Transcript
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Competition and Complementary Activities: Lessons from

Radiological Diagnosis and Treatment

Justin Lenzo ∗

September 2010

1 Introduction

We consider in this work how competition among hospitals affects their positioning decisionsacross the complementary, technologically intensive activities of diagnostic radiology and andradiation therapy. We view a hospital as jointly deciding whether to position itself on the tech-nological frontier of each field. The objective of our analysis is to characterize the manner inwhich these two positioning decisions interact both with each other and with the positioningdecisions of rival hospitals. Is frontier positioning in diagnosis substitutable or complementarywith frontier positioning in treatment? Are a hospital’s positioning decisions strategically sub-stitutable or strategically complementary with those of rivals? As we will discuss, the context ofhospitals exhibits a number of departures from standard competitive models that inhibit definitepredictions on these interactions. Our approach is therefore to let the data speak for itself anduse our estimates of these interactions to learn something about how hospitals compete in theseservice areas.

We identify a set of frontier technologies in each field and examine how tendency towardjoint adoption varies with market-level effects, hospital characteristics, and beliefs about rivalpositioning strategy. We employ a modified bivariate probit specification that allows us to es-timate payoff interactions separately from correlation in unobserved hospital preferences acrossthe two fields. We handle the endogeneity of rival behavior using the Nested-Pseudo Likelihoodapproach developed by Aguirregabiria (2004) and Aguirregabiria and Mira (2007). Our prelimi-nary results suggest complementarity does exist between the two services, that it is stronger forteaching hospitals, that hospitals are more threatened by rivalry in diagnostics than by rivalryin treatment, and that hospital systems coordinate more in the provision of frontier treatmentservices than in frontier diagnostics.

∗Department of Management and Strategy, Kellogg School of Management, Northwestern University, e-mail:[email protected]

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Diagnostic imaging has been the subject of several studies of hospital behavior, includingTrajtenberg (1989), Baker (2001), and Schmidt-Dengler (2006). Trajtenberg (1989) examinesthe market for computed tomography (CT) scanners and estimates welfare gains that stem frominnovation. Baker (2001) studies the timing of magnetic resonance imaging (MRI) by hospitals,with a focus on the role of HMO penetration in the hospital’s market. Schmidt-Dengler (2006)also studies MRI, but focuses on strategic factors stemming from the oligopoly structure of mosthospital markets. Radiotherapy has not seen quite the attention from economists.

In a previous paper, Lenzo (2008) studies the interaction between hospital decisions to adopttwo substitutable nuclear medicine diagnostic imaging machines: single-photon emission tomog-raphy (SPECT) and positron emission tomography (PET). That paper shares with the presentwork an emphasis on how competition among hospitals may drive changes in complementarity.However, with SPECT and PET, two imperfectly substitutable technologies for patients andphysicians, the complementarity is driven largely by economies of scope. Competition has theeffect of diminishing the self-cannibalization in revenues that a hospital faces when adoptingboth machines, and thereby increases the degree to which the technologies are complements inprofits.

In contrast, we are investigating complementary services here. Given that diagnostic radiol-ogy and radiation therapy are distinct departments within the hospital that manage their ownservices and equipment, and given that supply-sides of the respective markets are disjoint, theredoes not appear to be substantial sources of economies of scope. Complementarity is insteaddriven driven by revenue considerations or indirect benefits of “technological excellence,” suchas attractiveness to physicians. If there are patients (or referring physicians) that are sensitiveto the additional quality that the frontier technologies provide, and if preferences for high-techmedical services correlate positively across diagnostics and therapy, then a hospital positionedon the frontier in one sector should be able to attract these patients away from rival hospitalsthat have only conventional varieties.

We should note that the flow of patients between diagnosis and treatment goes in bothdirections. While diagnostics may be the entry point to the feedback loop between the twoservices, the need for follow-up scans to discern whether treatment was effective ensures thata patient travels back and forth between the two, potentially several times. This bidirectionalflow of patients between these services undergirds a feedback loop in payoffs to their provision.A hospital positioned on the frontier in one service may attract patients or physicians whoare sensitive to the added quality of the frontier technology. However, if consumer (patient orphysician) preferences are strongly correlated, then a rival can attract these consumers awayfrom the hospital by positioning on the frontier of both services.

The remainder of this paper proceeds as follows. We discuss the technological services understudy and provide intuition for how they may interact in Section 2. We describe the econometricspecification in Section 3 and the data used to estimate it in Section 4. Then we present some

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preliminary results in Section 5. Please note that this version of the paper is preliminary andincomplete.

2 Diagnostic Radiology and Radiation Therapy

A relevant patient typically starts a diagnostic-treatment process by undergoing one or morescans conducted by the diagnostic radiology provider. If a malady is detected, the patient is thensent to either radiation therapy or an alternative treatment service. If the patient undergoesradiation therapy, then the patient is invariably sent for one or more follow-up diagnostic scans,possibly iterating through diagnosis and treatment several times until the process terminates.

Most hospitals can provide a basic level of both diagnostic and treatment service for cancerpatients. On the diagnostic side, nearly all observed hospitals in 2006 had conventional scanningtechnologies like CT, ultrasound, and film mammography, and most hospitals provided MRIservice either in-house or through an external partner. On the treatment side, conventionalradiotherapy technologies or substitutes such as chemotherapy were provided by two-thirds ofobserved hospitals.

2.1 Frontier Technologies in Radiology and Radiotherapy in 2006

A hospital differentiates itself in diagnostic radiology or radiation therapy service by adoptingone or more of the frontier technologies of each field. In 2006, the frontier diagnostic technolo-gies were multi-slice computed tomography (MSCT) with at least a 64-slice scanner, positronemission tomography with computed tomography (PET/CT), or full-field digital mammogra-phy (FFDM). The frontier treatment technologies are shaped-beam radiation therapy (BEAM),intensity-modulated radiation therapy (IMRT), or image-guided radiation therapy (IGRT).1 Allsix technologies were introduced to the market after 2000 and the AHA started recording thepresence of each facility within two years prior to the sample.

One can think of each of the three technologies on the diagnostic frontier as the latest gen-eration of a diagnostic subspecialty. Multi-slice CT follows a long lineage of X-ray and CTscanners, machines that shoot beams of radiation through the patient and capture an imageon the patient’s opposite side.2 PET/CT, on the other hand, falls under the nuclear medicineumbrella, where radioactive material is injected into the patient and the machine’s camera mea-sures emissions from the material’s decay. One should note that PET/CT also represents a trendtoward hybridization of diagnostic imaging in that it combines a conventional CT scanner and aPET scanner into a single machine. Both MSCT and PET/CT are general-purpose diagnostic

1The abbreviations MSCT, PET/CT, FFDM, IMRT, and IGRT are all standard in the industry and academicliterature on these technologies. There is no such standard abbreviation for shaped-beam radiation therapy, sothe author has chosen that used in the AHA Survey data. One should note that SBRT, often discussed alongsideIMRT and IGRT, stands for “stereotactic body radiotherapy,” a somewhat different treatment technology fromshaped-beam RT and one that is not included in the AHA survey.

2Information on MSCT and PET/CT is taken from Busse (2006).

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imaging technologies applied across a number of specialties including oncology, cardiology, andneurology. FFDM, on the other hand, was the latest in mammography machines, and hence hasspecialized application to breast cancer diagnosis. However, breast cancer is of sufficiently highincidence that it constitutes a significant portion of radiation therapy applications (AmericanCancer Society, 2005).

All three of the treatment technologies have applications in the treatment several differenttypes of cancer. As the name suggests, IGRT uses imaging technology in the course of radiationdelivery. Automated image-guided delivery of radiation allows for higher precision in where theradiation hits. One should note that since IGRT facilities use only conventional imaging tech-nologies (mainly CT), the bundling of imaging and treatment with this service is not a directsource of complementarity between the frontier positions. IMRT differs from conventional radi-ation therapy technologies in that it allows pre-planned, computerized variation in the intensityof radiation used over an area.3 Shaped-beam radiation therapy is similar in spirit to IMRT,but is based around a different type of beam technology.

2.2 Interaction within hospital positioning decisions

Complementarity in the present setting refers to the additional gain from being positioned onboth technological frontiers. Let π00, π10, π01, and π11 be the payoffs to a hospital respectivelyof positioning on neither frontier, on only the diagnostic frontier, on only the treatment frontier,and on both frontiers. Given that we can only identify payoff differences in discrete choicesettings, consider the payoff difference to positioning on both frontiers expressed as:

(π11 − π00) = (π10 − π00) + (π01 − π00) + Γ (1)

where Γ is the complementarity term, which may in principle positive or negative, betweenthe positions. From this expression, we easily see the intimate connection between complemen-tarity and supermodularity of hospital payoffs in positioning. That is, frontier positioning iscomplementary if and only if:

Γ = π11 − π10 − π01 + π00 > 0 (2)

A natural place to start the analysis of complementarity between the positioning decisionsis whether there are likely substantial economies of scope between being on the diagnosticfrontier and the treatment frontier. In the setting of Lenzo (2008), economies of scope betweenPET and SPECT imaging services were driven largely by shared fixed costs from commoninfrastructure used by the two technologies. In the present setting, however, the potential forsuch shared infrastructure is much less. The diagnostic and treatment machines are often housedand maintained in separate hospital departments. Furthermore, the operating staff is distinct

3Information on IGRT and IMRT is taken from Meyer (2007)

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and are not typically trained in the other services.4

Even without organizational sources such as common infrastructure, economies of scope canbe driven by the existence of common suppliers in some settings. If the same set of firmsproduced both diagnostic and treatment machines, these firms could bundle the sale of themachines, generating economies of scope in the form of lower joint fixed costs. Somewhatsurprisingly perhaps, the sets of suppliers to the U.S. market in the technologies under studyare distinct. On the diagnostic side, GE Healthcare and Siemens produce all three machines,Phillips produces MSCT and PET/CT, and Toshiba Medical Systems produces only MSCT. Onthe treatment side, Varian is the market leader in all three technologies. In addition to Varian,there is Elekta, which purchased Phillips’s radiation therapy division in 1997, BrainLab, andAccuray. One should note that Siemens does produce IMRT machines for overseas markets, butdoes not sell these machines in the U.S.5

Another potential source of complementarity between the frontiers is the use of diagnostics toinduce demand for treatment services. For example, if the cutting-edge diagnostic technologiescould reveal tumors that are invisible to conventional scanners, then the adoption of thesemachines could stimulate demand for the hospital’s radiation therapy services. For this effect totranslate to complementarity between the frontier positioning decisions, the hospital would haveto see more benefit in treating these additional patients with cutting-edge treatment services thanwith conventional treatment services. In particular, if tumors that are more difficult to detecttend to be more difficult to treat, then this source of complementarity could arise. However, ifthe primary benefit of frontier diagnostic technology is to detect tumors earlier than possiblewith conventional technologies, and if tumors detected earlier tend to be easier to treat, thenhaving frontier diagnostics could be a disincentive to investing in frontier treatment technology.

Patient sorting may be another source of complementarity between the positioning decisions.Suppose that patients are heterogeneous in their sensitivity to the additional quality providedby the frontier technologies in each area. If patient sensitivities across the two dimensionsare positively correlated, then a hospital may be able to stimulate demand for both high-techservices by attracting these technology-sensitive patients. Note that this same pattern wouldbe generated by analogous preference variation among physicians that refer patients to the twodepartments. The necessary condition for patient sorting to generate complementarity is thatpositioning on both frontiers is required for the hospital to attract a sufficient amount of thesesensitive patients. This condition is more likely to be met if there are other hospitals in themarket positioned on one or both frontiers.

It is important to note that, when combined with high switching costs for patients to choosedifferent hospitals for diagnosis and treatment, this patient sorting could have the opposite effectof making the frontier positioning decisions substitutable in markets where frontier positioning is

4Linton (1997) describes the development of the two specialties from a combined ancestor. Both specialtiesexhibit continuing trends toward sub-specialization, as described in Smith et al. (2009).

5Information presented here was gathered from the listed companies’ websites.

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rare. Suppose that a hospital is considering its positioning decision in the face of patient sortingas described above and suppose that no other hospital is positioned on the frontier of either field.By adopting a position on only one of the frontiers (say diagnosis), the hospital will attract mostof these sensitive patients. By also positioning on the treatment frontier, the hospital will notattract many additional patients. That is, most patients who would be drawn to the hospitalbecause of its position on the treatment frontier were already drawn to the hospital becauseof its position on the diagnostic frontier. While the hospital would derive additional benefitsfrom serving these tech-sensitive patients with high-tech treatment, it would not be the casethat the value of being on the treatment frontier was increased by also being on the diagnosticsfrontier (in fact, the value is decreased). Note, however, in markets where rivals are positionedon one or both of the frontiers, patient sorting could indeed generate complementarity becausethe redundancy in the attracted patients would not be as strong.

Finally, a important source of trade-offs between the two positioning decisions for a hospitalcould be the scarcity of investment capital. Such budget constraints may force the centralhospital management to prioritize capital investment by one of the departments over the other.

2.3 Interaction across hospital positioning decisions

At a first glance, we might expect frontier positioning decisions to be strategic substitutes. If arival positions itself on the frontier, then this would seem to make it less profitable for a hospitalto be on that frontier. However, in analyzing interaction among the decisions of rival hospitals,we have to consider that competition among hospitals may be quite different than competitionamong profit-maximizing firms in conventional industries.

First, the objectives of a hospital may depart from profit maximization. Most hospitals inthe sample are not-for-profits. We might view a not-for-profit hospital as gaining benefit fromquantity beyond its effect on profits. As modeled in Gaynor and Vogt (2003), not-for-profitswould face “behavioral marginal costs” that are effectively lower than their marginal costs wouldbe if they were pure profit-maximizers. The implication for the present context is that a hospitalunder behavioral marginal costs may position itself on a frontier to attract patients even if rivalpositioning have reduced the direct profits of being on that frontier.

Furthermore, if hospitals compete in “technological excellence” or some other quality dimen-sion supported by frontier positioning, then frontier positioning may be strategically comple-mentary. The “Medical Arms Race” is a common concern in the health economics literature (seefor example Dranove, Shanley, and Simon (1992) or Gaynor and Vogt (2003)). An arms race intechnology may result for competition among hospitals for physician affiliations. If physiciansare attracted to hospitals with cutting-edge technology, hospitals may continually strive to outdoone another in this dimension, independent of the direct profitability of the affected services

Another feature of hospital services that is relevant here is rigid pricing. Because of the role ofMedicare reimbursements and third-party payment generally, service prices are unlikely to vary

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significantly across positioning profiles in a market. Among Medicare patients at least, prices fora frontier service are effectively fixed, regardless of how many rivals are also positioned on thefrontier. In this setting, benefit to the hospital is determined by utilization rates. As described inWinter and Ray (2008), the Medicare reimbursement system accentuates a hospital’s incentiveto use its diagnostic imaging facilities at capacity. Even without the reimbursement rules, thehigh fixed cost relative to marginal cost in diagnostic imaging (even compared to the frontiertreatment services) generates stronger economies of scale in diagnostic imaging. These gainsto scale, all else equal, would suggest that frontier positioning in diagnostics is strategicallysubstitutable. On the other hand, if hospitals can effectively induce their own demand fordiagnostic services, they can diminish or even neutralize the impact of rival positioning.

Our estimates of a hospital’s reaction to positioning decisions should inform us as to whichof these various effects dominate. For example, a pure Medical Arms Race would suggest thatthe decisions are all strategically complementary. Furthermore, complementarity in the payoffsto each frontier in an arms race scenario would tend to increase with rival frontier positioningbecause of the need to outdo, or at least match, competitors in frontier positioning. Findingstrategic substitutes among the decisions would suggest either some price competition in frontierservices or strong gains to high utilization rates.

3 Econometric Specification

The main focus of this work is on the relationship between the two positioning decisions byhospitals with respect to diagnostic and treatment services. Several factors could plausiblyexplain a tendency toward joint positioning in the data. Regional variation in standards of careand in the use of technology is substantial in hospital services. If regions that exhibit greater useof high-tech diagnosis also exhibit greater use of high-tech treatment, then the decisions couldappear correlated without there being actual interaction between them. Furthermore, correlationin preferences for technology at the hospital level can also induce such correlation, again withoutany complementarity between the two decisions. Estimating how competition interacts with thisrelationship requires an bivariate econometric specification that permits separate estimation ofcomplementarity in payoffs from correlation in unobserved preferences for technology or qualityleadership across the two fields. At the same time, measurement of strategic interaction amongrival hospital decisions requires that we handle the endogeneity that arises among the decisions.The econometric approach wrestles with each of these issues to arrive at a valid estimate of theinteractions under study.

We first explain the approach used in the preliminary univariate analysis that we conductto rule out regional variation as the driving factor behind the observed correlations. In doingso, we describe the method we use to treat the endogeneity problem in rival decisions in boththe univariate and bivariate analysis. The treatment of endogeneity and regional effects carries

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over to the bivariate analysis. We then describe the approach used in the bivariate positioninganalysis, focusing on the issue of separating complementarity from mere correlation.

3.1 Univariate Analysis

As a preliminary step, we examine the positioning decisions in isolation. The primary goal of thisanalysis is to determine whether the tendency toward joint positioning seen is fully explainedby regional variation. That is, it is possible that the raw correlation in frontier status observedin the data is a product neither of complementarity nor correlation in hospital-level preferences.Rather, it could merely be an artifact variation in market-level variables that drive the provisionof both frontier services. If this is the case, we should be able to explain the covariance betweendiagnostic frontier status and treatment frontier status by controlling for market-level effects.

For each of the diagnostic and therapeutic fields, we run a probit specification on the hos-pital’s decision to position itself on the frontier in that field. We include market-level fixedeffects to absorb unobserved market-level variation that may explain the differences in choicepatterns.6 To gage reactions to rival positioning, we include an endogenous variable measuringthe log number of rivals positioned on the field’s frontier. The specification for hospital i withregard to either positioning decision takes the form:

ai = 1 if πi > 0 and 0 otherwise

πi = αm + x′iβ + γci + δ ln(1 + ri) + εi (3)

where αm is the market-level effect, xi are the exogenous characteristics, ci is an indicator ofwhether the hospital is positioned on the frontier of the other field, ri is the number of rivalspositioned on the frontier, and εi is the Gaussian error term.7 The variable ci, according tothe intuition discussed in Section 2, is likely endogenous. The endogeneity of these decisionsmotivates the use of a bivariate specification in the main analysis. For the preliminary analysis,however, we treat the hospital’s position in the other field as exogenous to determine whetherwe can explain away the tendency toward joint positioning through regional variation and com-petitor interactions.

For hospitals that are part of a hospital system, we include separate terms for rivals (hospitals6As discussed in Greene (2002) and many other places, probit models with panel data and fixed-effects are

known to exhibit an incidental parameters problem that renders the maximum-likelihood estimator inconsistent.One might wonder whether the same issue arises with the market-level fixed effects used here. Unlike in panel datacontexts, where the number of fixed-effects approaches infinity at the same rate as the number of observations,the number of markets in the present context is fixed. The incidental parameters problem is therefore not presentwith respect to the estimator’s consistency. It remains possible that the same factors that generate finite-samplebias in panel data contexts apply here. However, logit versions of the regressions presented in this section do notsuffer from the incidental parameters problem and generate the same qualitative results. We report the probitresults because they are more comparable to the results presented in the bivariate analysis.

7As is normally the case with probit and logit models, the parameters are only identified up to a scaling bythe variance of the error term. As is standard, we assume a normalization of the error variance to one.

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in the same market, but different system) and allies (hospitals in the same market and samesystem). A hospital would likely react differently to a positioning decision by an ally, withwhom coordination is more feasible, than it would to a rival. We omit these ally variables fromthe discussion in this section merely for notational simplicity, as we use the same method toestimate reactions to allies as we use for rival hospitals. This treatment continues still assumesthat positioning decisions are made at the individual hospital level, rather than by central systemmanagement. As a robustness check, we consider a specification based around the alternativeassumption that positioning decisions among allies within a market are coordinated centrally.

The use of a logarithmic version of the rival count is motivated by the intuition that the effectof additional rivals on a frontier should diminish as the number of rivals increases. Withoutfurther treatment, our estimates would be subject to an endogeneity bias. In markets withfavorable unobserved conditions for high-tech service, the same unobserved factors that makea hospital more likely to position itself on a frontier also make its rivals more likely to do so.These factors induce correlation between the ri term and εi and lead to omitted variable bias.

To handle the endogeneity of the competitive term, we specify the interaction of hospitalsas a game of incomplete information, as in Seim (2006). Rather than assuming that a hospitalobserves the exact number of rivals it would face on the frontier, we assume that a hospitaldevelops beliefs about how likely each rival is to position itself on the frontier. With thesebeliefs, each hospital forms an expected number of rivals that it would face on the frontier.A Bayes-Nash equilibrium occurs where each hospital’s beliefs are correct and each hospitalpositions itself on the frontier if its private preference term exceeds the threshold defined by thecharacteristics and parameter values. Equation (3) then becomes:

πi = αm + x′iβ + γci + δE[ln(1 + ri)

∣∣P−i

]+ εi (4)

where P−i denotes hospital i’s beliefs about its rivals’ choice probabilities.In principle, one can conduct the parameter search using a Nested-Fixed Point algorithm as

in Rust (1987). To reduce the computational complexity of the search, we employ the NestedPseudo-Likelihood (NPL) algorithm developed in Aguirregabiria and Mira (2007). The NPL isbest thought of as an EM-algorithm tailored to maximum likelihood estimation of games. In themaximization step of each NPL iteration, the algorithm uses the current guess of the equilibriumchoice probabilities to compute the expected values of the endogenous variables, and then findsthe parameters that maximize a pseudo-likelihood function where these values are fixed. In theexpectation step, we solve the the fixed-point system generated by the current parameter valuesto update our estimate of the choice probabilities. The choice of initial choice probabilities tostart the search can matter in some contexts. In the author’s experience, for static models withsmooth error distributions and sufficient variation in exogenous characteristics (conditions thatmatch the present setting), the starting point for the choice probabilities does not appear tomatter. We generate initial choice probabilities randomly and we perform each NPL search a

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number of times at different starting points to ensure the validity of the terminating point.

3.2 Bivariate Analysis

As mentioned above, the indicator for frontier positioning in the other field in (3) is endogenous.We expect that the frontier positioning decisions are ultimately made jointly by the hospitaladministration. If there is complementarity between the positions or correlation in unobservedpreferences over frontier positioning, then the two positioning decisions are not separable. Fur-thermore, there is no a priori ordering to the positioning decisions that would allow us toconstruct a reduced form of the joint decision model.

The most substantial complication in the bivariate analysis is the need to separate comple-mentarity between the frontier positions from correlation in unobserved hospital preferences overthe two decisions. The issue may appear subtle at first, but it is can be a crucial difference in howwe interpret the interaction between these two activities. As a simple example of the difference,consider a social planner with an objective to stimulate adoption of frontier cancer treatmenttechnologies.8 If there is substantially positive complementarity in being on the two frontiers,then the planner has more tools by which to stimulate adoption of treatment technologies. Thatis, in addition to stimulus policies that directly affect the payoffs to treatment technologies, theplanner can stimulate adoption of treatment technologies indirectly by encouraging adoption ofdiagnostic technologies. If the tendency toward joint adoption arises solely because of correlatedfactors, then encouraging frontier positioning in diagnostics will not actually change the payoffsto adopting frontier treatment technologies, and hence indirect stimulus is not possible.

As we discuss below, estimating the complementarity and correlation as separate parametersposes challenges both in identification and in computation. Consider the following bivariatespecification with a constant complementarity term:

aji = 1 if πji > 0 and 0 otherwise

π1i = α1m + x′1iβ1 + y′1iδ1 + 12Γa2i + ε1i (5)

π2i = α2m + x′2iβ2 + y′2iδ2 + 12Γa1i + ε2i (6)

Most of the terms in (5) and (6) have analogs in (4). Note that we assume the same groupingof observations across the two equations for the purpose of fixed effects; however, we allow thefixed effect coefficients to differ across the them, at the cost of greatly expanding the dimensionof the parameter space. The exogenous characteristics vectors x1i and x2i will share many of thesame variables. However, select exclusion restrictions among these characteristics will be the

8Another example of the difference arises if we think of ourselves as a supplier of technology to both the diag-nostic and treatment fields (again, the present setting lacks such common suppliers). If there is complementaritybetween the two decisions, we may be able to effectively employ cross-subsidization to increase profits beyondwhat linear pricing would deliver. If the joint tendency is a result only of correlation, then no potential gainsfrom cross-subsidization exist.

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cornerstone of the identification strategy, as discussed below. The vectors y1i and y2i containthe endogenous variables of the same form as the ln(1 + ri) term discussed in Section 3.1. In thebivariate analysis, however, we include counts of rivals and allies not only on the same frontier,but on the other frontier as well. The inclusion of expected rivalry on the other frontier iswarranted because of how patients are fed back and forth between the two fields. The errorterms (ε1i, ε2i) are assumed to follow a bivariate normal distribution with unit variances and acorrelation parameter ρ. It is this parameter ρ that will capture phenomena such as correlationin unobservable preferences for technology or the overall desire to be a “center of excellence”among hospitals.

The parameter Γ in (5) and (6) captures complementarity between the positioning decisions.It is because of the presence of this parameter that the specification deviates from the standardbivariate probit model. To see the effect, simplify Equations (5) and (6) to π1 = u+ca2 +ε1 andπ2 = v + ca1 + ε2, where (ε1, ε2) are the bivariate normal error terms. If c is restricted to 0, wehave the standard bivariate probit framework and the areas of integration for computation ofthe probability of each choice pair are all rectangular, as shown in Figure 1.9 Integration of thebivariate normal PDF over rectangular areas is fairly straightforward and widely implementedin statistics and mathematical programming software.10

Figure 1: Areas of integration in the standard bivariate probit model.

When the complementarity term is non-zero, the shapes of the areas of integration becomenon-rectangular. Figure 2 depicts the cases where c < 0 (left) and c > 0 (right). When c > 0(right graph), we see that the regions relevant to the choice pairs (0, 1) and (1, 0) are still

9Note that in Figures 1 and 2, the correlation parameter does not affect the shape of the areas of integration.Rather, the correlation parameter affects the eccentricity of the elliptical contours of the bivariate normal PDF,which for simplicity are not shown on the figures.

10For a good reference on precise calculation of bivariate normal probabilities, see Genz (2004) and the sourcescited therein.

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rectangular and can be computed as in the standard case. However, the regions of integrationfor the other two choices are no longer rectangular and so the standard evaluation methods cannot be used.

Figure 2: Areas of integration in the bivariate probit model with complementarity (c < 0 left,c > 0 right).

One approach to the the non-standard areas of integration is to employ the mixed-logitframework developed by McFadden and Train (2000). This approach has been used to estimatebivariate discrete choice models with complementarity by Gentzkow (2007) and Lenzo (2008).The estimation of mixed-logit models requires simulated methods such as those discussed inGourieroux and Monfort (1996). However, the combination of the simulated integrals, thefixed-point searches, and the high-dimensional parameter space brought by the market-leveleffects makes estimation of a mixed-logit specification difficult, if not intractable, for the presentcontext.

Instead, we construct a method to compute the volume over the non-rectangular areas usingtriangular decomposition and simple geometric transformations. Our method has its ancestryin the early statistics literature, in particular Nicholson (1943) and Owen (1956), where similartechniques were used to compute tables of bivariate normal probabilities. The method allows usto quickly and accurately compute the probability over any triangular area in the plane. The firststep is make the bivariate PDF is circular-symmetric using the well-known transformation ν1 =ε1, ν2 = (ε2 − ρε1)/

√1− ρ2. Any rectangular portions of the transformed area of integration

can of course be computed as in the conventional case. Remaining areas are decomposed intoright triangles with one of the acute vertices at the origin. Because the transformed spaceis circular symmetric, these triangles can be rotated around the origin without changing thevalue of the integral over them. We rotate each component triangle so that it has vertices{(0, 0), (h, 0), (h, q)}.11 The volume over this triangle is then given by Owen’s T -function (see

11If the rotation results in q < 0, then the triangle is reflected over the horizontal axis, again without changingthe volume.

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Owen (1956)):

T (h, q) ≡ 12π

∫ q

0

exp{−1

2h2(1 + x2)

}1 + x2

dx (7)

The integral T (h, q) is easily and accurately computed using Gauss-Legendre quadrature. Whilethe geometric transformations used are not without computational cost compared to the stan-dard bivariate probit without complementarity, the method used comes at enormous savingscompared to either raw two-dimensional quadrature or simulated maximum likelihood. Readersinterested in the details of the method should refer to the appendix (forthcoming).

Beyond the computational challenge, there is an identification problem in the simultaneousestimation of the complementarity and correlation parameters. This problem is well-known andhas been discussed in various contexts by Manski (1993), Athey and Stern (1998), Augereau,Greenstein, and Rysman (2005), Gentzkow (2007), and Lenzo (2008). The identification strategyrests on finding exclusion restrictions that allow us to isolate the effect of complementarity fromcorrelation. In particular, identification requires that we have variables that may affect thepayoffs of only one of the frontiers directly.

To understand the intuition, consider the example of how the provision of cardiac surgeryaffects the payoffs to positioning on the frontier of diagnostics and radiation therapy. Cardiacsurgery is available at 26% of the hospitals in the sample. Frontier diagnostic imaging (MSCTin particular) is useful to hospitals in deciding which cardiac patients require invasive surgery.12

Hence, the presence of cardiac surgery at the hospital should affect the payoff to the adoptionof MSCT. However, there does not appear to be a substantive source of interaction betweenradiation therapy and cardiac surgery. While radiation therapy (especially IMRT) and cardiacsurgery may be substitutes for treatment of heart cancer, the incidence of cases where both wouldeffectively be options appears to be too rare for the presence of one service to substantively affectthe payoffs to the other. We can therefore reasonably exclude cardiac surgery from the payoffto frontier positioning in radiotherapy. If we then observe that the provision of cardiac surgeryaffects the likelihood of frontier positioning in radiation therapy, we can be confident that theeffect comes indirectly through the interaction with diagnostic positioning.

If we are able to arrive at a valid estimate of the parameter Γ as it appears in Equations (5)and (6), then we can also estimate Γ as a function of covariates. In particular, we augment thebasic specification by setting

Γ = γ + x′3iβ3 + y′3iδ3 (8)

In specifying complementarity this way, we can measure whether certain factors affect the de-gree to which the two frontiers are complementary. We are especially interested in how the

12One may of course ask whether we should be treating the decision to provide cardiac surgery jointly withMSCT adoption and take into account complementarity between them. While complementarity is likely sub-stantial, the decision to provide cardiac surgery is most likely predetermined by the time MSCT hit the market.Furthermore, these services likely interact less because the patient flow is less bidirectional than in the case ofnon-invasive radiation therapy.

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strategic terms in y3i affect the complementarity. Note that we do not include fixed-effects oran idiosyncratic error term in the complementarity equation.

4 Data

U.S. non-federal, general medical and surgical hospitals in 2006 serve as the setting for ourempirical analysis. The primary data set used is the 2006 AHA Annual Survey of Hospitals.The data provides characteristics such as size, affiliation, and location as well as the set offacilities that a hospital provides. The AHA survey data is used widely in the literature onhospitals. From the full set of over 6000 hospitals listed, we select non-federal, general medicaland surgical hospitals that responded to the survey. The selection omits specialty hospitalssuch as psychiatric or children’s hospitals and institutional hospitals such as military or prisonhospitals.

We use the Hospital Referral Regions defined by the Dartmouth Atlas (1996) as the marketdefinitions for our analysis. The Dartmouth Atlas delineated 306 such regions in the U.S.by examining Medicare patient flows for major cardiovascular surgery and neurosurgery. Theregions better reflect markets for tertiary care, which are broader than alternative definitionssuch as the narrower Health Service Areas also defined by the Atlas. While HRRs are allowedto straddle state borders, most are contained within a single state. Included fixed effects at theHRR level should therefore absorb state-level variation that arises from institutional differences,in particular Certificate of Need laws.

To identify the market-level effect for diagnostic frontier positioning for a market (i.e. theα1m term in Equation (5)), we must observe at least one hospital in that market that positionsitself on the frontier and at least one hospital that does not.13 The analogous requirement holdsfor the treatment positioning decisions. Hence, we have to eliminate forty-seven markets thateither do not contain observed hospitals or that have a degenerate choice distribution alongeither frontier. Note that since we do not include market-level effects in the complementarityequation, we do not need all four choice combinations to appear in each market. We are leftwith 3562 hospitals in 259 markets in the final data set.

As described above, we define a hospital as positioned on the frontier of diagnostic radiologyservice if it offers at its hospital at least one of multi-slice computed tomography (MSCT) with atleast a 64-slice scanner, positron emission tomography with computed tomography (PET/CT),or full-field digital mammography (FFDM). We define a hospital as positioned on the frontierof radiation therapy if it offers at its hospital at least one of shaped-beam radiation therapy(BEAM), intensity-modulated radiation therapy (IMRT), or image-guided radiation therapy

13If a market contains no hospital that positions itself on the diagnostic frontier, then the value of likelihoodfunction can always be increased by decreasing the value of the market-level effect for that market. Likewise, ifall hospitals in a market position themselves on the diagnostic frontier, then we can always improve the likelihoodvalue by increasing the effect parameter.

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(IGRT). These criteria for frontier status are based on study of the radiology industry literature,discussions with physicians, and frequency analysis of the data. The AHA Survey started toask about IMRT and shaped beam radiation in 2004 and about the other services in 2005.Furthermore, the services exhibit a greater discrepancy in frequencies between teaching hospitalsand non-teaching hospitals than the older diagnostic imaging or radiation therapy facilities, anindication that they are on the technological frontier at the time. Table 1 provides the frequenciesfor these services for the sample overall, teaching hospitals, and non-teaching hospitals. Nearlyall hospitals in a market can offer conventional diagnostic services such as CT, MRI, ultrasound,and film mammography and around two-thirds offer conventional radiation therapy and othertherapeutic services such as chemotherapy.

Using our definition of frontier diagnostic service, we have that 40% of the overall sample,66% of the teaching hospitals, and 32% of the non-teaching hospitals are positioned on thediagnostic frontier. We have that 28% of the overall sample, 58% of the teaching hospitals, and19% of the non-teaching hospitals are on the treatment frontier.

Facility Overall Teaching Non-TeachingMSCT 0.228 0.423 0.165

PET/CT 0.168 0.340 0.113FFDM 0.211 0.379 0.157IGRT 0.124 0.278 0.075IMRT 0.246 0.530 0.155BEAM 0.199 0.457 0.117

Table 1: Frequencies of Facility Provision (2006 AHA Annual Survey)

A feature of many health care markets is the presence of associations between hospitals thatfoster coordination on matters including the provision of diagnostic and treatment services. TheAHA Guide (2006) provides listings for two such association types: systems and networks. Asystem is defined by the Guide as “two or more hospitals that are owned, leased, sponsored,or contract managed by a central organization.” A network is defined as “a group of hospitals,physicians, other providers, insurers, and/or community agencies that work together to coordi-nate and deliver a broad spectrum of services to their community.” As discussed in Bazzoli etal. (1999), while both associations reflect coordination among hospitals that may diminish oreliminate the extent to which we may consider them “rivals,” the crucial distinction betweenthe two is the common ownership of assets among hospitals within the same system. We con-sider two hospitals as rivals if they operate in the same market and are not part of the samesystem. This assumption of coordination at the system level and not at the network level isconsistent findings on joint pricing behavior by Burgess, Carey, and Young (2005), in their studyof California hospital networks. Given the strong correlation in the composition of systems andnetworks (many networks in fact consist only or primarily of hospitals within the same system),it is unlikely that grouping by systems or by networks will make a substantial difference.

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In addition to allowing allied hospitals in the same market to react to each other differentlythan rival hospitals do, we use system membership to construct log-counts of system-wide adop-tion of the frontier technologies. If expertise is shared within a system, then we would expectthat start-up costs for a frontier service are reduced for a hospital that is part of a system whereother hospitals position themselves on the frontier. We count only those hospitals outside anobservation’s market to avoid endogeneity problems with these variables. The assumption isthat the market definitions are correct: that is, patients largely do not flow across two hospitalsin different markets, even if these hospitals are within the same system. Under this assumption,the adoption of frontier technologies by system-mates outside a hospital’s market may affectcosts, but should not otherwise affect utilization or prices. We can therefore exclude the cross-effects from each payoff: system-wide adoption of diagnostic frontier technology affects only thediagnostic payoff directly, and likewise for system-wide treatment adoption. These variables canthen serve to help separately identify complementarity from correlation.

Among the exogenous variables included in the payoff to diagnostic frontier positioning, wealso exclude cardiac surgery, the presence of a diagnostic radioisotope facility, and computer-assisted orthopedic surgery from the payoff to the treatment frontier.14 The intuition behind theexclusion of computer-assisted orthopedic surgery is akin to that of cardiac surgery explained inSection 3.2. A diagnostic radioisotope facility is a supplier to radiology departments, but not toradiation therapy departments, so it should have no direct effect on the treatment positioning.

The only other exclusion restriction in place in the payoff to diagnostic imaging is whetherthe hospital’s cancer program is certified by the American College of Surgeons. Hospitals withthis certification (30% of the sample) undergo quality assurance conducted by the ACS in theirradiotherapy services. The assumption is that patients would be more willing to undergo newerradiation therapy methods (and physicians more willing to refer their patients to these therapies)if the hospital has a strong reputation for quality in its radiotherapy services. The ACS does notappear to have an equivalent role in signaling the reputation for quality of a hospital’s diagnosticservices.

Table 2 presents descriptive statistics on variables used in the regressions. Note that teachinghospitals represent 24% of the hospitals in the sample and that for-profit hospitals compose 12%.

5 Results

5.1 Univariate Regressions

Tables 3 and 4 present the results of the standalone probit regressions on diagnostic and treat-ment frontier status respectively. Looking at the positioning decisions in isolation, we find that

14It may still be the case that an unobserved preference for technology among central hospital management isdriving the decision to provide the whole array of technological services. Ideally, we would like to find variablesthat themselves are not a function of hospital management decisions. The identification strategy remains an areaof continued work.

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regional variation clearly does not explain all of the correlation between the positions observedin the data. In particular, after controlling for market-level fixed effects and other covariates,being on the treatment frontier appears to significantly increase the probability that the hospitalwill choose to position itself on the diagnostic frontier. The analogous result for the treatmentfrontier decision also holds. Mapping the coefficient estimates to marginal effects, we find thatbeing on the treatment frontier is associated with between a 7% and 14% increase in the proba-bility of choosing to position on the diagnostic frontier (which has an unconditional probabilityof 40% in the data).15 Likewise, being on the diagnostic frontier is associated with between a5% and 10% increase in the probability of choosing to position on the treatment frontier (whichhas an unconditional probability of 28% in the data).

Another result of note from the univariate regressions is the lack of a substantial teachingeffect. Once we control for hospital size, the presence of other facilities, and being positionedon the other frontier, a teaching hospital is not significantly more likely to position itself on thediagnostic frontier than a non-teaching hospital. The coefficient on teaching status is positiveand significant at the 90% level in the treatment regression, with a marginal effect of up to5%. These results are consistent with our finding from the bivariate analysis (discussed below)that teaching status does not affect the standalone payoffs to either frontier, but does affect thedegree to which frontier positioning is complementary.

The univariate regressions also suggest that the NPL approach does correct for endogeneitybias. The tables provide the results for both a naive regression, where the log count of rivals isincluded as it appears in the data, as well as the results from the NPL regression. One can seethat the estimated coefficients hospital-level covariates (other than the endogenous variables) arestable across the two regressions. The estimated coefficients on the endogenous counts, however,are quite different. This suggests that the results in naive regression exhibit substantial biasfrom the omission of unobserved factors that influence the decisions of hospitals similarly.

5.2 Bivariate Regressions

Table 5 presents the parameter estimates from the main hospital-level regression. As in othermultivariate discrete choice models such as the multinomial logit, the implied marginal effectshere can differ in significance and even in sign from the analogous coefficient estimate. In thepresent context, the fact that the choice occurs over combinations of positioning decisions furthercomplicates the mapping from coefficients to effects on decisions. Table 6 presents the averagemarginal effects implied by the estimates in Table 5. The reader should note that first two columngroups in Table 6 refer to the overall effect on frontier positioning in diagnostics and treatmentrespectively. For example, doubling the number of beds in a hospital increases the probability ofpositioning on the diagnostic frontier, with or without positioning on the treatment frontier, bybetween 3% and 7% on average with a confidence level of 90%. The third column group refers to

15Marginal effect confidence intervals reported for the 90% confidence level.

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the effects on the choice of positioning on both frontiers. In particular, the effects presented inthe third column group contribute to the effects in the first two.16 Further note that a variablecan have an unambiguous effect on the probability of joint adoption even if it has little effect oncomplementarity through simultaneous effects on the standalone payoffs. For example, doublingoutpatient visits has a positive effect of between 1% and 3% on the probability of positioning onboth frontiers even though the estimated coefficient in the complementarity equation is negative.

The effects of hospital size, in terms of beds and outpatient visits, on the positioning decisionsare not surprising. Bigger hospitals expect to have a larger base of patients over which to spreadthe substantial fixed costs of acquiring and maintaining the frontier technological services. Withdiagnostic imaging in particular, profitability likely increases with utilization, and hospitals witha larger pool from which to draw patients can expect greater utilization than smaller hospitals,all else equal. The results do not suggest that larger hospitals are able to exploit interactionbetween frontier positions any better than smaller hospitals. A similar result holds for thepresence of other complementary facilities. These services generally have the expected effectson payoffs and choice probabilities; however, those included in the complementarity equation donot appear to have significant effects on complementarity between the frontiers.

One of the more interesting results from the regression in Table 5 is the effect of teachingstatus. Teaching status does not appear to be associated with higher standalone payoffs topositioning on either frontier ceteris paribus. However, it appears to have a significantly pos-itive effect on the complementarity between the frontiers. These results suggest that teachinghospitals are less driven to adopt frontier technologies because they are more profitable in iso-lation than they are by a greater ability to exploit interaction between complementary services.Likewise, for-profit hospitals do not appear to earn less standalone profits from either frontierdiagnostics or treatment services; rather, they appear to either be less able to exploit interactionin these services or face stronger trade-offs between the positioning decisions. Taken together,these results suggest that perhaps budget constraints in capital investment are preventing somehospitals from benefiting from complementarity between the frontier services. A hospital facing atight budget constraint would be less able to make the needed investments in developing strongpositions on both frontiers. Teaching hospitals, which typically are less resource constrained,may not confront such trade-offs as often.

The primary results of interest are those related to the reaction of hospitals to others inits market. Many of the estimated coefficients are not statistically significant, so inference islimited. However, the coefficient estimates in the payoff equations present a picture of how ahospital profits from the frontier positions in different ways. In the complementarity equation,rival positioning on the diagnostic frontier appears to have a significantly negative impact. Thisresult indicates that there exists substantive price competition among frontier diagnostic serviceproviders. Such price competition may impede a provider’s ability use price discrimination in

16Because of how we have computed the first two column groups, it is not the case that the omitted effect onthe probability of positioning on neither frontier is one minus the sum of the effects along each row.

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imaging and similar tactics to extract value from technology-sensitive patients lured by thehospital’s treatment services. Standalone profits to treatment services appear most affected bythe number of rivals positioned on both frontiers, suggesting that profits to treatment servicesare more dependent on being positioned on both frontiers, for example, to attract patientsseeking the “center of excellence” in the market.

The marginal effects show that diagnostic frontier positions are strategic substitutes amongrivals, confirming the interpretation of price competition among rival diagnostic providers or thedependence on high utilization rates.17 The marginal effects also exhibit patterns of reactionamong allied hospitals that are not apparent from the coefficient estimates. For one, treatmentfrontier positions are strategic substitutes among allies. While technically this could indicatethat allies still compete in price in frontier treatment services, it seems more plausible thatit reflects a tendency to centralize frontier treatment services within a hospital system over amarket. On the other hand, allied provision of frontier diagnostic imaging appears to stimulate ahospital’s provision of frontier treatment services. Diagnostic imaging tends to be more profitablethe higher the utilization rate. Hospitals may be induced to provide frontier treatment servicesto stimulate not only demand for their own frontier diagnostic services (if they provide any),but also demand system-wide.

These interactions among allied hospitals suggest that there is some degree of central coor-dination among hospitals that share a market and system. One might wonder whether decisionsare actually made at the system level, rather than by hospitals individually, as has been assumedup to this point. Tables 10 and 11 present the results of a modified version where the position-ing decisions are assumed to be made at the system level (among hospitals within a market).In this specification, a system is considered positioned on a frontier in a market if any of itsmember hospitals are so positioned. In particular, a system is positioned on both frontiers if ithas at least one member positioned on each, even if there are no members positioned on both.An assumption underlying this specification is that migration costs for patients across hospitalswithin a system are minimal. The rival variables also count systems, not individual hospitals(obviously, there are no allies once hospitals are aggregated into systems).

The most striking result in Table 11 is that rival system positioning on the treatment frontierappears to stimulate a system’s positioning on the diagnostic frontier. Table 10 shows that theeffect occurs because rival positioning on the treatment frontier shifts the system’s standalonepayoff to positioning on the diagnostic frontier. One might attribute this effect to a propensityof systems to differentiate into “diagnostic specialists” and “treatment specialists.” However,other results call this interpretation into question. In particular, if systems were differentiatingin this way, we would expect the cross-effects to be similar; however, rival positioning on the

17Because we have included the rival count in logarithmic form, we have to be careful with how we interpretthe marginal effect. For example, let x be the estimated marginal effect of a change in ln(1 + v) for some variablev. This then means that increasing v increases the relevant choice probability by x/(1 + v). In particular, theeffect of an additional rival on a frontier attenuates as the number of rivals on that frontier increases.

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diagnostic frontier decreases the probability that a system positions on the treatment frontier.Another variation on the main regression above is to look only at the decisions of non-teaching

hospitals. The main regression views a teaching hospital and non-teaching hospital as rivals inthe same way as two teaching hospitals or two non-teaching hospitals. However, it may bethe case that the teaching and non-teaching segments are sufficiently differentiated that a non-teaching hospital reacts differently to non-teaching rivals (and allies) than it reacts to teachingrivals (and allies). We consider here a specification where teaching hospitals are eliminated fromthe sample. We construct additional counts of rival and allied teaching hospital positioningthat are included as exogenous variables in the decisions of the non-teaching observations. Thisversion reflects an assumption of Stackelberg leadership among teaching hospitals, though wefocus only on the decision of the non-teaching followers. Tables 12 and 13 hold the results forthis specification.

The results for non-teaching hospitals confirm and accentuate patterns found in the mainregression. Furthermore, they indicate that non-teaching hospitals react to other non-teachinghospitals more than they react to teaching hospitals, confirming our intuition that these segmentsare rather differentiated from each other. The strategic substitutability in treatment frontierpositioning among allies is confirmed by the marginal effects in Table 13. Furthermore, alliedfrontier positioning in diagnostics again stimulates frontier positioning in treatment. Amongnon-teaching hospitals, allied adoption of frontier treatment technology appears to diminish thelikelihood of frontier diagnostics, which is the not case for the full sample. This effect may reflecta greater tendency among non-teaching hospitals to centralize “technological excellence” with ahospital system in a market.

5.2.1 Effects of Allowing for Complementarity

Part of the motivation of this paper is to illustrate the importance of allowing for complemen-tarity in multivariate discrete choice models that are likely to exhibit it. To this end, Tables 7and 8 provide results of regressions where complementarity is handled more restrictively. Table7 contains estimates from a standard bivariate probit on the two frontier decisions. We cansee from this table that when the complementarity term is restricted to zero, much of its effectis pushed into the correlation estimate. The difference goes beyond a biased estimate of thecorrelation parameter, however. Table 9 presents a comparison of the marginal effects from thestandard bivariate probit with those from the main regression. For those variables excludedfrom the complementarity equation or those that do not substantively affect complementarity,the marginal effects are fairly close. However, the estimated effects of those variables that havean effect on complementarity exhibit a noticeable bias. The bias arises because there is no wayfor the standard bivariate probit to account for the indirect effects of variation in the char-acteristics. In the standard bivariate probit, the two decisions are linked only by correlationin unobserved preferences. If an additional rival positions itself on the diagnostic frontier, the

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bivariate probit can only reflect the direct effect on an observation’s propensity to position it-self on the diagnostic frontier. It can not see that this same change affects the propensity toposition on the treatment frontier and that this feeds back through the complementarity termto the diagnostics decision.

6 Conclusion

While much remains to be done, we have shown several patterns regarding frontier positioningin diagnostic radiology and radiation therapy. First, there exists complementarity betweenthe positioning decisions, despite the apparent absence of economies of scope. In particular,the tendency toward joint adoption exhibited by the data is not merely an artifact of regionalvariation nor is it induced by segmentation across hospital types. Second, hospitals appear reactin a limited fashion to the positioning decisions of other hospitals and appear to react more toallied hospitals than to rivals. Third, rival frontier positioning appears to threaten profits infrontier diagnostics more than it does in frontier treatment. In contrast, payoffs to frontiertreatment appear more affected by being superseded in technological excellence by hospitals onboth frontiers. Finally, omission of complementarity effects where they are relevant can biasempirical results.

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A Tables

Table 2: Descriptive Statistics

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Table 3: Probit Estimates on Diagnostic Frontier Status

Table 4: Probit Estimates on Treatment Frontier Status

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Table 5: Hospital-level Bivariate Probit with Complementarity Equation

Table 6: Marginal Effects from Bivariate Probit with Complementarity Equation

27

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Table 7: Hospital-level Bivariate Probit

Table 8: Hospital-level Bivariate Probit with Complementarity Parameter

28

Page 29: Competition and Complementary Activities: Lessons from ... · the market for computed tomography (CT) scanners and estimates welfare gains that stem from innovation. Baker (2001)

Table 9: Marginal Effects Comparison between Bivariate Probits with and without Complemen-tarity

29

Page 30: Competition and Complementary Activities: Lessons from ... · the market for computed tomography (CT) scanners and estimates welfare gains that stem from innovation. Baker (2001)

Table 10: System-level Bivariate Probit with Complementarity Equation

Table 11: Marginal Effects from System-level Bivariate Probit with Complementarity Equation

30

Page 31: Competition and Complementary Activities: Lessons from ... · the market for computed tomography (CT) scanners and estimates welfare gains that stem from innovation. Baker (2001)

Table 12: Hospital-level Bivariate Probit with Complementarity Equation among Non-TeachingHospitals

Table 13: Marginal Effects from Bivariate Probit with Complementarity Equation among Non-Teaching Hospitals

31


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