Advances and Applications in Statistics, Feb 1, 2010
Many inferential procedures for generalized linear models rely on the asymptotic normality of the... more Many inferential procedures for generalized linear models rely on the asymptotic normality of the maximum likelihood estimator (MLE). Fahrmeir & Kaufmann (1985, Ann. Stat., 13, 1) present mild conditions under which the MLEs in GLiMs are asymptotically normal. Unfortunately, limited study has appeared for the special case of binomial response models beyond the familiar logit and probit links, and for more general links such as the complementary log-log link, and the less well-known complementary log link. We verify the asymptotic normality conditions of the MLEs for these models under the assumption of a fixed number of experimental groups and present a simple set of conditions for any twice differentiable monotone link function. We also study the quality of the approximation for constructing asymptotic Wald confidence regions. Our results show that for small sample sizes with certain link functions the approximation can be problematic, especially for cases where the parameters are close to the boundary of the parameter space.
Many inferential procedures for generalized linear models rely on the asymptotic normality of the... more Many inferential procedures for generalized linear models rely on the asymptotic normality of the maximum likelihood estimator (MLE). Fahrmeir & Kaufmann (1985, Ann. Stat., 13, 1) present mild conditions under which the MLEs in GLiMs are asymptotically normal. Unfortunately, limited study has appeared for the special case of binomial response models beyond the familiar logit and probit links, and for more general links such as the complementary log-log link, and the less well-known complementary log link. We verify the asymptotic normality conditions of the MLEs for these models under the assumption of a fixed number of experimental groups and present a simple set of conditions for any twice differentiable monotone link function. We also study the quality of the approximation for constructing asymptotic Wald confidence regions. Our results show that for small sample sizes with certain link functions the approximation can be problematic, especially for cases where the parameters are ...
In 2003, the Social Vulnerability Index (SOVI), which utilizes Principal Component Analysis (PCA)... more In 2003, the Social Vulnerability Index (SOVI), which utilizes Principal Component Analysis (PCA), was created by Cutter, Boruff, and Shirley to examine spatial patterns of social vulnerability at the county level in the United States. This paper seeks to identify the sensitivity of this approach to changes in its construction, the scale at which it is applied, and to various geographic contexts. To determine the impact of scalar changes, the SOVI was calculated for multiple aggregation levels in the state of South Carolina. To examine the sensitivity of the algorithm to changes in construction, and determine if that sensitivity was constant in various geographic contexts, census data was collected at a sub-metropolitan level for portions of three U.S. cities. Fifty-four unique variations of the SOVI were calculated for each area. Each set of indexes was then evaluated using factorial analysis to see if substantial changes in assigned values occurred. These results were then compared across study areas to evaluate the impact of changing geographic context. While decreases in the scale of aggregation were found to result in decreases in the variability explained by the PCA, and increases in the variance of the resulting index values, the subjective interpretations yielded from the SOVI remained fairly stable. The algorithm was found to be sensitive to certain changes in index construction, which differed somewhat between study areas. Understanding the impacts of changes in index construction and scale are crucial in increasing confidence in attempts to represent the extremely complex phenomenon of social vulnerability.
Advances and Applications in Statistics, Feb 1, 2010
Many inferential procedures for generalized linear models rely on the asymptotic normality of the... more Many inferential procedures for generalized linear models rely on the asymptotic normality of the maximum likelihood estimator (MLE). Fahrmeir & Kaufmann (1985, Ann. Stat., 13, 1) present mild conditions under which the MLEs in GLiMs are asymptotically normal. Unfortunately, limited study has appeared for the special case of binomial response models beyond the familiar logit and probit links, and for more general links such as the complementary log-log link, and the less well-known complementary log link. We verify the asymptotic normality conditions of the MLEs for these models under the assumption of a fixed number of experimental groups and present a simple set of conditions for any twice differentiable monotone link function. We also study the quality of the approximation for constructing asymptotic Wald confidence regions. Our results show that for small sample sizes with certain link functions the approximation can be problematic, especially for cases where the parameters are close to the boundary of the parameter space.
Many inferential procedures for generalized linear models rely on the asymptotic normality of the... more Many inferential procedures for generalized linear models rely on the asymptotic normality of the maximum likelihood estimator (MLE). Fahrmeir & Kaufmann (1985, Ann. Stat., 13, 1) present mild conditions under which the MLEs in GLiMs are asymptotically normal. Unfortunately, limited study has appeared for the special case of binomial response models beyond the familiar logit and probit links, and for more general links such as the complementary log-log link, and the less well-known complementary log link. We verify the asymptotic normality conditions of the MLEs for these models under the assumption of a fixed number of experimental groups and present a simple set of conditions for any twice differentiable monotone link function. We also study the quality of the approximation for constructing asymptotic Wald confidence regions. Our results show that for small sample sizes with certain link functions the approximation can be problematic, especially for cases where the parameters are ...
In 2003, the Social Vulnerability Index (SOVI), which utilizes Principal Component Analysis (PCA)... more In 2003, the Social Vulnerability Index (SOVI), which utilizes Principal Component Analysis (PCA), was created by Cutter, Boruff, and Shirley to examine spatial patterns of social vulnerability at the county level in the United States. This paper seeks to identify the sensitivity of this approach to changes in its construction, the scale at which it is applied, and to various geographic contexts. To determine the impact of scalar changes, the SOVI was calculated for multiple aggregation levels in the state of South Carolina. To examine the sensitivity of the algorithm to changes in construction, and determine if that sensitivity was constant in various geographic contexts, census data was collected at a sub-metropolitan level for portions of three U.S. cities. Fifty-four unique variations of the SOVI were calculated for each area. Each set of indexes was then evaluated using factorial analysis to see if substantial changes in assigned values occurred. These results were then compared across study areas to evaluate the impact of changing geographic context. While decreases in the scale of aggregation were found to result in decreases in the variability explained by the PCA, and increases in the variance of the resulting index values, the subjective interpretations yielded from the SOVI remained fairly stable. The algorithm was found to be sensitive to certain changes in index construction, which differed somewhat between study areas. Understanding the impacts of changes in index construction and scale are crucial in increasing confidence in attempts to represent the extremely complex phenomenon of social vulnerability.
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