Study on Bivariate Normal Distribution

Let (X, Y) be bivariate normal random vectors which represent the responses as a result of Treatment 1 and Treatment 2. The statistical inference about the bivariate normal distribution parameters involving missing data with both treatment samples is considered. Assuming the correlation coefficient ρ of the bivariate population is known, the MLE of population means and variance (ξ, η, and σ2) are obtained. Inferences about these parameters are presented. Procedures of constructing confidence interval for the difference of population means ξ – η and testing hypothesis about ξ – η are established. The performances of the new estimators and testing procedure are compared numerically with the method proposed in Looney and Jones (2003) on the basis of extensive Monte Carlo simulation. Simulation studies indicate that the testing power of the method proposed in this thesis study is higher.

Application of a Bivariate Probit Model to Investigate the Intended Evacuation from Hurricane

With evidence of increasing hurricane risks in Georgia Coastal Area (GCA) and Virginia in the U.S. Southeast and elsewhere, understanding intended evacuation behavior is becoming more and more important for community planners. My research investigates intended evacuation behavior due to hurricane risks, a behavioral survey of the six counties in GCA under the direction of two social scientists with extensive experience in survey research related to citizen and household response to emergencies and disasters. Respondents gave answers whether they would evacuate under both voluntary and mandatory evacuation orders. Bivariate probit models are used to investigate the subjective belief structure of whether or not the respondents are concerned about the hurricane, and the intended probability of evacuating as a function of risk perception, and a lot of demographic and socioeconomic variables (e.g., gender, military, age, length of residence, owning vehicles).

A Study on the Correlation of Bivariate And Trivariate Normal Models

Suppose two or more variables are jointly normally distributed. If there is a common relationship between these variables it would be very important to quantify this relationship by a parameter called the correlation coefficient which measures its strength, and the use of it can develop an equation for predicting, and ultimately draw testable conclusion about the parent population. This research focused on the correlation coefficient ρ for the bivariate and trivariate normal distribution when equal variances and equal covariances are considered. Particularly, we derived the maximum Likelihood Estimators (MLE) of the distribution parameters assuming all of them are unknown, and we studied the properties and asymptotic distribution of . Showing this asymptotic normality, we were able to construct confidence intervals of the correlation coefficient ρ and test hypothesis about ρ. With a series of simulations, the performance of our new estimators were studied and were compared with those estimators that already exist in the literature. The results indicated that the MLE has a better or similar performance than the others.

Multivariate orthogonal polynomials and integrable systems

Multivariate orthogonal polynomials in D real dimensions are considered from the perspective of the Cholesky factorization of a moment matrix. The approach allows for the construction of corresponding multivariate orthogonal polynomials, associated second kind functions, Jacobi type matrices and associated three term relations and also Christoffel-Darboux formulae. The multivariate orthogonal polynomials, their second kind functions and the corresponding Christoffel-Darboux kernels are shown to be quasi-determinants as well as Schur complements of bordered truncations of the moment matrix; quasi-tau functions are introduced. It is proven that the second kind functions are multivariate Cauchy transforms of the multivariate orthogonal polynomials. Discrete and continuous deformations of the measure lead to Toda type integrable hierarchy, being the corresponding flows described through Lax and Zakharov-Shabat equations; bilinear equations are found. Varying size matrix nonlinear partial difference and differential equations of the 2D Toda lattice type are shown to be solved by matrix coefficients of the multivariate orthogonal polynomials. The discrete flows, which are shown to be connected with a Gauss-Borel factorization of the Jacobi type matrices and its quasi-determinants, lead to expressions for the multivariate orthogonal polynomials and their second kind functions in terms of shifted quasi-tau matrices, which generalize to the multidimensional realm, those that relate the Baker and adjoint Baker functions to ratios of Miwa shifted tau-functions in the 1D scenario. In this context, the multivariate extension of the elementary Darboux transformation is given in terms of quasi-determinants of matrices built up by the evaluation, at a poised set of nodes lying in an appropriate hyperplane in R^D, of the multivariate orthogonal polynomials. The multivariate Christoffel formula for the iteration of m elementary Darboux transformations is given as a quasi-determinant. It is shown, using congruences in the space of semi-infinite matrices, that the discrete and continuous flows are intimately connected and determine nonlinear partial difference-differential equations that involve only one site in the integrable lattice behaving as a Kadomstev-Petviashvili type system. Finally, a brief discussion of measures with a particular linear isometry invariance and some of its consequences for the corresponding multivariate polynomials is given. In particular, it is shown that the Toda times that preserve the invariance condition lay in a secant variety of the Veronese variety of the fixed point set of the linear isometry.

On polynomials with given Hilbert function and applications

Using Macaulay's correspondence we study the family of Artinian Gorenstein local algebras with fixed symmetric Hilbert function decomposition. As an application we give a new lower bound for the dimension of cactus varieties of the third Veronese embedding. We discuss the case of cubic surfaces, where interesting phenomena occur.

Some Remarks on the Approximation of Transitional Density in Stochastic Differential Equations

Aijt-Sahalia (2002) introduced a method to estimate transitional probability densities of di®usion processes by means of Hermite expansions with coe±cients determined by means of Taylor series. This note describes a numerical procedure to ¯nd these coe±cients based on the calculation of moments. One advantage of this procedure is that it can be used e®ectively when the mathematical operations required to ¯nd closed-form expressions for these coe±cients are otherwise infeasible.

Investigating the links between muscle strength, sun exposure, dietary vitamin D intake and the vitamin D status of ambulatory older adults in South East Queensland

Vitamin D deficiency and insufficiency are now seen as a contemporary health problem in Australia with possible widespread health effects not limited to bone health1. Despite this, the Vitamin D status (measured as serum 25-hydroxyvitamin D (25(OH)D)) of ambulatory adults has been overlooked in this country. Serum 25(OH)D status is especially important among this group as studies have shown a link between Vitamin D and fall risk in older adults2. Limited data also exists on the contributions of sun exposure via ultraviolet radiation and dietary intake to serum 25(OH)D status in this population. The aims of this project were to assess the serum 25(OH)D status of a group of older ambulatory adults in South East Queensland, to assess the association between their serum 25(OH)D status and functional measures as possible indicators of fall risk, obtain data on the sources of Vitamin D in this population and assess whether this intake was related to serum 25(OH)D status and describe sun protection and exposure behaviors in this group and investigate whether a relationship existed between these and serum 25(OH)D status. The collection of this data assists in addressing key gaps identified in the literature with regard to this population group and their Vitamin D status in Australia. A representative convenience sample of participants (N=47) over 55 years of age was recruited for this cross-sectional, exploratory study which was undertaken in December 2007 in south-east Queensland (Brisbane and Sunshine coast). Participants were required to complete a sun exposure questionnaire in addition to a Calcium and Vitamin D food frequency questionnaire. Timed up and go and handgrip dynamometry tests were used to examine functional capacity. Serum 25(OH)D status and blood measures of Calcium, Phosphorus and Albumin were determined through blood tests. The Mean and Median serum 25-Hydroxyvitamin D (25(OH)D) for all participants in this study was 85.8nmol/L (Standard Deviation 29.7nmol/L) and 81.0nmol/L (Range 22-158nmol/L), respectively. Analysis at the bivariate level revealed a statistically significant relationship between serum 25(OH)D status and location, with participants living on the Sunshine Coast having a mean serum 25(OH)D status 21.3nmol/L higher than participants living in Brisbane (p=0.014). While at the descriptive level there was an apparent trend towards higher outdoor exposure and increasing levels of serum 25(OH)D, no statistically significant associations between the sun measures of outdoor exposure, sun protection behaviors and phenotypic characteristics and serum 25(OH)D status were observed. Intake of both Calcium and Vitamin D was low in this sample with sixty-eight (68%) of participants not meeting the Estimated Average Requirements (EAR) for Calcium (Median=771.0mg; Range=218.0-2616.0mg), while eighty-seven (87%) did not meet the Adequate Intake for Vitamin D (Median=4.46ug; Range=0.13-30.0ug). This raises the question of how realistic meeting the new Adequate Intakes for Vitamin D is, when there is such a low level of Vitamin D fortification in this country. However, participants meeting the Adequate Intake (AI) for Vitamin D were observed to have a significantly higher serum 25(OH)D status compared to those not meeting the AI for Vitamin D (p=0.036), showing that meeting the AI for Vitamin D may play a significant role in determining Vitamin D status in this population. By stratifying our data by categories of outdoor exposure time, a trend was observed between increased importance of Vitamin D dietary intake as a possible determinant of serum 25(OH)D status in participants with lower outdoor exposures. While a trend towards higher Timed Up and Go scores in participants with higher 25(OH) D status was seen, this was only significant for females (p=0.014). Handgrip strength showed statistically significant association with serum 25(OH)D status. The high serum 25(OH)D status in our sample almost certainly explains the limited relationship between functional measures and serum 25(OH)D. However, the observation of an association between slower Time Up and Go speeds, and lower serum 25(OH)D levels, even with a small sample size, is significant as slower Timed Up and Go speeds have been associated with increased fall risk in older adults3. Multivariable regression analysis revealed Location as the only significant determinant of serum 25(OH)D status at p=0.014, with trends (p=>0.1) for higher serum 25(OH)D being shown for participants that met the AI for Vitamin D and rated themselves as having a higher health status. The results of this exploratory study show that 93.6% of participants had adequate 25(OH)D status-possibly due to measurement being taken in the summer season and the convenience nature of the sample. However, many participants do not meet their dietary Calcium and Vitamin D requirements, which may indicate inadequate intake of these nutrients in older Australians and a higher risk of osteoporosis. The relationship between serum 25(OH)D and functional measures in this population also requires further study, especially in older adults displaying Vitamin D insufficiency or deficiency.

An investigation into the barriers to the implementation of automation and robotics technologies in the construction industry

The rising problems associated with construction such as decreasing quality and productivity, labour shortages, occupational safety, and inferior working conditions have opened the possibility of more revolutionary solutions within the industry. One prospective option is in the implementation of innovative technologies such as automation and robotics, which has the potential to improve the industry in terms of productivity, safety and quality. The construction work site could, theoretically, be contained in a safer environment, with more efficient execution of the work, greater consistency of the outcome and higher level of control over the production process. By identifying the barriers to construction automation and robotics implementation in construction, and investigating ways in which to overcome them, contributions could be made in terms of better understanding and facilitating, where relevant, greater use of these technologies in the construction industry so as to promote its efficiency. This research aims to ascertain and explain the barriers to construction automation and robotics implementation by exploring and establishing the relationship between characteristics of the construction industry and attributes of existing construction automation and robotics technologies to level of usage and implementation in three selected countries; Japan, Australia and Malaysia. These three countries were chosen as their construction industry characteristics provide contrast in terms of culture, gross domestic product, technology application, organisational structure and labour policies. This research uses a mixed method approach of gathering data, both quantitative and qualitative, by employing a questionnaire survey and an interview schedule; using a wide range of sample from management through to on-site users, working in a range of small (less than AUD0.2million) to large companies (more than AUD500million), and involved in a broad range of business types and construction sectors. Detailed quantitative (statistical) and qualitative (content) data analysis is performed to provide a set of descriptions, relationships, and differences. The statistical tests selected for use include cross-tabulations, bivariate and multivariate analysis for investigating possible relationships between variables; and Kruskal-Wallis and Mann Whitney U test of independent samples for hypothesis testing and inferring the research sample to the construction industry population. Findings and conclusions arising from the research work which include the ranking schemes produced for four key areas of, the construction attributes on level of usage; barrier variables; differing levels of usage between countries; and future trends, have established a number of potential areas that could impact the level of implementation both globally and for individual countries.

What lies beneath? : The pattern and abundance of the subterranean tuber bank of the invasive liana cat’s claw creeper, Macfadyena unguis-cati (Bignoniaceae)

Cat’s claw creeper, Macfadyena unguis-cati (L.) Gentry (Bignoniaceae) is a major environmental weed of riparian areas, rainforest communities and remnant natural vegetation in coastal Queensland and New South Wales, Australia. In densely infested areas, it smothers standing vegetation, including large trees, and causes canopy collapse. Quantitative data on the ecology of this invasive vine are generally lacking. The present study examines the underground tuber traits of M. unguis-cati and explores their links with aboveground parameters at five infested sites spanning both riparian and inland vegetation. Tubers were abundant in terms of density (~1000 per m2), although small in size and low in level of interconnectivity. M. unguis-cati also exhibits multiple stems per plant. Of all traits screened, the link between stand (stem density) and tuber density was the most significant and yielded a promising bivariate relationship for the purposes of estimation, prediction and management of what lies beneath the soil surface of a given M. unguis-cati infestation site. The study also suggests that new recruitment is primarily from seeds, not from vegetative propagation as previously thought. The results highlight the need for future biological-control efforts to focus on introducing specialist seed- and pod-feeding insects to reduce seed-output.

Trends in demographic, health behaviour factors, and self-perceived weight status : influences on obesity in Australia from 1995 to 2005

Overweight and obesity are two of the most important emerging public health issues in our time and regarded by the World Health Organisation [WHO] (1998) as a worldwide epidemic. The prevalence of obesity in the USA is the highest in the world, and Australian obesity rates fall into second place. Currently, about 60% of Australian adults are overweight (BMI „d 25kg/m2). The socio-demographic factors associated with overweight and/or obesity have been well demonstrated, but many of the existing studies only examined these relationships at one point of time, and did not examine whether significant relationships changed over time. Furthermore, only limited previous research has examined the issue of the relationship between perception of weight status and actual weight status, as well as factors that may impact on people¡¦s perception of their body weight status. Aims: The aims of the proposed research are to analyse the discrepancy between perceptions of weight status and actual weight status in Australian adults; to examine if there are trends in perceptions of weight status in adults between 1995 to 2004/5; and to propose a range of health promotion strategies and furth er research that may be useful in managing physical activity, healthy diet, and weight reduction. Hypotheses: Four alternate hypotheses are examined by the research: (1) there are associations between independent variables (e.g. socio -demographic factors, physical activity and dietary habits) and overweight and/or obesity; (2) there are associations between the same independent variables and the perception of overweight; (3) there are associations between the same independent variables and the discrepancy between weight status and perception of weight status; and (4) there are trends in overweight and/or obesity, perception of overweight, and the discrepancy in Australian adults from 1995 to 2004/5. Conceptual Framework and Methods: A conceptual framework is developed that shows the associations identified among socio -demographic factors, physical activity and dietary habits with actual weight status, as well as examining perception of weight status. The three latest National Health Survey data bases (1995 , 2001 and 2004/5) were used as the primary data sources. A total of 74,114 Australian adults aged 20 years and over were recruited from these databases. Descriptive statistics, bivariate analyses (One -Way ANOVA tests, unpaired t-tests and Pearson chi-square tests), and multinomial logistic regression modelling were used to analyse the data. Findings: This research reveals that gender, main language spoken at home, occupation status, household structure, private health insurance status, and exercise are related to the discrepancy between actual weight status and perception of weight status, but only gender and exercise are related to the discrepancy across the three time point s. The current research provides more knowledge about perception of weight status independently. Factors which affect perception of overweight are gender, age, language spoken at home, private health insurance status, and diet ary habits. The study also finds that many factors that impact overweight and/or obesity also have an effect on perception of overweight, such as age, language spoken at home, household structure, and exercise. However, some factors (i.e. private health insurance status and milk consumption) only impact on perception of overweight. Furthermore, factors that are rel ated to people’s overweight are not totally related to people’s underestimation of their body weight status in the study results. Thus, there are unknown factors which can affect people’s underestimation of their body weight status. Conclusions: Health promotion and education activities should provide education about population health education and promotion and education for particular at risk sub -groups. Further research should take the form of a longitudinal study design ed to examine the causal relationship between overweight and/or obesity and underestimation of body weight status, it should also place more attention on the relationships between overweight and/or obesity and dietary habits, with a more comprehensive representation of SES. Moreover, further research that deals with identification of characteristics about perception of weight status, in particular the underestimation of body weight status should be undertaken.

New insights into appetite, inflammation and the use of fish oil in hemodialysis patients

Protein-energy wasting (PEW) is commonly seen in patients with chronic kidney disease (CKD). The condition is characterised by chronic, systemic low-grade inflammation which affects nutritional status by a variety of mechanisms including reducing appetite and food intake and increasing muscle catabolism. PEW is linked with co-morbidities such as cardiovascular disease, and is associated with lower quality of life, increased hospitalisations and a 6-fold increase in risk of death1. Significant gender differences have been found in the severity and effects of several markers of PEW. There have been limited studies testing the ability of anti-inflammatory agents or nutritional interventions to reduce the effects of PEW in dialysis patients. This thesis makes a significant contribution to the understanding of PEW in dialysis patients. It advances understanding of measurement techniques for two of the key components, appetite and inflammation, and explores the effect of fish oil, an anti-inflammatory agent, on markers of PEW in dialysis patients. The first part of the thesis consists of two methodological studies conducted using baseline data. The first study aims to validate retrospective ratings of hunger, desire to eat and fullness on visual analog scales (VAS) (paper and pen and electronic) as a new method of measuring appetite in dialysis patients. The second methodological study aims to assess the ability of a variety of methods available in routine practice to detect the presence of inflammation. The second part of the thesis aims to explore the effect of 12 weeks supplementation with 2g per day of Eicosapentaenoic Acid (EPA), a longchain fatty acid found in fish oil, on markers of PEW. A combination of biomarkers and psychomarkers of appetite and inflammation are the main outcomes being explored, with nutritional status, dietary intake and quality of life included as secondary outcomes. A lead in phase of 3 months prior to baseline was used so that each person acts as their own historical control. The study also examines whether there are gender differences in response to the treatment. Being an exploratory study, an important part of the work is to test the feasibility of the intervention, thus the level of adherence and factors associated with adherence are also presented. The studies were conducted at the hemodialysis unit of the Wesley Hospital. Participants met the following criteria: adult, stage 5 CKD on hemodialysis for at least 3 months, not expected to receive a transplant or switch to another dialysis modality during the study, absence of intellectual impairment or mental illness impairing ability to follow instructions or complete the intervention. A range of intermediate, clinical and patient-centred outcome measures were collected at baseline and 12 weeks. Inflammation was measured using five biomarkers: c-reactive protein (CRP), interleukin-6 (IL6), intercellular adhesion molecule (sICAM-1), vascular cell adhesion molecule (sVCAM-1) and white cell count (WCC). Subjective appetite was measured using the first question from the Appetite and Dietary Assessment (ADAT) tool and VAS for measurements of hunger, desire to eat and fullness. A novel feature of the study was the assessment of the appetite peptides leptin, ghrelin and peptide YY as biomarkers of appetite. Nutritional status/inflammation was assessed using the Malnutrition-Inflammation Score (MIS) and the Patient-Generated Subjective Global Assessment (PG-SGA). Dietary intake was measured using 3-day records. Quality of life was measured using the Kidney Disease Quality of Life Short Form version 1.3 (KDQOL-SF™ v1.3 © RAND University), which combines the Short-Form 36 (SF36) with a kidney-disease specific module2. A smaller range of these variables was available for analysis during the control phase (CRP, ADAT, dietary intake and nutritional status). Statistical analysis was carried out using SPSS version 14 (SPSS Inc, Chicago IL, USA). Analysis of the first part of the thesis involved descriptive and bivariate statistics, as well as Bland-Altman plots to assess agreement between methods, and sensitivity analysis/ROC curves to test the ability of methods to predict the presence of inflammation. The unadjusted (paired ttests) and adjusted (linear mixed model) change over time is presented for the main outcome variables of inflammation and appetite. Results are shown for the whole group followed by analyses according to gender and adherence to treatment. Due to the exploratory nature of the study, trends and clinical significance were considered as important as statistical significance. Twenty-eight patients (mean age 61±17y, 50% male, dialysis vintage 19.5 (4- 101) months) underwent baseline assessment. Seven out of 28 patients (25%) reported sub-optimal appetite (self-reported as fair, poor or very poor) despite all being well nourished (100% SGA A). Using the VAS, ratings of hunger, but not desire to eat or fullness, were significantly (p<0.05) associated with a range of relevant clinical variables including age (r=-0.376), comorbidities (r=-0.380) nutritional status (PG-SGA score, r=-0.451), inflammatory markers (CRP r=-0.383; sICAM-1 r=-0.387) and seven domains of quality of life. Patients expressed a preference for the paper and pen method of administering VAS. None of the tools (appetite, MIS, PG-SGA, albumin or iron) showed an acceptable ability to detect patients who are inflamed. It is recommended that CRP should be tested more frequently as a matter of course rather than seeking alternative methods of measuring inflammation. 27 patients completed the 12 week intervention. 20 patients were considered adherent based on changes in % plasma EPA, which rose from 1.3 (0.94)% to 5.2 (1.1)%, p<0.001, in this group. The major barriers to adherence were forgetting to take the tablets as well as their size. At 12 weeks, inflammatory markers remained steady apart from the white cell count which decreased (7.6(2.5) vs 7.0(2.2) x109/L, p=0.058) and sVCAM-1 which increased (1685(654) vs 2249(925) ng/mL, p=0.001). Subjective appetite using VAS increased (51mm to 57mm, +12%) and there was a trend towards reduction in peptide YY (660(31) vs 600(30) pg/mL, p=0.078). There were some gender differences apparent, with the following adjusted change between baseline and week 12: CRP (males -3% vs females +17%, p=0.19), IL6 (males +17% vs females +48%, p=0.77), sICAM-1 (males -5% vs females +11%, p=0.07), sVCAM-1 (males +54% vs females +19%, p=0.08) and hunger ratings (males 20% vs females -5%, p=0.18). On balance, males experienced a maintainence or reduction in three inflammatory markers and an improvement in hunger ratings, and therefore appeared to have responded better to the intervention. Compared to those who didn’t adhere, adherent patients maintained weight (mean(SE) change: +0.5(1.6) vs - 0.8(1.2) kg, p=0.052) and fat-free mass (-0.1 (1.6) vs -1.8 (1.8) kg, p=0.045). There was no difference in change between the intervention and control phase for CRP, appetite, nutritional status or dietary intake. The thesis makes a significant contribution to the evidence base for understanding of PEW in dialysis patients. It has advanced knowledge of methods of assessing inflammation and appetite. Retrospective ratings of hunger on a VAS appear to be a valid method of assessing appetite although samples which include patients with very poor appetite are required to confirm this. Supplementation with fish oil appeared to improve subjective appetite and dampen the inflammatory response. The effectiveness of the intervention is influenced by gender and adherence. Males appear to be more responsive to the primary outcome variables than females, and the quality of response is improved with better adherence. These results provide evidence to support future interventions aimed at reducing the effects of PEW in dialysis patients.

Modeling corneal surfaces with rational functions for high-speed videokeratoscopy data compression

High-speed videokeratoscopy is an emerging technique that enables study of the corneal surface and tear-film dynamics. Unlike its static predecessor, this new technique results in a very large amount of digital data for which storage needs become significant. We aimed to design a compression technique that would use mathematical functions to parsimoniously fit corneal surface data with a minimum number of coefficients. Since the Zernike polynomial functions that have been traditionally used for modeling corneal surfaces may not necessarily correctly represent given corneal surface data in terms of its optical performance, we introduced the concept of Zernike polynomial-based rational functions. Modeling optimality criteria were employed in terms of both the rms surface error as well as the point spread function cross-correlation. The parameters of approximations were estimated using a nonlinear least-squares procedure based on the Levenberg-Marquardt algorithm. A large number of retrospective videokeratoscopic measurements were used to evaluate the performance of the proposed rational-function-based modeling approach. The results indicate that the rational functions almost always outperform the traditional Zernike polynomial approximations with the same number of coefficients.

Modeling videokeratoscopic height data with spherical harmonics

Purpose: All currently considered parametric models used for decomposing videokeratoscopy height data are viewercentered and hence describe what the operator sees rather than what the surface is. The purpose of this study was to ascertain the applicability of an object-centered representation to modeling of corneal surfaces. Methods: A three-dimensional surface decomposition into a series of spherical harmonics is considered and compared with the traditional Zernike polynomial expansion for a range of videokeratoscopic height data. Results: Spherical harmonic decomposition led to significantly better fits to corneal surfaces (in terms of the root mean square error values) than the corresponding Zernike polynomial expansions with the same number of coefficients, for all considered corneal surfaces, corneal diameters, and model orders. Conclusions: Spherical harmonic decomposition is a viable alternative to Zernike polynomial decomposition. It achieves better fits to videokeratoscopic height data and has the advantage of an object-centered representation that could be particularly suited to the analysis of multiple corneal measurements.

Describing ocular aberrations with wavefront vergence maps

A common optometric problem is to specify the eye’s ocular aberrations in terms of Zernike coefficients and to reduce that specification to a prescription for the optimum sphero-cylindrical correcting lens. The typical approach is first to reconstruct wavefront phase errors from measurements of wavefront slopes obtained by a wavefront aberrometer. This paper applies a new method to this clinical problem that does not require wavefront reconstruction. Instead, we base our analysis of axial wavefront vergence as inferred directly from wavefront slopes. The result is a wavefront vergence map that is similar to the axial power maps in corneal topography and hence has a potential to be favoured by clinicians. We use our new set of orthogonal Zernike slope polynomials to systematically analyse details of the vergence map analogous to Zernike analysis of wavefront maps. The result is a vector of slope coefficients that describe fundamental aberration components. Three different methods for reducing slope coefficients to a spherocylindrical prescription in power vector forms are compared and contrasted. When the original wavefront contains only second order aberrations, the vergence map is a function of meridian only and the power vectors from all three methods are identical. The differences in the methods begin to appear as we include higher order aberrations, in which case the wavefront vergence map is more complicated. Finally, we discuss the advantages and limitations of vergence map representation of ocular aberrations.