Pseudo R² Probablity Measures, Durbin Watson Diagnostic Statistics and Einstein Summations for Deriving Unbiased Frequentistic Inferences and Geoparameterizing Non-Zero First-Order Lag Autocorvariate Error in Regressed Multi-Drug Resistant Tuberculosis Time Series Estimators

Benjamin G. Jacob^1, , Daniel Mendoza², Mario Ponce³, Semiha Caliskan¹, Ali Moradi⁴, Eduardo Gotuzzo³, Daniel A. Griffith⁵ and Robert J. Novak¹

¹Department of Global Health, College of Public Health, University of South Florida, Tampa Fl

²Department of Environmental and Occupational Health, College of Public Health, University of South Florida, Tampa Fl

³Instituto de Medicina Tropical Alexander Von Humboldt-Universidad Peruana Cayetano Heredia, Lima, Peru

⁴LECOM School of Medicine 5000 Lakewood Ranch Blvd Bradenton, FL

⁵School of Economic, Political and Policy Sciences, The University of Texas at Dallas, 800 West Campbell Road, Richardson, TX

Cite this paper

Benjamin G. Jacob, Daniel Mendoza, Mario Ponce, Semiha Caliskan, Ali Moradi, Eduardo Gotuzzo, Daniel A. Griffith and Robert J. Novak. Pseudo R² Probablity Measures, Durbin Watson Diagnostic Statistics and Einstein Summations for Deriving Unbiased Frequentistic Inferences and Geoparameterizing Non-Zero First-Order Lag Autocorvariate Error in Regressed Multi-Drug Resistant Tuberculosis Time Series Estimators. American Journal of Applied Mathematics and Statistics. 2014; 2(5):252-301. doi: 10.12691/AJAMS-2-5-1

Abstract

In randomized clinical trials using clustered multi-drug resistant tuberculosis (MDR-TB) data, groups of human population are routinely assigned to treatments; whereas, observations are taken on the individual subjects using clinically-oriented explanatory covariate coefficient estimates for identifying sites of hyperendemic transmission. Further, standard methods for data analyses of clinical MDR-TB data postulate models relating observational parameters to the response variables without accurately quantitating varying observational intra-cluster error coefficient effects. Implicit in this assumption is that the effect of these error coefficient estimates are identical. However, non-differentiation of varying and constant residual within-cluster covariate coefficient uncertainty effects in a time-series clinical MDR-TB endemic transmission model can lead to misspecified forecasted predictors of endemic transmission zones (e.g., mesoendemic). In this research we constructed multiple georeferenced autoregressive hierarchical models accompanied by non-generalized predictive residual uncertainty non-normal diagnostic tests employing multiple covariate coefficient estimates clinically-sampled in San Juan de Lurigancho Lima, Peru. Initially, a SAS-based hierarchical agglomerative polythetic clustering algorithm was employed to determine high and low MDR-TB clusters stratified by prevalence data. Univariate statistics and Poisson regression models were then generated in R and PROC NL MIXED, respectively. Durbin-Watson statistics were derived. A Bayesian probabilistic estimation matrix was then constructed employing normal priors for each of the error coefficient estimates which revealed both spatially structured (SSRE) and spatially unstructured effects (SURE). The residuals in the high MDR-TB explanatory prevalent cluster revealed two major uncertainty estimate interactions: 1) as the number of bedrooms in a house in which infected persons resided increased and the percentage of isoniazid-sensitive infected persons increased, the standardized rate of tuberculosis tended to decrease; and, (2) as the average working time and the percentage of streptomycin-sensitive persons increased, the standardized rate of MDR-TB tended to increase. In the low MDR-TB explanatory time series cluster single marital status and building material used for house construction were important predictors. Latent explanatory non-normal error probabilities in empirically regressed MDR-TB clinical-sampled covariate estimates can be robustly spatiotemporally quantitated employing a first-order autoregressive resdiualized model and a Bayesian diagnostic uncertainty estimation matrix.

Keywords

Multi-Drug Resistant Tuberculosis (MDR-TB), Durbin-Watson statistics, Bayesian, San Juan de Lurigancho

Copyright

This work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/

References

[1]	Anselin, L. 1996. The Moran Scatterplot as an ESDA Tool to Assess Local Instability in Spatial Association. In M. Fischer, H. Scholten, and D. Unwin (eds.), Spatial Analytical Perspectives on GIS. London: Taylor and Francis.
[2]	Becerra, M.C., Bayona, J., Freeman, J., Farmer, P.E. and Kim, J.Y., 2000. Redefining MDR-TB transmission 'hot spots'. Int J Tuberc Lung Dis, 4(5): 387-94.
[3]	Besag, J., Green, P., Higdon, D. and Kengersen, K., 1995. Bayesian computation and stochastic system, Stat. Sci., pp. 3-66.
[4]	Besag, J., York, J. and Mollie, A., 1991. Bayesian image restorarion, with two applications in spatial statistics. Annals of the Institute of Statistical Mathematics, 43(1).
[5]	Besag, J., Newell J. The detection of clusters in rare diseases. Journal of the Royal Statistics Society A. 1991; 154: 143-155.
[6]	Blake, K. S., Kellerson, R. L., & Simic, A. (2007). Measuring Overcrowding in Housing: U.S. Department of Housing and Human Development.
[7]	CDC, 1999. Reported TB in the United States, 1998: TB Surveillance report, Atlanta.
[8]	Cegielski, C.G., Hall, D.J. and Rebman, C., 2006. Enterprise resource planning systems implementation success. International Journal of Information Systems and Change Management, 1(3): 301-317.
[9]	Clayton, D.G. and Kaldor, J., 1987. Empirical Bayes estimates of age-standardized relative risks for use in disease mapping. Biometrics, 43(3): 671-681.
[10]	Fahrmeir, L. and Lang, S., 2001. Bayesian semiparametric regression analysis of multicategorical time-space data. Annals of the Institute of Statistical Mathematics, 53(1): 11-30.
[11]	Ferreira, J.T.A.S., Denison, D.G.T. and Holmes, C.C., 2002. Partition Modelling.
[12]	Frieden, T.R. et al., 1993. Emergence of vancomycin-resistant enterococci in New York City. Lancet, 342(8863): 76-9.
[13]	Gamerman, D., 1997. Sampling from the posterior distribution in generalized linear mixed models. Statistics and Computing, 7(1): 57-68.
[14]	Gandhi, N. et al., 2006. Extensively drug-resistant tuberculosis as a cause of death in patients co-infected with tuberculosis and HIV in a rural area of South Africa. The Lancet, 368(9547): 1575-1580.
[15]	Gelman, A., Chew, G.L. and Shnaidman, M., 2004. Bayesian Analysis of Serial Dilution Assays. Biometrics, 60(2): 407-417.
[16]	Getis, A. and Griffith, D.A., 2002. Comparative spatial filtering in regression analysis. Geographical Analysis, 34: 130-140.
[17]	Ghosh, S. et al., 1999. Type 2 diabetes: evidence for linkage on chromosome 20 in 716 Finnish affected sib pairs. Proc Natl Acad Sci U S A, 96(5): 2198-203.
[18]	Godoy, P. et al., 2004. Characteristics of tuberculosis patients with positive sputum smear in Catalonia, Spain. Eur J Public Health, 14(1): 71-5.
[19]	Goodchild M (1986) Spatial autocorrelation (CATMOG 47). GeoBooks, Norwich.
[20]	Griffith, D., 2004. A spatial filtering specification for the autologistic model. Environment and Planning A
[21]	Griffith, D. and Peres-Neto, P.R., 2006. Spatial modeling in ecology: The flexibility of eigenfunction spatial analysis. Ecology, 87(10): 2603-2613.
[22]	Griffith, D.A., 2000. A linear regression solution to the spatial autocorrelation problem. J of Geogr Syst, 2(2): 141-156.
[23]	Griffith, D.A., 2003. Spatial autocorrelation on spatial filtering. Springer.
[24]	Griffith, D.A., 2005. A comparison of six analytical disease mapping techniques as applied to West Nile Virus in the coterminous United States. International Journal of Health Geographics, 4: 18.
[25]	Hastie, T.J. and Tibshirani, R.J., 1990. Generalized Additive Models. Chapman and Hall.
[26]	Hosmer, D.W. and Lemeshow, S., 2000. Applied logistic regression. Wiley.
[27]	Jacob, B.G. et al., 2007. Environmental abundance of Anopheles (Diptera: Culicidae) larval habitats on land cover change sites in Karima Village, Mwea Rice Scheme, Kenya. Am J Trop Med Hyg, 76(1): 73-80.
[28]	Kulldorff, M., Heffernan, R., Hartman, J., Assuncao, R. and Mostashari, F., 2005. A space-time permutation scan statistic for disease outbreak detection. PLoS Med, 2(3): e59.
[29]	Kulldorff, M. and Nagarwalla, N., 1995. Spatial disease clusters: Detection and inference. Statistics in Medicine, 14: 799-810.
[30]	Le, N.D. Petkau, A.J., Rosychuk, R.J. Surveillance of clustering near point sources. Statistics in Medicine 1996;15:727-740.
[31]	Le Gallo J, Ertur C (2003) Exploratory spatial data analysis of the distribution of regional per capita GDP in Europe, 1980-1995. Papers in Regional Science 82:175-201
[32]	Oeltmann, J.E. et al., 2008. Multidrug-resistant tuberculosis outbreak among US-bound Hmong refugees, Thailand, 2005. Emerg Infect Dis, 14(11): 1715-21.
[33]	Pearson, M.L. et al., 1992. Nosocomial transmission of multidrug-resistant Mycobacterium tuberculosis. Annals of Internal Medicine, 117(3): 191-196.
[34]	Rosychuk, R.J., Huston, C, Prasad, NGN. Spatial event cluster detection using a compound Poisson distribution. Biometrics. 2006; 62: 465–470
[35]	Rushton, G. and Lolonis, P., 1996. Exploratory spatial analysis of birth defects rates in an urban population. Statistics in Medicine, 15(7-9): 717-726.
[36]	Shah, H.N., Jain, P. and Chibber, P.J., 2006. Renal tuberculosis simulating xanthogranulomatous pyelonephritis with contagious hepatic involvement. Int J Urol, 13(1): 67-8.
[37]	Spiegelhalter, D.J., Best, N.G., Carlin, B.P. and van der Linde, A., 2002. Bayesian measures of model complexity and fit. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 64(4): 583-639.
[38]	Smith T (2001) Aggregation bias in maximum likelihood estimation of spatial autoregressive processes. Paper presented to the North American Regional Science Association, Charleston, 15-17, 2001 November.
[39]	Waller, L.A. and Zelterman, D., 1997. Log-linear modeling with the negative multinomial distribution. Biometrics, 53(3): 971-82.
[40]	Zhang, T. and Lin, G., 2009. Spatial scan statistics in loglinear models. Computational Statistics & Data Analysis, 53(8): 2851-2858.
[41]	Zignol, M. et al., 2006. Global incidence of multidrug-resistant tuberculosis. J Infect Dis, 194(4): 479-85.