Identifying extremely large values using composite sample data

Quantifying a composite sample results in a loss of information on the values of the constituent individual samples. As a consequence of this information loss, it is impossible to identify individual samples having large values, based on composite sample measurements alone. However, under certain circumstances, it is possible to identify individual samples having large values without exhaustively measuring all individual samples. In addition to composite sample measurements, a few additional measurements on carefully selected individual samples are sufficient to identify the individual samples having large values. In this paper, we present a statistical method to recover extremely large individual sample values using composite sample measurements. An application to site characterization is used to illustrate the method.The paper has been prepared with partial support from the United States Environmental Protection Agency Number CR815273. The contents have not been subject to Agency review and therefore do not necessarily reflect the views or policies of the Agency and no official endorsement should be inferred.     

A Simple Model to Determine In Situ Mixing Height Growth from Surface Measurements

A generic In Situ Mixing Height Growth (IMG) model, capable of predicting the real-time growth of the mixed layer and its diurnal evolution from routinely observed simple surface meteorological is developed. The algorithm for the determination of temporally growing daytime mixing height includes both convective and mechanical turbulence contributions. It accounts for neutral as well as height varying potential temperature gradients above the mixed layer. For thermally stable and mechanically dominated unstable night time Atmospheric Boundary Layer (ABL) the module uses similarity formulae based on the wind velocity [1]), the Monin—Obukhov length [2], and the Coriolis parameter. In the convective case simple slab model is integrated, based on initial lapse rate and the surface heat flux. The lapse rate is evaluated on the basis of vertical atmospheric stability, surface type and surface temperature. This differentiates the IMG model from other existing mixing height models that need initial measured lapse rate for calculation. IMG model is site specific as it calculates the radiative incoming heat flux depending on the solar declination estimates based on-site latitude and longitude. The IMG model is applied to calculate mixing height for India by using surface data (viz. wind speed, surface temperature, surface type) from 152 surface meteorological stations. Results have been evaluated with radiosonde mixing height data procured from 18 upper air stations. Sensitivity analysis of the model with respect to various parameters is performed. The model is formulated after reviewing presently available radiosonde mixing height data in India and can satisfactorily provide an alternative means of estimating mixing height for air pollution dispersion models.     

Qualitative and quantitative assessment of the risk from the exposure to fetotoxic chemical compounds

A new mathematical dose-response model for the expected probability of toxic response and also for the expected measure of the overdispersion parameter for the reproductive and developmental risk assessment is proposed. The model for the expected probability of toxic response is an improvised Weibull dose-response model incorporating the litter-size effect while the model for the overdispersion parameter is a polynomial function of the dose level. A beta-binomial distribution for the number of offspring showing toxic responses in a litter satisfactorily accounts for the extra-binomial variation and the intralitter correlation of responses of these pups. Confidence limits for low-dose extrapolation are based on the asymptotic distribution of the likelihood ratio. The safe dose for human exposure is then calculated by simple linear extrapolation. The model for overdispersion allows us to obtain the estimates of the overdispersion parameter at these dosages. This was not possible in the earlier models. The proposed model is illustrated by an application to a study on the effect of exposure to diethylhexylphthalate in mice. The results are compared with those obtained by Chen and Kodell (1989) who have applied the simple Weibull dose-response model to the same data set.This paper was prepared with partial support from the United States Environmental Protection Agency under a Cooperative Agreement Number CR-815273. The contents have not been subject to Agency review and therefore do not necessarily reflect the views or policies of the Agency and no official endorsement should be inferred.     

Continuous dose-response modeling and risk analysis with the gamma and reciprocal gamma distributions

Kodell and West (1993) describe two methods for calculating pointwise upper confidence limits on the risk function with normally distributed responses and using a certain definition of adverse quantitative effect. But Banga et al. (2000) have shown that these normal theory methods break down when applied to skew data. We accordingly develop a risk analysis model and associated likelihood-based methodology when the response follows either a gamma or reciprocal gamma distribution. The model supposes that the shape (index) parameter k of the response distribution is held fixed while the logarithm of the scale parameter is a linear model in terms of the dose level. Existence and uniqueness of the maximum likelihood estimates is established. Asymptotic likelihood-based upper and lower confidence limits on the risk are solutions of the Lagrange equations associated with a constrained optimization problem. Starting values for an iterative solution are obtained by replacing the Lagrange equations by the lowest order terms in their asymptotic expansions. Three methods are then compared for calculating confidence limits on the risk: (i) the aforementioned starting values (LRAL method), (ii) full iterative solution of the Lagrange equations (LREL method), and (iii) bounds obtained using approximate normality of the maximum likelihood estimates with standard errors derived from the information matrix (MLE method). Simulation is used to assess coverage probabilities for the resulting upper confidence limits when the log of the scale parameter is quadratic in the dose level. Results indicate that coverage for the MLE method can be off by as much as 15% points and converges very slowly to nominal coverage levels as the sample size increases. Coverage for the LRAL and LREL methods, on the other hand, is close to nominal levels unless (a) the sample size is small, say N < 25, (b) the index parameter is small, say k 1, and (c) the direction of adversity is to the left for the gamma distribution or to the right for the reciprocal gamma distribution.     

Direct calculation of likelihood-based benchmark dose levels for quantitative responses

A benchmark dose (BMD) for quantitative responses is a lower confidence limit (LCL) on the effective dose corresponding to a specified risk level r. A commonly adopted method for calculating the BMD is to obtain a pointwise upper confidence curve U(d) on the risk function and then invert this relationship by solving the equation U(d)=r. The solution d is taken to be the BMD. Sciullo et al. (2000) have shown that the coverage achieved by this inversion method is at least as great as the coverage achieved by U (·) but that there is otherwise no general relationship between the two coverage probabilities. The present paper develops a method for direct calculation of the BMD based on the asymptotic distribution of the likelihood ratio statistic. It is further shown that the direct method and the inversion method are equivalent when U (·) is also based on the likelihood ratio. Since the direct method is known to be asymptotically correct, it follows that the LR-based inversion method is also asymptotically correct. However, the direct method is computationally faster and easier to program. Finally, some simulation studies are conducted to assess the small sample coverage probabilities of the direct method when responses follow either a normal or a gamma distribution.     

Chemical mass balance model for source apportionment of aerosols in Bombay

Aerosol samples collected within an industrial region of Bombay were analyzed for elemental concentrations using inductively coupled plasma emission spectroscopy, ultraviolet/visible spectrophotometry and X-ray fluorescence spectroscopy. Nineteen elements were selected as tracers of identified sources of aerosol in the region. The U.S. EPA chemical mass balance model was employed for source apportionment. Seven major source types were identified and the performance of the model was evaluated at different sampling locations. Model results were unsatisfactory at highly polluted sites in the study regions. It was found that U.S. EPA source profiles are not suitable for such regions in India and site-specific source profiles should be used in the application of chemical mass balance for source apportionment.     

A Statistical Analysis of Particulate Data Sets for Jawaharlal Nehru Port and Surrounding Harbour Region in India

The concentrations of total suspended particulate matter (TSP) and particulate matter less than 10 microns (PM10) were measured at various locations in a Jawaharlal Nehru port and surrounding harbour region. Meteorological data was also collected to establish the correlation with air pollutant concentration. The results are analysed from the standpoint of monthly and seasonal variations, annual trends as well as meteorological effects. The monthly mean concentration of TSP was in the range of 88.2 to 199.3 microg m(-3). The maximum and minimum-recorded value of PM10 was 135.8 and 20.3 microg m(-3), respectively. The annual average concentration of PM10 was 66.1 microg m(-3). There are clear associations between TSP and PM10 data set at all the measured three sites with a correlation coefficient of 0.89, 0.69 and 0.81, respectively. PM10 data appears to be a constant fraction of the TSP data throughout the year, indicating common influences of meteorology and sources. Particle size analysis showed PM10 to be 47% of the total TSP concentration, which is lower than reported for industrial area and traffic junctions in Mumbai. Anthropogenic sources contribute significantly to the PM10 fraction in an industrial region, while contributions from natural sources are more in a port and harbour area. Statistical analysis of air quality data shows that TSP is strongly correlated with wind speed but weakly correlated with temperature. There appears to be a simple inverse relationship between TSP and wind speed data, indicating the dilution and transport by winds.     

Statistical analyses of coastal water quality for a port and harbour region in India

A long-term study of temperature, pH, turbidity, suspended solid, salinity, dissolved oxygen, biochemical oxygen demand and ammonia nitrogen has been performed in a port and harbour region in India for four years from December 1996 to November 2000. Marine water quality results showed no regular trend. The mean monthly values of temperature, pH, turbidity, suspended solid, salinity, dissolved oxygen, biochemical oxygen demand and ammonia nitrogen were in the range of 22.64 ± 0.4 to 29.05 ± 1.37 °C; 7.65 ± 0.04 to 7.81 ± 0.13; 28.8 ± 14.7 to 64.2 ± 32.0 NTU; 283.5 ± 81.8 to 356.0 ± 159.7 mg/L; 29.78 ± 7.18 to 29.78 ± 1.04 ppt; 4.67 ± 0.50 to 6.01 ± 1.02 mg/L; 5.41 ± 1.92 to 7.56 ± 2.1 mg/L; and 0.25 ± 0.07 to 0.63 ± 0.49 mg/L, respectively. The results of correlation analysis showed that biochemical oxygen demand (BOD) was inversely correlated dissolved oxygen (DO) and poorly correlated with all other parameters. Turbidity and suspended solid were moderately correlated with each other while salinity was moderately correlated with other water quality parameters. In factor analysis, four factors were drawn out of the eight variables, which represented 74% of the variance of the original data. Factor I was related to suspended solid and turbidity. Factor II represented mainly temperature and DO showing inverse relation between these two. Factor III implied the degree of pollution at any monitoring station. Factor IV included pH and salinity. It could be concluded that the factor model represented almost all the variables.     

A performance indicator for ranked set sampling using ranking error probability matrix

Ranked set sampling (RSS) is a sampling procedure that has been shown to provide more efficient procedures than simple random sampling, in particular the Mann-Whitney-Wilcoxon (MWW) statistic and the empirical distribution function (EDF). We briefly review the work of Bohn (1992) and Stokes and Sager (1988) on the effect of imperfect ranking on the RSS-based MWW test and on the RSS-based EDF, respectively. We propose a model for a ranking error probability matrix which we hope will become a useful tool for evaluating RSS-based statistical procedures