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## Reducing Suicide: A National Imperative (2002) Institute of Medicine (IOM)

### Citation Manager

. "Appendix A: Statistical Details." Reducing Suicide: A National Imperative. Washington, DC: The National Academies Press, 2002.

 Page 441

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Reducing Suicide: A National Imperative

### A Statistical Details

#### MIXTURE MODELS

One can naturally model the unobserved population heterogeneity or extra-population heterogeneity via mixture models. The simplest and most natural occurrence of the mixture model arises when one samples from a super population g which is a mixture of a finite number, say m, of populations (g1,…, gm), called the components of the population. Suppose a sample from a super population g is recorded as data (Yi, J i) for i = 1, …, n, where Yi = yi is a measurement on the ith sampled unit and Ji = ji indicates the index number of the component to which the unit belongs. If sampling is done from the jth component, an appropriate probability model for the sampling distribution is given by

P(Yi = yi|Ji = j) = fj(yi; θj)

The function fj(yi; θj) represents a density function, called the component density for the ith observation yi and the parameter θj Θj dj called the component parameter, that describes the specific attributes of the jth component population, gj.

We treat the Ji as missing data and define the latent random variable Φ= (Ф1,…, Фn) to be the values of the parameter θ 1, …,θm} corresponding to the sampled components J1, …, Jn; i.e., if the ith observation came from the jth component, then define Фi = θj. Then the Фi are a random sample from the discrete probability measure Q that assigns a positive probability πj to the support point θj for j = 1, …, m. That is, the latent

 Page 441
 Front Matter (R1-R16) Executive Summary (1-16) 1 Introduction (17-32) 2 Magnitude of the Problem (33-68) 3 Psychiatric and Psychological Factors (69-118) 4 Biological Factors (119-156) 5 Childhood Trauma (157-192) 6 Society and Culture (193-228) 7 Medical and Psychotherapeutic Interventions (229-272) 8 Programs for Suicide Prevention (273-330) 9 Barriers to Effective Treatment and Intervention (331-374) 10 Barriers to Research and Promising Approaches (375-422) 11 Findings and Recommendations (423-438) Appendix A: Statistical Details (439-468) Appendix B: Consultants (469-471) Appendix C: Workshop Agendas (472-474) Index (475-496)