Optimal designs when the variance is a function of the mean

We develop locally D-optimal designs for nonlinear models when the variance of the response is a function of its mean. Using the two-parameter Michaelis-Menten model as an example, we show that the optimal design depends on both the type of heteroscedasticity and the magnitude of the variation. In addition, our results suggest that the homoscedastic D-optimal design has high efficiency under a broad class of heteroscedastic patterns and that it is fairly insensitive to nominal values of the parameters.     

Bayesian optimal designs for Phase I clinical trials

A broad approach to the design of Phase I clinical trials for the efficient estimation of the maximum tolerated dose is presented. The method is rooted in formal optimal design theory and involves the construction of constrained Bayesian c- and D-optimal designs. The imposed constraint incorporates the optimal design points and their weights and ensures that the probability that an administered dose exceeds the maximum acceptable dose is low. Results relating to these constrained designs for log doses on the real line are described and the associated equivalence theorem is given. The ideas are extended to more practical situations, specifically to those involving discrete doses. In particular, a Bayesian sequential optimal design scheme comprising a pilot study on a small number of patients followed by the allocation of patients to doses one at a time is developed and its properties explored by simulation.     

D-optimal design applied to binding saturation curves of an enkephalin analog in rat brain

The D-optimal design, a minimal sample design that minimizes the volume of the joint confidence region for the parameters, was used to evaluate binding parameters in a saturation curve with a view to reducing the number of experimental points without loosing accuracy in binding parameter estimates. Binding saturation experiments were performed in rat brain crude membrane preparations with the opioid mu-selective ligand [3H]-[D-Ala2,MePhe4,Gly-ol5]enkephalin (DAGO), using a sequential procedure. The first experiment consisted of a wide-range saturation curve, which confirmed that [3H]-DAGO binds only one class of specific sites and non-specific sites, and gave information on the experimental range and a first estimate of binding affinity (Ka), capacity (Bmax) and non-specific constant (k). On this basis the D-optimal design was computed and sequential experiments were performed each covering a wide-range traditional saturation curve, the D-optimal design and a splitting of the D-optimal design with the addition of 2 points (+/- 15% of the central point). No appreciable differences were obtained with these designs in parameter estimates and their accuracy. Thus sequential experiments based on D-optimal design seem a valid method for accurate determination of binding parameters, using far fewer points with no loss in parameter estimation accuracy.     

Retrospective analysis of sequential dose-finding designs

The continual reassessment method (CRM) is a dose-finding design using a dynamic sequential updating scheme. In common with other dynamic schemes the method estimates a current dose level corresponding to some target percentile for experimentation. The estimate is based on all included subjects. This continual reevaluation is made possible by the use of a simple model. As it stands, neither the CRM, nor any of the other dynamic schemes, allow for the correct estimation of some target percentile, based on retrospective data apart from the exceptional situation in which the simplified model exactly generates the observations. In this article we focus on the very specific issue of retrospective analysis of data generated by some arbitrary mechanism and subsequently analyzed via the continual reassessment method. We show how this can be done consistently. The proposed methodology is not restricted to that particular design and is applicable to any sequential updating scheme in which dose levels are associated with percentiles via model inversion.     

Minimax D-optimal designs for the logistic model

We propose an algorithm for constructing minimax D-optimal designs for the logistic model when only the ranges of the values for both parameters are assumed known. Properties of these designs are studied and compared with optimal Bayesian designs and Sitter's (1992, Biometrics, 48, 1145-1155) minimax D-optimal kk-designs. Examples of minimax D-optimal designs are presented for the logistic and power logistic models, including a dose-response design for rheumatoid arthritis patients.     

Experimental design for the identification of macrokinetic models and model discrimination

An experimental design method for the identification of macrokinetic models was developed applying an extended D-optimal design criterion. The D-optimal design criterion was modified to consider variable measurement variances as well as multivariate macrokinetic models. The macrokinetics of formate dehydrogenase (FDH) production with Candida boidinii were thus identified within 10 steady state experiments in a labscale continuous stirred tank reactor (10 model parameters). Closed loop control (nutristat) was applied to set-up the operating states suggested by this experimental design method. After each set of steady state experiments the quality of macrokinetic parameters was characterized statistically. For model discrimination a parameter discrimination algorithm based on entropy formulations was adapted. Again a multivariate criterion considering variable measurement variances was developed. This discrimination algorithm was applied to discriminate the macrokinetic model of FDH production with Candida boidinii out of 10 different macrokinetic approaches. An unequivocal discrimination result could be obtained calculating model specific probabilities. These were compared with commonly used sum of squares values. (c) 1997 John Wiley & Sons, Inc. Biotechnol Bioeng 56: 564-576, 1997.     

A genetic algorithm approach to the inverse problem of treatment planning for intensity-modulated radiotherapy

A genetic algorithm optimization approach for designing treatment plans in intensity-modulated radiotherapy is proposed. The approach determines the beam intensities of the pencil-beam dose model such that the optimized dose distribution closely matches the prescribed dose distribution. The approach indirectly inverts the ill-conditioned dose-projection matrix, which can be very large and extremely sparse. The beam intensities are treated as chromosomes that are encoded as binary strings. The approach was used to design treatment plans for two deceptive clinical test cases. In both case, cancerous tissues in the planning target region received at least 98% of the prescribed dose level while dose levels delivered to the organs at risk were well within safe limits, with a maximum exposure of 2.5 and 52.5% of the prescribed tolerance level for the brain and prostrate cancer cases, respectively. Dose levels delivered to the healthy tissues were small with a mean exposure of 22.8 and 23.5% of the prescribed tolerance level.     

A statistical framework for quantile equivalence clinical trials with application to pharmacokinetic studies that bridge from HIV-infected adults to children

SUMMARY: Bridging clinical trials are sometimes designed to evaluate whether a proposed dose for use in one population, for example, children, gives similar pharmacokinetic (PK) levels, or has similar effects on a surrogate marker as an established effective dose used in another population, for example, adults. For HIV bridging trials, because of the increased risk of viral resistance to drugs at low PK levels, the goal is often to determine whether the doses used in different populations result in similar percentages of patients with low PK levels. For example, it may be desired to evaluate that a proposed pediatric dose gives approximately 10% of children with PK levels below the 10th percentile of PK levels for the established adult dose. However, the 10th percentile for the adult dose is often imprecisely estimated in studies of relatively small size. Little attention has been given to the statistical framework for such bridging studies. In this article, a formal framework for the design and analysis of quantile-based bridging studies is proposed. The methodology is then developed for normally distributed outcome measures from both frequentist and Bayesian directions. Sample size and other design considerations are discussed.     

Modeling the effect of dose on the lifetime tumor rate from an animal carcinogenicity experiment

The proportion of tumor-bearing animals in a chronic bioassay is often used as a basis for assessing risk at human exposure levels that are below the experimental range. However, such a crude experimental tumor rate can be biased by dose-related differences in the nontumor mortality. To adjust tumor rates for competing mortality, Kodell, Gaylor, and Chen (1986, Biometrics 42, 867-873) propose a standardized tumor rate, calculated separately in each dose group from dose-specific estimates of the tumor prevalence and mortality functions. This paper extends the approach of Kodell et al, by developing a method of modeling the standardized rate as a function of dose. An advantage of this approach is that it leads to estimates that are monotone in dose. In addition, by modeling the lifetime risk as a function of dose directly, it is possible to obtain estimates for the risk at dose levels outside the experimental range, such as for low-dose extrapolation, and estimation of the "tumorigenic dose 50" (TD50). A semiparametric model is developed, as well as the nonparametric alternative of isotonic regression. Simulations are provided to show the relative bias and precision of the two approaches to that of Kodell et al.     

DESIGN: computerized optimization of experimental design for estimating Kd and Bmax in ligand binding experiments. I. Homologous and heterologous binding to one or two classes of sites

We have developed a versatile computer program for optimization of ligand binding experiments (e.g., radioreceptor assay system for hormones, drugs, etc.). This optimization algorithm is based on an overall measure of precision of the parameter estimates (D-optimality). The program DESIGN uses an exact mathematical model of the equilibrium ligand binding system with up to two ligands binding to any number of classes of binding sites. The program produces a minimal list of the optimal ligand concentrations for use in the binding experiment. This potentially reduces the time and cost necessary to perform a binding experiment. The program allows comparison of any proposed experimental design with the D-optimal design or with assay protocols in current use. The level of nonspecific binding is regarded as an unknown parameter of the system, along with the affinity constant (Kd) and binding capacity (Bmax). Selected parameters can be fixed at constant values and thereby excluded from the optimization algorithm. Emphasis may be placed on improving the precision of a single parameter or on improving the precision of all the parameters simultaneously. We present optimal designs for several of the more commonly used assay protocols (saturation binding with a single labeled ligand, competition or displacement curve, one or two classes of binding sites), and evaluate the robustness of these designs to changes in parameter values of the underlying models. We also derive the theoretical D-optimal design for the saturation binding experiment with a homogeneous receptor class.     

Changes in cardiovascular disease risk factors from age 9 to 19 and the influence of television viewing

Objective:

The purpose of this study was to determine if time spent watching television is associated with changes in cardiovascular disease (CVD) risk factors from age 9 to 19.

Design and Methods:

Participants were girls enrolled in the NHLBI Growth and Health Study (n = 1,702), and CVD risk factors were measured at ages 10, 12, 14, 16, and 19. Longitudinal quantile regression was used to determine if television viewing was associated with changes in CVD risk factors over time at the 10th, 25th, 50th, 75th, and 90th percentiles of the CVD risk factors.

Results:

In black girls, television viewing (h/wk) was positively associated with sum of skinfolds (mm) at the 75th (0.22, 99% CI, 0.06‐0.38) and 90th percentiles (0.21, 99% CI, 0.05‐0.36), but not at the 10th, 25th, or 50th percentiles. In white girls, television viewing (h/wk) was positively associated with sum of skinfolds (mm) with the strength of the associations progressively stronger toward the upper tail of the skinfold distribution, adjusting for physical activity and other covariates (10th percentile: 0.08, 99% CI, 0.03‐0.13; 90th percentile: 0.42, 99% CI, 0.24‐0.59). All associations were adjusted for physical activity, dietary factors, sleep, and maturation. No associations were observed between television viewing and changes in systolic blood pressure, triglycerides, or cholesterol levels in black or white girls.

Conclusion:

Girls who spent more time watching television had greater increases in sum of skinfolds from age 9 to 19, independent of physical activity levels and other covariates. The associations observed were stronger at the upper tail of the skinfold distribution.     

A Probabilistic Ecological Risk Assessment of Tributyltin in Surface Waters of the Chesapeake Bay Watershed

The goal of this study was to conduct a probabilistic ecological risk assessment for tributyltin (TBT) in surface waters of the Chesapeake Bay watershed. Ecological risk was characterized by comparing the probability distributions of environmental exposure concentrations with the probability distributions of species response data determined from laboratory studies. The overlap of these distributions was a measure of risk to aquatic life. Tributyltin exposure data from the Chesapeake Bay watershed were available from over 3600 water column samples from 41 stations in nine basins from 1985 through 1996. Most of the stations were located in the Virginia waters of Chesapeake Bay, primarily the James, Elizabeth and York Rivers. In Maryland waters of the Bay, various marina, harbor and river systems were also sampled. As expected, the highest environmental concentrations of tributyltin (based on 90th percentiles) were reported in and near marina areas. The sources of TBT causing these high concentrations were primarily boat hulls and painting/depainting operations. Lower concentrations of TBT were reported in open water areas, such as the Potomac River, Choptank River and C and D Canal, where the density of boats was minimal. Temporal data from a ten year data base (1986-1996) from two areas in Virginia showed that TBT water column concentrations have declined since 1987 legislation prohibited the use of TBT paints on recreation boats (<25?m). Acute saltwater and freshwater TBT toxicity data were available for 43 and 23 species, respectively. Acute effects for saltwater species were reported for concentrations exceeding 420?ng/L; the lowest acute value for a freshwater species was 1110?ng/L. The acute 10th percentiles for all saltwater and freshwater species were 320 and 103?ng/L, respectively. The order of sensitivity from most to least sensitive for saltwater trophic groups and corresponding acute 10th percentiles were as follows: zooplankton (5?ng/L), phytoplankton (124?ng/L), benthos (312?ng/L) and fish (1009?ng/L). For freshwater species, the order of sensitivity from most to least sensitive trophic groups and corresponding acute 10th percentiles were: benthos (44?ng/L), zooplankton (400?ng/L), and fish (849?ng/L). Chronic data for both saltwater and freshwater species were limited to a few species in each water type. Based on these limited data, the saltwater and freshwater chronic 10th percentiles were 5 and 102?ng/L, respectively. Limited mesocosm and microcosm studies in saltwater suggested that TBT concentrations less than 50?ng/L did not impact the structure and function of biological communities. The saltwater acute (320?ng/L) and chronic (5?ng/L) 10th percentiles were used to determine ecological risk because all exposure data were from saltwater areas of the Chesapeake Bay watershed. Highest ecological risk was reported for marina areas in Maryland waters of Chesapeake Bay and for areas in Virginia such as the Elizabeth River, Hampton Creek and Sarah Creek. Low ecological risk was reported for areas such as the Potomac River, Choptank River, C and D Canal and Norfolk Harbor. Regulation of TBT on recreational watercraft in 1987 has successfully reduced water column concentrations of this organometallic compound. However, various studies have showed that TBT may remain in the sediment for years and continue to be source for water column exposures.     

All MoMs are not equal: some statistical properties associated with reporting results in the form of multiples of the median.

The statistical procedure for discriminating between a Down syndrome or neural tube defect (NTD) fetus and a normal fetus relies, to a great extent, on the reporting of maternal serum alpha-fetoprotein (MSAFP), hCG, and uE3 results in the form of multiples of the median (MoMs). Further, threshold MoMs values for MSAFP, such as 2.5 MoMs, are often used to define a reference range to identify an NTD fetus. We show that a constant threshold-MoMs cutoff for MSAFP values actually refers to different percentiles of MSAFP levels at different gestational ages and that the combining of MoMs values between centers and gestational ages, such as suggested by Wald et al. for deriving a patient-specific risk index, is highly questionable. The results presented in this paper are quite general and will apply to all situations where MoMs are used.     

A Quantile Regression Approach Can Reveal the Effect of Fruit and Vegetable Consumption on Plasma Homocysteine Levels

Introduction

A reduction in homocysteine concentration due to the use of supplemental folic acid is well recognized, although evidence of the same effect for natural folate sources, such as fruits and vegetables (FV), is lacking. The traditional statistical analysis approaches do not provide further information. As an alternative, quantile regression allows for the exploration of the effects of covariates through percentiles of the conditional distribution of the dependent variable.

Objective

To investigate how the associations of FV intake with plasma total homocysteine (tHcy) differ through percentiles in the distribution using quantile regression.

Materials and Methods

A cross-sectional population-based survey was conducted among 499 residents of Sao Paulo City, Brazil. The participants provided food intake and fasting blood samples. Fruit and vegetable intake was predicted by adjusting for day-to-day variation using a proper measurement error model. We performed a quantile regression to verify the association between tHcy and the predicted FV intake. The predicted values of tHcy for each percentile model were calculated considering an increase of 200 g in the FV intake for each percentile.

Results

The results showed that tHcy was inversely associated with FV intake when assessed by linear regression whereas, the association was different when using quantile regression. The relationship with FV consumption was inverse and significant for almost all percentiles of tHcy. The coefficients increased as the percentile of tHcy increased. A simulated increase of 200 g in the FV intake could decrease the tHcy levels in the overall percentiles, but the higher percentiles of tHcy benefited more.

Conclusions

This study confirms that the effect of FV intake on lowering the tHcy levels is dependent on the level of tHcy using an innovative statistical approach. From a public health point of view, encouraging people to increase FV intake would benefit people with high levels of tHcy.     

Radiation injury: some aspects of the oncogenic effects.

The late effects or irradiation stem from cell killing, mutation, and malignant transformation. Cancer is the major somatic late effect of exposure to low dose levels of radiation, and estimates of risk of cancer in man after irradiation are based entirely on human experience. The data for dose-response relationships for the induction of tumors by external irradiation in man have been obtained from a single exposure or a small number of exposures delivered at high dose rates. In contrast, exposure to environmental irradiation is mainly protracted over a long period of time and is delivered at a low dose rate. As yet no allowance has been made for the effect of protraction of the exposure time in estimating the risk of cancer, although an adjustment has been made in the case of estimates of genetic risk. Incidence of tumors has been the only parameter used for risk estimates, but latent period and degree of malignancy, which are probably both dose and dose-rate dependent, influence the nature of the risk from radiation. As the knowledge about the effects of low-level radiation has been accumulated and assimilated over the last 70 years, so has the concern for reasonable standards of safety. There are still problems in the estimation of radiation risks, but at least many of the relevant questions can now be framed. The problems of estimating risks for chemical carcinogens are clearly greater, but the experience gained from radiation studies should help in the design of the necessary experiments.     

Solar UV exposure of children in a summer school in Valencia,Spain

Ultraviolet (UV) exposure is the major environmental factor involved in the development of skin cancers and occurs mainly during outdoor activities. During summer schools, children receive regular and significant solar ultraviolet erythemal radiation (UVER) while practising outdoor activities. Personal dosimeters (VioSpor) were attached to the shoulders of schoolchildren and used to quantify their exposure to UVER. The study took place in Valencia, Spain, during July 2008, with three age groups (7–8, 9–10 and 11–12 years old) and involved about 15 schoolchildren. The median (25, 75 percentiles) twice-daily UV exposure values for all groups was 5.49 (3.59, 8.00) standard erythemal doses (SEDs), where 1 SED is defined as effective 100 Jm⁻² when weighted with the CIE erythemal response function. Exposure ratio (ER) is defined as the ratio between the personal dose on a selected body site and the corresponding ambient dose received on a horizontal plane during the same exposure period. The median (25, 75 percentiles) ER value for all groups in the study was 5.9% (4.1, 8.7).     

Exposure estimates of chromeplaters in India: an exploratory study

The literature has a paucity of knowledge on the exposure and effect estimates of chromeplaters in India. In an exploratory endeavour on chromium (Cr) exposure risk assessment, blood and urinary Cr levels plus the DNA-protein crosslink content were analysed in peripheral blood lymphocytes of chromeplaters (n=24). A cross-sectional study design was selected. Non-chromeplaters (n=35) were taken as the matching control. The results show that levels of blood and urinary Cr were greater in chromeplaters. A significant increase in DNA-protein crosslink coefficients of peripheral blood lymphocytes and urinary Cr levels was observed. The results demonstrate higher exposure estimates in chromeplaters and reveal exposure to a biologically effective dose of the toxic metal. The study also validated the employed biomarkers for Cr exposure risk assessment.     

Risk‐Group‐Specific Dose Finding Based on an Average Toxicity Score

Summary We propose a Bayesian dose‐finding design that accounts for two important factors, the severity of toxicity and heterogeneity in patients' susceptibility to toxicity. We consider toxicity outcomes with various levels of severity and define appropriate scores for these severity levels. We then use a multinomial‐likelihood function and a Dirichlet prior to model the probabilities of these toxicity scores at each dose, and characterize the overall toxicity using an average toxicity score (ATS) parameter. To address the issue of heterogeneity in patients' susceptibility to toxicity, we categorize patients into different risk groups based on their susceptibility. A Bayesian isotonic transformation is applied to induce an order‐restricted posterior inference on the ATS. We demonstrate the performance of the proposed dose‐finding design using simulations based on a clinical trial in multiple myeloma.     

Identification of variables influencing resistin serum levels in patients with type 1 and type 2 diabetes mellitus.

BACKGROUND: Resistin, a peptide hormone, has been discussed controversially as a missing link between obesity and insulin resistance. In contrast to resistin mRNA expression in adipose tissue, data on human serum levels in obesity and diabetes mellitus is scarce. The physiological range of serum resistin levels, reference values or adjusted percentiles have not yet been determined, making the interpretation of serum resistin concentrations quite difficult. METHODS: Resistin serum concentrations were measured systematically by ELISA in 216 healthy controls, 555 patients with type 2 diabetes and 114 patients with type 1 diabetes. Mean values, median, and range were determined, and BMI-, gender-, and disease-adapted percentiles were calculated for all subgroups. RESULTS: Age and gender did not have any influence on resistin levels. BMI and resistin levels were positively correlated in healthy controls (p = 0.02), albeit with a weak correlation coefficient. This correlation was absent in patients with type 1 and type 2 diabetes. In both genders, healthy controls had significantly higher resistin levels than patients with type 1 and type 2 diabetes (7.9 +/- 0.2 ng/ml vs. 5.7 +/- 0.2 ng/ml and 5.5 +/- 0.1 ng/ml, respectively; p < 0.0001). There was no correlation between resistin levels and occurrence of diabetic retinopathy or nephropathy. CONCLUSIONS: Serum resistin levels can be measured by ELISA over a broad range from 0.6 ng/ml up to 27.7 ng/ml, suggesting that percentiles might be helpful in the interpretation of an individuals resistin value. While age and gender do not influence resistin levels, BMI and occurrence of diabetes have to be considered.