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Interdependence among mental health care providers: evidence from a spatial dynamic panel data model with interactive fixed effects

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Abstract

Understanding the pricing and operation mechanism of community mental health (CMH) providers is important for designing effective health care policies as the CMH system has been playing an essential role in providing mental health services. This paper studies the delivery of community-based mental health care and interdependence among CMH providers by investigating the pricing patterns of CMH providers in the Detroit-Wayne County of Michigan. We employ a spatial dynamic panel data model with interactive fixed effects to identify the strategic interactions among the providers, along both dimensions of time-series and cross-section spatial variations. We find evidence for significant spatial interdependence among the CMH providers, even after controlling for the interactive fixed effects. In contrast, the coefficient on the spatial-dynamic term is insignificant, suggesting that no time-space simultaneous effects exist. We also find the pure dynamic effect to be positive and significant. These findings provide important implications for mental health care policy and practice decisions.

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Notes

  1. The time-space simultaneous effects refer to the effects of the spatial-dynamic term \(W_N Y_{N,T-1}\), which capture the time lagged spatial spillover effect from neighboring providers.

  2. Anselin et al. (2008) provides a list of spatial panel data models and presents the corresponding likelihood functions. Lee and Yu (2010a), Lee and Yu (2015) review some recent development in econometric specification and estimation of spatial panel data models for both static and dynamic cases and investigate asymptotic properties of estimators.

  3. The five most costly conditions were: heart disease, trauma-related disorders, cancer, mental disorders and asthma. See Figure 3 for the expenditure data for the five most costly conditions.

  4. Please check http://www.nasmhpd.org for more information.

  5. Currently Detroit Wayne Mental Health Authority.

  6. The two primary sources of public mental health funding are Medicaid and state general funds, accounting for 90% of the system on average. The rest 10% is funded by Medicare, federal block grant funds, which are not tracked under the Agency (NAMI 2010).

  7. Fiscal year (FY). A fiscal year is used because the funding budget is made by both the federal and state governments based on the calendar of a fiscal year which begins on October 1 of the previous calendar year and ends on September 30 of the year which is numbered.

  8. Shi and Lee (2018) employ a similar model to study the effects of gun control on crime rates among US states.

  9. In contrast, the usual time fixed effect specification assumes the effect of time trend to be constant across individual units. And compared with the random effects specification which assumes no correlations between the common factor components and the regressors, the fixed effects approach allows unknown and flexible correlations between them.

  10. Many studies have provided evidence for heterogeneous time-varying factors in a variety of settings. In particular, Shi and Lee (2017) suggest that states with and without right-to-carry laws may have different political preferences and other time-varying social economic characteristics, and thus may follow different time trends. Kim and Oka (2014) point out that unobserved heterogeneity such as demographic structure, stigma of divorce, religious beliefs and other cultural factors may develop over time and vary to a large extent across states. As a result, these factors may generate heterogeneous impacts on both divorce rates and divorce law reforms across states. Bai (2009) argues that unobservable characteristics including innate ability, perseverance, and motivation may vary across age cohorts and affect individuals heterogeneously.

  11. In the literature, several approaches have been proposed, including the information criteria in Bai and Ng (2002), the eigenvalue differences criterion in Onatski (2010), as well as the methods proposed in Alessi et al. (2010), to name a few. However, as demonstrated in Ahn and Horenstein (2013), the eigenvalue ratio and growth ratio criteria have very nice property, generally outperforming competing estimators. In particular, the estimators proposed in Bai and Ng (2002) are sensitive to the choice of \(r_{\mathrm{max}}\).

  12. In our sample, N=182, T=4, so we set \(r_{\mathrm{max}}=3\).

  13. In the estimation, we assign a weight of 0 for any sites that are further than 150 miles away. But the results are robust against a variety of specifications as demonstrated in Tables 4 and 5.

  14. The service codes follow the Healthcare Common Procedure Coding System code set issued annually by the Centers for Medicare & Medicaid Services.

  15. ICD-9 codes.

  16. Medicaid and state general fund dollars are two primary sources of funding for public mental health services. In the state of Michigan, individuals who are eligible for Supplemental Security Income are automatically eligible for Medicaid. State mental health budgets are primarily funded by state general fund dollars to provide state hospital and inpatient care, crisis services and community mental health services for individuals with mental disorders.

  17. This revenue-based measure is comparable to that used in several studies of hospital pricing, such as Dranove and Ludwick (1999), Keeler et al. (1999), Lynk (1995), and Mobley (2003).

  18. CMHSP Sub-element Cost Reports for Section 404, available at http://www.michigan.gov/mdch.

  19. Available at http://www.hcup-us.ahrq.gov/toolssoftware/ccs/ccs.jsp.

  20. ICD-9 codes 293.83, 296.2x, 296.3x, 300.40, 311.00.

  21. ICD-9 codes 293.81, 293.82, 295.xx, 297.00, 297.10, 297.20, 297.30, 297.80, 297.90, 298.00, 298.10, 298.20, 298.30, 298.40, 298.80, 298.90.

  22. As pointed out in Brueckner (2003), such spatial interdependence among individual agents may arise from either spillover effect or the resource flow across agents.

  23. We follow LeSage and Pace (2009) to calculate the direct effect for spatial models. Specifically, for provider i, the direct effect of changing the \(k^{\mathrm{th}}\) explanatory variable \(x_{\mathrm{ik}}\) on \(y_i\) includes both own effect and the feedback effects from paths where the unit i affects unit j which in turn also affects unit i, i.e., \(i \rightarrow j \rightarrow i\), as well longer loops such as \(i \rightarrow h \rightarrow j \rightarrow i\), and so on, which can be calculated as: \(\frac{\partial y_i}{\partial x_{\mathrm{ik}}}=S_N (\lambda )_{\mathrm{ii}} \beta _k\), where \(S_N (\lambda )_{\mathrm{ii}}\) denotes the i th diagonal element of the matrix \(S_N (\lambda )=(I_N-\lambda W_N)^{-1}\). Therefore, the average direct effect is given by \(\frac{1}{N} tr(S_N (\lambda )) \beta _k\). It turns out that the corresponding marginal effect of the Number of patients served based on this calculation is roughly the same as the estimated coefficient of 0.0007.

  24. To save space, we only report the results for the SDPD with interactive fixed effects.

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Authors and Affiliations

  1. Department of Economics, Virginia Tech, 3016 Pamplin Hall, 880 West Campus Drive, Blacksburg, VA, 24061, USA

    Xu Lin

  2. Optum, Eden Prairie, MN, 55344, USA

    Lizi Wu

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  1. Xu Lin
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Correspondence to Xu Lin.

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Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

We thank the editor, Janet E. Kohlhase, and one anonymous referee for very helpful comments. We also thank the Detroit Wayne Mental Health Authority for giving us consent to use the data and adequate resources to conduct the analysis. We are grateful to Allen Goodman for valuable comments and to Shi Wei and Jihai Yu for help on Matlab codes. All remaining errors are our own.

X. Lin: Most of the work was done during the author’s visit to Zhongnan University of Economics and Law. L. Wu: At the time this study was mostly completed, Lizi Wu was a PhD candidate at Wayne State University, and she is currently Data Analyst at Optum.

Appendix

Appendix

See Tables 7, 8, 9 and Figs. 3, 4.

Table 7 Percent distribution of mental health and all-health expenditures by payer: 2009, 2014 and 2020.
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Table 8 Service codes submitted by D-WCCMHA providers, FY2008–FY2012
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Table 9 Diagnosis categories present in the claims data
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Fig. 3
figure 3

Source: Center for Financing, Access, and Cost Trends, AHRQ, Household Component of the Medical Expenditure Panel Survey, 2002 and 2012, https://meps.ahrq.gov/data_files/publications/st470/stat470.pdf#search=%20Five%20Most%20Costly%20Conditions

Expenditures for the five most costly conditions, 2002 and 2012.

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Fig. 4
figure 4

Distribution of mental health expenditures by type of service, 1986 & 2014. Exclude spending on insurance administration. Substance Abuse and Mental Health Services Administration https://store.samhsa.gov/system/files/sma14-4883.pdf

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Lin, X., Wu, L. Interdependence among mental health care providers: evidence from a spatial dynamic panel data model with interactive fixed effects. Ann Reg Sci 67, 131–165 (2021). https://doi.org/10.1007/s00168-020-01043-w

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