Reducing Suicide A National Imperative (2002) / Chapter Skim
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Appendix A: Statistical Details
Appendix A: Statistical Details
Pages 439-468
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Appendixes
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Suppose a sample from a super population g is recorded as data (Yi, J i)
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considered nonparametric likelihood estimation of a mixing distribution and Lindsay (1983) linked its general theory, developed from a study of the geometry of the mixture likelihood, to the problem of estimating a discrete mixing distribution.
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The complete data log likelihood function becomes log LCom (a; y, Z)
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There are other techniques for identifying the number of components of a mixing distribution including a variety of graphical techniques (Lindsay and Roeder, 1992)
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The gradient function therefore creates a natural stopping rule for iterative algorithms such as the EM when the final log likelihood is unknown, although it is more stringent than the ideal stopping rule. Illustration of Poisson Mixture Model To illustrate use of the Poisson mixture model we analyzed countylevel suicide data for the US for the time period of 1996-1998.
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Unlike the earlier attempt at fitting a mixture distribution to temporal suicide data collected in Cook county (Gibbons et al., 1990) that was designed to identify "suicide epidemics" and failed to do so, the results of this analysis has focused on the spatial distribution of suicide and has identified qualitatively distinct geographic groupings of suicide rates across the United States.
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Then the Poisson regression model Pi is At )
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Then the conditional density function,f~yi; Bi) , of the ni suicide rates in cluster i is written as: ni f(Yi; ~i)
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The scoring solution is often used in cases where the matrix of second derivatives is difficult to obtain. Note that when the expected value and the actual value of the Hessian matrix coincide, the Fisher scoring method and the Newton-Raphson method reduce to the same algorithm.
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, is given by Var(Bi I Yi ~ = he ~ io(Bi - Bi ~ f(Yi I pi ~g(~) d~ Generalized Estimating Equations - GEE An alternative approach to the analysis of clustered suicide rate data is based on Generalized estimating equations (GEEs)
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A The maximum likelihood estimator ,B can be obtained by solving the above estimating equations iteratively: -1 Fr+1 = Fr—| ~ ;3 i Vi 1 JUi ~ l I nc a ' (lo) Illustration of Poisson Regression Model Returning to the national suicide data from the previous section, we now illustrate how Poisson regression models can be used to estimate the effects of age, race, and sex on clustered (i.e., within counties)
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for the Clustered Poisson Regression Model MMLE GEE Effect Estimate SE Prob Estimate SE Prob Intercept -4.331 0.040 <0.0001 -4.408000 0.048000 <0.0001 Female -1.009 0.076 <0.0001 -1.009000 0.073000 <0.0001 Black vs Other -0.292 0.103 0.0046 -0.279000 0.112000 0.0127 15-24 vs 05-14 2.788 0.041 <0.0001 2.787000 0.043000 <0.0001 25-44 vs 05-14 3.030 0.040 <0.0001 3.021000 0.043000 <0.0001 45-64 vs 05-14 2.971 0.041 <0.0001 2.975000 0.046000 <0.0001 65+ vs 05-14 3.371 0.041 <0.0001 3.390000 0.047000 <0.0001 Female x Black -0.345 0.036 <0.0001 -0.345000 0.041000 <0.0001 Female x 15-24 -0.664 0.080 <0.0001 -0.663000 0.077000 <0.0001 Female x 25-44 -0.355 0.077 <0.0001 -0.357000 0.074000 <0.0001 Female x 45-64 -0.223 0.077 0.0038 -0.223000 0.075000 0.0029 Female x 65+ -0.938 0.078 <0.0001 -0.945000 0.078000 <0.0001 Black x 15-24 0.065 0.106 0.5434 0.068000 0.107000 0.5297 Black x 25-44 -0.139 0.105 0.1829 -0.141000 0.107000 0.1877 Black x 45-64 -0.473 0.107 <0.0001 -0.486000 0.111000 <0.0001 Black x 65+ -0.740 0.112 <0.0001 -0.748000 0.117000 <0.0001 County Variance 0.280 0.003 <0.0001 displays observed and expected annual suicide rates for both methods of estimation, broken down by age, sex, and race. Inspection of Table A-2 reveals several interesting results.
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Second, we can use the Bayes estimates directly to obtain county-level suicide rates adjusted for the effects of race, sex, and age. In the case of a Poisson model, the Bayes estimate for a given county is a multiple of the national suicide rate adjusted for the case mix in that county (i.e., race, sex and age)
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454 TABLE A-3 Observed and Expected Number of Suicides for 100 Randomly Selected Counties APPENDIX A Observed Expected Observed Expected State County # of Deaths # of Deaths State County # of Deaths # of Deaths 56 7 16 10.3 51 75 5 5.9 53 63 169 170.4 20 159 6 4.5 5 91 14 13.6 21 103 14 8.7 47 111 13 9.4 40 45 1 1.6 54 93 1 2.8 31 61 1 1.5 28 5 9 5.5 31 91 0 0.3 27 141 22 21.7 40 73 9 6.3 38 47 0 1.0 20 203 1 1.0 21 59 25 27.5 1 5 8 8.2 48 451 54 50.2 38 51 0 1.4 48 87 3 1.4 38 39 2 1.3 18 97 391 385.3 21 95 11 12.0 35 7 8 6.2 48 73 32 25.8 48 383 2 1.5 23 11 38 39.6 31 41 3 4.3 55 101 58 59.5 18 7 2 3.4 17 107 9 10.8 45 61 6 5.9 47 127 1 1.9 19 1 4 3.4 55 55 28 28.1 8 119 13 9.9 21 223 5 3.4 36 3 22 20.9 30 97 1 1.4 19 93 2 2.9 19 51 2 3.0 12 95 303 314.1 47 129 4 6.4 49 49 89 91.1 48 9 2 3.0 28 45 30 24.1 19 5 9 6.7 18 107 19 16.9 28 69 3 3.0 5 1 6 6.8 46 19 6 4.1 17 167 64 64.8 27 171 31 30.8 39 89 43 44.7 28 53 1 2.4 28 1 9 9.5 20 31 2 3.1 53 23 0 0.9 51 47 18 15.0 48 213 30 28.8 46 123 3 2.7 55 107 9 7.1 12 81 135 129.6 31 177 6 6.7 2 122 22 20.4 17 49 12 12.2 21 61 0 3.4 55 75 18 17.5 21 25 7 6.3 8 55 3 2.7 26 13 6 4.0 29 121 4 5.4 18 157 42 44.1 40 125 34 30.2 21 165 2 2.2 20 73 3 3.2 48 447 2 0.8 21 109 7 5.5 13 265 3 0.6 29 197 2 1.8 2 100 2 1.0 51 133 7 5.0 55 11 4 5.2 29 105 8 10.0 48 81 0 1.3 47 181 14 9.7 21 229 6 4.6 51 103 9 5.5 55 35 32 32.3 16 15 3 2.2 39 151 112 113.5 38 55 2 3.5 28 89 17 18.4 39 17 91 93.6 19 167 4 8.2 55 51 3 2.8 23 27 17 15.7 19 119 3 4.2 38 1 0 1.1
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where there is a high concentration of counties with the lowest suicide rates, there are a few counties that exhibit the highest suicide rates. As an example, Table A-4 displays the Bayes estimates, observed and expected numbers of suicides and suicide rates for all counties in Alaska and New Mexico and the 8 counties with estimated adjusted suicide rates less than or equal to half of the national average (national average is 12.33 suicides per 100,000 during the period of 1996-1998~.
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456 TABLE A-4 Observed and Expected rates per 100,000 for Alaska, New Mexico, and Counties with BE <= 0.50 APPENDIX A Observed Expected Observed State County Name # of Suicides # of Suicides Rate/100, ALASKA ALEUTIANS EAST 1 0.84 15.66 ALASKA ALEUTIANS WEST 2 1.92 16.27 ALASKA ANCHORAGE 118 115.66 16.90 ALASKA BETHEL 22 10.17 52.75 ALASKA BRISTOL BAY 1 0.58 26.75 ALASKA DILLINGHAM 4 1.46 34.00 ALASKA FAIRBANKS NORTH STAR 52 46.63 22.51 ALASKA HAINES 2 0.96 32.20 ALASKA JUNEAU 13 11.95 15.56 ALASKA KENAI PENINSULA 22 20.38 16.65 ALASKA KETCHIKAN GATEWAY 4 4.91 10.28 ALASKA KODIAK ISLAND 5 5.17 12.42 ALASKA LAKE AND PENINSULA 4 0.63 87.45 ALASKA MATANUSKA-SUSITNA 29 25.80 19.23 ALASKA NOME 14 5.02 58.89 ALASKA NORTH SLOPE 10 3.35 53.11 ALASKA NORTHWEST ARCTIC 15 3.87 87.41 ALASKA PRINCE OF WALES-OUTER 3 2.64 15.29 ALASKA SITKA 6 3.71 25.55 ALASKA SKAGWAY-HOONAH-ANGOON 1 1.34 9.36 ALASKA SOUTHEAST FAIRBANKS 6 2.69 38.23 ALASKA VALDEZ-CORDOVA 1 3.31 3.47 ALASKA WADE HAMPTON 18 4.37 105.40 ALASKA WRANGELL-PETERSBURG 8 3.73 41.81 ALASKA YAKUTAT 0 0.29 0.00 ALASKA YUKON-KOYUKUK 16 5.91 73.37 ILLINOIS MC LEAN 17 25.62 4.31 NEW JERSEY HUNTERDON 13 22.23 3.86 NEW JERSEY MORRIS 80 88.20 6.27 NEW JERSEY UNION 80 88.66 5.73 NEW MEXICO BERNALILLO 303 295.93 20.70 NEW MEXICO CATRON 1 1.18 12.76 NEW MEXICO CHAVES 24 23.46 13.88 NEW MEXICO CIBOLA 12 9.84 16.81 NEW MEXICO COLFAX 8 6.16 20.83 NEW MEXICO CURRY 23 20.28 17.94 NEW MEXICO DE BACA 2 1.05 29.93 NEW MEXICO DONA ANA 63 62.45 13.81 NEW MEXICO EDDY 30 26.30 20.25 NEW MEXICO GRANT 23 17.83 26.42 NEW MEXICO GUADALUPE 2 1.64 17.59 NEW MEXICO HARDING 1 0.40 38.83 NEW MEXICO HIDALGO 1 2.14 5.77 NEW MEXICO LEA 15 16.90 9.68
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STATISTICAL DETAILS elected Observed Expected Bayes Suicides Rate/100,000 Rate/100,000 Population Estimate 34 15.66 13.08 2,128 1.01 92 16.27 15.61 4,098 1.00 66 16.90 16.56 232,752 1.24 17 52.75 24.37 13,902 2.58 58 26.75 15.50 1,246 1.03 46 34.00 12.44 3,921 1.22 63 22.51 20.19 76,998 1.50 96 32.20 15.52 2,070 1.08 95 15.56 14.30 27,850 1.07 38 16.65 15.43 44,035 1.11 91 10.28 12.62 12,967 0.92 17 12.42 12.85 13,415 0.97 63 87.45 13.69 1,524 1.31 30 19.23 17.10 50,276 1.26 02 58.89 21.11 7,924 2.05 35 53.11 17.80 6,276 1.70 37 87.41 22.54 5,720 2.45 64 15.29 13.45 6,541 1.02 71 25.55 15.80 7,827 1.19 34 9.36 12.58 3,561 0.97 69 38.23 17.13 5,232 1.30 31 3.47 11.51 9,597 0.82 37 105.40 25.59 5,692 3.00 73 41.81 19.47 6,377 1.40 29 0.00 12.45 764 0.97 91 73.37 27.08 7,269 2.24 62 4.31 6.49 131,572 0.48 23 3.86 6.59 112,392 0.46 20 6.27 6.91 425,476 0.49 66 5.73 6.35 465,455 0.49 93 20.70 20.22 487,809 1.48 18 12.76 15.02 2,613 0.98 46 13.88 13.57 57,623 1.02 34 16.81 13.78 23,799 1.17 16 20.83 16.03 12,799 1.15 28 17.94 15.82 42,733 1.22 05 29.93 15.64 2,227 1.08 45 13.81 13.69 152,009 1.02 30 20.25 17.76 49,371 1.31 33 26.42 20.48 29,022 1.49 64 17.59 14.42 3,790 1.02 40 38.83 15.70 858 1.05 14 5.77 12.35 5,775 0.90 90 9.68 10.91 51,633 0.84 continues 457
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The traditional ordinal regression model is parameterized such that regression coefficients or do not depend on k, i.e., the model asumes that the relationship between the explanatory variables and the cumulative logits does not depend on k. McCullagh (1980)
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have described an extension of the random effects proportional odds model to allow for non-proportional odds for a set of explanatory variables. For example, it sems reasonable to assume that the effects of the model covariates on suicidal ideation and suicidal attempts may not be the same.
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Also, ~ is the p x 1 vector of unknown fixed regression parameters, and ,Bi is the r x 1 vector of unknown random effects for the level-2 unit i. Since the regression coefficients, or do not depend on k, the model assumes that the relationship between the explanatory variables and the cumulative logits does not depend on k.
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cumulative logits. This extension of the model follows similar extensions of the ordinary fixed-effects ordinal logistic regression model discussed by Peterson and Harrell (1990)
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is described in detail by Hedeker and Gibbons (1994~. Illustration To illustrate application of the mixed-effects ordinal logistic regression model, we reanalyzed longitudinal data from Rudd et al.
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Anxiety 0.00970 0.00462 2.09768 0.03593 (2) Random effect variance term (standard deviation)
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Finally, a nonproportional odds model, in which the treatment by time interaction was allowed to be category-specific, also failed to identify any positive treatment related effects on suicide ideation or attempts. INTERVAL ESTIMATION Poisson Confidence Limits A (1- oc)
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The prediction limit reveals that we can have 95% confidence that no more than four suicides will occur in the next month. Poisson Tolerance Limits The uniformly most accurate upper tolerance limit for the Poisson distribution is given by Zacks (1970~.
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+ 2] = 2.3421 The 99% upper tolerance limit is obtained by finding the smallest nonnegative integer j such that: %20~ [2 j + 2]
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1994. A random effects ordinal regression model for multilevel analysis.
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1990. Partial proportional odds models for ordinal response variables.
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