THE USE OF MULTIVARIATE ANALYSIS OF COVARIANCE IN EVALUATING THE COMPARABILITY OF LABORATORY DETERMINATIONS IN COOPERATIVE STUDIES

The role of clinical chemistry has traditionally been to evaluate acutely ill or hospitalized patients. Traditional statistical methods have serious drawbacks in that they use univariate techniques. To demonstrate alternative methodology, a multivariate analysis of covariance model was developed and applied to the data from the Cooperative Study of Sickle Cell Disease.^ The purpose of developing the model for the laboratory data from the CSSCD was to evaluate the comparability of the results from the different clinics. Several variables were incorporated into the model in order to control for possible differences among the clinics that might confound any real laboratory differences.^ Differences for LDH, alkaline phosphatase and SGOT were identified which will necessitate adjustments by clinic whenever these data are used. In addition, aberrant clinic values for LDH, creatinine and BUN were also identified.^ The use of any statistical technique including multivariate analysis without thoughtful consideration may lead to spurious conclusions that may not be corrected for some time, if ever. However, the advantages of multivariate analysis far outweigh its potential problems. If its use increases as it should, the applicability to the analysis of laboratory data in prospective patient monitoring, quality control programs, and interpretation of data from cooperative studies could well have a major impact on the health and well being of a large number of individuals. ^

Black/white disparities in infant mortality rates in the U.S.: A quality of care perspective

The infant mortality rate for non-Hispanic Black infants in the U.S. is 13.63 deaths per 1,000 live births while the IMR for non-Hispanic White persons in the U.S. is 5.76 deaths per 1,000 live births. Black women are 2 times as likely as White women to deliver preterm infants and Black women are 2 times as likely as White women to deliver low birth weight infants (weighing less than 2,500 grams at birth). Differential underlying risk factors among mothers of different racial/ethnic groups for delivering pre-term and low birth weight infants have been historically accepted as the cause of racial disparities in IMRs. However, differential underlying risk status may not be the only major causative factor. Differential or unequal access to and provision of care is widely speculated to be a leading contributing factor to the wide racial disparity in infant mortality.2 This paper conducts a systematic review of existing literature investigating racial disparities in obstetrical care provided by healthcare practitioners to evaluate whether inequities in healthcare services provided to pregnant mothers and their neonates exist. The search terms "racial disparities obstetrical care," "racial differences quality of prenatal care," and "infant mortality racial disparities" were entered into the EBSCO Medline, Ovid Medline, PubMed, and Academic Search Complete databases, and articles between years 1990–2011 were selected for abstract review. The only articles included were those that used statistical methods to assess whether racial inequalities were present in the obstetrical services provided to pregnant women. My literature search returned 5 articles. Four of the five studies yielded significant racial differences in obstetrical care. However, the one study that used a large, nationally representative valid sample did not represent significant differences. Thus, this review provides initial evidence for racial disparities in obstetrical care, but concludes that more studies are needed in this area. Not all of the studies reviewed were consistent in the use and measurement of services, and not all studies were significant. The policy and public health implications of possible racial disparities in obstetrical care include the need to develop standard of care protocols for ALL obstetrical patients across the United States to minimize and/or eliminate the inequities and differences in obstetrical services provided.^

A hybrid method in combining treatment effects from matched and unmatched studies

In the biomedical studies, the general data structures have been the matched (paired) and unmatched designs. Recently, many researchers are interested in Meta-Analysis to obtain a better understanding from several clinical data of a medical treatment. The hybrid design, which is combined two data structures, may create the fundamental question for statistical methods and the challenges for statistical inferences. The applied methods are depending on the underlying distribution. If the outcomes are normally distributed, we would use the classic paired and two independent sample T-tests on the matched and unmatched cases. If not, we can apply Wilcoxon signed rank and rank sum test on each case. ^ To assess an overall treatment effect on a hybrid design, we can apply the inverse variance weight method used in Meta-Analysis. On the nonparametric case, we can use a test statistic which is combined on two Wilcoxon test statistics. However, these two test statistics are not in same scale. We propose the Hybrid Test Statistic based on the Hodges-Lehmann estimates of the treatment effects, which are medians in the same scale.^ To compare the proposed method, we use the classic meta-analysis T-test statistic on the combined the estimates of the treatment effects from two T-test statistics. Theoretically, the efficiency of two unbiased estimators of a parameter is the ratio of their variances. With the concept of Asymptotic Relative Efficiency (ARE) developed by Pitman, we show ARE of the hybrid test statistic relative to classic meta-analysis T-test statistic using the Hodges-Lemann estimators associated with two test statistics.^ From several simulation studies, we calculate the empirical type I error rate and power of the test statistics. The proposed statistic would provide effective tool to evaluate and understand the treatment effect in various public health studies as well as clinical trials.^

Classification of cellular malignancy associated changes in cervical cancer

Cervical cancer is the leading cause of death and disease from malignant neoplasms among women in developing countries. Even though the Pap smear has significantly decreased the number of deaths from cervical cancer in the past years, it has its limitations. Researchers have developed an automated screening machine which can potentially detect abnormal cases that are overlooked by conventional screening. The goal of quantitative cytology is to classify the patient's tissue sample based on quantitative measurements of the individual cells. It is also much cheaper and potentially can take less time. One of the major challenges of collecting cells with a cytobrush is the possibility of not sampling any existing dysplastic cells on the cervix. Being able to correctly classify patients who have disease without the presence of dysplastic cells could improve the accuracy of quantitative cytology algorithms. Subtle morphologic changes in normal-appearing tissues adjacent to or distant from malignant tumors have been shown to exist, but a comparison of various statistical methods, including many recent advances in the statistical learning field, has not previously been done. The objective of this thesis is to use different classification methods applied to quantitative cytology data for the detection of malignancy associated changes (MACs). In this thesis, Elastic Net is the best algorithm. When we applied the Elastic Net algorithm to the test set, we combined the training set and validation set as "training" set and used 5-fold cross validation to choose the parameter for Elastic Net. It has a sensitivity of 47% at 80% specificity, an AUC 0.52, and a partial AUC 0.10 (95% CI 0.09-0.11).^

Proceedings of the Twenty-Sixth Annual Biochemical Engineering Symposium

This volume contains the Proceedings of the Twenty-Sixth Annual Biochemical Engineering Symposium held at Kansas State University on September 21, 1996. The program included 10 oral presentations and 14 posters. Some of the papers describe the progress of ongoing projects, and others contain the results of completed projects. Only brief summaries are given of some of the papers; many of the papers will be published in full elsewhere. A listing of those who attended is given below. ContentsForeign Protein Production from SV40 Early Promoter in Continuous Cultures of Recombinant CHO Cells - Gautam Banik, Paul Todd, and Dhinakar Kampala Enhanced Cell Recruitment Due to Cell-Cell Interactions - Brad Farlow and Matthias Nollert The Recirculation of Hybridoma Suspension Cultures: Effects on Cell Death, Metabolism and Mab Productivity - Peng Jin and Carole A. Heath The Importance of Enzyme Inactivation and Self-Recovery in Cometabolic Biodegradation of Chlorinated Solvents - Xi-Hui Zhang, Shanka Banerji, and Rakesh Bajpai Phytoremediation of VOC contaminated Groundwater using Poplar Trees - Melissa Miller, Jason Dana, L.C. Davis, Murlidharan Narayanan, and L.E. Erickson Biological Treatment of Off-Gases from Aluminum Can Production: Experimental Results and Mathematical Modeling - Adeyma Y. Arroyo, Julio Zimbron, and Kenneth F. Reardon Inertial Migration Based Separation of Chlorella Microalgae in Branched Tubes - N.M. Poflee, A.L. Rakow, D.S. Dandy, M.L. Chappell, and M.N. Pons Contribution of Electrochemical Charge to Protein Partitioning in Aqueous Two-Phase Systems - Weiyu Fan and Charles C. Glatz Biodegradation of Some Commercial Surfactants Used in Bioremediation - Jun Gu, G.W. Preckshot, S.K. Banerji, and Rakesh Bajpai Modeling the Role of Biomass in Heavy Metal Transport Ln Vadose Zone - K.V. Nedunuri, L.E. Erickson, and R.S. Govindaraju Multivariable Statistical Methods for Monitoring Process Quality: Application to Bioinsecticide Production by 73 89 Bacillus Thuringiensis - c. Puente and M.N. Karim The Use of Polymeric Flocculants in Bacterial Lysate Streams - H. Graham, A.S. Cibulskas and E.H. Dunlop Effect of Water Content on transport of Trichloroethylene in a Chamber with Alfalfa Plants - Muralidharan Narayanan, Jiang Hu, Lawrence C. Davis, and Larry E. Erickson Detection of Specific Microorganisms using the Arbitrary Primed PCR in the Bacterial Community of Vegetated Soil - X. Wu and L.C. Davis Flux Enhancement Using Backpulsing - V.T. Kuberkar and R.H. Davis Chromatographic Purification of Oligonucleotides: Comparison with Electrophoresis - Stephen P. Cape, Ching-Yuan Lee, Kevin Petrini, Sean Foree, Micheal G. Sportiello and Paul Todd Determining Singular Arc Control Policies for Bioreactor Systems Using a Modified Iterative Dynamic Programming Algorithm - Arun Tholudur and W. Fred Ramirez Pressure Effect on Subtilisins Measured via FTIR, EPR and Activity Assays, and Its Impact on Crystallizations - J.N. Webb, R.Y. Waghmare, M.G. Bindewald, T.W. Randolph, J.F. Carpenter, C.E. Glatz Intercellular Calcium Changes in Endothelial Cells Exposed to Flow - Laura Worthen and Matthias Nollert Application of Liquid-Liquid Extraction in Propionic Acid Fermentation - Zhong Gu, Bonita A. Glatz, and Charles E. Glatz Purification of Recombinant T4 Lysozyme from E. Coli: Ion-Exchange Chromatography - Weiyu Fan, Matt L. Thatcher, and Charles E. Glatz Recovery and Purification of Recombinant Beta-Glucuronidase from Transgenic Corn - Ann R. Kusnadi, Roque Evangelista, Zivko L. Nikolov, and John Howard Effects of Auxins and cytokinins on Formation of Catharanthus Roseus G. Don Multiple Shoots - Ying-Jin Yuan, Yu-Min Yang, Tsung-Ting Hu, and Jiang Hu Fate and Effect of Trichloroethylene as Nonaqueous Phase Liquid in Chambers with Alfalfa - Qizhi Zhang, Brent Goplen, Sara Vanderhoof, Lawrence c. Davis, and Larry E. Erickson Oxygen Transport and Mixing Considerations for Microcarrier Culture of Mammalian Cells in an Airlift Reactor - Sridhar Sunderam, Frederick R. Souder, and Marylee Southard Effects of Cyclic Shear Stress on Mammalian Cells under Laminar Flow Conditions: Apparatus and Methods - M.L. Rigney, M.H. Liew, and M.Z. Southard

Reseña sobre las metodologías de campo, analíticas y estadísticas empleadas en la determinación y manejo de datos de los elementos de tierras raras en el sistema suelo-planta

El presente artículo es una revisión detallada de estudios científicos publicados que tratan el tema relacionado con la determinación de los elementos de las tierras raras (REEs) en el sistema suelo-planta. Los estudios han sido llevados a cabo principalmente en países europeos y asiáticos. Cabe señalar que la investigación en los países latinoamericanos es muy escasa; sin embargo, es creciente el interés de analizar la aportación de estos elementos al suelo y la planta, lo cual se debe a la aplicación de fertilizantes que contienen dosis elevadas de estos elementos en su composición. Diversas técnicas de muestreo, experimentación y análisis han sido empleadas para la determinación de los REEs. No obstante, se considera que el manejo de los datos ha sido incorrecto estadísticamente. El contenido del presente artículo aborda: (i) las generalidades de los REEs; (ii) el análisis de la bibliografía disponible con el fin de conocer las metodologías de muestreo y análisis más utilizadas en 37 artículos en total, señalando algunos puntos que se consideran todavía deficientes; (iii) dos ejemplos de la aplicación de técnicas estadísticas (intervalo de confianza de la media y pruebas de significancia de la relación F de Fisher y t de Student) utilizando datos reportados en dos artículos. Los resultados mostraron, con los datos del primer artículo analizado, que: a) no se aplicó una metodología estadística para evaluar la calidad de datos; b) al aplicar estadística se encontró que existen diferencias sistemáticas entre los datos determinados en el laboratorio y los certificados. En el segundo artículo analizado se demostró, mediante pruebas de significancia, que existen diferencias significativas en las medias de Ce y Eu (los dos elementos tomados como ejemplos) en las plantas de un sitio a otro.

Environmental variables and multivariate seabed classification maps via k means clustering of Potter Cove, Antarctica, link to input and result files in GeoTIFF format

This study subdivides the Potter Cove, King George Island, Antarctica, into seafloor regions using multivariate statistical methods. These regions are categories used for comparing, contrasting and quantifying biogeochemical processes and biodiversity between ocean regions geographically but also regions under development within the scope of global change. The division obtained is characterized by the dominating components and interpreted in terms of ruling environmental conditions. The analysis includes in total 42 different environmental variables, interpolated based on samples taken during Australian summer seasons 2010/2011 and 2011/2012. The statistical errors of several interpolation methods (e.g. IDW, Indicator, Ordinary and Co-Kriging) with changing settings have been compared and the most reasonable method has been applied. The multivariate mathematical procedures used are regionalized classification via k means cluster analysis, canonical-correlation analysis and multidimensional scaling. Canonical-correlation analysis identifies the influencing factors in the different parts of the cove. Several methods for the identification of the optimum number of clusters have been tested and 4, 7, 10 as well as 12 were identified as reasonable numbers for clustering the Potter Cove. Especially the results of 10 and 12 clusters identify marine-influenced regions which can be clearly separated from those determined by the geological catchment area and the ones dominated by river discharge.

Seasonal production patterns of planktonic foraminifera in the NE Atlantic Ocean

Sediment trap samples from OMEX 2 (49°N, 13°W) provide a continuous record of the seasonal succession of planktonic foraminifera in the midlatitude North Atlantic and reveal a complex relationship between periods of production and specific hydrographic conditions. Neogloboquadrina pachyderma dextral coiling (d.), Globigerina bulloides, and Globorotalia inflata are found in great numbers during both the spring and summer seasons, whereas Globigerina quinqueloba, Globorotalia hirsuta, Globorotalia scitula, and Globigerinita glutinata are associated predominantly with the increase in productivity during the spring bloom. Globigerinella aequilateralis, Orbulina universa, and Globigerinoides sacculifer are restricted to late summer conditions following the establishment of a warm, well-stratified surface ocean. An annually integrated fauna from the sediment trap, comprising ~13,000 individuals, is used to evaluate the accuracy of five faunal-based statistical methods of paleotemperature estimation. All of the temperature reconstruction techniques produce estimates of ~16°C and ~11°C for summer and winter surface temperature, respectively, which are in excellent agreement with regional hydrographic data and suggest that the sediment trap assemblage is well represented in the core top faunas. Analysis of the key species that dominate the OMEX 2 sediment trap fauna, G. bulloides, G. inflata, and N. pachyderma d., based on d18O derived temperatures from North Atlantic core top samples, suggests that seasonal variations in planktonic foraminiferal production are nonuniform across the midlatitudes and that this is likely to complicate reconstructing past seasonal hydrographic dynamics using these taxa.

Early Cenozoic Atlantic and Pacific benthic foraminiferal isotopes from 12 sites

Oxygen and carbon isotope records are important tools used to reconstruct past ocean and climate conditions, with those of benthic foraminifera providing information on the deep oceans. Reconstructions are complicated by interspecies isotopic offsets that result from microhabitat preferences (carbonate precipitation in isotopically distinct environments) and vital effects (species-specific metabolic variation in isotopic fractionation). We provide correction factors for early Cenozoic benthic foraminifera commonly used for isotopic measurements (Cibicidoides spp., Nuttallides truempyi, Oridorsalis spp., Stensioina beccariiformis, Hanzawaia ammophila, and Bulimina spp.), showing that most yield reliable isotopic proxies of environmental change. The statistical methods and larger data sets used in this study provide more robust correction factors than do previous studies. Interspecies isotopic offsets appear to have changed through the Cenozoic, either (1) as a result of evolutionary changes or (2) as an artifact of different statistical methods and data set sizes used to determine the offsets in different studies. Regardless of the reason, the assumption that isotopic offsets have remained constant through the Cenozoic has introduced an 1-2°C uncertainty into deep sea paleotemperature calculations. In addition, we compare multiple species isotopic data from a western North Atlantic section that includes the Paleocene-Eocene thermal maximum to determine the most reliable isotopic indicator for this event. We propose that Oridorsalis spp. was the most reliable deepwater isotopic recorder at this location because it was best able to withstand the harsh water conditions that existed at this time; it may be the best recorder at other locations and for other extreme events also.

Introducing Nuclear Energy to High-School Students: The Spanish Young Generation in Nuclear (Jóvenes Nucleares) Lectures

One of the main goals of Spanish Young Generation (JJNN) is to spread knowledge about nuclear energy, not only pointing out its advantages and its role in our society, but also trying to correct some of the ideas that are due to the biased information and to the lack of knowledge. With this goal in mind, lectures were given in several high schools, aimed at students ranging from 14 to 18 years old. This paper explains the experience accumulated during those talks and the conclusions that can be drawn, so as to better focus the communication about nuclear energy, especially the one aimed at a young public. In order to evaluate the degree of knowledge and information on a specific topic of a given group of individuals, statistical methods must be used. At the beginning of each lecture (and sometimes at the end, in order to evaluate the impact of the talk) the students were submitted to a short survey conducted by Spanish Young Generation. It consisted in eight questions, dealing with the relation between the main environmental issues (global warming, acid rain, radioactive waste…) and nuclear energy. The answers can be surprising, especially for professionals of the nuclear field who, since they are so familiar with this topic, often forget that this is just the case of a minority of people. A better knowledge of the degree of information of a given group enables to focus and personalize the communication. Another communication tool is the direct contact with students: it starts with their questions, which can then lead to a small debate. If the surveys inform about the topics they are unaware of, the direct exchange with them enables to find the most effective way to provide them the information. Of course, it depends a lot on the public attending the talk (age, background…) and on the debate following the talk: a good communication, adapted to the public, is necessary. Therefore, the outcome of the performed exercise is that Spanish teenagers have still a lack of knowledge about nuclear energy. We can learn that items that are evident for nuclear young professionals are unknown for high school teenagers

Optimización de la succión de sólidos sumergidos utilizando bombas centrífugas con velocidad variable

La acumulación de material sólido en embalses, cauces fluviales y en zonas marítimas hace que la extracción mecánica de estos materiales por medio de succión sea cada vez mas frecuente, por ello resulta importante estudiar el rendimiento de la succión de estos materiales analizando la forma de las boquillas y los parámetros del flujo incluyendo la bomba. Esta tesis estudia, mediante equipos experimentales, la eficacia de distintos dispositivos de extracción de sólidos (utilizando boquillas de diversas formas y bombas de velocidad variable). El dispositivo experimental ha sido desarrollado en el Laboratorio de Hidráulica de la E.T.S.I. de Caminos, C. y P. de la Universidad Politécnica de Madrid. Dicho dispositivo experimental incluye un lecho sumergido de distintos tipos de sedimentos, boquillas de extracción de sólidos y bomba de velocidad variable, así como un elemento de separación del agua y los sólidos extraídos. Los parámetros básicos analizados son el caudal líquido total bombeado, el caudal sólido extraído, diámetro de la tubería de succión, forma y sección de la boquilla extractora, así como los parámetros de velocidad y rendimiento en la bomba de velocidad variable. Los resultados de las medidas obtenidas en el dispositivo experimental han sido estudiados por medio del análisis dimensional y con métodos estadísticos. A partir de este estudio se ha desarrollado una nueva formulación, que relaciona el caudal sólido extraído con los diámetros de tubería y boquilla, caudal líquido bombeado y velocidad de giro de la bomba. Así mismo, desde el punto de vista práctico, se han analizado la influencia de la forma de la boquilla con la capacidad de extracción de sólidos a igualdad del resto de los parámetros, de forma que se puedan recomendar que forma de la boquilla es la más apropiada. The accumulation of solid material in reservoirs, river channels and sea areas causes the mechanical extraction of these materials by suction is becoming much more common, so it is important to study the performance of the suction of these materials analyzing the shape of the nozzles and flow parameters, including the pump. This thesis studies, using experimental equipment, the effectiveness of different solids removal devices (using nozzles of different shapes and variable speed pumps). The experimental device was developed at the Hydraulics Laboratory of the Civil University of the Polytechnic University of Madrid. The device included a submerged bed with different types of sediment solids, different removal nozzles and variable speed pump. It also includes a water separation element and the solids extracted. The key parameters analyzed are the total liquid volume pumped, the solid volume extracted, diameter of the suction pipe, a section of the nozzle and hood, and the parameters of speed and efficiency of the variable speed pump. The basic parameters analyzed are the total liquid volume pumped, the removed solid volume, the diameter of the suction pipe, the shape and cross-section of the nozzle, and the parameters of speed, efficiency and energy consumed by the variable speed pump. The measurements obtained on the experimental device have been studied with dimensional analysis and statistical methods. The outcome of this study is a new formulation, which relates the solid volume extracted with the pipe and nozzle diameters, the pumped liquid flow and the speed of the pump. Also, from a practical point of view, the influence of the shape of the nozzle was compared with the solid extraction capacity, keeping equal the rest of the parameters. So, a recommendation of the best shape of the nozzle can be given.

Selección de métodos estadísticos para la interpretación de intercomparaciones en ensayos de áridos finos

El objeto de este proyecto es seleccionar los métodos estadísticos adecuados para el tratamiento de los resultados obtenidos en un ejercicio de intercomparación para ensayos de áridos para construcción con el fin de cumplir la normativa para dicho ejercicio. Los ensayos estudiados son: - Ensayo Azul de Metileno - Equivalente de arena - Contenido total en azufre - Contenido en humus - Reactividad álcali-carbonato Los criterios aplicables para el desarrollo de los trabajos son los exigidos por la Entidad Nacional de Acreditación (ENAC) para laboratorios acreditados conforme a la Norma UNE-EN ISO/IEC 17025 y para proveedores acreditados conforme a la Norma UNE-EN ISO/IEC 17043. ABSTRACT: The purpose of this Project is to select appropriate statistical methods for the treatment of the results obtained from an intercomparison exercise for testing of aggregates for construction with the target of comply the rules for this exercise. The studied assays are: - Methylene Blue Assay - Sand equivalent - Total Sulfur Content - Humus content - Alkali-carbonate reactivity The criteria for the development of the work are those required by the National Accreditation Body (ENAC) for laboratories accredited in accordance with the UNE-EN ISO / IEC 17025 and for suppliers in accordance with the UNE-EN ISO / IEC 17043.

A proposal for improving safety in construction projects by strengthening coordinators’ competencies in health and safety issues

The figure of the coordinator in health and safety issues in the construction sector first appeared in our legislation through the incorporation of the European Directives (in our case Royal Decree 1627/97 on the minimum health and safety regulations in construction works), and is viewed differently in different countries of the European Union regarding the way they are hired and their role in the construction industry. Coordinating health and safety issues is also a management process that requires certain competencies that are not only based on technical or professional training, but which, taking account of the work environment, require the use of strategies and tools that are related to experience and personal skills. Through a piece of research that took account of expert opinions in the matter, we have found which competencies need to be possessed by the health and safety coordinator in order to improve the safety in the works they are coordinating. The conclusions of the analyses performed using the appropriate statistical methods (comparing means and multivariate analysis techniques), will enable training programmes to be designed and ensure that the health and safety coordinators selected have the competencies required to carry out their duties.

A proposal for improving safety in construction projects by strengthening coordinators’ competencies in health and safety issues

The figure of the coordinator in health and safety issues in the construction sector first appeared in our legislation through the incorporation of the European Directives (in our case Royal Decree 1627/97 on the minimum health and safety regulations in construction works), and is viewed differently in different countries of the European Union regarding the way they are hired and their role in the construction industry. Coordinating health and safety issues is also a management process that requires certain competencies that are not only based on technical or professional training, but which, taking account of the work environment, require the use of strategies and tools that are related to experience and personal skills. Through a piece of research that took account of expert opinions in the matter, we have found which competencies need to be possessed by the health and safety coordinator in order to improve the safety in the works they are coordinating. The conclusions of the analyses performed using the appropriate statistical methods (comparing means and multivariate analysis techniques), will enable training programmes to be designed and ensure that the health and safety coordinators selected have the competencies required to carry out their duties.

SMS design and aberration theory

The SMS, Simultaneous Multiple Surfaces, design was born to Nonimaging Optics applications and is now being applied also to Imaging Optics. In this paper the wave aberration function of a selected SMS design is studied. It has been found the SMS aberrations can be analyzed with a little set of parameters, sometimes two. The connection of this model with the conventional aberration expansion is also presented. To verify these mathematical model two SMS design systems were raytraced and the data were analyzed with a classical statistical methods: the plot of discrepancies and the quadratic average error. Both the tests show very good agreement with the model for our systems.