentist performs most procedures. Although state laws will no doubt continue to require that dentists supervise the work of all assistants, it is anticipated that the EDDA will have some autonomy and will be able to free the dentist to perform more tasks for which he alone is authorized [2]. Recent studies of the impact of the EDDA on dentist productivity have been of two principal types: actual experiments, in which one or more EDDAs are introduced into a laboratory setting [3-10], and digital computer experiments, in which detailed data on the input characteristics of a few real practices are the basis for a model that simulates the production behavior of a variety of types of dental practices. The experimental research efforts of Baird et al. [3], Ludwick et al. [6], Roemke [9], and Soricelli [10] show that EDDAs can increase dentist proResearch supported by the University of North Carolina Health Services Research Center through Research Grant No. 5 Pis HS 00239-03 from the U.S. Department of

Health, Education, and Welfare. Address communications and requests for reprints to Joseph Lipscomb, Ph.D., Institute of Policy Sciences, Duke University, Durham, NC 27706.

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ductivity, but these studies evolved from specific experiments conducted in special clinical settings. It was not their purpose to predict dental practice productivity with different staffing or case mix proportions. Macro level studies by Maurizi [11] and P. Feldstein [12] indicate that increased use of assistants would raise dental productivity. However, Boulier [13] adapted Reinhardt's neoclassical production function methodology to the study of dentistry and found that-given the hours worked by dentists, the prices of dental services, and the wage rates of auxiliaries-dentists on average hire about the optimal number of aides for maximizing their hypothesized utility function (about 1.5 aides per dentist). These studies did not investigate the productivity potential of the EDDA. The best-known simulation analysis is by Kilpatrick et al. [14]. The approach taken in the present study is similar to that of Kilpatrick et al. in that microproduction cost and revenue data are used to study the productive potential of the EDDA in different types of practices. However, because that simulation and this study employed different procedure time distributions, assumptions on patient scheduling, and values for a number of input parameters, the similar boosts in productivity cannot be used to cross-validate the two methods. This article describes the results of taking a third approach to examining the economic impact of adding EDDAs-developing an optimization model of a dental practice that uses information on the number and performance characteristics of inputs and suggests how EDDAs can be employed to maximize net revenue. This approach differs from laboratory studies and simulation experiments, both of which are essentially descriptive, in that it is concerned with how practices might modify their current use of paramedical personnel. The model is used to examine the potential productivity gains of introducing EDDAs into different kinds of dental practices and to explore production possibilities, legal and economic constraints, and output implications. An activity analysis using a linear programming model of dental care delivery is presented that uses cost, revenue, and time-and-motion data on the functioning of dentists and EDDAs. Data for the model were generated at the University of North Carolina Dental Demonstration Practice (DDP), where four dentists and two EDDAs served a patient population typical for private practices in a small urban setting [4]. Data for the analysis were collected during an 11-month period in 1972, when the DDP was fully operative. In developing the linear programming model, the following issues were considered: (1) the marginal physical, marginal revenue, and marginal net revenue productivity resulting from the addition of EDDAs; (2) the expected gains in output and net revenue from the addition of EDDAs and how the gains vary according to the number of full-time dentists in the practice; (3) the impact of different case mix requirements on practice output, cost, and profitability as the number of EDDAs is varied; (4) the cost to a dental practice in reduced net revenue, and to a community in reduced services, of state laws restricting the delegation of dental tasks; and (5) the output and profit

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losses associated with increased dentist leisure and the extent to which the use of EDDAs would reduce these costs. The Model An attempt to develop an optimization model for something as complex as the practice of dentistry requires a number of initial assumptions, particularly about the behavior of actors in the model. In developing a model for an efficient, profitable dental practice, it was assumed that the objectives of such a practice would be to maximize net revenue, subject to resource, technological, legal, and case mix constraints. The maximand implies that the dentist is guided, in part, by the profit motive; there is evidence in the literature to support this assumption. Benham et al. found that, despite strict state licensing laws, dentists "displayed a tropism for higher incomes that has caused them to migrate with the effective demand for their services" [15]. Several sociological studies have pointed to the profit-making orientation of dentists. Linn [16] found that when dental students were asked to rank order the values held by their classmates, they ranked economics first. Manhold et al. [17] concluded that dental students were more economically oriented than medical students, while Heist [18] found dental students to be highly utilitarian and pragmatic. Nevertheless, it may be argued that leisure time is a relevant element in the maximand of a dental practice, especially in light of studies that indicate a dentist could employ EDDAs, reduce his chairside time, and still maintain or increase his income [14]. Two possible approaches in developing the model were considered: either the dentist maximizes some combination of net income and leisure, subject to all constraints, or he minimizes practice time, subject to achieving an acceptable minimum net income and subject to all other constraints. The model employs the former approach because the sociological and economic studies cited place considerable emphasis on the pecuniary motives of dentists and dental students; because virtually all state dental practice laws require that EDDAs work under the direct supervision of dentists, implying that the relevant trade-off may not be between work and leisure but rather between chairside and management duties; because the former approach is a natural framework within which to examine legal restrictions on task delegation; and because there is little other empirical or theoretical evidence discriminating between the two approaches. The four dentists participating in the project were each scheduled to work one day at the DDP each week. They were paid a base salary with the opportunity to earn additional income after expenses were met. This profit-oriented structure prevents doing a retrospective study now to test whether dentists would prefer the option of minimizing their practice time (and earning, perhaps, less money than they could). Lacking other evidence, the approach taken in developing the model was: given some prior division of the dentist's time between work and lei16

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sure, what is the maximum revenue that can be generated by an efficiently operating practice? The model consists of a revenue-generating component, which estimates the maximum revenue, and a cost-generating component, which estimates the total cost per unit of practice time, given the staffing composition of the practice, the skill level of auxiliaries, and the case mix. It is assumed that a practice contemplating hiring one or more EDDAs will adjust its other inputs to economically optimal levels considering the expanded staff and output. Since the cost estimates are essentially long-run, the overall model should be viewed as a planning tool with which to explore the economics of various scales and types of dental practices. Revenue Component The revenue component is designed to predict a practice's maximum gross revenue, subject to the following constraints: * That the total amount of dentist and EDDA service time not exceed the number of working hours for which these inputs are available. * That whereas the dentist may perform any procedure, the EDDA is only permitted to undertake sanctioned tasks. * That a dental service is defined as the performance of a specific sequence of tasks (if the model required that some number, N, of a service be performed per unit of practice time, each component task of the service would be done exactly N times). * That the quantities of services produced must reflect a predetermined proportional relationship. This is tantamount to requiring that the practice maximize net revenue, subject to a demand composition that is exogenously determined. (The economic impact of manipulating demand composition toward more profitable services may be investigated by adjusting the a priori proportional relationships governing service composition.) The general form of the revenue-generating component is given in Eqs. 1 through 5 below. This part of the model is a linear program, the objective function of which is Total revenue = [(Y/r)Ald + (Y/r)Ale + (Y/r)A2d + (Y/r)A2e +± * * + (Y/r)Ard + (Y/r)Are] + * + [ (W/s) Qld + ( W/s) Qe + (W/s) Q2d +

(W/s)Q2e + * * * + (W/s)Q8d + (W/S)Q8e]

(1)

where each bracketed expression is the total revenue generated by the performance of the set of procedures that together define a particular dental service; the practice produces a number of services, two of which are represented in the above expression and are called A and Q. Service A is characterized by r procedures and Q by s procedures. The superscript d on a variable indicates that the procedure may be performed only by the dentist; the e superscript means that the task may also be performed by the EDDA. Thus Are represents the level or intensity of the rth procedure of service A performed by the

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EDDA per time unit (in the results below, per month). The revenue coefficients, Y/r and W/s, are constructed so that the performance of one unit of each service means that exactly (Y/r)r = Y dollars and (W/s)s = W dollars of gross revenue are recorded. (Dental services defined for this study are listed in Table 1, p. 20.) The labor input constraints mentioned above are specified as DT > aldAld + a2dA2d + * *

+ aidQid + a2dQ2d +

+ ardArd ...

+ q8dQgd

ET > aleAle + a2eA2e + * *+ areAre + q,eQ,e + q2eQ2e + * * + q8eQ8e

(2)

(3) unit of

where a is the expected amount of time required to perform one procedure A and q is the time required to perform one unit of procedure Q. DT and ET are the total amounts of dentist and EDDA time effectively available during a given period. (Values for DT and ET must accurately reflect downtime due to missed appointments, scheduling difficulties, and miscellaneous problems.) Constraints for other resources of the practice (regular dental assistants, instruments, etc.) are not given since it is assumed that there are no limits on the availability of other labor and capital inputs and that dentists and EDDAs have equal access to these resources. According to DDP staff, these are realistic conditions that did generally prevail during the study period. The EDDA utilization constraint discussed earlier is already incorporated into the objective function by omission of EDDA procedure variables where state laws limit task delegation. The stipulation that services consist of well-defined sets of dental procedures may be fulfilled by imposing the constraints that (for instance) A1d + Ale = A2d

A2= A3d + A3e

Ar-1d + Arle = Ard + Are

Q1d + Qle = Q2d + Q2e

Qsd + Q,_1e

=

Qd

(4)

The r -1 equality constraints ensure that the r procedures defining service A are performed an equal number of times regardless of the number of units of

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A that are given in the solution of the linear program. The s- 1 constraints likewise require that the procedures defining Q are produced in equal number. If the practice must comply with a specific case mix constraint, so that A and Q are to be produced in the ratio t :1, a sufficient condition to ensure this is (for instance) (5) tQ8d A2 = 0 That is, for every Q8d, and thus for each unit of service Q produced, the practice must produce t units of A2d and t units of service A. (Empirical specification of the case mix ratios is given in Table 2, p. 21.) The programming algorithm, which is well defined [14], selects the particular vector of procedure output levels

[Ald A1 A2, A2, .

,

Ard Are

Q 1d, Qle, Q2d, Q2 , . . . , Q8d, Qse]

that maximizes total revenue subject to all constraints discussed above. Cost Component The cost component yields estimates of both line-item and total long-run cost of producing a particular service load. Specific elements of long-run cost include dental supplies, laboratory fees, laundry, EDDA salary, dental assistant salary, payroll taxes, office supplies, patient statements, postage, rent, telephone, subscriptions, insurance, maintenance and repairs, accounting fees, professional meetings and dues, depreciation of dental equipment, and depreciation of furniture and fixtures. Long-run average cost (LRAC) is defined as the total cost incurred in adding an EDDA divided by the number of procedures produced as a result. Empirical Basis for the Model Service Categories Each dental practice modeled produces some mix of the following service types: diagnosis, extraction and surgery, endodontics, crown and bridge services, prosthetics, preventive care, and restoration. Each category consists of one or more specific dental procedures (Table 1, p. 20). Where several services include similar procedures, only one representative service is shown. For instance, under restoration, the model designates procedures for six different types of metal fillings: one-, two-, and three-surface metal fillings and one-, two-, and three-surface metal fillings preceded by temporary fillings. Variations in the number of surfaces treated are handled by adjusting the sampling estimates for the two procedures, amalgam preparation and amalgam restoration. Table 1 also lists the dental procedures that can be assigned to an EDDA under most state laws.

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Table 1. Delegation Guidelines for Dental Procedures Procedure

Service Diagnosis.

May also be delegated

Dentist only

to EDDA

Panorex Periapical radiograph Bite wing radiograph Charting

Examination

Extraction and

surgery ..............

Endodontics ............ Crown and bridge services ..............

Anesthesia Extraction Postoperative treatment Curettage, root planing, periodontal scaling Anesthesia Extirpation, instrumentation, fill Anesthesia Fitting of crown or bridge Try in of crown or bridge Cementation

Prosthetics .............

Partial denture (drill) Denture repair and adjustment

Preventive care .........

Scaling

Restoration .............

Anesthesia

Amalgam preparation (drill) Adaptic preparation (drill)

Clean off cement, pack string Precision partial Place/remove temporary crown or bridge Impression for denture reline Impression for partial

denture Oral health instruction Flossing and brushing techniques Prophylaxis Rubber dam placement Amalgam restoration (fill) Occlusal adjustment Finish and polish restoration

Adaptic restoration (fill)

Procedure Performance Times Sample means and standard deviations of the times (in hours) required by the four dentists, as a group, and the two EDDAs, as a group, to perform the procedures are available from the first author on request. These means are unbiased estimators of the a and q input coefficients in Eqs. 2 and 3. Distributions were calculated separately for the dentist and for the EDDA since the differences in training and experience between the two personnel types might have resulted in differences in procedure performance times. The striking result was that for many procedures, EDDA mean time was less than that for the dentists; it is possible that the EDDAs concentrated on rela20

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Table 2. Percentage Composition of Case Mix by Type of Practice Service category

Practice

Extraction Endo

Crown

Pros-

Preven- Resto-

nosisag-

and surgery

dontics

and bridge

thetics

tive care

ration

14

3

2

20

1

7

53

...............

6

20

0

0

10

1

63

Cosmetic care ...............

10

9

14

35

1

4

27

General practice

..........

Primary care

tively few duties and therefore attained exceptional efficiency in performing these functions. Studies conducted at the DDP indicate that efficiencies gained through the use of EDDAs do not come at the expense of quality of care, as measured both by independent professional judgment of clinical skills and in posttreatment patient interviews [4]. Case Mix Three specific case mixes were defined: general practice (GP), reflecting the case mix of the DDP; primary care (PC), reflecting the case mix of a (perhaps relatively new) practice facing the pent-up and emergency demands of a poor population; and cosmetic care (CC), with a case mix emphasizing nonemergency, often elective, services. The service composition of each case mix is given in Table 2; case mix requirements are incorporated in the model through equality constraints similar to Eq. 5. There is experimental evidence [4] that the DDP case mix, which represents a balance in restorative care, preventive care, crown and bridge services, prosthetics, and diagnostic care, is typical of small urban practices. The PC category includes a high percentage of emergency services, extractions, and basic restorative services delivered in many neighborhood health care centers [19]. The CC mix, heavily weighted toward crown and bridge services, prosthetics, and endodontics, characterizes a "Madison Avenue" type of practice. Information on the number of each type of dental practice is not available. Ratios of service types were specified within each service category from observations and staff consultations at the DDP. The following intracategory case mix ratios were determined: diagnosis: diagnosis with alginate impression = 4:1; extraction : periodontal surgery = 56:1; gold crown : porcelain crown: gold bridge: porcelain bridge = 43:27:14:16; metal filling: composite filling = 15: 1; and (regular) metal filling: metal filling with temporary = 20 : 1. (Equal numbers of one-, two-, and three-surface fillings were required within each category of metal filling.) To simplify matters, it was decided to apply

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the above intracategory ratios to all three general case mix types. The ratios were incorporated into the model through constraints in the basic form of Eq. 5. Labor Input The revenue estimates for each practice are for a month including 22 working days; the EDDA, if present, works chairside 8 hr per day and the dentist works 7 hr. Following Kilpatrick et al. [14], the model assumes that patient demand is sufficient to fully employ the practice during the 22 days. This full-demand assumption is not unrealistic and facilitates comparison among dental practices by allowing examination of the maximum expected productivity attainable, on the average, from each type of dental practice [20]. The maximum possible values of DT and ET for one dentist and one EDDA are, respectively, 154 hr and 176 hr, but these figures do not allow for broken appointments, scheduling difficulties, and other downtime. Special studies at the DDP indicated that, on the average, a dentist or EDDA will be effectively unemployed about 12 percent of the time because of disruptions in production; thus the maximum DT and ET values were reduced by 12 percent. Clinic data also showed that, with the introduction of EDDAs, some dentist time shifts from direct delivery of services to supervision of auxiliaries. Hence each dentist's effective availability was reduced 8 min per day per EDDA. To modify the number of dentists and EDDAs within a practice, one need only adjust the values of DT and ET in the model. One advantage of a period of operation as long as a month is that it allows consideration of more complicated dental services such as crown and bridge services and temporary metal fillings, which require multiple patient visits. Costs Based on cost accounting operations at the DDP, cost information [21,22], and the judgment of DDP staff, the following operational assumptions were made: laboratory fees are 38 percent of total revenue generated by crown and bridge services and prosthetics; supplies (including drugs and equipment repairs) are 58 percent of laboratory fees; there is one regular dental assistant for each dentist and one for each EDDA; there is one fully equipped chair for each dentist and each EDDA in the practice (minimum of three chairs); and the chairs are depreciated monthly in a straight-line fashion over a 7-yr period. At present there are few EDDAs in private practice [4], and no market wage rate has emerged; the model therefore assumes the EDDA is paid the salary prevailing in 1972 at the DDP, $6200 per year. For each type of practice, values for other cost items were obtained by adjusting the estimates by using multipliers chosen by the clinic staff that reflected the impact of practice expansion on costs. For many components the expansion was proportional to output. Operationally, for any type of practice studied, total cost is the simple 22

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sum of costs associated with each element of cost listed above in the section describing the cost component of the model (p. 19). Service Fees Fees were based on charges thought to be standard for dental services in North Carolina. Service fees are represented by the parameters Y and W in Eq. 1. The revenue coefficients Y/r and W/s are not fees as such, since services are billed rather than procedures. This coefficient specification is simply a convenient way to ensure that each rendered service yields the appropriate revenue. Model Verification and Validation It had been planned to verify and validate the linear programming model on a prospective basis at the DDP, by determining whether the process of production in a dental practice could be represented as a linear program and whether the chosen program would replicate practice output. Unfortunately, the DDP ended its dentist-EDDA project before this article was completed. Further, a decision in the spring of 1974 by the North Carolina State Board of Dental Examiners to forbid in experimental settings the delegation of tasks not authorized by state law has precluded any such prospective study. Nevertheless, data gathered during the 11-month observation period at the DDP can be used to validate aspects of the model and to test whether the model couldgiven appropriate input specification-replicate the behavior of output over that period.

Testing Assumptions of the Model Putting the optimization model in the form of a linear program implies certain assumptions concerning the nature of returns to scale in the practice of dentistry, the relationship between patient scheduling and productivity, the divisibility of inputs and outputs, and the stochastic nature of the production technology. Returns to Scale. Use of a linear programming model implies that the production process exhibits constant return to scale; that a given percentage change in the quantity of all continuously variable factors of production (factors that may be easily adjusted) yields the same percentage change in output. If each procedure requires a fixed amount of dentist and/or assistant time, if the intervals between procedures are constant (so that the time cost of manpower substitution is not a function of output), and if the intervals between services are constant, a dental practice will exhibit constant returns to scale. Correspondingly, constant return to scale is not a reasonable assumption if task time distributions or the ease with which EDDAs and dentists may be substituted for each other is functionally related to the volume of output. A task time distribution would be altered because of significant, systematic changes in either its mean, its variance, or its covariance with other procedure

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times. Fluctuations in covariance with output could be a relevant concern if the levels of factor inputs in each constraint were not strictly additive, implying the existence of interaction effects among inputs. In order to test for significant relationships between mean procedure times and the volume of output produced per time unit, Pearson correlation coefficients were computed between the mean monthly times for placing amalgam restorations (a common, well-defined procedure) and total chairside time per month for each dentist. The analysis was then repeated using charting, another common, well-defined procedure. XVith four dentists and two procedures, there were eight correlation coefficients between mean task times and hours worked. Of these, only one exceeded 0.50 in absolute value, and it was the only one that was statistically significant at the 0.05 level. (In fact, it was significant at the 0.01 level.) These figures should be treated with some caution since there was no attempt to control for the myriad factors also affecting the two time series. But the results do indicate that fluctuations in hours worked have little impact on the average time required by dentists to perform tasks. This would support the assumption that mean task times are stable. It was not possible to perform correlation analysis using total chairside time and other indexes of production within the practice because 1972 data on task times were no longer contained in the computer (although the total number of each procedure performed each month and the time spent by dentist and EDDA can be determined). The relative stability of the mean times for amalgam restoration does give some indication that the variance in that case was not fluctuating perversely over time. Such stability may be reasonably expected because of the nature of dental practice. A dentist spends the majority of his time performing a relatively small number of tasks requiring considerable precision. Standards of care for many tasks can be enumerated in checklist form [4,5], and other studies of medical manpower at the micro level accept the assumption of task time distribution stability [14,23]. Another assumption made here (and by Kilpatrick et al. [14] and Smith et al. [23]) is that the covariance structure between task times does not vary functionally with output. This may not be a reasonable assumption, but the problem has not been explored. The procedure times of dentists and EDDAs working jointly must vary, depending on their professional and personal characteristics. In no sense can observations on procedures for a given service be paired as if they had been generated by a single multivariate stochastic process. It may be that as output expands, labor inputs interact at chairside in response to increased demand. The analysis did not assume absence of economies of scale. Rather, for the purpose of assessing the profitability of adding EDDAs incrementally, it was assumed that all factors of production (except the number of dentists) are variable. Since the physical plant and large capital items are inputs that need to be incremented only at certain discrete points as output expands, total cost did not vary linearly with total revenue. On the other hand, because the behavior of average cost as all inputs expanded proportionately was not ex24

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plored, the analysis here does not, strictly speaking, focus on economies of scale as traditionally defined. However, in studying these "scale" effects for practices with differing numbers of dentists, all relevant dentist-EDDA combinations (for up to three dentists) are examined here. Another problem in verifying constant returns to scale is whether ease of input substitution is physically affected by the volume of output. Although DDP clinic staff indicated that output volume had no effect over the range of output rates produced at the DDP, it cannot be assumed that there is no time cost associated with switching manpower types during the delivery of a given service or between successive services. Downtime distributions have been computed from observations at a DHEW pilot program being conducted at a Cleveland, TN facility, and these data are now being used by Kilpatrick and Mackenzie [1]. Observers at the DDP recording downtime estimated a 12percent average reduction in input availability and an 8-min per day allowance for dentist supervision of EDDAs. Thus manpower-switching costs are important, and they have not been ignored by researchers, but more work is needed in relating these estimates to the degree and composition of demand facing a given practice. Patient Scheduling vs. Productivity. The relationship between patient scheduling and productivity can be stated as follows: aside from specific downtime allowances and minimum intervals between services, the optimum solution requires a sequencing of patients that utilizes personnel to the maximum. This is the reason that it was claimed above that the linear programming representation in the model is geared to indicate the maximum feasible output obtainable from a given set of inputs and prior constraints. However, such an optimization model does not explicitly indicate what the maximizing sequence of patients should be. Computer simulation models have an advantage here in that they are able to test the output effects of alternative scheduling sequences. (This implies that optimization and simulation could work well in concert, the former suggesting the optimum solution for given inputs and the latter testing the sensitivity of that solution to various input structures.) Divisibility of Inputs and Outputs. Specification of the input availability values DT and ET in Eqs. 2 and 3 reflects an integral combination of dentists and EDDAs. Thus Reinhardt's suggestion [24] is being followed that allowance be made for the probable divergence between input time purchased and time actually used. A linear programming model, unless restricted to produce integer-valued outputs, will generally yield fractional solutions. If the optimum number of procedures produced is large, as in this case, rounding these solutions to nearest whole-unit values provides a reasonable approximation to the corresponding integer programming solution. But in modeling the production process for dental services delivered on a daily basis, one of two additional assumptions must be made: either that fractional parts of patients can be seen during a given work day (based on 1/22 of the value of DT and ET) or that the practice can juggle its schedule so that each work day ends as the final patient

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procedure is completed. Strictly speaking, neither assumption is realistic. If one assumes that DT and ET are as inflexible as the model portrays them, a linear program will tend to slightly overestimate productivity. If dentists and EDDAs finished treating that last, fractional patient, the programming results would then represent a slight underestimate of practice productivity. Regardless of how a real practice would handle the end-of-day patient, the margin of error from application of the model should be small relative to total output. The Stochastic Nature of Production Technology. The linear programming model treats the a and q coefficients of Eqs. 2 and 3 as if they were known, fixed constants, which is an unrealistic assumption. Stochastic programming problems in which the technological coefficients are assumed to be random have not been treated in depth in the literature, and methods for solving other than special cases have not been developed [25]. The central theoretical problem can be characterized as follows: the stochastic nature of the a and q coefficients implies that the right-hand sides of the inequality constraints in Eqs. 2 and 3 are well-defined multivariate distributions. The resource availabilities, DT and ET, act to truncate these distributions, so that the corresponding distribution of optimal solutions is not easily derived theoretically. Therefore, if one uses mean procedure times, calculates the optimum vector of output (as has been done here repeatedly), and then draws observations from the resulting multivariate constraint distributions, this "optimum" output vector will in fact be feasible only a certain (unpredictable) part of the time. (This observation applies to all such constrained optimization models.) A related question is the degree to which the randomness of the technological coefficients imparts variability to the right-hand sides in Eqs. 2 and 3 over the period modeled. The programming solution summarized in Table 3, p. 28, for a practice with one dentist, one EDDA, and facing a GP case mix provides a good example. It was found that all of the dentist's 132.79 scheduled hours and 137.81 of the EDDA's 153.12 scheduled hours were utilized in producing $14 134 in monthly gross revenue. Standard deviations were then calculated for each procedure in the model. (Because of the problem of grouped observations, each procedure distribution was assumed to be approximately normal and a standard deviation estimator was used that can be shown to be efficient [26].) Given the individual task time distributions and the optimum vector of procedures for a GP case mix, the standard deviation of the multivariate normal right-hand side of the dentist constraint was 12.05 hr. The corresponding right-hand-side standard deviation for the EDDA constraint was 17.87 hr. These results suggest that in order to produce the optimum output vector (i.e., maintaining the same task delegation pattern), the amount of dentist chairside time required will fall between 120.14 and 144.55 hr about 68 percent of all months. Likewise, for about 68 percent of all months the amount of EDDA chairside time required will be between 119.93 and 156.68 hr. Viewed another way, for only 16 percent of all practice months will a dentist have to work more than 144.55 hr and an EDDA have to work more than 156.68 hr to produce the optimum monthly procedure vector. 26

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Thus, although stochastic variations are not insignificant, they do not appear to seriously discredit the results obtained using mean values. This tentative conclusion receives some support from an unrelated study by Kilpatrick and Mackenzie [1] in which the results of simulation experiments were compared to those of a mean-value model in which all stochastic elements were removed, i.e., all probability distributions were reduced to their expected values. With both models adjusted to serve the same number of patients during a large number of equal-length work days, utilization statistics on dentist, assistant, receptionist, and hygienist were quite similar. Figures on chair utilization differed somewhat, primarily because the mean value model did not consider patient waiting time in the chair.

Testing the Predictive Ability of the Model The second major aspect of model validation involves determining whether the model is able to replicate the output behavior of real dental practices. Although it was impossible to make this determination on a prospective basis, partial validation of the model was accomplished by using DDP productivity statistics [4], data on chairside time of dentists and EDDAs, and patient case mix information. Douglass et al. [4] found that during the base-line period of 1972 (February-May), when four dentists were employed for four full days each week without EDDAs, the practice earned an average of $249 per day. During the next phase of the study (June-November), when the equivalent of one full-time EDDA was added, average gross revenue was $278 per day, a 12-percent increase. During the base-line period, the average observed monthly value of DT was 110.6 hr and the service category ratio was prosthetics: diagnosis: extraction: endodontics: crown and bridge: prevention: restoration = 1:19:3:1:7:4: 65. Similar analysis of the EDDA phase indicates that the monthly mean of DT was 80.5 hr, the monthly mean of ET (time spent on expanded functions only) was 28.4 hr, and the service category ratio was 1:14: 2:1:18: 6: 58. With these parameters, the model predicted an average daily revenue of $231 for the base-line period and $289 for the EDDA phase, a productivity boost of 25 percent. Therefore the model underestimated base-line revenue by about $18 per day and overestimated EDDA-phase revenue by $11 per day. No test of statistical significance is possible, but the closeness of observed and predicted revenue figures is encouraging. In particular, after bringing only a few basic input parameters, such as DT, ET, and the case mix constraints, into agreement with practice data, the model was able to yield fairly accurate revenue estimates. The performance of the linear programming model in the partial validation appears to warrant preliminary acceptance, but a more extensive evaluation in a laboratory setting is clearly desirable. Given a certain case mix ratio, detailed information on task assignments, staffing patterns, work hours, and physical plant features, the object of such a study should be to compare the practice and the linear programming solution for gross revenue, net revenue, personnel and equipment utilization, and the degree and type of task delegation emerging.

Spring 1975

27

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Health Services Research

EDDAs

IN

DENTAL PRACTICES

Results and Discussion

Addition of one EDDA to a dental practice has a sizable impact on productivity because the new auxiliary performs a significant number of tasks and because the dentist is released to engage in nondelegatable tasks, which are often components of relatively high-fee services. With the addition of one EDDA, a practice is able to produce more routine (e.g., restoration, diagnosis) and complex (e.g., crown and bridge work and endodontics) services with a given case mix. However, the fixed case mix of most one-dentist and twodentist practices is the reason that such practices are generally unable to use more than one EDDA efficiently. Several EDDAs could be fully and effectively utilized, for instance, if the model permitted unlimited handling of diagnostic and preventive services-both highly EDDA-intensive. But the realistic requirement that a practice produce a certain proportion of dentist-intensive services results in exhaustion of the dentist's available time before the EDDA is fully used. With the addition of one EDDA, or a similar input level change, there is a discrete parametric variation in the left-hand-side resource constraint elements in Eqs. 2 and 3. After allowance for downtime loss, the ET value for one EDDA is 153.12 hr per month, 306.24 hr per month for two EDDAs, etc. Similarly, the value of DT for one dentist (no EDDA) is 135.72 hr per month, 271.44 hr per month for two dentists, etc. As the number of EDDAs increases, a given DT value is further reduced to allow for the expanded management role of the dentist. Finally, as DT or ET changes, there are corresponding modifications in the value of the optimal solution of the revenue-generating model. The new optimal solution has a particular associated total cost that must be calculated separately. This parametric analysis was used to obtain the following results.

Productivity Addition of a single expanded-duty assistant to a general dental practice results in sizable gains in output and gross revenue. For the solo dental practitioner with a GP case mix, output and revenue are both increased by about 104 percent. (Since service fees are constant, the percentage changes in output and revenue are equal.) Productivity gains from hiring a second or third EDDA are modest for two- and three-dentist GP groups; for the solo GP dentist, output and revenue decline when a second EDDA is added, apparently for two reasons: (1) the practice is unable to use fully even one auxiliary, so that, in effect, the second EDDA is not used at all; and (2) with the addition of each EDDA, DT is reduced to allow for increased supervisory time. Thus, with the addition of a second EDDA, the total personnel time that can be used to produce services is reduced and output and revenue decline. The full extent of the predicted gains in productivity can be seen in Table 3. Trends in total revenue for GP groups are shown in Fig. 1 (p. 30). In that

Spring 1975

29

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figure, the shapes of the curves that trace the change in total revenue depend on how the optimal basis of the corresponding linear program changes in response to the parametric variation of ET. The curves in Fig. 1 would then be piecewise linear, with the nondifferentiable points occurring for those values of ET where the optimal basis shifts. Calculating these curves precisely is beyond the scope of this article. For the most part, though, a practice will only have the choice of hiring an integral number of expanded-duty assistants. For a detailed analysis of the impact of labor input indivisibilities on productivity and efficiency in a primary medical care setting, see Golladay et al. [27]. There is considerable evidence in the literature to support the finding that the EDDA may have a significant impact on productivity. In their simulation of alternative practice configurations, Kilpatrick et al. [14] estimated that when one EDDA was added to a solo practice of general dentistry, revenue increased about 97 percent. Lotzkar et al. [5], in a study on team dentistry in Louisville, found that dentists employing four EDDAs were able to achieve productivity increases ranging from 110 to 133 percent and that dentists adding three EDDAs increased productivity from 62 to 84 percent. The most significant

divergence between this study's results and results of studies employing different methods is the finding stated above that one EDDA tends to be optimal for a solo dentist. Although variations in a number of factors (relating to demand volumes, patient scheduling, case mix, fees, input prices, tasks comprising services, etc.) could have brought about these divergent conclusions, it would appear that the most crucial differentiating assumption concerns the proportion of his time the dentist is presumed to spend chairside. In the Louisville study [5], about 40 percent of the work was delegated to EDDAs; during much of the work day, the dentists rotated among EDDAs performing nondelegatable tasks. The Louisville study is perhaps the best study yet of team dentistry, but the approach used in that study differs from the one taken in this analysis in 30

Health Services Research

EDDAs IN DENTAL PRACTICES

that, except for downtime allowances, the present model assumes that the dentist employing EDDAs does not reduce his chairside time significantly and does not delegate to the EDDA those expanded-duty tasks that it is technically efficient for him to perform. In addition, the DDP case mix differs somewhat from the one in the Louisville study [5] in that a greater percentage of nondelegatable tasks is required here-in particular, that more crown and bridge and endodontic work be performed. The tasks comprising these services were found to be highly time-intensive for the dentist. Net Revenue Addition of one EDDA, either to a solo dental practice or to a group practice, resulted in large increases in profits, but gains from introducing a second EDDA were relatively small or nonexistent. These results are summarized in Table 3 and illustrated in Fig. 2 for practices with GP case mix.

Long-Run Average Cost Within any one case mix, output is homogeneous; therefore LRAC comparisons may be made within a given case mix. (Note, however, that this formulation of LRAC differs slightly from the classical conception, in which scale expansion presumably proceeds through adjustment of all inputs. In order to concentrate on the economic 5 impact of adding EDDAs here, the number of dentists in a practice is held fixed for any given scale expansion.) Solo

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and that, for any output level and num31 ber of EDDAs employed, LRAC tends j to decrease as practice scale (number of 2 O dentists) increases. Number of EDDAs Figure 3 shows long-run average Fig. 3. Long-run average cost per costs for practices with GP case mix procedure with GP case mix. with the addition of one, two, and three EDDAs. The true LRAC for a given mix may not be at a minimum at a value of ET corresponding exactly to the employment of one full-time EDDA; yet lower LRAC may be achieved by using smaller increments of EDDA time. The results of the present analysis are consistent with those of Golladay et al. [27]. EDDA Utilization The most striking conclusion concerning EDDA utilization is that the solo GP dentist is able to use 90 percent of one EDDA's available time to perform

Spring 1975

31

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about 75 percent of all procedures. The two- and three-dentist practices used one EDDA fully, and in each case the auxiliary did more than 50 percent of procedures performed during the month. These results for GP case mix are displayed in Figs. 4 and 5. In all practices the available dentist time was fully utilized.

The Costs of Dentist Leisure Assuming excess demand for services, a dentist's decision to work fewer than 35 hr per week will result in a loss in net revenue to the practice and in a reduction of services to the community. To what extent would the addition of EDDA time compensate for this loss in dentist time? In order to facilitate the analysis (and because of the legal requirement that EDDAs be supervised), the question is rephrased in terms of the data presented in Table 3: if each member of a two-dentist GP group without EDDAs decided to work half time rather than full time and also agreed to hire an EDDA for the practice, how would total output and net revenue be affected? Similarly, if a three-dentist GP group without EDDAs decided to reduce the work load of each dentist by one-third and to add an EDDA to the practice, what would happen to output and profits? These two queries suggest a general line of investigation in which dentist time (DT) is parametrically reduced in the model-with and without the presence of EDDAs-and the effects on output and profits are noted. In this way the theoretical trade-off between net revenue and leisure is generated. The output and profit loss from each of two full-time dentists deciding to work half time is equal to the output difference between a two-dentist practice and a solo practice, given a particular case mix and the absence of EDDAs. To determine the impact of one EDDA, the output and profits of a two-dentist practice without an EDDA may be compared to those for a one-dentist practice with one EDDA. In the case of the GP mix, the two-dentist group without EDDA produces 1857 procedures and a net revenue of $6410 per month, whereas the solo practice with one EDDA turns out 1965 procedures and 32

Health Services Research

EDDAs

IN

DENTAL PRACTICES

$6412 in net revenue. Thus, with a GP case mix, productivity actually increases when the two dentists work half time and add an EDDA. The results are not surprising when one considers that personnel time availability for the solo GP dentist with one EDDA exceeds that for the two-dentist practice and that the one EDDA performs about 75 percent of all procedures in the solo practice. (The EDDA in the solo practice works a 40-hr week and is 90 percent utilized. Thus effective service time for the EDDA exceeds the 35 hr lost to the practice each week through dentist leisure.) For the GP case mix, the three-dentist group without an EDDA produces 2885 procedures and $10 163 each month and the two-dentist group with an EDDA delivers 3202 procedures for a net revenue of $12 004. As shown in Table 3, both practices would find it financially advantageous to take the leisure and hire an EDDA. Case Mix Variations Dental practices with GP case mix realize the greatest percentage gains in output and net revenue from introduction of EDDAs, but cosmetic care practices realize the greatest absolute increases. Primary care practices show the smallest absolute gains in output and profits as EDDA use expands. (These results are described in more detail in appendixes available from the first author.) That the GP groups experience the greatest relative increases is due to the fact that some procedures for which EDDAs are specially suited are prominent in GP group service output but are omitted or deemphasized in PC and CC practices. Addition of an EDDA to a solo general practice increases output and revenue by 104 percent; the corresponding gain is 85 percent for a solo CC practice and 57 percent for a solo PC practice. Gains from hiring a second or third EDDA are small for all practices. For solo practices and the two-dentist CC practices, output and revenue decline when a second EDDA is added. Similar trends are observed for net revenue. Assuming a 48-week work year, the annual net revenue per dentist in any PC practice in North Carolina is only about $14 300, or 40 percent of the average for dentists in general practice in the state. Adding one EDDA to any PC practice raises net revenue per dentist to between $35 000 and $40 000, the exact amount depending on the number of dentists. These findings suggest that the inclusion of expanded-duty auxiliaries in PC practices would not only increase productivity significantly but also would increase dentists' incomes so that they would be comparable with those of GP dentists. Long-run average cost tends to be highest for the CC mix, which requires a large percentage of costly crown and bridge services, and lowest for the PC mix, which requires none of these services. Cost of Restrictions on Task Delegation The degree of task delegation assumed in the model-and also being taught currently in several dental training centers-is greater than that permitted now in many states. As Johnson and Holz [2] point out, there is little

Spring 1975

33

Lipscomb & Scheffler

uniformity among states on what tasks may be delegated. For this study, a restricted EDDA was defined as an auxiliary who may perform only the following functions: charting, Panorex, periapical radiograph, bite wing radiograph (within diagnosis) and rubber dam placement and finish and polish restoration (within restoration). The economic performance of the various types of practices using none, one, two and (sometimes) three restricted EDDAs is detailed in appendixes available from the first author. The results of this analysis indicate that the costs of legal restrictions on the EDDA are high. For the solo general practice, a restricted EDDA is utilized only 19 percent as much as an unrestricted EDDA and performs only 39 percent as many tasks. With the addition of a restricted EDDA, monthly revenue rises to $8188 and net revenue to $3162, compared with increases to $14 134 and $6412 achieved by hiring an unrestricted EDDA. As might be expected, output does not increase as much as total cost with the introduction of a restricted EDDA; therefore LRAC rises from $4.43 to $4.55 per procedure. Finally, it must be remembered that the focus here has been the production potential of the EDDA. In practice environments with significant slack in the demand for services involving expanded-duty auxiliaries, where legal restrictions are severe, or where auxiliary management and patient scheduling are inefficient, the value of the EDDA is doubtless diminished. One should bear these provisos in mind in drawing policy conclusions from this analysis. Acknowledgments. The authors wish to thank Chester Douglass and an anonymous referee for a number of instructive comments, which have been incorporated into this article. Any remaining errors are the authors'. REFERENCES

1. Kilpatrick, K. E. and R. S. Mackenzie. Dental Task Assignment Via Computer Simulation: Second Interim Report. Health Systems Research Division, J. Hillis Miller Health Center, University of Florida, 1973. 2. Johnson, D. W. and F. M. Holz. Legal Provisions on Expanded Functions for Dental Hygienists and Assistants: Summarized by State, 1972. Division of Dental Health, Bureau of Health Manpower Education, National Institutes of Health, DHEW. Washington: Government Printing Office, 1973. 3. Baird, K. M., E. C. Purdy, and D. H. Protheroe. Pilot study on advanced training and employment of auxiliary dental personnel in the Royal Canadian Dental Corps: Final report. J. Can. Dent. Ass. 29:779 Dec. 1963. 4. Douglass, C. W., R. L. Lindcahl, D. B. Gillings, and S. S. Moore. Laboratory model of private practice: Method for studying new systems of health care delivery in dental schools. J. Dent. Ed. 37:8 Dec. 1973. 5. Lotzkar, S., D. W. Johnson, and M. B. Thompson. Experimental program in expanded functions for dental assistants: Phase 3 experiment with dental teams. JADA 82: 1067 May 1971. 6. Ludwick, W. E., E. 0. Schnoebelen, and D. J. Knoedler. Greater Utilization of Dental Technicians. I. Report of Training. Dental Research Facility, U.S. Naval Training Center, Great Lakes, IL, 1963. 7. Pelton, W. J., 0. H. Embry, G. A. Overstreet, and J. B. Dilworth. Economic implications of adding two expanded-duty dental assistants to a practice. JADA 87:604 Sept. 1973. 8. Retig, D., M. Snyder, G. Nevitt, and J. Tocchini. Expanded-duty dental auxiliaries in four private dental offices: The first year's experience. JADA 85:969 May 1974.

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Health Services Research

EDDAs

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DENTAL PRACTICES

9. Roemke, R. G. Island hygienists boost productivity. J. Can. Dent. Ass. 37:50 Feb. 1971. 10. Soricelli, D. A. Implementation of the delivery of dental services by auxiliaries-the Philadelphia experience. Am. J. Pub. Health 62:177 Aug. 1972. 11. Maurizi, A. Economic Essays on the Dental Profession. College of Business Administration, University of Iowa, 1969. 12. Feldstein, P. Financing Dental Care: An Economic Analysis. Lexington, MA: Heath, 1973. 13. Boulier, B. Two Essays in the Economics of Dentistry: A Production Function for Dental Services and an Examination of the Effects of Licensure. Doctoral dissertation, Princeton University, 1974. 14. Kilpatrick, K. E., R. S. Mackenzie, and A. G. Delaney. Expanded-function auxiliaries in general dentistry: A computer simulation. Health Serv. Res. 7:288 Winter 1972. 15. Benham, L., A. Maurizi, and M. W. Reder. Migration, location, and remuneration of medical personnel: Physicians and dentists. Rev. Econ. & Stat. 50:332 Aug. 1968. 16. Linn, E. L. Service to others and economic gain as professional objectives of dental students. J. Dent. Ed. 32:75 Mar. 1968. 17. Manhold, J. H., L. Shatin, and B. S. Manhold. Comparison of interests, needs, and selected personality factors of dental and medical students. JADA 67:601 Oct. 1963. 18. Heist, P. Personality characteristics of dental students. Ed. Record 41:240 July 1960. 19. Shimoni, K. The Charlotte model cities demonstration medical-dental insurance program. Inquiry 10:39 June 1973. 20. Feldstein, M. S. The rising price of physicians' services. Rev. Econ. & Stat. 52:121 May 1970. 21. Bureau of Economic Research and Statistics, American Dental Association. Cost of conducting a dental practice, 1958-1970: Part II. JADA 86:427 Feb. 1973. 22. Division of Educational Measurements (Council on Dental Education). Expanded Functions Delegated to Dental Hygienists and Dental Assistants. Chicago: American Dental Association, 1974. 23. Smith, K. R., M. Miller, and F. L. Golladay. An analysis of the optimal use of inputs in the production of medical services. J. Human Resources 7:208 Spring 1972. 24. Reinhardt, U. E. Manpower substitution and productivity in medical practice: Review of research. Health Serv. Res. 8:200 Fall 1973. 25. Sengupta, J. K. Stochastic Programming: Methods and Applications. Amsterdam: North Holland, 1972. 26. Lipscomb, J. and P. Schmidt. Efficient estimation of the parameters of a normal distribution from grouped data. Unpublished paper, University of North Carolina at Chapel Hill, 1974. 27. Golladay, F. L., M. E. Manser, and K. R. Smith. Scale Economies in the Delivery of Medical Care: A Mixed Integer Programming Analysis of Efficient Manpower Utilization. Discussion Paper No. 2-73, Center for Medical Sociology and Health Services Research, Health Economics Research Center, University of Wisconsin, 1973.

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