healthmedicinet

Optimization

We identified recommended staffing mixes through the process of optimization. Optimization
involves searching through a set of feasible solutions for one that achieves the best
value of an objective function. In our case, our objective was to maximize net benefits
per patient within each type of SUD treatment program. Each solution corresponds to
a staffing mix, the percentage of staffing hours provided by each staff type to a
patient in a particular treatment program. A feasible staffing mix is one that satisfies
a set of constraints to ensure that it is reasonable and conforms to the ranges observed
in practice. Once potential solutions, objective function, and constraints are defined,
standard algorithms can be used to determine the best feasible solution. We constructed
the optimization problems in Microsoft Excel 2010 and solved them by using the solver
function with the Generalized Reduced Gradient (GRG) algorithm to determine recommended
staffing mixes for each type of treatment program. The detailed descriptions of the
GRG algorithm can be found in the following literatures 6]–8].

Optimization problems have been formulated to tackle diverse health care issues. Some
examples include scheduling for bladder cancer patients 9], 10], determining resource allocation in HIV prevention programs 11]–13], and creating a portfolio of screening and contact tracing for endemic diseases 14]. Optimization problems also have been formulated to identify a better staffing mix
for high technology companies 15]–17], call centers 18], 19], and healthcare providers 20], 21]. However, we could not find any study that addressed optimal mixes of different types
of health professionals for the context of SUDTPs.

We set the percentages of 12 different types of staff (e.g., psychiatrist, addiction
therapists, and clerks, Table 1) in a treatment program as the decision variable matrix (x). We hypothesized that varying the composition of staffing x alters the benefits from the treatment program B(x) in addition to changing staffing costs C(x). We only considered staffing costs as a cost factor because other costs, such as
facility overhead, can be regarded as fixed and unaffected by the decision variables
x and staffing costs are the major cost drivers for SUDTPs. We determined the recommended
staffing mix x* by solving the following optimization problem:

(1)

(2)

(3)

(4)

(5)

(6)

(7)

Table 1. Staff types and FY01-FY03 average wages

Equation 1 indicates that this optimization problem aims to find a staffing mix to maximize
net benefits per patient from a treatment program. The term W denotes conversion factors to transform benefits derived from treating a patient
into monetary values. Some benefits, such as increased employment earnings, are already
expressed in dollar values and thus the conversion factor is 1. However, other benefits,
like reduced days with medical problems, need to be converted to monetary values.
The monetary conversion factors were obtained from the literature and are summarized
in Table 222].

Table 2. Benefit prediction functions

The constraints for the optimization problem are stated in Eqs 2, 3, 4, 5, 6 and 7. Equation 2 indicates that the sum of all staff proportions should equal to 1. Equation 3 specifies that a program’s proportion for each staff type should be in the range
of those observed in treatment programs, denoted by x _observed
. As two examples, standard outpatient program were run by a few as one staff member,
an addiction therapist, and most standard outpatient programs did not have licensed
vocational nurses as more than 10 % of total staff. Equation 4 specifies that the ratio of trainees (e.g., resident MDs or other paid trainees)
to supervisors (e.g., MDs or psychologists) should fall within the observed ranges.
In Eqs 5 and 6, L and H refer to treatment length in terms of days and treatment intensity in terms of hours
per day, respectively, and we limited them to be within observed program ranges. We
hypothesized that treatment length and intensity may change depending on staffing
mix, because each type of staff provides different treatment services requiring different
treatment lengths and intensities. Equation 7 restricts treatment benefits to within possible ranges. For example, if a patient
suffers from drug problems for 20 days per month at admission, the treatment cannot
possibly reduce the patient’s days of drug problems more than 20 days per month, nor
can patients’ days of drug problems worsen to more than 30 days per month.

Data

We derived prediction functions for benefits B(x) based on a VA Outcomes Monitoring Project (OMP) 23] database. The OMP was deemed an exempt project by the VA Palo Alto Health Care System/Stanford
University IRB. The OMP sought to collect representative patient outcome data by randomly
selecting a sample of programs and samples of their patients, and collected baseline
and “6-month” follow-up data on patients in VA SUDTPs in three annual cohorts from
FY01 to FY03 23]. Compared with the previously mandated system-wide monitoring system, the OMP achieved
a higher follow-up rate without paying patients for their participation (67 vs. 15–21
%) 23]. The actual follow-up point averaged 7.4 months (SD?=?2.4 months) and the follow-up
rate was 65.2 %. In all, 5548 patients in 55 standard outpatient, 36 intensive outpatient,
39 residential and 14 inpatient programs were assessed at baseline, and the patients
in the methadone programs were excluded from the analysis. A brief self-report form
24] of the Addiction Severity Index (ASI) 25] was used to assess problems over the past 30 days in the seven ASI domains: alcohol
use, drug use, psychiatric, medical, legal, family/social relationship, and employment
problems. A cost-benefit analysis guideline for addiction treatment using ASI recommends
including the following variables as benefits: reduced number of days in a controlled
environment (medical, psychiatric, residential or hospital substance abuse treatment
program), reduced number of days experiencing medical or psychiatric problems, increased
income received from employment, reduced money spent on alcohol or drugs, and reduced
number of days engaged in illegal activities 22].

Unfortunately, the self-report ASI form did not include the items to assess some key
benefit variables, such as money spent on drugs and number of days engaged in illegal
activities of the original ASI form. Thus, the analysis here may underestimate the
benefits from treatment programs and may explain why the net benefits are estimated
negative for some existing programs. In addition, the self-report form did not specify
the type of controlled environment for each period in an environment; instead, it
asked about how long a respondent stays in all types of controlled environments and
whether the respondent stays in one or more controlled environment during the last
30 days. Thus, we could not use separate monetary conversion factors for each type
of controlled environments, and we needed to calculate a ‘composite’ conversion factor
for all types of controlled environments. We counted responses to each type of controlled
environments and calculated changes in those responses between baseline and follow-up.
We used the response changes for each type of controlled environment as weights to
calculate a weighted arithmetic mean of the monetary conversion factors of all types
of controlled environments, a ‘composite’ factor to convert reductions in days in
all types of controlled environments into monetary benefits from a treatment.

The OMP also assessed VA SUDTPs’ characteristics, staff mix and services delivered
via a program survey 23]. The survey gathered additional information on staff (e.g., time spent in group and
individual treatment) to better assess program costs 23].

Influence diagrams

Influence diagrams 26], 27] were used to specify likely potential relationships among variables affected by staffing
mix and highlight each treatment program’s characteristics. In influence diagrams,
a decision, such as staffing mix, is represented as a rectangle, and an uncertainty
quantity, such as patient health status, is represented as an oval. A double oval
represents a variable that is a deterministic function of its inputs, such as hourly
staffing cost, and a diamond refers to the objective function, such as net benefit.
Arcs represent possible conditional dependence among quantities 26], 27]. For example, having no arc from staffing mix to total staffing cost per patient
indicates that they are conditionally independent, given hourly staffing costs.

We formulated the hypothetical influence diagram in Fig. 1 based on the variables that are assessed in the OMP database assuming no conditional
independence. The hypothetical model shows how staffing mix is believed to affect
patient status after treatment. We allowed treatment length and intensity to change
depending on the staffing mix. We also allowed treatment intensity and length to affect
patient status after treatment and those treatment factors to be tailored to the baseline
characteristics of patients (e.g., more severe patients would be treated longer).
Treatment length was allowed to depend on treatment intensity (e.g., more intense
treatments might be provided for shorter periods). Total staffing costs were calculated
by multiplying hourly staffing cost, treatment intensity, and treatment length. Benefits
from treating a patient were calculated by subtracting baseline patient status from
patient status at follow-up, and total benefits were calculated by multiplying benefits
with the average follow-up point (e.g., 7.4 months) 22]. Net benefits are total benefits less total staffing costs.

Fig. 1. Influence diagram for VA SUD treatment program assuming no conditional independence

Based on the hypothetical influence diagram, we examined the correlations among variables
for each type of program, and whenever there was any nonzero correlation between two
variables at the significance level of 0.05, we included an arc between them. The
correlations among variables also can be found from prediction functions in the following
section; however, influence diagrams graphically present the relations among variables
more clearly.

Prediction functions

We derived prediction functions for benefits (Table 2) by using a stepwise method. The same covariates were included in prediction functions
derived from other methods (e.g., backward method), and coefficients of the covariates
were almost identical. For some types of treatment programs, baseline patient status
and staff variables were the only covariates to predict benefits, but for others treatment
intensity and length have predictive power as presented in Fig. 2. We did not find any significant correlations among the staff variables and did not
include any interactions between staff variables in deriving prediction functions.
For all benefit prediction functions, baseline patient status appeared to be a major
benefit driver, and if a patient came to treatment when he or she was in a more serious
status, changes in patient status between before and after treatments were likely
to be larger (as would be expected from regression to the mean). For the base analysis,
we used average patient status values and then varied them in sensitivity analyses.
We also derived prediction functions for staffing cost factors (treatment intensity
and treatment length) (Table 3) by using a stepwise method and verified them with other methods. Hourly staffing
cost is a weighted arithmetic mean of different types of staffs’ actual costs 28], and only treatment intensity and treatment length needed to be predicted with staff
and baseline patient status covariates (Table 3). Treatment length for standard and intensive outpatient programs could not be predicted
with enough predictive power (e.g., adjusted R
²
less than 0.10); thus, the average constant values of those cost factors were used
for the analysis. All statistical analyses were conducted using PASW Statistics 18,
Release Version 18.0.0 (SPSS, Inc., 2009, Chicago, IL, www.spss.com).

Fig. 2. Influence diagrams of VA SUD treatment programs

Table 3. Prediction functions for treatment intensity and treatment length