Smallest Deviance Research Articles

Nonparametric regression estimator of multivariable Fourier Series for categorical data

In recent years, Fourier Series estimators in nonparametric regression for quantitative data have received significant attention. However, in reality, there is often a relationship between response and predictor, where the response is categorical data. Some methods developed today to address the case of qualitative response data use only certain approaches. No Fourier Series estimator can handle categorical response data. This paper introduces a new method that uses response variable in the form of categorical data. This study aimed to develop a multivariable Fourier Series nonparametric regression estimator for categorical data. The research methods used are literature and theoretical studies. To apply this method, we used two application data. The results obtained indicate that the nonparametric regression of the Fourier Series provides significantly better estimation results and accuracy for both data applications, due to the small deviance value and larger AUC and Press'Q values. The highlights of this research are summarized below.•The Fourier Series method for categorical data assumes a relationship between the logit function and predictor variables that has a repeating pattern.•The estimator was obtained through Maximum Likelihood Estimation and Newton–Raphson method.•The Fourier Series nonparametric regression method provides better estimation than binary logistic regression.

Determinants of modern contraceptive utilization among married women in sub-Saharan Africa: multilevel analysis using recent demographic and health survey

BackgroundFamily planning is a low-cost, high-impact public health and development strategy to improve child and maternal health. However, there is a lack of evidence on modern contraceptive use and determinants in sub-Saharan Africa. Hence, this study aimed at determining the pooled prevalence and determinants of modern contraceptive utilization among married women of sub-Saharan Africa.MethodsThirty-six sub-Saharan African countries' demographic and health survey (DHS) data were used for pooled analysis. A total weighted sample of 322,525 married women was included. Cross tabulations and summary statistics were done using STATA version 14 software. The pooled prevalence of modern contraceptive utilization with a 95% Confidence Interval (CI) was reported. Multilevel regression analysis was used to identify the determinants of modern contraceptive use among married women. Four models were fitted to select the best-fitted model using the Likelihood Ratio (LLR) and Deviance test. Finally, the model with the highest LLR and the smallest deviance was selected as the best-fitted model.ResultsThe pooled estimate of modern contraception use in sub–Saharan African countries was 18.36% [95% CI: 18.24, 18.48], with highest in Lesotho (59.79%) and the lowest in Chad (5.04%). The odds of modern contraception utilization were high among women living in East Africa [AOR = 1.47 (1.40, 1.54)], urban areas [AOR = 1.18 (1.14, 1.24)], and women with primary [AOR = 1.49 (1.44, 1.55)] and secondary and above educational level [AOR = 1.66 (1.58, 1.74)]. Moreover, husbands with primary educational level [AOR = 1.38 (1.33, 1.42)], middle [AOR = 1.17, (1.14, 1.21)], rich wealth status [AOR = 1.29 (1.25, 1.34)], media exposure [AOR = 1.25 (1.22, 1.29)], and postnatal care (PNC) utilization [AOR = 1.25 (1.22, 1.29)] had higher odds of modern contraceptive utilization compared with their counter parts. Furthermore, deliver at health facility [AOR = 1.74 (1.69, 1.79)] and birth order 2–4 [AOR = 1.36 (1.31, 1.41)] had higher odds of modern contraceptive utilization. On the other hand, women living in Central [AOR = 0.23 (0.22, 0.24)], Western regions [AOR = 0.46 (0.40, 0.54)], women who decided with husband [AOR = 0.90 (0.87, 0.93)], and decisions by husband alone [AOR = 0.73 (0.71, 0.75)] decreased the odds of modern contraceptive utilization.ConclusionThe uptake of modern contraception in sub-Saharan Africa is low. Modern contraceptive utilization is affected by different factors. More attention needs to be given to rural residents, illiterate women, and communities with low wealth status.

Normative reference values for estimated cardiorespiratory fitness in apparently healthy British men and women.

ObjectivesTo develop normative reference standards for estimated cardiorespiratory fitness (eCRF) measured from treadmill-based incremental exercise testing in ~12 000 British men and women.MethodsCross-sectional study using retrospectively collected eCRF data from five preventative health screening clinics in the United Kingdom. Reference centiles were developed using a parametric approach by fitting fractional polynomials. We selected the ‘best’ powers by considering both the smallest deviance, and clinical knowledge from the following set of a priori decided powers (-2,-1,-0.5, 0, 0.5,1,2,3). A series of fractional polynomials (FPs) were investigated with three-parameters (median, standard deviation and skewness). The following reference centiles were plotted (3, 5, 10, 25, 50, 75, 90, 95, 97).ResultsWe included 9 204 males (median [25th,75th centiles] age 48 [44, 53] years; BMI 27 {25, 29] kg∙m-2; peak VO2 36.9 [30.5, 44.7] ml∙kg-1∙min-1) and 2 687 females (age 48, [41, 51] years; BMI 24 {22, 27] kg∙m-2; peak VO2 36.5 [30.1, 44.8] ml∙kg-1∙min-1) in our analysis to develop the normative values.ConclusionReference values and nomograms for eCRF were derived from a relatively large cohort of preventative health care screening examinations of apparently healthy British men and women. Age- and sex-specific eCRF percentiles were similar to data from international cohort studies. The adoption of submaximal exercise testing protocols reduces individual risk when exercise history is unknown and testing is conducted in a community-based setting. Our findings can be used by health professionals to help guide clinical decision making.

445-P: Network Meta-analysis of Seven Glucagon-Like Peptide-1 Receptor Agonists: A Focus on Cardiovascular Outcome Trials Data

Objective: To indirectly compare glucagon like peptide-1 receptor agonists (GLP-1RA) cardiovascular outcome trials (CVOTs) using network meta-analysis (NMA). Methods: We searched Medline to retrieve CVOTs that involved GLP-1RA. We conducted NMAs using binomial likelihood logit link model for the following outcomes: CV death, myocardial infarction (IM), stroke, and death from any cause. After running both fixed effects and random effects models for each outcome, we identified the best fit model that has the smallest deviance information and the average posterior residual deviance. We followed the preferred reporting items for systematic reviews and meta-analyses (PRISMA-NMA) in reporting this NMA. Results: After screening, a total of seven GLP-1RA CVOTs were included. The NMA results showed that oral semaglutide was statistically better than exenatide (HR 0.47, CI: 0.21-0.99), dulaglutide (HR 0.46, CI: 0.20-0.97), albiglutide (HR 0.45, CI: 0.19-0.97), and lixisenatide (HR 0.43, CI: 0.19-0.92) in reducing CV death events. No significant differences were detected between most of the treatments regarding reducing death from any cause, MI and stroke events. The ranking results showed that oral semaglutide had the highest probability to be ranked first (>90%) in reducing CV death and death from any cause. Moreover, once weekly semaglutide had the highest probability to be ranked first in reducing MI and stroke events. Conclusion: The indirect comparison of GLP-1RAs using CVOTs data showed that oral semaglutide is preferred over most of the GLP-1RAs in reducing CV death and death from any cause events. The ranking probability showed that once weekly semaglutide and oral semaglutide are the most likely to be ranked first in reducing the outcomes of interest. Disclosure O. Alfayez: None. O. Alkhezi: None. A.R. Almutairi: None.

Use of Space Time Model for Forecasting Mortality due to Malaria: A Case of Ifakara and Rufiji Health and Demographic Surveillance System Sites

Malaria is a leading cause of morbidity and mortality in developing countries especially in rural areas where local resources are limited. Accurate disease forecasts can provide information to public and clinical health services to design targeted interventions for malaria control that make effective use of limited resources. Using verbal autopsy data, space-time model was used to forecast mortality due to malaria. The study used longitudinal data which were collected from Rufiji and Ifakara Health Demographic Surveillance System (HDSS) sites for the period of 1999 to 2011 and 2002 to 2012 respectively to assess models. The models included environmental factors and mosquito net ownership as predictor variables for mortality due to malaria. Deviance information criteria (DIC), logarithm score and root mean square error (RMSE) were used to assess the goodness of fit and forecasting accuracy of the models. The results indicate that the model included spatial and temporal random effect terms had small deviance information criteria, logarithm score and root mean square error. This model was the best model for forecasting and prediction of mortality due to malaria in both HDSS sites. In addition, mosquito net ownership and rainfall were significantly associated with mortality due to malaria. The model with spatial and temporal random effect terms is useful tool to provide reasonably reliable forecasts for mortality due to malaria. This might help to design appropriate strategies for targeting malaria control. On the other hand, including spatially and temporal varying random terms in the model is necessary and good strategy for modelling mortality due to malaria.

Predicting velum movement from acoustic features—A regression approach

Although acoustical methods have been widely used in the nasality literature, a direct link between the acoustical measurements and velum movement during speech production is yet to be established. In this study, we propose a model through which the vertical movements of the velum are inferred from an acoustic feature set. An X-ray Microbeam data set collected at University of Tokyo are used for the modeling. The data recorded the vertical movements of the velum of 11 American English speakers saying both isolated words and sentences. Velum positions are recorded from tracing a metal pallet placed on top of the velum. 40 MFCC (Mel-frequency Cepstral Coefficient) features are extracted from the accompanying acoustic signal at each time frame. MFCCs of ten frames before the current frame, together with the current frame, consist of the feature vector for predicting the velum movement of the current frame. Elasticnet regression is used to reduce the dimensionality of the feature vector. In general, MFCCs from higher frequencies are penalized during model selection. The selected features are then fitted to a stepwise logistic model. For each individual speaker, the inferred velum movements in the validation set are a good fit to the actual observation as judged by the high accuracy in identifying locations of peaks and valleys and small deviance from the response. However, there exists large inter-speaker variation in terms of both movement pattern and model performance.

Testing Homogeneity of Variance

This paper compares empirical type I error and power of different tests that have been proposed to assess the homogeneity of within-group variances. The main questions answered in the paper are: (1) Which tests are most effective (powerful and robust) when testing the homogeneity of variances in various conditions of heterogeneity, and (2) When these conditions exist, do any of the tests of homogeneity of variance prove more effective in the detection of heterogeneity than others? An introductory simulation study established that Anova is extremely susceptible to heterogeneity of the variances. This occurs both in normally distributed and non-normal datasets. Even small deviances from homogeneity to that of heteroscedasticity results in inflated type I error. A subsequent simulation study was conducted. The purpose of this study was to determine which of the tests of homogeneity of variances preformed best in unfavourable conditions. These conditions included small sample sizes and a number of non-normal distributions. In these conditions, Bartlett's or Box's tests perform well. Conversely, Cochran's test and the Log Anova test exhibited low power with small to moderate sample sizes even with normally distributed datasets. A decision tree has been created for these results that can be used to determine the most effective test strategy.

When prosody kicks in: The intricate interplay between segments and prosody in perceptions of foreign accent

This study examines the relative contribution of prosody and segments in the perception of foreign-accented speech. Although both prosody and segments have previously been shown to contribute to the degree of foreign accentedness, results for their relative contribution have proven inconclusive and little is known about their interaction. We recorded read speech of five German learners of English – with and without previous exposure to the Belfast English accent – and one native speaker of Belfast English. Through acoustic manipulations that allow separation of prosodic and segmental information in the speech signal, we created combinations of native segments with non-native prosody and vice versa. The resulting stimuli were then presented to 60 native English listeners, who rated their foreign accentedness, in order to establish whether the impact of prosody depends on the degree of foreign accent at the segmental level and/or the other way round. Group analyses of the obtained foreign accent ratings are presented, supplemented by phonetic analyses of the manipulated stimuli and individual L2 speakers. Results confirm that whilst both segments and prosody contributed to foreign accent perception, listeners were more influenced by segments than prosody. In addition, it was found that the impact of prosody on perceived foreign accentedness varied depending on the origin of the segments: listeners’ sensitivity to small prosodic differences from the native norm was compromised when mixed with non-native segments, whereas sensitivity to small prosodic differences from the non-native norm was compromised when mixed with native segments. In light of this, we propose that there needs to be a degree of correspondence between segments and prosody for small deviances in prosody to be perceived. The study thus contributes to research on foreign accent perception and suggests that the interplay between segments and prosody in such perceptions is more intricate than previously thought.

Phase Formation Study of Ca-Terephthalate MOF-Type Materials

Ca(BDC)(DMF)(H2O) with square lattice topology (sql) and rhombic-shaped channels crystallizes under solvothermal conditions (H2O/DMF mixture) in orthorhombic (Ca-BDC-orth) and triclinic (Ca-BDC-tric) conformations. The latter structure is solved and described here for the first time. The main difference between the two phases lies in the orientation of the coordinated DMF molecule located within the channels of the structure, influencing their porosity properties. In spite of the small deviances in crystal structure, the formation of individual phases can be controlled by the amount of base agent (TEA) and the temperature of crystallization. Triclinic phase is formed at lower synthesis temperature, and it is transformed into orthorhombic phase at increased temperatures. Systematic phase formation study varying TEA/Ca2+ molar ratios and the synthesis temperature showed that higher temperature of the synthesis is required for the Ca-BDC-orth phase to occur from Ca-BDC-tric using the TEA/Ca2+ ratio between 0.63–1.30 compared to the syntheses with TEA/Ca2+ values out of this range. Within the range of 0.63–1.30 the dissociation of the base is in the buffer region indicating that stability of triclinic phase is related to the thermodynamics at equilibrium conditions.

Effect of Changes in Stimulus Site on Activation of the Posterior Parietal Cortex

A previous functional magnetic resonance imaging study elucidated the specific activity of the inferior parietal lobe (IPL) during a two-point discrimination task compared with that during an intensity discrimination task Akatsuka et al. (Neuroimage 40: 852-858, 2008). If the posterior parietal cortex (PPC), including IPL, is responsible for detecting changes in stimulus sites, PPC activity depends on the level of change at stimulus sites. The aim of this study was to clarify whether a particular site exists that could detect changes in stimulus sites using the oddball paradigm. Somatosensory-evoked magnetic fields were recorded in 10 right-handed subjects. Three oddball conditions were performed by all subjects, with the probability of deviant and standard stimuli being 20 and 80%, respectively, under all three conditions. Deviant stimuli were always presented to the second digit of the hand and standard stimuli were presented to the first (small deviance: SD) and fifth digits (medium deviance: MD) of the hand and the first digit of the toe (large deviance: LD). Inter-stimulus intervals were set at 500ms. A brain electrical source analysis showed that activities of areas 1 and 3b elicited by the deviant stimuli were not significantly different among the three conditions. In contrast, PPC activity was significantly greater for LD than for SD and MD. PPC activity tended to increase with greater deviance at stimulus sites, but activities of areas 1 and 3b did not differ. These findings suggest that PPC may have a functional role in automatic change detection systems with regard to deviance of stimulus sites.

Detection of Micrometastases in Peritoneal Washings of Pancreatic Cancer Patients by the Reverse Transcriptase Polymerase Chain Reaction

ObjectivePancreatic cancer patients with positive (+) peritoneal cytology have a prognosis similar to stage IV patients. We studied the ability of quantitative real time-polymerase chain reaction (RT-PCR) to detect micrometastases in patients undergoing staging laparoscopy. MethodsPeritoneal washes were obtained prospectively from 35 consecutive patients with pancreatic adenocarcinoma undergoing staging laparoscopy and 16 patients undergoing laparoscopy for benign disease. Each sample was assessed by cytologic examination and RT-PCR analysis for tumor markers: CEA, CK7, Kras2, and MUC1. Markers and their combinations were evaluated on the basis of their deviance from the ideal marker. ResultsPathologic stages for pancreatic cancer patients were: 1A-1 (3%), IB-1 (3%), IIA-5 (15%), IIB-13 (38%), III-5 (15%), IV-9 (26%). Eight patients were cytology (+) and stages IIA-1, IIB-2, IV-5. Twenty-five patients were RT-PCR (+). The optimal threshold for cycle amplification was 35 based on a receiver operating characteristic curve. CEA had the best profile of sensitivity, specificity, PPV, NPV, and the smallest deviance. ConclusionRT-PCR using a panel of tumor markers, including CEA, was comparable in sensitivity, specificity, PPV, and NPV to cytology. RT-PCR could represent a more sensitive method for detection of subclinical peritoneal tumor dissemination; this may be useful in patient selection for operative management and clinical trials.

Is the failure to detect stimulus deviance during sleep due to a rapid fading of sensory memory or a degradation of stimulus encoding?

The mismatch negativity (MMN) is thought to reflect the outcome of a system responsible for the detection of change in an otherwise repetitive, homogenous acoustic environment. This process depends on the storage and maintenance of a sensory representation of the frequently presented stimulus to which the deviant stimulus is compared. Few studies have been able to record the MMN in non-rapid eye movement (NREM) sleep. This pattern of results might be explained by either a rapid fading of sensory memory or an inhibition of stimulus input prior to entry into the cortical MMN generator site. The present study used a very rapid rate of presentation in an attempt to capture mismatch-related negativity prior to the fading of sensory memory. Auditory event-related potentials were recorded from 12 subjects during a single sleep period. A 1000 Hz standard stimulus was presented every 150 ms. At random, on 6.6% of the trials, the standard was changed to either a large 2000 Hz or a small 1100 Hz deviant. In wakefulness, the large deviant elicited an extended negativity that was reduced in amplitude following the presentation of the small deviant. This negativity was also apparent during REM sleep following the presentation of the large deviant. These deviant-related negativities (DRNs) were probably a composite of N1 and MMN activity. During NREM sleep (stage 2 and slow-wave sleep), only the large deviant continued to elicit a DRN. However this DRN might be overlapped by the initial activity of a component that is unique to sleep, the N350. There was little evidence of the DRN or the MMN during sleep following the presentation of the small deviant. A rapid rate of presentation, therefore, does not preserve the MMN following small deviance within sleep. It is possible that inhibition of sensory input occurs before entry into the MMN generating system in the temporal cortex.

Effects of sleep onset on the mismatch negativity (MMN) to frequency deviants using a rapid rate of presentation

This study examined the effects of sleep onset—the transition from a waking, conscious state to one of sleep and unconsciousness—on the mismatch negativity (MMN) following frequency deviants when a rapid rate of stimulus presentation is employed. The MMN is thought to reflect a brief-lasting sensory memory. Rapid rates of stimulus presentation should guard the sensory memory from fading. A 1000 Hz standard stimulus was presented every 150 ms. At random, on 6.6% of the trials, the standard was changed to either a large 2000 or a small 1100 Hz deviant. During alert wakefulness (when subject ignored the stimuli and read a book), the large deviant elicited a larger deviant related negativity (DRN) than did the small deviant. This negativity may be a composite of both N1 and MMN activity while that following the small deviant is probably a ‘true’ MMN. The large deviant continued to elicit a DRN in relaxed wakefulness (eyes closed) and Stages 1 and 2 of sleep, although it was much reduced in amplitude. A significant MMN was recorded for the small deviant only in alert wakefulness. The failure to observe an MMN to small deviance and the attenuation of the DRN to large deviance at sleep onset therefore is probably not due to a decay of sensory memory. It is more likely that cortical encoding of both the standard and deviant is weakened during sleep onset because of prior thalamic inhibition of sensory input.

Sagittal abdominal diameter is associated with insulin sensitivity in Chinese hypertensive patients and their siblings.

The purpose of the study is to observe the relation between anthropometric measurements, focusing on sagittal abdominal diameter (SAD), and insulin sensitivity indices in Chinese hypertensive patients and their siblings. In total, 907 participants, 537 hypertensive and 370 nonhypertensive, from 311 Taiwanese families were drawn from the Stanford Asia and Pacific Program for Hypertension and Insulin Resistance for the study. The participants received anthropometric measurements and 75-g oral glucose tolerance tests after an overnight fast. Fasting insulin, homeostasis model assessment for insulin resistance (HOMA-IR), and the insulin sensitivity index ISI(0,120) were chosen as surrogate measures of insulin sensitivity. In addition to Pearson and partial correlations, we used generalized estimating equations (GEEs) to examine the association between anthropometric measurements and insulin sensitivity indices. A small deviance in the GEEs indicates the goodness of model fit, irrespective of the independence among variables. The hypertensive patients were older in age, wider in waist circumference (WC), larger in body mass index (BMI) and SAD, and more insulin resistant than the nonhypertensive counterparts. The logarithmic transformation of fasting insulin, HOMA-IR, and ISI(0,120) significantly correlated with SAD, WC, and BMI before and after adjustments for age and sex. The deviances of SAD in the GEEs were similar to those of WC in all subjects, while BMI had smaller deviances than SAD and WC in the hypertensive patients. Our results suggest that the performance of SAD in predicting insulin sensitivity is comparable with WC in Chinese hypertensive patients and their siblings. BMI, however, seems to have better association with insulin sensitivity than SAD and WC in the patients with hypertension.

The posterior probability of Bayes nets with strong dependences

Stochastic independence is an idealized relationship located at one end of a continuum of values measuring degrees of dependence. Modeling real-world systems, we are often not interested in the distinction between exact independence and any degree of dependence, but between weak ignorable and strong substantial dependence. Good models map significant deviance from independence and neglect approximate independence or dependence weaker than a noise threshold. This intuition is applied to learning the structure of Bayes nets from data. We determine the conditional posterior probabilities of structures given that the degree of dependence at each of their nodes exceeds a critical noise level. Deviance from independence is measured by mutual information. The arc probabilities are set equal to the probability that the mutual information, provided by the neighbors of a node, exceeds a given threshold. A χ2 approximation for the probability density function of mutual information is used. A large number of network structures in which the arc probabilities are evaluated, is generated by stochastic simulation. Finally, the probabilities of the whole graph structures are obtained by combining the individual arc probabilities with the number of possible construction sequences compatible with the partial ordering of the graph. While selecting models with large deviance from independence results in simple networks with few but important links, selecting models with small deviance results in highly connected networks also containing less important links.

Behavioral and electrophysiological effects of task-irrelevant sound change: a new distraction paradigm

A distraction paradigm was utilized that is suited to yield reliable auditory distraction on an individual level even with rather small frequency deviances (7%). Distraction to these tiny deviants was achieved by embedding task-relevant aspects and task-irrelevant, distracting aspects of stimulation into the same perceptual object. Event-related potential (ERP) and behavioral effects of this newly developed paradigm were determined. Subjects received tones that could be of short or long duration equiprobably. They were instructed to press a response button to long-duration tones (targets). In oddball blocks, tones could be of standard frequency or of low-probability ( p=0.1), deviant frequency. The task-irrelevant frequency deviants elicited MMN, N2b, and P3a components, and caused impoverished behavioral performance to targets. The usage of tiny distractors permits an interpretation of auditory distraction in terms of attention switching due to a particular memory-related change-detection process. On the basis of the results from an additional condition in which tones were of 10 different frequencies (involving those frequencies which served as standard and deviant in oddball blocks), it is argued that one important prerequisite for linking the neural mechanisms reflected in change-related brain waves to behavioral distraction effects may be regarded as fulfilled. The robustness of the distraction effects to tiny deviations was confirmed in two control experiments.

S.03.04 - Magnetoencephalography (MEG) in the study of human brain functions

A distraction paradigm was utilized that is suited to yield reliable auditory distraction on an individual level even with rather small frequency deviances (7%). Distraction to these tiny deviants was achieved by embedding task-relevant aspects and task-irrelevant, distracting aspects of stimulation into the same perceptual object. Event-related potential (ERP) and behavioral effects of this newly developed paradigm were determined. Subjects received tones that could be of short or long duration equiprobably. They were instructed to press a response button to long-duration tones (targets). In oddball blocks, tones could be of standard frequency or of low-probability (p=0.1), deviant frequency. The task-irrelevant frequency deviants elicited MMN, N2b, and P3a components, and caused impoverished behavioral performance to targets. The usage of tiny distractors permits an interpretation of auditory distraction in terms of attention switching due to a particular memory-related change-detection process. On the basis of the results from an additional condition in which tones were of 10 different frequencies (involving those frequencies which served as standard and deviant in oddball blocks), it is argued that one important prerequisite for linking the neural mechanisms reflected in change-related brain waves to behavioral distraction effects may be regarded as fulfilled. The robustness of the distraction effects to tiny deviations was confirmed in two control experiments.

Dynamics of an epidemic in a closed population

A simple model for the intensity of infection during an epidemic in a closed population is studied. It is shown that the size of an epidemic (i.e. the number of persons infected) and the cumulative force of an epidemic (i.e. the amount of infectiousness that has to be avoided by a person that will stay uninfected during the entire epidemic) satisfy an equation of balance. Under general conditions, small deviances from this balance are, in large populations, asymptotically mixed normally distributed. For some special epidemic models the size of an asymptotically large epidemic is asymptotically normally distributed.

Dynamics of an epidemic in a closed population

A simple model for the intensity of infection during an epidemic in a closed population is studied. It is shown that the size of an epidemic (i.e. the number of persons infected) and the cumulative force of an epidemic (i.e. the amount of infectiousness that has to be avoided by a person that will stay uninfected during the entire epidemic) satisfy an equation of balance. Under general conditions, small deviances from this balance are, in large populations, asymptotically mixed normally distributed. For some special epidemic models the size of an asymptotically large epidemic is asymptotically normally distributed.