Sample Percentiles Research Articles

Greater Socioeconomic Disadvantage as Measured by the Area Deprivation Index Is Associated With Failure of Healing Following Arthroscopic Repair of Massive Rotator Cuff Tears but Not With Clinical Outcomes

To analyze the relationship between Area Deprivation Index (ADI) and preoperative status and short-term postoperative clinical outcomes among patients who underwent arthroscopic rotator cuff repair (ARCR) of massive rotator cuff tears (MRCTs). A retrospective review was conducted on prospectively maintained data on patients who underwent ARCR of MRCTs defined as tear size ≥5 cm or complete tear of at least 2 tendons, with a minimum 2-year follow-up and a valid home address between January 2015 and December 2018. Each patient's home address was mapped to the ADI to determine neighborhood disadvantage. This composite index is composed of 17 census-based indicators, including income, education, employment, and housing quality to quantify the level of socioeconomic deprivation. Ratings were recorded and categorized based on the sample's percentile. Patients were then divided into 2 groups: upper quartile (ie, most disadvantaged [≥75th percentile]) and lower 3 quartiles (ie, least disadvantaged [<75th percentile]). Bivariate analysis was performed to associate ADI with patient-reported outcomes (PROs) and range of motion pre- and postoperatively, as well as complications, healing rate, satisfaction, and return to work. Patients reaching or exceeding the minimal clinically important difference for visual analog scale (VAS), American Shoulder and Elbow Surgeons, Veterans Rand 12-Item questionnaire, and subjective shoulder value were recorded for both cohorts. Ninety-nine patients were eligible for study analysis. Preoperative PROs and range of motion were similar, except for a greater VAS for pain (6.3 vs 4.3; P < .01) and lower American Shoulder and Elbow Surgeons score (32.2 vs 45.1; P= .01) in the most disadvantaged group. Both groups showed similar postoperative PROs scores, but greater VAS improvement was seen in the upper quartile group (Δ 4.2 vs Δ 3.0; P= .04). In contrast, only the least-disadvantaged group significantly improved in internal rotation (P= .01) and forward flexion (18°; P < .01) from baseline. Although satisfaction, complications, and return to work were comparable (P > .05), failure of healing occurred more frequently in the most disadvantaged group (21% vs 6%; P= .03). Patients with MRCTs residing in the most disadvantaged neighborhoods as measured by the ADI have more pain and functional limitations before undergoing ARCR but demonstrate similar postoperative functional improvements to patients from other socioeconomic backgrounds. Failure of healing of MRCTs may be more common in disadvantaged groups. Furthermore, both groups reported similar rates of clinically important functional improvement. Level III, retrospective cohort comparison.

Improving the robustness of spectral estimation to loud transients with a truncated order statistics filter

Underwater acoustic recordings often include loud transients from human or natural sources. The transients cause a positive bias for Welch's average periodogram spectral estimator when estimating the power spectral density of the background environment. Estimators based on single order statistics (e.g., the sample median) avoid the bias caused by outliers at the cost of a higher variance than the sample mean. Schwock and Abadi (2021) showed that, for exponential random variables, the estimator based on the 80th sample percentile has the lowest variance among any unbiased estimator based on a single order statistics. This work tests a hybrid approach between Welch's and order statistics estimators by performing a weighted sum of the quietest subset of ordered samples of the periodograms. By discarding the loudest samples of the periodogram, the truncated linear order statistics filter (TLOSF) reduces the bias caused by loud transients. By combining multiple order statistics into the estimate, the TLOSF achieves a lower variance than the 80th percentile estimator. The TLOSF reduced the MSE by 1dB compared to Schwock &amp; Abadi's 80th percentile estimator for a mixture combining an exponential distribution with 1% outliers 23 dB above the background. On periodograms of shallow water hydrophone recordings, the TLOSF yielded a lower output power in the frequency bins where both the Welch's and 80th percentile estimators had a positive bias due to loud transients. [Work supported by ONR Code 321US.]

Measuring Individual Benefits of Medical Treatments Using Longitudinal Hospital Data with Non-Ignorable Missing Responses Caused by Patient Discharge: Application to the Study of Benefits of Pain Management Post Spinal Fusion

Electronic health records (EHR) provide valuable resources for longitudinal studies and understanding risk factors associated with poor clinical outcomes. However, they may not contain complete follow-ups, and the missing data may not be at random since hospital discharge may depend in part on expected but unrecorded clinical outcomes that occur after patient discharge. These non-ignorable missing data requires appropriate analysis methods. Here, we are interested in measuring and analyzing individual treatment benefits of medical treatments in patients recorded in EHR databases. We present a method for predicting individual benefits that handles non-ignorable missingness due to hospital discharge. The longitudinal clinical outcome of interest is modeled simultaneously with the hospital length of stay using a joint mixed-effects model, and individual benefits are predicted through a frequentist approach: the empirical Bayesian approach. We illustrate our approach by assessing individual pain management benefits to patients who underwent spinal fusion surgery. By calculating sample percentiles of empirical Bayes predictors of individual benefits, we examine the evolution of individual benefits over time. We additionally compare these percentiles with percentiles calculated with a Monte Carlo approach. We showed that empirical Bayes predictors of individual benefits do not only allow examining benefits in specific patients but also reflect overall population trends reliably.

Outlier-robust tri-percentile parameter estimation of compound-Gaussian clutter with lognormal distributed texture

Compound-Gaussian model with lognormal distributed texture (CG-LNT) is one of effective amplitude distribution models to characterize high-resolution sea clutter and therefore it is important for high-resolution maritime radars to robustly estimate the parameters of the CG-LNT model from raw radar data, a mixture of sea clutter data and a small number of outliers. Existing estimators, such as moment-based and [zlog(z)]-based ones, are sensitive to outliers and fail to use in practical radars. In this paper, an outlier-robust tri-percentile estimator is proposed to realize the robust parameter estimation of the CG-LNT model of sea clutter. It is shown that the ratio of two percentiles is a monotonic function of the shape parameter and independent of the scale parameter, though without analytical expressions. In this way, the shape parameter is estimated from the ratio of the two sample percentiles by the look-up table method. Later, the scale parameter is estimated by the third sample percentile that is determined by the estimated shape parameter. Moreover, it is found that the positions of the three percentiles affect the performance and thus two empirical formulae are given to improve the estimate precision by numerical computation. Finally, the experimental results using simulated and real sea clutter data show that the proposed tri-percentile estimator implements robust and fast estimation of the parameters of the CG-LNT distribution.

Physical fitness of children and youth with asthma in comparison to the reference population

BackgroundPhysical fitness is an essential marker of health. The literature regarding the question of whether individuals with asthma have reduced physical fitness compared to their non-asthmatic peers is inconsistent and focuses on the cardiorespiratory endurance dimension. This study provides a comparison of different dimensions of physical fitness in individuals with and without asthma on the basis of the German population-based study “KiGGS” (German Health Interview and Examination Survey for Children and Adolescents) and its in-depth study “MoMo” (2009–2012: wave 1 and 2014–2017: wave 2).MethodsIn total, 7731 individuals aged 6–30 years were included in this cross-sectional analysis at two measurement waves, including 353 individuals with and 7378 without asthma. The 12-month prevalence of physician-diagnosed asthma was assessed by interview. Physical fitness was measured by six test items of the MoMo test profile. “Cardiorespiratory endurance” was measured by an ergometric test, “muscular strength” by standing long jump, push-ups and sit-ups and “coordination” by jumping sideways and balancing backwards. Because of the broad age range of the sample, age- and sex-specific percentiles were used. Physical activity, age, gender and general state of health were assessed by questionnaire.ResultsThe individuals with asthma reported a poorer general state of health at both measurement waves. However, the results of the fitness tests indicated that they were as physically fit as their peers without asthma in relation to cardiorespiratory endurance and muscular strength. The mean percentiles were all within the same range. The results of the comparisons of coordination performance were inconsistent. At wave 1 they were within the same range, at wave 2 individuals with asthma showed a poorer coordination performance (p = 0.041; HL = 4.125, CI of HL 0.155–8.125).ConclusionsTo the best of our knowledge, this is the first study to compare the physical fitness of individuals with and without asthma by considering several dimensions of physical fitness. The study demonstrates that cardiorespiratory endurance and muscular strength are not reduced in individuals with asthma. The results of the comparisons at the two measurement waves were remarkably stable.

1309Breastfeeding and weight retention in women with Gestational Diabetes Mellitus (GDM): LINDA-Brasil Cohort

Abstract Background Breastfeeding seems to be inversely related to postpartum weight retention (PPWR), but this relationship is unclear in women with GDM. We aimed to describe the association between breastfeeding and PPWR at 1 year postpartum in women with GDM. Methods Multicentre cohort study of 524 women with GDM enrolled in prenatal clinics of the Brazilian National Health System and followed by telephone up to one year postpartum. Socio-demographics and pre-gestational weight were obtained during recruitment. Breastfeeding and post-partum maternal weight through telephone calls. PPWR was defined as 1-year postpartum weight minus pre-pregnancy weight and categorized as above and below the 80th sample percentile of retention. Breastfeeding was categorized as duration greater or less than 6 months. Adjusted Poisson regression was performed to assess relative risks (RR). Results Most women were 30 to 39 years old (50.6%), multiparous (71.0%), 39.5% completed high school, 60.1% were non-white (60.1%). Mean pre-gestational BMI was 30.42 (± 6.7) kg/m² and the mean weight gain during pregnancy was 9.5 (± 7.1) kg. After adjustments for previous variables, breastfeeding for 6 months or more reduced by 35% (RR 0.65; 95% CI 0.47; 0.90, p &amp;lt; 0.01) the incidence of 1-year PPWR up to 80th percentile (≥7Kg) compared to breastfeeding less than 6 months. Conclusions Breastfeeding up to 6 months might be encouraged in women with GDM to help preventing higher levels of 1-year PPWR. Key messages Breastfeeding up to 6 months was associated with less PPWR at 1 year after delivery in women with GDM.

Assessing the environmental urban exposure to erbium and gadolinium in a healthy group of Spanish adolescents

BACKGROUND AND AIM: High-technology rare earth elements (REEs) are contaminants of emerging concern. We conducted a study to determine the environmental presence and exposure to two REEs frequently (Gadolinium; Gd) and not frequently (Erbium; Er) studied in topsoils and a teenage population group living in Alcalá de Henares (Spain). METHODS: Scalp hair was collected from 96 adolescents (13-16 years-old; 28 boys and 68 girls) born and residing in Alcalá. Er and Gd were analysed by ICP-MS after appropriate removal of exogenous contamination using Triton X-100/bath sonication. Additionally, both REEs were monitored in 70/167 topsoil samples randomly sampled from different urban parks across the city. Data was processed using statistical methods applied to censored data available in the ‘NADA’ statistical package. RESULTS:The limits of detection in hair and soils, respectively, were (in µg/g): Er (0.0004; 0.0005) and Gd (0.001; 0.0007). Both REEs were detected only in a few hair samples monitored: Er (8.2%) and Gd (1%). Owing to the high levels of censored data obtained, high sample percentiles were determined by means of the methods available in ‘NADA’. 95th percentile and range for Er in scalp hair in µg/g was 0.001 (0.0004-0.001); meanwhile the range for Gd was 0.001-0.002 µg/g. However, both REEs were detected in all topsoil samples monitored; median and percentiles in mg/kg are: Er 0.93 (0.64, 1.06), Gd 2.95 (2.07, 3.46). CONCLUSIONS:Our results could suggest that Alcalá’s teenage population would have not been significantly exposed to Er and Gd, as hair has been suggested a good matrix to quantitatively measure the absorption of REEs from environmental exposure. However, the high environmental presence and distribution of these two REEs detected in soils from Alcalá might represent a potential health risk for the population in the long-term owing to the increasing anthropic uses of REEs. KEYWORDS: Erbium, gadolinium, human hair, monitoring, Spanish teenagers, exposure.

Statistical Considerations in the Evaluation of Continuous Biomarkers.

Discovery of biomarkers has been steadily increasing over the past decade. Although a plethora of biomarkers has been reported in the biomedical literature, few have been sufficiently validated for broader clinical applications. One particular challenge that may have hindered the adoption of biomarkers into practice is the lack of reproducible biomarker cut points. In this article, we attempt to identify some common statistical issues related to biomarker cut point identification and provide guidance on proper evaluation, interpretation, and validation of such cut points. First, we illustrate how discretization of a continuous biomarker using sample percentiles results in significant information loss and should be avoided. Second, we review the popular "minimal-P-value" approach for cut point identification and show that this method results in highly unstable P values and unduly increases the chance of significant findings when the biomarker is not associated with outcome. Third, we critically review a common analysis strategy by which the selected biomarker cut point is used to categorize patients into different risk categories and then the difference in survival curves among these risk groups in the same dataset is claimed as the evidence supporting the biomarker's prognostic strength. We show that this method yields an exaggerated P value and overestimates the prognostic impact of the biomarker. We illustrate that the degree of the optimistic bias increases with the number of variables being considered in a risk model. Finally, we discuss methods to appropriately ascertain the additional prognostic contribution of the new biomarker in disease settings where standard prognostic factors already exist. Throughout the article, we use real examples in oncology to highlight relevant methodologic issues, and when appropriate, we use simulations to illustrate more abstract statistical concepts.

Outlier-robust tri-percentile parameter estimation of K-distributions

K-distribution is one of the most popular sea clutter amplitude models where the scale and shape parameters reflect the power level and non-Gaussianity of the clutter. In real oceanic environments, the two spatial-temporally varying parameters are estimated from sea clutter data with outliers. Outliers are probably from sea-surface ships, reefs, or abnormal scattering phenomena. Existing moment-based, numerical maximum likelihood, and [zlog(z)]-based estimators are sensitive to outliers, which results in abrupt degradation of parameter estimation precision in real clutter environments. This paper proposes an outlier-robust tri-percentile parameter estimator of K-distributions. It is shown that the ratio of two percentiles is a monotonically decreasing function of the shape parameter and independent of the scale parameter. In this way, the shape parameter is estimated from the ratio of two sample percentiles by the look-up table method and interpolation method. Moreover, an empirical formula on the optimal setup of the two percentiles is given by numerical computation. Next, the scale parameter is estimated by the third sample percentile whose position depends on the estimated shape parameter. Finally, the tri-percentile estimator is evaluated by using simulated and measured data, showing that it is rather robust to outliers in simulated data and raw sea clutter data.

A percentile-based estimator for the log-logistic function: Application to forestry

Abstract Developing a simplified estimation method without compromising the performance of the distribution is germane to forest modelling. Few estimation methods exist for the Log-Logistic distribution and are relatively complex. A simplified estimator for the Log-Logistic parameters will increase its application in diameter distribution yield systems. Therefore, in this study, a percentile-based estimator was applied for the Log-Logistic distribution. The Kolmogorov-Smirnov, Anderson-Darling and Cramer-von Mises statistics were used to evaluate the method in two natural forest stands and two monospecific plantations of Gmelina arborea Roxb. and Tectona grandis Linn. f. in Nigeria. The parameter recovery model (PRM) and parameter prediction model (PPM) were used to predict the diameter distributions of independent stands of G. arborea and T. grandis. The results showed that the percentile estimator did not compromise the quality of fits of the Log-Logistic function across the four forest stands and are comparable to the maximum likelihood estimator. The 25th and 75th, and 40th and 80th were the best sample percentiles for the estimator. The predicted diameter distributions of G. arborea and T. grandis stands from the PRM and PPM were reasonable and compare well with the observed distribution. Thus, either of the models can be incorporated into the growth and yield system of forest stand management.

A pooled percentile estimator for parallel simulations

ABSTRACT Percentile is an important risk measure quantifying the stochastic system random behaviours. This paper studies a pooled percentile estimator, which is the sample percentile of detailed simulation outputs after directly pooling independent sample paths together. We derive the asymptotic representation of the pooled percentile estimator and further prove its normality. By comparing with the classical percentile estimator used in stochastic simulation, both theoretical and empirical studies demonstrate the advantages of the proposal under the context of parallel simulation.

Percentile-based grain size distribution analysis tools (GSDtools) – estimating confidence limits and hypothesis tests for comparing two samples

Abstract. Most studies of gravel bed rivers present at least one bed surface grain size distribution, but there is almost never any information provided about the uncertainty in the percentile estimates. We present a simple method for estimating the grain size confidence intervals about sample percentiles derived from standard Wolman or pebble count samples of bed surface texture. The width of a grain size confidence interval depends on the confidence level selected by the user (e.g., 95 %), the number of stones sampled to generate the cumulative frequency distribution, and the shape of the frequency distribution itself. For a 95 % confidence level, the computed confidence interval would include the true grain size parameter in 95 out of 100 trials, on average. The method presented here uses binomial theory to calculate a percentile confidence interval for each percentile of interest, then maps that confidence interval onto the cumulative frequency distribution of the sample in order to calculate the more useful grain size confidence interval. The validity of this approach is confirmed by comparing the predictions using binomial theory with estimates of the grain size confidence interval based on repeated sampling from a known population. We also developed a two-sample test of the equality of a given grain size percentile (e.g., D50), which can be used to compare different sites, sampling methods, or operators. The test can be applied with either individual or binned grain size data. These analyses are implemented in the freely available GSDtools package, written in the R language. A solution using the normal approximation to the binomial distribution is implemented in a spreadsheet that accompanies this paper. Applying our approach to various samples of grain size distributions in the field, we find that the standard sample size of 100 observations is typically associated with uncertainty estimates ranging from about ±15 % to ±30 %, which may be unacceptably large for many applications. In comparison, a sample of 500 stones produces uncertainty estimates ranging from about ±9 % to ±18 %. In order to help workers develop appropriate sampling approaches that produce the desired level of precision, we present simple equations that approximate the proportional uncertainty associated with the 50th and 84th percentiles of the distribution as a function of sample size and sorting coefficient; the true uncertainty in any sample depends on the shape of the sample distribution and can only be accurately estimated once the sample has been collected.

When large n is not enough – Distribution-free interval estimators for ratios of quantiles

Ratios of sample percentiles or of quantiles based on a single sample are often published for skewed income data to illustrate aspects of income inequality, but distribution-free confidence intervals for such ratios are to our knowledge not in the literature. Here we derive and compare two large-sample methods for obtaining such intervals. They both require good distribution-free estimates of the quantile density at the quantiles of interest, and such estimates have recently become available. Simulation studies for various sample sizes are carried out for Pareto, lognormal and exponential distributions, as well as fitted generalized lambda distributions, to determine the coverage probabilities and widths of the intervals. Robustness of the estimators to contamination or a positive proportion of zero incomes is examined via influence functions. The motivating example is Australian household income data where ratios of quantiles measure inequality, but of course these results apply equally to data from other countries.

Weibull lagged effect step-stress model with competing risks

ABSTRACTIn many survival analysis studies, failure can come from one of several competing risks. Additionally, where survival times are lengthy, researchers can increase stress levels to cause units to fail faster. One type of accelerated testing is a step-stress test where the increase is presented in quantum jumps at predetermined time points. If the impact of the increase is not immediately attained, an interim lag period is modeled. In this article, we propose a two-competing risk step-stress model with a lag period where each independent risk follows a Weibull lifetime distribution, the interim lag period is linear, and the attainment point is assumed known. We obtain the maximum likelihood estimators and the observed information matrix; we construct confidence intervals and provide estimates of coverage probabilities using large sample theory, percentile bootstrap, and bias-corrected accelerated (BCa) bootstrap methods.

On Asymptotic Distributions and Confidence Intervals for LIFT Measures in Data Mining

A LIFT measure, such as the response rate, lift, or the percentage of captured response, is a fundamental measure of effectiveness for a scoring rule obtained from data mining, which is estimated from a set of validation data. In this article, we study how to construct confidence intervals of the LIFT measures. We point out the subtlety of this task and explain how simple binomial confidence intervals can have incorrect coverage probabilities, due to omitting variation from the sample percentile of the scoring rule. We derive the asymptotic distribution using some advanced empirical process theory and the functional delta method in the Appendix. The additional variation is shown to be related to a conditional mean response, which can be estimated by a local averaging of the responses over the scores from the validation data. Alternatively, a subsampling method is shown to provide a valid confidence interval, without needing to estimate the conditional mean response. Numerical experiments are conducted to compare these different methods regarding the coverage probabilities and the lengths of the resulting confidence intervals.

An Improved Weise’s Rule for Efficient Estimation of Stand Quadratic Mean Diameter

The main objective of this study was to explore the accuracy of Weise’s rule of thumb applied to an estimation of the quadratic mean diameter of a forest stand. Virtual stands of European beech (Fagus sylvatica L.) across a range of structure types were stochastically generated and random sampling was simulated. We compared the bias and accuracy of stand quadratic mean diameter estimates, employing different ranks of measured stems from a set of the 10 trees nearest to the sampling point. We proposed several modifications of the original Weise’s rule based on the measurement and averaging of two different ranks centered to a target rank. In accordance with the original formulation of the empirical rule, we recommend the application of the measurement of the 6th stem in rank corresponding to the 55% sample percentile of diameter distribution, irrespective of mean diameter size and degree of diameter dispersion. The study also revealed that the application of appropriate two-measurement modifications of Weise’s method, the 4th and 8th ranks or 3rd and 9th ranks averaged to the 6th central rank, should be preferred over the classic one-measurement estimation. The modified versions are characterised by an improved accuracy (about 25%) without statistically significant bias and measurement costs comparable to the classic Weise method.

Statistical Evaluation of Several Methods for Cut-Point Determination of Immunogenicity Screening Assay

The cut point of the immunogenicity screening assay is the level of response of the immunogenicity screening assay at or above which a sample is defined to be positive and below which it is defined to be negative. The Food and Drug Administration Guidance for Industry on Assay Development for Immunogenicity Testing of Therapeutic recommends the cut point to be an upper 95 percentile of the negative control patients. In this article, we assume that the assay data are a random sample from a normal distribution. The sample normal percentile is a point estimate with a variability that decreases with the increase of sample size. Therefore, the sample percentile does not assure at least 5% false-positive rate (FPR) with a high confidence level (e.g., 90%) when the sample size is not sufficiently enough. With this concern, we propose to use a lower confidence limit for a percentile as the cut point instead. We have conducted an extensive literature review on the estimation of the statistical cut point and compare several selected methods for the immunogenicity screening assay cut-point determination in terms of bias, the coverage probability, and FPR. The selected methods evaluated for the immunogenicity screening assay cut-point determination are sample normal percentile, the exact lower confidence limit of a normal percentile (Chakraborti and Li, 2007) and the approximate lower confidence limit of a normal percentile. It is shown that the actual coverage probability for the lower confidence limit of a normal percentile using approximate normal method is much larger than the required confidence level with a small number of assays conducted in practice. We recommend using the exact lower confidence limit of a normal percentile for cut-point determination.

Serum liver enzymes are associated with all‐cause mortality in an elderly population

Little is known about the association of serum liver enzymes with long-term outcome in the elderly. We sought to clarify the association of serum gamma-glutamyltransferase (GGT), alkaline phosphatase (ALP), alanine aminotransferase (ALT) and aspartate aminotransferase (AST) with all-cause and cause-specific mortality in an elderly population. This study was embedded in the Rotterdam Study, a large population-based cohort of persons aged 55 years or older. Cox-regression analyses were performed to examine the association of baseline serum GGT, ALP, and aminotransferase levels with mortality, adjusted for age, sex, education, smoking status, alcohol intake, hypertension, diabetes mellitus, body mass index and total cholesterol levels. Liver enzyme levels were categorized according to sample percentiles; levels <25th percentile were taken as a reference. During a follow-up of up to 19.5 years, 2997 of 5186(57.8%) participants died: 672 participants died of causes related to cardiovascular diseases (CVD) and 703 participants died of cancer. All serum liver enzymes were associated with all-cause mortality (all P < 0.001). Moreover, GGT was associated with increased CVD mortality (P < 0.001), and ALP and AST with increased cancer-related mortality (P = 0.03 and P = 0.005 respectively). Participants with GGT and ALP in the top 5% had the highest risk for all-cause mortality (HR1.55; 95%CI 1.30-1.85 and HR1.49; 95%CI 1.25-1.78 respectively). AST and ALT <25th percentile were also associated with a higher risk of all-cause mortality. All serum liver enzymes were positively associated with long-term mortality in this elderly population. Why participants with low ALT and AST levels have higher risk of mortality remains to be elucidated.

Who is stressed? Comparing cortisol levels between individuals

Cortisol is the most commonly used biomarker to compare physiological stress between individuals. Its use, however, is frequently inappropriate. Basal cortisol production varies markedly between individuals. Yet, in naturalistic studies that variation is often ignored, potentially leading to important biases. Identify appropriate analytical tools to compare cortisol across individuals and outline simple simulation procedures for determining the number of measurements required to apply those methods. We evaluate and compare three alternative methods (raw values, Z-scores, and sample percentiles) to rank individuals according to their cortisol levels. We apply each of these methods to first morning urinary cortisol data collected thrice weekly from 14 cycling Mayan Kaqchiquel women. We also outline a simple simulation to estimate appropriate sample sizes. Cortisol values varied substantially across women (ranges: means: 1.9-2.7; medians: 1.9-2.8; SD: 0.26-0.49) as did their individual distributions. Cortisol values within women were uncorrelated. The accuracy of the rankings obtained using the Z-scores and sample percentiles was similar, and both were superior to those obtained using the cross-sectional cortisol values. Given the interindividual variation observed in our population, 10-15 cortisol measurements per participant provide an acceptable degree of accuracy for across-women comparisons. The use of single raw cortisol values is inadequate to compare physiological stress levels across individuals. If the distributions of individuals' cortisol values are approximately normal, then the standardized ranking method is most appropriate; otherwise, the sample percentile method is advised. These methods may be applied to compare stress levels across individuals in other populations and species.

Software for Weibull Inference

ABSTRACT Exact inference, that is, confidence limits and hypothesis tests for the Weibull distribution parameters and percentiles based on maximum likelihood estimation in complete or type II censored samples requires the determination via simulation of percentage points of the distribution of certain pivotal quantities. Tables have been published for a limited range of sample sizes and are scattered through the literature. The program Pivotal.exe described in this article performs these calculations for user-selected sample sizes and percentiles of interest. The software is also applicable to inference on series systems of identical Weibull distributed components and for analyzing the results of sudden death tests. The software allows the output of 10,000 paired values of the ML estimates of the Weibull shape and scale parameters that can be postprocessed using spreadsheet-based software to provide Operating characteristic (OC) curves for hypothesis tests on a Weibull percentile. Confidence intervals on reliability. Prediction intervals for a future value.