Chapter 7. A Hybrid Cost Function for Outpatient Nonmethadone Substance Abuse Treatment Facilities

Cost Function Specification

Economists often estimate cost functions to determine whether or not economies of scale can be realized in a particular industry. An industry is said to exhibit economies of scale if the average cost of producing a unit of output declines as more output is produced. If scale economies exist, the industry would be more efficient if it consisted of a few large firms rather than many small firms, other things being equal and given other constraints. In the case of substance abuse treatment, such a constraint might be ensuring adequate access. Economists also use cost functions to estimate the effect on costs of firm characteristics, such as the effect of teaching programs on hospital costs. In the substance abuse treatment industry, policy makers may wish to examine, for example, whether for-profit substance abuse treatment facilities have higher costs than nonprofit facilities, or the effect on costs of offering a variety of special programs. The purpose of this study is to explore these issues by specifying and estimating a cost function for a nationally representative sample of outpatient nonmethadone substance abuse treatment facilities.

Although, to our knowledge, no one has estimated an economic cost function for substance abuse treatment facilities, economists have estimated cost functions for other health care facilities, such as hospitals, nursing homes, and physician practices.¹ These studies usually have used a functional form that is less restrictive than the standard textbook economic cost function. The standard textbook cost function, which is derived from the economic theory of the firm, models cost as a function of the volume of output and input prices only (e.g., see Silberberg, 1978, pp. 173–213). For an industry with N outputs (Y) and K input prices (w), the cost function would be

C = C (Y₁, Y₂, Y₃, ... Y_N; w₁, w₂, w₃, ..., w_K), D (1)

where C is the cost of production. To obtain unbiased and efficient parameter estimates from a textbook cost function requires, among other things, that all firms in an industry use identical inputs and produce identical outputs, and that data on the quantity of all outputs produced and the prices of all inputs used by each firm are entered into the model. Economists use textbook cost functions to determine the economic properties of an industry's cost function, such as economies of scale and scope and the degree of input substitutability.² When using data on industries that conform to the required assumptions, coefficient restrictions implied by the economic theory of the firm can be imposed on the estimation to improve statistical efficiency.

Economists estimating cost functions for health care facilities rarely use the textbook cost function for a variety of reasons, many of which apply to the outpatient drug-free substance abuse treatment industry. First, and perhaps most important, health care firms within an industry generally are not identical in terms of their inputs or outputs. Second, complete data outputs or input prices are rarely available. Finally, the focus of these studies often is the effect on costs of facility characteristics beyond output and input prices, such as whether for-profit providers have higher costs than nonprofit providers, or rural providers have lower costs than urban providers.

Instead, health economists often estimate what has been called a "hybrid" cost function (Rosko & Broyles, 1988). A hybrid cost function models costs as a function of measurable outputs and input prices, as well as other facility characteristics that may affect costs. We have

C = C(Y, w, X, F, A), D (2)

where, as before, Y is a vector of outputs and w is a vector of input prices. To these standard cost function variables we add a vector of measures of the case mix or severity of the facility's clients (X), a vector of facility characteristics (F), such as the number of services it offers, and a vector of characteristics describing the facility's location (A). In this chapter, we estimate a model similar to equation (2). The one difference is that because we have only one output, Y will be a scalar rather than a vector.

To estimate a cost function, the model must be written as a specific functional form that allows for the possibility for economies of scale. A linear function does not allow for economies of scale because it constrains the relationship between cost and the quantity of output to be the same over the entire output range, rather than allowing it to decline, as would be the case under economies of scale, or increase, as in the case of diseconomies of scale. We follow Vitaliano (1987) who used the following logarithmic estimating equation derived from a theoretical model proposed by Nerlove:

D (3)

where C, Y, w, X, F, and A are as defined before; , , , , and are coefficients to be estimated; and is a random, normally, and identically distributed error term. Like Vitaliano (1987), we used Ramsey's Regression Specification Error Test (RESET) to test the suitability of this functional form, as well as other popular functional forms, including the quadradic and translog specifications. Our testing revealed that the logarithmic model was the only functional form for which the null hypothesis of no specification error could not be rejected at conventional levels of significance.

Data

We used data from the Alcohol and Drug Services Study (ADSS) on 222 outpatient nonmethadone treatment facilities. ADSS was conducted under the auspices of the Office of Applied Studies (OAS) at the Substance Abuse and Mental Health Services Administration (SAMHSA) (OAS, 2000a). Among the objectives of ADSS was the collection of detailed information on the characteristics of a random, nationally representative sample of substance abuse treatment facilities and of clients discharged from those facilities. Such data allow development of better estimates of the costs of treatment than had been available using previously existing datasets. The study consisted of three phases, two of which we draw from for this analysis.

Phase I of ADSS was a telephone interview with a nationally representative, stratified probability proportionate to size (pps) sample of 2,395 substance abuse treatment facilities, representing a 91.4 percent response rate from the 2,621 eligible facilities. Facility administrators were asked about the characteristics of the facility and its clients. Phase I missing data were imputed according to standard statistical procedures (OAS, 2000b). Final facility weights included a nonresponse adjustment, and methods used to impute missing values included logical imputation, imputation from external sources, regression, and random within-class hot-deck procedures.

Phase II facilities were selected from among Phase I facilities located in 1 of 62 primary sampling units (PSUs) again according to a stratified pps design. Excluded from Phase II were hospital inpatient facilities and facilities that treated only those with alcohol use disorders. These facilities were considered out-of-scope. Phase II data were initially collected during a site visit to 280 facilities. These 280 facilities included 234 of the 294 eligible facilities that agreed to be surveyed, as well as 46 shadow³ facilities for the 60 that chose not to participate. Thus, the overall response rate for cost study facilities was approximately 86 percent. The site visit consisted of an in-person interview with the facility director or administrator to collect data on the facility's characteristics, including expense and revenue information; compilation of a sampling frame and selection of a representative sample of client records; and collection of client-level data from the sample of client records at each facility (OAS, 2000a). In addition to the variables that were imputed on the final Phase II facility file, we used the same methods to impute additional missing variables that we needed for our analysis (Krenzke & Mohadjer, 2002). The variables and the number of observations we imputed are noted below.

Phase II administrator data were used as the basis for the ADSS cost study, the main source of data for this chapter. Data on facilities' expenses, revenues, and client volumes were entered into a data audit spreadsheet developed by Capital Consulting Corporation (CCC) to check for accuracy (OAS, 2003). CCC had developed this audit instrument after intensive study of some 400 substance abuse treatment facilities using their Cost Allocation Methodology (CCC, 1998; CCC, The Lewin Group, & Caliber Associates, 1998). In that study, CCC sent professional accountants to facilities to collect cost data. From this information, CCC developed data reliability and validation procedures to test the accuracy of other provider-supplied data. Those procedures are at the heart of the data audit spreadsheet used in the ADSS cost study. The facilities' original responses were entered into the audit instrument, and key financial ratios were examined. Facilities with anomalous results on their Phase II cost, client volume, or personnel variables were contacted and given the opportunity to change their responses. Some 96 percent of the facilities required a callback for at least one variable. Of those, most were able to either verify the originally reported data or provide new information, much of it documented with information from the facility's financial information system. Some facilities either could not be contacted or could not provide more information on their expenses. For those facilities, which amounted to about 26 percent of the outpatient nonmethadone facilities studied here, data were imputed. In most cases, almost 22 percent of all outpatient nonmethadone facilities, missing expense data were imputed using the facility's own volume information and the data audit instrument, or the facility's Phase I response to the expense question. In 4.5 percent of the facilities, expenses were imputed using information from other facilities (OAS, 2003). The cost study data file includes final Phase II facility weights, which were adjusted for facility nonresponse using a raking procedure (OAS, 2000a). In addition to using the ADSS data, we used some supplementary data, as described below.

The variables included in the analysis are described in the following paragraphs.

Costs: Our measure of costs, the dependent variable, was the natural logarithm of the total substance abuse treatment costs in the facility, as reported by the administrator in the Phase II administrator interview and verified or edited as described above.

Output: We estimated a single output model, where we considered a unit of output as a treatment episode for a given client within a given type of care within a given facility. However, due to the characteristics of the substance abuse treatment industry and its clients, and the measures available in the data, we estimated the models using two different measures to determine whether they exhibited different relationships with costs. The number of annual admissions was the first measure of output we considered. Many substance abuse treatment facilities have to keep track of admissions because they are required to report them to their State substance abuse treatment agencies (OAS, 2000c). However, the number of admissions may overstate the annual output of a treatment center because a large number of clients who are admitted do not finish treatment. In the ADSS Phase I sample, for example, administrators reported that, on average, more than 40 percent of the clients in outpatient nonmethadone facilities failed to complete their planned treatment. In our study, we entered the facility's completion rate as an explanatory variable to partially control for that issue. We also considered using discharges, an output measure commonly used in hospital studies. However, that measure also has its limitations. As mentioned earlier, a large number of clients leave programs without finishing their treatment plan. Their discharge often is not documented until some time period elapses, usually 30 days, during which they have received no treatment. Furthermore, some facilities do not formally discharge anyone, due to the relapsing nature of the substance use disorder. In these facilities, clients are not discharged and are allowed to return for more care if needed. Therefore, discharge numbers may underestimate treatment output. In this study, we estimated two models—one with admissions as the measure of output and one with discharges. Because the results were virtually identical, we report only the results of the admissions regressions here. Given that many facilities must report these routinely to State substance abuse treatment authorities, we believe they may be more accurately reported than are discharges.

Input Prices: The inputs for which we had prices were various categories of labor and office space. To measure wages, the input price of labor, we used the Metropolitan and Balance of State Area Occupational Employment and Wage Estimates for selected occupations, collected by the Bureau of Labor Statistics (2003b) as part of their Occupational Employment Statistics Survey (OESS). The three occupations we included were (a) Substance Abuse Counselors, (b) Senior Administrative (includes five job categories: administrative services manager, general managers and other top executives, all other managers, and financial managers), and (c) Administrative Clerical Workers (includes seven job categories: file clerks, general office clerks, payroll and timekeeping clerks, and secretaries [excluding legal and medical]). Although the latter two occupation categories are not specific to substance abuse treatment facilities, we believe the average wages across administrative workers in all industries should serve as a good proxy for administrative wages in substance abuse treatment facilities. Even though the ADSS cost study collected facility-specific wage information, we used the OESS estimates of the market wage rates to overcome two possible problems in estimation. The first was the problem of missing observations. Not all facilities employ all types of workers, so we would not have wage information for all possible types of employees for all facilities, which is required for estimating a cost function. The second potential problem was endogeneity. If we were to use a weighted average of each facility's wages as its wage variable, to overcome the missing values problem, we would no longer just be measuring the effect of exogenous factors, such as location in a high cost or highly unionized area, on the facility's cost function. Rather, we would be measuring both the prevailing wage rate and the facility's choice of inputs, which is endogenous. For example, a facility's higher costs may be attributed to its being in a higher wage area, when, in reality, its costs are higher because it has an inefficient staffing mix. To account for this possibility, economists often use local area wage rates (e.g., see Custer & Wilke, 1991; Salkever et al., 1986; Zuckerman et al., 1994), as was our plan here. Because the OESS does not collect data on all occupations for each location each year, for some variables we used data from later years, deflated using the average Consumer Price Index (CPI) representing changes in the prices of all goods and services purchased for consumption by urban households (Bureau of Labor Statistics, 2003a).

We also included a variable measuring the cost of office space in the facility's area, again turning to an external source, this time because the variable was not collected in ADSS. However, it too is potentially endogenous. Unfortunately, to our knowledge, no data exist on commercial real estate rental rates throughout the Nation. So, instead, we used the four-bedroom Section 8 Fair Market Rent as defined by the U.S. Department of Housing and Urban Development's Office of Policy Development and Research (1995) as a proxy for commercial rents. This is the variable that is used in SAMHSA's Substance Abuse Prevention and Treatment (SAPT) block grant formula to measure the rental component of the cost of doing business in each State (U.S. Department of Health and Human Services [DHHS], 1996).

Case Mix Adjustment for Substance Abuse Treatment Facilities

Adjusting for case mix in analyses of substance abuse treatment facilities has become more common in the past few years as governments and other payers demand accountability from service providers. One way to determine whether a provider is accountable is to compare that provider's performance with that of other providers. For the comparison to be fair and meaningful, however, it must account for differences across providers in the mix of clients they treat. Several recent studies of substance abuse treatment providers have used the same basic method of case mix adjustment to study outcomes of substance abuse treatment (Koenig, Fields, Dall, Ameen, & Harwood, 2000; Moos, Moos, & Andrassy, 1999; Phibbs, Swindle, & Recine, 1997; Phillips et al., 1995), access to substance abuse treatment (Deck, McFarland, Titus, Laws, & Gabriel, 2000), and participation in self-help groups among those in treatment (Ouimette et al., 2001). Although the results from these studies are informative and useful in outcome studies, they unfortunately provide little guidance on how to measure case mix complexity as it relates to the cost of treatment at a given facility.

The study conducted by Moos et al. (1999) is illustrative. These researchers analyzed outcomes of four treatment approaches (therapeutic community, psychosocial rehabilitation, 12–step, and undifferentiated) in 88 residential programs. The outcomes they investigated were posttreatment abstinence, substance use disorders, distress, mental disorder symptoms, arrests, and employment. They presented results from separate regressions run with each outcome as the dependent variable and the client as the unit of analysis. Each regression included variables describing clients and the programs in which they had been treated. Client characteristics included age, marital status, prior mental disorder episode, mental disorder diagnosis, and the value of the outcome variable at intake. With the exception of the outcome variable measured at intake, no variable was significant in all regressions. Only one, prior mental disorder episode, had the same effect on all the outcomes for which it was significant.

This lack of consistency in results holds across the other studies cited as well. No variable had a similar effect on all outcomes in all of the studies in which it was included. In state-of-the-art studies of substance abuse treatment, case mix adjustment methods vary by the purpose of the analysis and the outcome and population being studied. To our knowledge, no one has attempted to adjust for the costliness of a provider's client population when comparing substance abuse treatment providers on the basis of costs.

This is in contrast to cost studies of the hospital industry in which case mix adjustment, based on data routinely collected by hospitals from patients' discharge abstracts, has been used for more than 20 years. Information on the clients' medical diagnoses, procedures, length of stay, age, and gender is abstracted from each patient's record, then standardized and combined with information on the amount the patient was charged for the stay to create an electronic discharge abstract record. Electronic databases containing these records then are forwarded to some 40 States across the country that require hospitals to submit these data, many of which make it available for researchers. Using such data, researchers create a case mix index that describes the costliness of treating each hospital's patients. The case mix index usually takes the following form:

CMI_h = hospital h's case mix index;

weight_j = a resource use weight, such as the total charge or the length of stay, assigned to homogeneous patient group j;

group_hj = proportion of hospital h's patients in group j; and

group_j = average proportion of patients in group j.

A hospital with a CMI greater than 1 has a case mix that is more costly to treat than the average hospital's; a hospital with values less than 1 has a case mix that is less costly to treat than the average hospital's.

When examining hospital costs, researchers often use groupings and weights created by the U.S. Government as part of Medicare's hospital inpatient prospective payment system. The groups, called diagnosis related groups (DRGs), are groupings of patients that, according to analysis of discharge abstract data, cost a similar amount to treat. The weights are the average costs incurred in treating cases in that DRG, relative to the average costs incurred in treating all DRGs. Each hospital discharge can be classified into a DRG based on the principal diagnosis, up to eight additional diagnoses, and up to six procedures performed during the stay, as well as age, gender, and discharge status of the patient. There are more than 500 DRGs, and their weights are reviewed each year and updated as necessary (Centers for Medicare & Medicaid Services [CMS], 2001).

Unfortunately, the data required to create a grouping system that could be used to construct weights for substance abuse treatment client groups cannot easily be obtained. Although States routinely require substance abuse treatment providers to submit data on individual clients, these are usually admissions data, which do not include any information on the costs of the client's treatment or the client's disposition at discharge. Sometimes these data elements are collected in special studies, as was done in ADSS. However, an examination of the ADSS Phase II client discharge abstract data reveals that the data are missing from a large number of clients. Total charge was missing or invalid for some 35 percent of the outpatient nonmethadone clients in ADSS, it was zero for another 12 percent, and it was on a sliding scale or otherwise reduced for another 17 percent. Without complete charge or cost information for each client, it is difficult to create groups of clients based on resource use. Although length of stay could conceivably be used, as it was in the early work on DRGs (Fetter, Shin, Freeman, Averill, & Thompson, 1980), the problems noted above about how discharges are documented and handled make using length of stay to measure resource use questionable.

For these reasons, we did not control for case mix among substance abuse treatment facilities using a single case mix index. Instead, we used a method that was employed in hospital studies before case mix indices were widely adopted. We included several summary measures of characteristics that we believe may make clients more costly to treat.

To determine which summary measures to use, we reviewed the substance abuse treatment literature relating treatment costs to client characteristics. Unfortunately, there appears to be a dearth of literature on this subject as well, and most of what is available is based on analyses of claims data, usually for specific time periods rather than by treatment episode at a given facility (Ettner, Frank, McGuire, Newhouse, & Notman, 1998; Goodman, Holder, Nishiura, & Hankin, 1992; Goodman, Nishiura, & Hankin, 1998; Goodman, Nishiura, Hankin, Holder, & Tilford, 1996; Holder & Blose, 1991; Huskamp, 1999; Salomé, French, Scott, Foss, & Dennis, 2003; Westermeyer, Eames, & Nugent, 1998).

Thus, we chose the following summary measures of client severity based on data availability and characteristics that we expected to affect the cost of treatment. Some clients have more complex substance use disorders and therefore require more resources to treat. These may include those who, at admission, used both drugs and alcohol, injected drugs (imputed for nine facilities), or who received supplemental security income (SSI) or social security disability insurance (SSDI) (imputed for two facilities). We included in our models measures representing the percentage of each facility's admissions with each of these characteristics. This information comes from the discharge abstracts, when available, or from the Phase I ADSS administrator survey (OAS, 2000a).

Other clients may be more costly to treat for other reasons. Those who are referred by the criminal justice system, for example, may be less compliant and ready for treatment and therefore require more resources to treat compared with those referred in some other way, so we included the percentage of clients who were referred by the criminal justice system in the model (imputed for one facility). Minority clients may be more costly to treat if they require more services due to their social and economic disadvantages (D'Aunno & Vaughn, 1995). To capture this, we entered four variables from the Phase I administrator survey: the administrator's estimate of the percentage of each facility's clients who were Hispanic, non-Hispanic black, other race, or unknown race on the point-prevalence date (October 1, 1996), with non-Hispanic white as the reference category (imputed for one facility). Ten facilities reported that the race of all of their clients was "unknown," an allowable response to the survey. It is unclear if these facilities truly did not know the race of all their clients, or chose not to respond to the question. If they truly did not know or ask, it could be that race/ethnicity plays no part in their treatment decisions and therefore resources required to treat these clients. Because this was unclear, we ran the model both including and excluding these 10 facilities and found that the results did not differ significantly. We report the results including all of the facilities and include "unknown" race as a category.

We ran versions of our model with other client mix variables we thought might affect facility costs, but they were not significant, and removing them from the model did not affect any of the coefficients on the remaining variables. The variables we considered but excluded were the percentage of clients who were homeless, unemployed, or had co-occurring substance use and mental disorders at admission, and the percentage who completed treatment.

Facility Characteristics

We included several facility characteristics in the model. We entered dummy variables to measure the extent of urbanization of the facility's location. These dummy variables indicated whether the facility was in a small- or medium-sized metropolitan area or a nonmetropolitan area, with large metropolitan area comprising the reference cell. Metropolitan area classifications were based on the Beale urbanicity codes, which were assigned based on the facility's zip code (Butler & Beale, 1994). Nonmetropolitan facilities are located in nonmetropolitan counties. Small and medium-sized metropolitan area facilities are located in metropolitan areas with fewer than 1 million people. Large metropolitan area facilities are located in metropolitan areas with more than 1 million people. We expected facilities in large urban areas to have higher costs than other facilities for input prices we did not measure (e.g., security). In addition, we entered a dummy variable indicating whether or not the facility was owned by a private for-profit entity. Government-owned and nonprofit facilities comprised the reference group. According to economic theory, in a competitive market, the profit motive forces firms to minimize the costs of doing business. Private nonprofit and government-owned facilities face no profit incentive, so they may be less likely to be cost minimizers.⁴ Previous research suggested that such differences may exist in the substance abuse treatment industry (Wheeler, Fadel, & D'Aunno, 1992). However, it also should be noted that, if our client characteristic variables did not adequately account for case mix, we also might have expected a profit/nonprofit differential—private for-profit facilities may admit less severely ill clients compared with private nonprofit and government-owned facilities, which may be more severely ill clients' last resort.

Another difference among facilities is the number of services and the number of special programs for specific populations they offer. A substantial number of those with substance use disorders also have other problems, such as homelessness, unemployment, or physical or mental illnesses that must be addressed for treatment to be successful (D'Aunno & Vaughn, 1995). Because, again, we could find no direction in the literature as to which services were most likely to affect costs, we included a variable that measured the number of the following services offered by the facility, based on the administrator's response to the Phase I ADSS survey: comprehensive assessment/diagnosis, transportation, individual therapy, relapse prevention, family counseling, employment counseling, academic education, HIV/AIDS education/counseling/support, combined substance abuse treatment and mental health services, tuberculosis (TB) screening, prenatal care, smoking cessation, acupuncture, aftercare, outcome follow-up, urine screens, alcohol and other drug tests, medical detox, mental health services, and medical treatment.

Finally, the diversity of a facility's clients may affect its cost due to the importance of offering culturally competent care. For example, facilities that serve clients who speak many different languages may need to hire a greater variety of counselors or counselors with special skills. To measure this, we computed a race/ethnicity index similar to the Herfindahl index commonly used in the industrial organization literature to measure market structure. The race/ethnicity index for facility t, R_t, is

where S_tj is the share of the i^th racial/ethnic group in facility t. We included the five groups identified earlier: non-Hispanic white, non-Hispanic black, Hispanic, other, and unknown. R_t is bounded by 0 and 1 and is inversely related to diversity. The greater R_t is, the less diverse is the facility; facilities with only one racial/ethnic group have R_t = 1. R_t declines as the number of racial/ethnic groups at the facility increases. It increases with rising inequality among any given number of racial/ethnic groups. We hypothesized that R_t is negatively related to costs.

Several other facility characteristics also may affect costs. One such characteristic is whether or not the facility is part of a larger organization. Although facilities were encouraged to report total costs, including dollar values of items supplied by a parent company, some facilities may have had trouble doing so. Furthermore, being part of a larger entity may allow the facility to obtain inputs at lower prices, or take part in other efficiencies, which may lower its costs. Therefore, we hypothesized that facilities that were part of a larger organization may have had lower costs. Because many substance abuse treatment facilities are nonprofit, they may received in-kind donations of goods or services, such as space, furniture, or volunteer help. Facilities that received in-kind donations should have had lower expenses, so we included a dummy variable from the ADSS Phase II survey that indicated that the facility received such donations (imputed for one facility).

We suspected other facility characteristics might have affected cost, such as the number of special programs offered by the facility (intensive outpatient treatment and special programs for women, pregnant women, adolescents, DWI/DUI clients, AIDS/HIV positive, and co-occurring disorder clients), the age of the facility, and a variable indicating that the facility was a multimodality facility. However, we excluded these variables from the final model because the estimated parameters were not significantly different from zero, and excluding these variables did not affect the other results.

Results

Table 7.1 displays the weighted mean estimates of the variables included in our analysis for our sample of outpatient nonmethadone substance abuse treatment facilities. After incorporating the final facility weights, the total population size represented was 9,166 facilities (N = 222, unweighted sample).

The mean of the log of total costs was approximately 12.13 in 1997, which translates into $185,350. The mean of the log of annual admissions was approximately 4.94, or 140 admissions, yielding a cost per admission of $1,324. Of the three different mean hourly wages included as input prices, the senior administrative wage was the highest ($24.29), followed by the substance abuse counselor wage ($12.55), and the clerical administrative wage ($9.97). The average of our proxy for office space costs was approximately $916 per month.

The average facility treated a client population that was somewhat racially diverse. Most clients in the facilities were non-Hispanic white, with the mean across facilities of 63 percent of clients. The second largest racial/ethnic group in most facilities was non-Hispanic black, with the mean across facilities of 22 percent of clients. Hispanics accounted for, on average, about 9 percent of a facility's client population. Clients of other races comprised about 2 percent of total clients, and those with an unknown race or ethnicity, approximately 4 percent. The derived race/ethnicity index had a weighted mean estimate of 0.7 across facilities, indicating at least some diversity in the average facility.

Almost half of the clients in the facilities were referred by the criminal justice system, with the mean across facilities of 42 percent. A majority of clients in the facilities received treatment for both alcohol and drug use disorders, with the mean across facilities of 54 percent. The average proportion of those who inject drugs within a facility was low (10 percent), and few clients reported receiving SSI or SSDI (mean across facilities of 8 percent of clients).

Most of the facilities in the sample were private nonprofit (59 percent). An estimated 27 percent were private for-profit, and 14 percent were government-owned. On average, each facility offered approximately 10 services. Finally, most facilities were located in a large-sized metropolitan area (55 percent), with 35 percent located in a small to medium-sized metropolitan area and 11 percent in a nonmetropolitan area.

Cost Function Results

As mentioned earlier, estimating standard errors using these data was not straightforward. In Table 7.2, we present the coefficient estimates, along with three estimates of the standard errors. As Table 7.2 shows, the standard errors estimated using the Taylor series method are much smaller than those estimated using the Jackknife method. Because we believe there is bias in both estimates—Taylor standard errors are biased downward and Jackknife estimates are biased upward—we adjusted the Taylor standard errors to create an adjusted standard error that falls between the two. We present this adjusted standard error as well and its p value. These are the results we discuss below.

Table 7.1 Means and Standard Errors of Analysis Variables, Outpatient Nonmethadone Substance Abuse Treatment Facilities, ADSS Phase II Administrator and Cost Studies, 1997 (N = 222; Weighted N = 9,166)
Variable	Mean	Standard Error
Log of Costs ($185,350 total costs)	12.13	0.09
Log of Admissions ($1,324 per admission, 140 admissions)	4.94	0.12
Log of Substance Abuse Counselor Wage ($12.55 per hour)	2.53	0.03
Log of Clerical Administrative Wage ($9.97 per hour)	2.30	0.02
Log of Senior Administrative Wage ($24.29 per hour)	3.19	0.03
Log of Office Space Cost ($916 per month)	6.82	0.04
% of Clients Who Are White	0.63	0.04
% of Clients Who Are Black	0.22	0.03
% of Clients Who Are Hispanic	0.09	0.02
% of Clients Who Are Another Race	0.02	0.01
% of Clients Whose Race Is Unknown	0.04	0.02
Herfindahl Index	0.70	0.03
% of Clients Referred by Criminal Justice System	0.42	0.05
% of Clients with Drug and Alcohol Use Disorders	0.54	0.05
% of Clients Who Injected Drugs	0.10	0.02
% of Clients Who Received SSI or SSDI	0.08	0.02
Private For-Profit Facility	0.27	0.07
Private Nonprofit Facility	0.59	0.07
Government Facility	0.14	0.04
Sum of Special Services	10.38	0.33
Part of a Larger Organization	0.57	0.07
Facility Receives In-Kind Goods or Services	0.45	0.08
Facility Located in a Large Metro Area	0.55	0.08
Facility Located in a Medium or Small Metro Area	0.35	0.08
Facility Located in a Nonmetro Area	0.11	0.06
SSI = supplemental security income. SSDI = social security disability income. Source: SAMHSA, Office of Applied Studies, Alcohol and Drug Services Study, Phase II administrator data, 1997 to 1999, and ADSS cost study, 1997.

Our results suggest that there are substantial economies of scale in the outpatient nonmethadone substance abuse treatment industry. We found that a 10 percent increase in the total number of admissions was associated with only a 6.7 percent increase in total costs (p < 0.0001). Because the estimate also was statistically different from 1 at the 0.01 percent level, the results suggest that the outpatient nonmethadone substance abuse treatment industry experiences economies of scale. Larger facilities were less costly on a per admission basis than smaller ones.

With one exception, the facility characteristics were insignificant. The only client mix variable that was significant was the percentage of clients with SSI or SSDI. Facilities with a larger percentage of clients on SSI or SSDI had higher costs (p = 0.0418).

**Table 7.2 Cost Function Results, Outpatient Nonmethadone Substance Abuse Treatment Facilities, ADSS Phase II Administrator and Cost Studies, 1997**
Variable	Coefficient	Taylor Series Standard Error	Jackknife Replicate Standard Error	Adjusted Standard Error	Adjusted T Statistic	Adjusted p Value
Intercept	6.71	1.41	2.25	1.91	3.51	0.0009
Log of Admissions	0.67	0.04	0.06	0.05	12.36	<0.0001
Log of Substance Abuse Counselor Wage	-0.20	0.42	0.67	0.57	-0.36	0.7201
Log of Clerical Administrative Wage	-1.09	1.06	1.84	1.44	-0.76	0.4502
Log of Senior Administrative Wage	0.92	0.58	0.91	0.79	1.17	0.2466
Log of Office Space Cost	0.23	0.41	0.61	0.56	0.41	0.6833
% of Clients Who Are Black	0.17	0.19	0.26	0.26	0.66	0.5118
% of Clients Who Are Hispanic	0.00	0.24	0.39	0.33	-0.01	0.9921
% of Clients Who Are Another Race	-1.66	0.84	2.88	1.14	-1.46	0.1495
% of Clients Whose Race Is Unknown	0.23	0.29	0.54	0.39	0.57	0.5708
Herfindahl Index	0.50	0.29	0.50	0.39	1.27	0.2090
% of Clients Referred by Criminal Justice System	-0.12	0.20	0.31	0.27	-0.44	0.6614
% of Clients with Drug and Alcohol Use Disorders	-0.12	0.15	0.24	0.20	-0.60	0.5508
% of Clients Who Injected Drugs	0.28	0.32	0.39	0.43	0.64	0.5246
% of Clients Who Received SSI or SSDI	0.65	0.23	0.35	0.31	2.08	0.0418
Private For-Profit Facility	-0.15	0.14	0.20	0.19	-0.80	0.4269
Sum of Special Services	0.04	0.02	0.03	0.03	1.52	0.1338
Part of a Larger Organization	-0.09	0.10	0.15	0.14	-0.67	0.5054
Facility Receives In-Kind Goods or Services	0.23	0.11	0.15	0.15	1.53	0.1313
Facility Located in a Medium or Small Metro Area	-0.21	0.12	0.18	0.16	-1.26	0.2126
Facility Located in a Nonmetro Area	-0.40	0.21	0.43	0.28	-1.40	0.1666
SSI = supplemental security income. SSDI = social security disability income. Source: SAMHSA, Office of Applied Studies, Alcohol and Drug Services Study, Phase II administrator data, 1997 to 1999, and ADSS cost study, 1997.

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Chapter 7. A Hybrid Cost Function for Outpatient Nonmethadone Substance Abuse Treatment Facilities

Introduction

Cost Function Specification

Data

Estimation

Results

Discussion

References

End Notes