Box-Cox Power Exponential Distribution Research Articles

Disparities with global standards about growth references of mid-upper arm circumference-for-age for Pakistani children aged 6-60 months.

Growth reference values about mid-upper arm circumference (MUAC) are vital for assessing children's nutritional status. However, Pakistan lacks these reference values and growth charts. This study aims to develop these for children aged 6-60 months and compare them with global standards. The data were acquired from the 2018 National Nutrition Survey of Pakistan, which was conducted by the United Nations Children's Fund (UNICEF) during 2018-2019. The final study cohort comprised 57 285 children, with 51% being boys. Percentile values and charts for MUAC-for-age were developed using generalised additive models for location, scale and shape with the Box-Cox power exponential distribution. The mean MUAC was 14.21 cm (±2.07 cm) and 14.13 cm (±2.12 cm) for the boys and girls, respectively. At 60 months of age, the P3 and P97 percentiles for girls were slightly higher than those for boys. The median percentiles of Pakistani children were smaller than the World Health Organisation 2007 standards and with international references. We observed disparities in MUAC-for-age growth references among Pakistani children compared to global standards, highlighting regional, age and gender variations. This underscores the need for developing countries like Pakistan to establish their growth references.

Kalman Filter Method for Handling Missing Values in Soil Moisture Sensor Data

Several imputation techniques have been developed specifically to deal with missing values. This research used data from soil moisture sensors for 34 days where there are missing values. Size of soil moisture sensor data to be quite large. So that with the missing value, it is difficult to determine how well the imputation technique is applied. Therefore, imputation technique is performed on the generated data based on the distribution of soil moisture sensor data so that an evaluation of the utilization of the imputation technique can be carried out on large data containing missing values. The method used in this study is Kalman Filter. Evaluation using the Mean Average Percentage Error (MAPE) after intentionally removed using the Missing Completely at Random (MCAR) technique with a missing rate of 5%, 10%, and 15%. The results showed that soil moisture data has a Box-Cox power exponential distribution. It was found that for generating data, Kalman Filter method has not much different MAPE value for 100 and 1000 data with missing rate was 5%, 10%, and 15%. The results of estimating missing values with the Kalman Filter on the soil moisture sensor data are in line with the soil moisture sensor data

Generation of percentile curves for strength and functional abilities for boys with Duchenne muscular dystrophy.

Considering the heterogeneity of the clinical manifestations of Duchenne muscular dystrophy (DMD), it is important to describe their various clinical profiles. Thus, in this study we aimed to develop percentile curves for DMD using a battery of measures to define the patterns of functional abilities, timed tests, muscle strength, and range of motion (ROM). This retrospective data analysis was based on the records of patients with DMD using the Motor Function Measure (MFM) scale, isometric muscle strength (IS), dorsiflexion ROM, 10-meter walk test (10 MWT), and 6-minute walk test (6 MWT). Percentile curves (25th, 50th, and 75th percentiles) with MFM, IS, ROM, 10 MWT, and 6 MWT on the y axis and patient age on the x axis were constructed using the generalized additive model for location, scale, and shape, with Box-Cox power exponential distribution. There were records of 329 assessments of patients between 4 and 18 years of age. The MFM percentiles showed a gradual reduction in all dimensions. Muscle strength and ROM percentiles showed that the knee extensors were the most affected from 4 years of age, and dorsiflexion ROM negative values were noted from the age of 8 years. The 10 MWT showed a gradual increase in performance time with age. For the 6 MWT, the distance curve remained stable until 8 years, with a subsequent progressive decline. In this study we generated percentile curves that can help health professionals and caregivers follow the trajectory of disease progression in DMD patients.

Statistical Confirmation of a Method of US Determination of Bone Age

Background There is limited literature on conventional US to assess bone age. Purpose To determine the diagnostic performance of US in the assessment of abnormal bone age in Chinese children. Materials and Methods In this prospective study, children and young adults aged between birth and 19 years from a large provincial teaching hospital were enrolled from January to November 2020. Children without clinical diseases potentially affecting skeletal growth were included in the normal-value group. Children with clinically suspected growth disturbances who were undergoing bone age evaluation were included in the validation group. Ossification ratios (ie, the ratio of the height of the epiphyseal ossification center to the entire epiphysis, including the cartilaginous component) of the radius, ulna, and femur from all the children were measured using US. Ultrasonic skeletal maturity scores (ie, the summation of ossification ratios of the radius, ulna, and femur multiplied by 100) collected from children in the normal-value group were used for score-for-age curve fitting through Box-Cox power exponential distribution. Test performance characteristics for the ability of US to help diagnose abnormal bone age were determined using radiographic bone age as a reference standard. Statistically significant difference between groups was determined by using a paired-sample t test. Results A total of 1089 children (median age, 9 years [interquartile range, 3-14 years]; 578 boys) were enrolled, including 929 children (mean age, 8 years [interquartile range, 4-12 years]; 515 boys) in the normal-value group and 160 children (mean age, 9 years [interquartile range, 7-11 years]; 63 boys) in the validation group. Ultrasonic bone ages in the validation group were evaluated with use of the lists of normal score-for-age values. With radiographic bone age as a reference standard, US could help diagnose abnormal bone age with high sensitivity (93% [14 of 15 participants; 95% CI: 66, 100] for boys, 100% [14 of 14 participants; 95% CI: 73, 100] for girls) and specificity (98% [47 of 48 participants; 95% CI: 88, 100] for boys, 98% [81 of 83 participants; 95% CI: 91, 100] for girls). Conclusion The US scoring system established can be used to evaluate bone age with high sensitivity and specificity. Clinical trial registration no. ChiCTR1900027917 © RSNA, 2021 Online supplemental material is available for this article.

Using an Algorithm to Fit a GAMLSS Model on Dry Matter Data from Brachiaria brizantha.

<b>Background and Objective:</b> Forage production in the tropics is generally asymmetrically distributed. Hence the need to use more complex models, especially when multiple comparisons are made and there are very large deviations from normality. The objective of this research is to fit a Generalized Additive Model for Location, Scale and Shape (GAMLSS) model on accumulated dry matter data from <i>Brachiaria brizantha</i> using a model selection algorithm. <b>Materials and Methods:</b> A Box-Cox Power Exponential (BCPE) distribution was adjusted on the dry matter from <i>Brachiaria brizantha</i> data implementing GAMLSS in R (programming language). The accumulated dry matter data for <i>B. brizantha</i> were obtained from a study carried out on a farm in the state of Portuguesa, Venezuela. The explanatory covariate x was the interval between cuts (21, 28, 35 and 42 days). <b>Results:</b> The dependent variable (dry matter) exhibited both skewness and kurtosis. GAMLSS allowed flexible modeling of both the distribution of the dry matter yield from <i>B. brizantha</i> and the dependence of all the parameters of the distribution on intervals between cuttings. For the dry matter yield from <i>B. brizantha</i>, which exhibited skewness and leptokurtosis, the BCPE distribution, provided the best fit. <b>Conclusion:</b> The interval between cuttings showed an effect that is reflected in the average yield of dry matter from <i>B. brizantha</i>. The interval between cuts affected the skewness and the kurtosis of the distribution.

Bias-Variance Trade-Off in Continuous Test Norming.

In continuous test norming, the test score distribution is estimated as a continuous function of predictor(s). A flexible approach for norm estimation is the use of generalized additive models for location, scale, and shape. It is unknown how sensitive their estimates are to model flexibility and sample size. Generally, a flexible model that fits at the population level has smaller bias than its restricted nonfitting version, yet it has larger sampling variability. We investigated how model flexibility relates to bias, variance, and total variability in estimates of normalized z scores under empirically relevant conditions, involving the skew Student t and normal distributions as population distributions. We considered both transversal and longitudinal assumption violations. We found that models with too strict distributional assumptions yield biased estimates, whereas too flexible models yield increased variance. The skew Student t distribution, unlike the Box–Cox Power Exponential distribution, appeared problematic to estimate for normally distributed data. Recommendations for empirical norming practice are provided.

A big-data approach to producing descriptive anthropometric references: a feasibility and validation study of paediatric growth charts

Both national and WHO growth charts have been found to be poorly calibrated with the physical growth of children in many countries. We aimed to generate new national growth charts for French children in the context of huge datasets of physical growth measurements routinely collected by office-based health practitioners. We recruited 32 randomly sampled primary care paediatricians and ten volunteer general practitioners from across the French metropolitan territory who used the same electronic medical records software, from which we extracted all physical growth data for the paediatric patients, with anonymisation. We included measurements from all children born from Jan 1, 1990, and aged 1 month to 18 years by Feb 8, 2018, with birthweight greater than 2500 g, to which an automated process of data cleaning developed to detect and delete measurement or transcription errors was applied. Growth charts for weight and height were derived by using generalised additive models for location, scale, and shape with the Box-Cox power exponential distribution. We compared the new charts to WHO growth charts and existing French national growth charts, and validated our charts using growth data from recent national cross-sectional surveys. After data cleaning, we included 1 458 468 height and 1 690 340 weight measurements from 238 102 children. When compared with the existing French national and WHO growth charts, all height SD and weight percentile curves for the new growth charts were distinctly above those for the existing French national growth charts, as early as age 1 month, with an average difference of -0·75 SD for height and -0·50 SD for weight for both sexes. Comparison with national cross-sectional surveys showed satisfactory calibration, with generally good fit for children aged 5-6 years and 10-11 years in height and weight and small differences at age 14-15 years. We successfully produced calibrated paediatric growth charts by using a novel big-data approach applied to data routinely collected in clinical practice that could be used in many fields other than anthropometry. The French Ministry of Health; Laboratoires Guigoz-General Pediatrics section of the French Society of Pediatrics-Pediatric Epidemiological Research Group; and the French Association for Ambulatory Pediatrics.

Percentile curves and reference values for 2000-m rowing ergometer performance time in international rowers aged 14-70 years

The aim of this study was to provide percentile curves and reference values for the performance in 2000-m maximal effort on rowing ergometer. A cross-sectional study was carried out with a non-probabilistic sample (n=15420) obtained from an on-line ranking of indoor rowing and made from results between 2010 and 2014 recorded in 2000-m official races. Percentile curves and reference values were calculated using Generalized Additive Models for Location, Scale and Shape (GAMLSS) with a transformation of data to Box-Cox Power Exponential distribution and cubic splines as smoothing technique with age as the explanatory variable. This study is the first to present percentile curves and reference data to evaluate 2000-m performance time (indirect measure of mechanical power) in rowing ergometer depending on age (14-70) for both sexes and body-mass classifications (light- and heavyweight rowers). These curves and values are of interest in assessing indoor rowing performance and in measuring the specific physical condition of rowers in 2000-m regattas on-water. Percentiles also can be useful to predict performance levels in oncoming ages.

Age-related normative changes in phasic orthostatic blood pressure in a large population study: findings from The Irish Longitudinal Study on Ageing (TILDA).

In this report, we provide the first normative reference data and prevalence estimates of impaired orthostatic blood pressure (BP) stabilization, initial orthostatic hypotension, and orthostatic hypotension based on beat-to-beat blood pressure methods in a population-representative sample. Participants were recruited from a nationally representative cohort study (≥50 years). Beat-to-beat systolic BP, diastolic BP, and heart rate records were analyzed among those who underwent an active stand test (n=4475). Normograms were estimated by use of generalized additive models for location, shape, and scale with Box-Cox power exponential distribution. Prevalence estimates of impaired BP stabilization, initial orthostatic hypotension, and orthostatic hypotension are reported. Orthostatic BP responses in adults aged 50 to 59 years stabilized within 30 seconds of standing, with older groups taking 30 seconds or longer. The total prevalence of impaired BP stabilization was 15.6% (95% confidence interval [CI], 14.1%-17.1%), increasing with age to 41.2% (95% CI, 30.0%-52.4%) in people ≥80 years old. Initial orthostatic hypotension occurred in 32.9% (95% CI, 31.2%-34.6%) of the population aged ≥50 years, with no age gradient evident. The prevalence of orthostatic hypotension was 6.9% (95% CI, 5.9%-7.8%) in the total population, increasing to 18.5% (95% CI, 9.0%-28.0%) in those aged ≥80 years old. Significant age-related differences exist in the time course of postural BP responses, with abnormal responses taking longer than 30 seconds to stabilize. Impaired BP stabilization is more common as we age, affecting more than two-fifths of the population aged ≥80 years, and may play a future role in the management of falls and syncope.

Time trends and forecasts of body mass index from repeated cross‐sectional data: a different approach

In this paper, we report a case study on a technical generalization of the Lee-Carter model, originally developed to project mortality, to forecast body mass index (BMI, kg/m2). We present the method on an annually repeated cross-sectional data set, the Dutch Health Survey, covering years between 1981 and 2008. We applied generalized additive models for location, scale and shape semi-parametric regression models to estimate the probability distribution of BMI for each combination of age, gender and year assuming that BMI follows a Box-Cox power exponential distribution. We modelled and extrapolated the distribution parameters as a function of age and calendar time using the Lee-Carter model. The projected parameters defined future BMI distributions from which we derived the prevalence of normal weight, overweight and obesity. Our analysis showed that important changes occurred not only in the location and scale of the BMI distribution but also in the shape of it. The BMI distribution became flatter and more shifted to the right. Assuming that past trends in the distribution of BMI will continue in the future, we predicted a stable or slow increase in the prevalence of overweight until 2020 among men and women. We conclude that our adaptation of the Lee-Carter model provides an insightful and flexible way of forecasting BMI and that ignoring changes in the shape of the BMI distribution would likely result in biased forecasts.

Modelling skewness and kurtosis with the BCPE density in GAMLSS

This paper illustrates the power of modern statistical modelling in understanding processes characterised by data that are skewed and have heavy tails. Our particular substantive problem concerns film box-office revenues. We are able to show that traditional modelling techniques based on the Pareto–Levy–Mandelbrot distribution led to what is actually a poorly supported conclusion that these data have infinite variance. This in turn led to the dominant paradigm of the movie business that ‘nobody knows anything’ and hence that box-office revenues cannot be predicted. Using the Box–Cox power exponential distribution within the generalized additive models for location, scale and shape framework, we are able to model box-office revenues and develop probabilistic statements about revenues.

Forecasting Film Revenues Using GAMLSS

This paper utilises the GAMLSS framework for the statistical modelling of movie box-office revenues. The dominant modelling paradigm of the film industry, traditionally exemplified by the nobody knows principle is based upon the infinite variance of the Pareto distribution. Using GAMLSS we have the flexibility to model up to 4 parameters of any distribution in terms of the avail- able explanatory variates, including a predictor that has smooth non-parametric functions. We here show that total box-office revenue can be better modelled by distributions with finite variance contradicting the Paretian hypothesis. Moreover the final version of the paper will illustrate that the Box-Cox power exponential distribution gives models where the parameters vary smoothly with an important explanatory variable, leading to the substantive conclusion that the post-opening revenue can, in fact, be explained by the opening box-office

Smooth centile curves for skew and kurtotic data modelled using the Box-Cox power exponential distribution.

The Box-Cox power exponential (BCPE) distribution, developed in this paper, provides a model for a dependent variable Y exhibiting both skewness and kurtosis (leptokurtosis or platykurtosis). The distribution is defined by a power transformation Y(nu) having a shifted and scaled (truncated) standard power exponential distribution with parameter tau. The distribution has four parameters and is denoted BCPE (mu,sigma,nu,tau). The parameters, mu, sigma, nu and tau, may be interpreted as relating to location (median), scale (approximate coefficient of variation), skewness (transformation to symmetry) and kurtosis (power exponential parameter), respectively. Smooth centile curves are obtained by modelling each of the four parameters of the distribution as a smooth non-parametric function of an explanatory variable. A Fisher scoring algorithm is used to fit the non-parametric model by maximizing a penalized likelihood. The first and expected second and cross derivatives of the likelihood, with respect to mu, sigma, nu and tau, required for the algorithm, are provided. The centiles of the BCPE distribution are easy to calculate, so it is highly suited to centile estimation. This application of the BCPE distribution to smooth centile estimation provides a generalization of the LMS method of the centile estimation to data exhibiting kurtosis (as well as skewness) different from that of a normal distribution and is named here the LMSP method of centile estimation. The LMSP method of centile estimation is applied to modelling the body mass index of Dutch males against age.