SEARCH · PubMed Health

Results for “Likelihood Functions”

Explore indexed PubMed citations for clinical trials, systematic reviews and public health research. Read source abstracts and follow each citation to its original PubMed record.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 1,063 records · Page 59Linked to original sources

PubMed2005

Natural selection and genetic variation for reproductive reaction norms in a wild bird population.

Many morphological and life-history traits show phenotypic plasticity that can be described by reaction norms, but few studies have attempted individual-level analyses of reaction norms in the wild. We analyzed variation in individual reaction norms between laying date and three climatic variables (local temperature, local rainfall, and North Atlantic Oscillation) of 1126 female collared flycatchers (Ficedula albicollis) with a restricted maximum likehood linear mixed model approach using random-effect best linear unbiased predictor estimates for the elevation (i.e., expected laying date in the average environment) and slope (i.e., adjustment in laying date as a function of environment) of females' reaction norms. Variation in laying date was best explained by local temperature, and individual females differed in both the elevation and the slope of their laying date-temperature reaction norms. As revealed by animal model analyses, there was weak evidence for additive genetic variance of elevation (h2 +/- SE = 0.09 +/- 0.09), whereas there was no evidence for heritability of slope (h2 +/- SE = 0.00 +/- 0.01). Selection analysis, using a female's lifetime production of fledglings or recruits as an estimate of her fitness, revealed significant selection for a lower phenotypic value and breeding value for elevation (i.e., earlier laying date at the average temperature). There was selection for steeper phenotypic values of slope (i.e., greater plasticity in the adjustment of laying date to temperature), but no significant selection on the breeding values of slope. Although these results suggest that phenotypic laying date is influenced by additive genetic factors, as well as by an interaction with the environment, selection on plasticity would not produce an evolutionary response.

Animals↗

PubMed2006

ROC curves predicted by a model of visual search.

In imaging tasks where the observer is uncertain whether lesions are present, and where they could be present, the image is searched for lesions. In the free-response paradigm, which closely reflects this task, the observer provides data in the form of a variable number of mark-rating pairs per image. In a companion paper a statistical model of visual search has been proposed that has parameters characterizing the perceived lesion signal-to-noise ratio, the ability of the observer to avoid marking non-lesion locations, and the ability of the observer to find lesions. The aim of this work is to relate the search model parameters to receiver operating characteristic (ROC) curves that would result if the observer reported the rating of the most suspicious finding on an image as the overall rating. Also presented are the probability density functions (pdfs) of the underlying latent decision variables corresponding to the highest rating for normal and abnormal images. The search-model-predicted ROC curves are 'proper' in the sense of never crossing the chance diagonal and the slope is monotonically changing. They also have the interesting property of not allowing the observer to move the operating point continuously from the origin to (1, 1). For certain choices of parameters the operating points are predicted to be clustered near the initial steep region of the curve, as has been observed by other investigators. The pdfs are non-Gaussians, markedly so for the abnormal images and for certain choices of parameter values, and provide an explanation for the well-known observation that experimental ROC data generally imply a wider pdf for abnormal images than for normal images. Some features of search-model-predicted ROC curves and pdfs resemble those predicted by the contaminated binormal model, but there are significant differences. The search model appears to provide physical explanations for several aspects of experimental ROC curves.

Algorithms↗

PubMed1999

Regression diagnostics for the class A regressive model with quantitative phenotypes.

Regression diagnostic methods are developed and investigated under the Class A regressive model proposed by Bonney [(1984) Am J Med Genet 18:731-749]. We call a family whose phenotypic distribution does not conform to the same genetic model as the majority of the families an etiotic family. The exact case-deletion approach for identifying etiotic families, based on examining the changes in each model parameter estimate by excluding one family at a time, is very time-consuming. We proposed three alternative diagnostic methods: the empirical influence function (EIF), the one-step approximation, and the approximated one-step approach. These methods can be computed efficiently and were incorporated into the existing software package S.A.G.E. A thorough Monte-Carlo investigation of the performance of the diagnostic methods was conducted and generally supports the EIF approach as the recommended alternative. The phenotypic variance is the parameter whose associated regression diagnostic most frequently and correctly identified etiotic families in the models that were examined. An analysis of body mass index data from 402 individuals in 122 Muscatine, Iowa families is used to illustrate the methods. A Class A regressive model with a recessive major locus and equal mother-offspring and father-offspring correlations provided the best-fitting model. The proposed regression diagnostics identified up to 7.4% of the 122 families as etiotic. As a result of this investigation, case-deletion diagnostic assessment is now a practical component in the analysis of quantitative family data.

Adolescent↗

PubMed2004

Application of tetranucleotide frequencies for the assignment of genomic fragments.

A basic problem of the metagenomic approach in microbial ecology is the assignment of genomic fragments to a certain species or taxonomic group, when suitable marker genes are absent. Currently, the (G + C)-content together with phylogenetic information and codon adaptation for functional genes is mostly used to assess the relationship of different fragments. These methods, however, can produce ambiguous results. In order to evaluate sequence-based methods for fragment identification, we extensively compared (G + C)-contents and tetranucleotide usage patterns of 9054 fosmid-sized genomic fragments generated in silico from 118 completely sequenced bacterial genomes (40 982 931 fragment pairs were compared in total). The results of this systematic study show that the discriminatory power of correlations of tetranucleotide-derived z-scores is by far superior to that of differences in (G + C)-content and provides reasonable assignment probabilities when applied to metagenome libraries of small diversity. Using six fully sequenced fosmid inserts from a metagenomic analysis of microbial consortia mediating the anaerobic oxidation of methane (AOM), we demonstrate that discrimination based on tetranucleotide-derived z-score correlations was consistent with corresponding data from 16S ribosomal RNA sequence analysis and allowed us to discriminate between fosmid inserts that were indistinguishable with respect to their (G + C)-contents.

Bacteria↗

PubMed2001

Parametric modelling of neonatal mortality in relation to size at birth.

We use a richly parameterized model to analyse the effects of shortness and thinness at birth on neonatal mortality in Swedish live-born singletons. The model captures the hazard of neonatal mortality according to the function alpha x exp(-gamma x t) + delta, where t represents age. Covariates are allowed to simultaneously influence the initial excess hazard, the rate of decay, and the long-term hazard. Among term newborns, birth length for gestational age had a stronger effect on the long-term hazard and a weaker effect on the initial excess hazard than the Ponderal index. The initial excess hazard associated with a low Ponderal index tended to decay quickly. Among preterm newborns, a higher birth length for gestational age was associated with lower initial and long-term hazards, and with a faster rate of decay of the initial excess hazard. In contrast, the Ponderal index was not associated with the long-term hazard. We discuss the interpretability of the model and its potential use in neonatology and medical demography. We also compare the model to a Cox model with time-dependent covariates.

Analysis of Variance↗

PubMedDate not supplied

SF-36 as a predictor of health states.

BACKGROUND: There are a number of claims that Medical Outcomes Study Short Form 36 (MOS SF-36) mean scores can be used to discriminate between healthy and nonhealthy persons and determine various levels of health. OBJECTIVES: The purpose of this study was to evaluate the ability of the SF-36 to predict whether or not respondents reported health problems. METHODS: We used structural equation modeling (SEM) techniques to evaluate the SF-36 and its ability to discriminate between those who reported health problems or reported physician-determined illness and those who did not in a sample from the 1990 National Survey of Functional Health Status (NHS). RESULTS: The correlation between physician-determined illness and Physical Health was -.404, resulting in 16.32% shared variance. The correlation between reported health problems and Physical Health was -.360, resulting in 12.96% shared variance. These correlations are markedly lower than those to the eight first-order scales or between Physical and Mental Health (r = .889). Mental Health could not predict physician-determined illness or reported health problems independent of Physical Health. CONCLUSIONS: The SF-36 is relatively poor at accounting for the health status of respondents. There are significant paths but the variance accounted for in absolute and relative terms is small. Physical Health does a much better job of accounting for general mental health than it does for perceived health problems or physician-determined illness. These findings suggest that the SF-36 may not discriminate well between healthy and nonhealthy groups and that objective measures of health status may be required in conjunction with the use of the SF-36.

Angina Pectoris↗

PubMed1996

Statistical methods for cost-effectiveness analyses.

A statistical framework is presented for examining cost and effect data on competing interventions obtained from an RCT or from an observational study. Parameters of the join distribution of costs and effects or a regression function linking costs and effects are used to define cost-effectiveness (c-e) measures. Several new c-e measures are proposed that utilize the linkage between costs and effects on the patient level. These measures reflect perspectives that are different from those of the commonly used measures, such as the ratio of expected cost to expected effect, and they can lead to different relative rankings of the interventions. The cost-effectiveness of interventions are assessed statistically in a two stage procedure that first eliminates clearly inferior interventions. Members of the remaining admissible set are then rank ordered according to a c-e preference measure. Statistical techniques, particularly in the multivariate normal case, are given for several commonly used c-e measures. These techniques provide methods for obtaining confidence intervals, for testing the hypothesis of admissibility and for the equality of interventions, and for ranking interventions. The ideas are illustrated for a hypothetical clinical trial of antipsychotic agents for community-based persons with mental illness.

Antipsychotic Agents↗

PubMed2005

Full transcriptome analysis of rhabdomyosarcoma, normal, and fetal skeletal muscle: statistical comparison of multiple SAGE libraries.

Rhabdomyosarcoma (RMS) is the most frequent soft tissue sarcoma in children. Improved treatment strategies have increased overall survival, but the response of approximately one-third of the patients is still poor. To increase the knowledge of RMS pathogenesis, we performed the first full transcriptome analysis of RMS using serial analysis of gene expression (SAGE). With a G-test for the simultaneous comparison of subsets of SAGE libraries of normal skeletal muscle, embryonal (ERMS) and alveolar (ARMS) RMS, we identified 251 differentially expressed genes. A literature-mining procedure demonstrated that 158 of these genes have not previously been associated with RMS or normal muscle. Gene Ontology (GO) analysis assigned 198 of the 251 genes to muscle-specific classes, including those involved in normal myogenic development, as well as tumor-related classes. Prominent GO classes were those associated with proliferation and actin reorganization, which are processes that play roles during early muscle development, muscle function, and tumor progression. Using custom microarrays, we confirmed the (up- or down-) regulation of 80% of 98 differentially expressed genes. Another SAGE library of 19- to 22-week-old fetal skeletal muscle was compared with the RMS and normal muscle transcriptomes. Cluster analysis showed that the RMS and fetal muscle SAGE libraries formed one cluster distinct from normal muscle samples. Moreover, the expression profile of 86% of the differentially expressed genes between normal muscle and RMS was highly similar in fetal muscle and RMS. In conclusion, the G-test is a robust tool for analyzing groups of SAGE libraries and correctly identifies genes marking the difference between fully differentiated skeletal muscle and RMS. This study not only substantiates the close association between embryonic myogenesis and RMS development but also provides a rich source of candidate genes to further elucidate the etiology of RMS or to identify diagnostic and/or prognostic markers.

Cell Adhesion↗

PubMed1995

Diagnostic plots to reveal functional form for covariates in multiplicative intensity models.

We show how plots based on the residuals from a proportional hazards model may be used to reveal the correct functional form for covariates in the model. A smoothed plot of the martingale residues was suggested for this purpose by Therneau, Grambsch, and Fleming (1990, Biometrika 77, 147-160); however, its consistency required that the covariates be independent. They also noted that the plot could be biased for large covariate effects. We introduce two refinements which overcome these difficulties. The first is based on a ratio of scatter plot smooths, where the numerator is the smooth of the observed count plotted against the covariate, and the denominator is a smooth of the expected count. This is related to the Arjas goodness-of-fit plot (1988, Journal of the American Statistical Association 83, 204-212). The second technique smooths the martingale residuals divided by the expected count, using expected count as a weight. This latter approach is related to a GLM partial residual plot, as well as to the iterative methods of Hastie and Tibshirani (1990, Biometrics 46, 1005-1016) and Gentleman and Crowley (1991, Biometrics 47, 1283-1296). Applications to survival data sets are given.

Analysis of Variance↗

PubMed1999

Apolipoprotein E-epsilon 4 frequency in affective disorder.

BACKGROUND: The epsilon 4 allele of apolipoprotein E (APOE) as well as affective disorder have been found to be associated with Alzheimer's disease, but it is unclear whether cognitive impairment in affective disorder or subtypes of affective disorder is mediated by the epsilon 4 allele of APOE. METHODS: The genotype of APOE was analyzed in 106 unipolar patients, 21 bipolar patients, and 46 controls and correlated with cognitive function in the euthymic phase as measured by the Mini-Mental State Examination, the Cambridge Cognitive Examination, the Mattis Dementia Rating Scale, the Gottfries-Bråne-Steen Dementia Rating Scale, and the Global Deterioration Scale. RESULTS: The frequency of APOE-epsilon 4 allele was approximately the same in unipolar patients (.189) and in bipolar patients (.167). Although patients showed more cognitive impairment than controls, no significant overall difference was found between the frequency of APOE-epsilon 4 allele in patients (.185) and controls (.131). In fact, the frequency of APOE-epsilon 4 allele did not correlate with cognitive impairment. It was not possible to identify subgroups of patients with an increased frequency of APOE-epsilon 4 allele, as no association was found with gender, age at onset, the number of affective episodes, the presence of psychotic features, or the prevalence of familial affective disorder. CONCLUSIONS: It seems that cognitive impairment in affective disorder can be attributed to pathways other than the APOE genotype.

Adult↗

PubMed2003

Lineage-specific selection in human mtDNA: lack of polymorphisms in a segment of MTND5 gene in haplogroup J.

Human mitochondrial DNA (mtDNA) is a nonrecombining genome that codes for 13 subunits of the mitochondrial oxidative phosphorylation system, 2 rRNAs, and 22 tRNAs. Mutations have accumulated sequentially in mtDNA lineages that diverged tens of thousands of years ago. The genes in mtDNA are subject to different functional constraints and are therefore expected to evolve at different rates, but the rank order of these rates should be the same in all lineages of a phylogeny. Previous studies have indicated, however, that specific regions of mtDNA may have experienced different histories of selection in different lineages, possibly because of lineage-specific interactions or environmental factors such as climate. We report here on a survey for lineage-specific patterns of nucleotide polymorphism in human mtDNA. We calculated molecular polymorphism indices and neutrality tests for classes of functional sites and genes in 837 human mtDNA sequences, compared the results between continent-specific mtDNA lineages, and used two sliding window methods to identify differences in the patterns of polymorphism between haplogroups. A general correlation between nucleotide position and the level of nucleotide polymorphism was identified in the coding region of the mitochondrial genome. Nucleotide diversity in the protein-coding sequence of mtDNA was generally not much higher than that found for many genes in nuclear DNA. A comparison of nonsynonymous/synonymous rate ratios in the 13 protein-coding genes suggested differences in the relative levels of selection between haplogroups, including the European haplogroup clusters. Interestingly, a segment of the MTND5 gene was found to be almost void of segregating sites and nonsynonymous mutations in haplogroup J, which has been associated with susceptibility to certain complex diseases. Our results suggest that there are haplogroup-specific differences in the intensity of selection against particular regions of the mitochondrial genome, indicating that some mutations may be non-neutral within specific phylogenetic lineages but neutral within others.

DNA, Mitochondrial↗

PubMed2005

Quantitative trait locus study design from an information perspective.

We examine the efficiency of different genotyping and phenotyping strategies in inbred line crosses from an information perspective. This provides a mathematical framework for the statistical aspects of QTL experimental design, while guiding our intuition. Our central result is a simple formula that quantifies the fraction of missing information of any genotyping strategy in a backcross. It includes the special case of selectively genotyping only the phenotypic extreme individuals. The formula is a function of the square of the phenotype and the uncertainty in our knowledge of the genotypes at a locus. This result is used to answer a variety of questions. First, we examine the cost-information trade-off varying the density of markers and the proportion of extreme phenotypic individuals genotyped. Then we evaluate the information content of selective phenotyping designs and the impact of measurement error in phenotyping. A simple formula quantifies the information content of any combined phenotyping and genotyping design. We extend our results to cover multigenotype crosses, such as the F(2) intercross, and multiple QTL models. We find that when the QTL effect is small, any contrast in a multigenotype cross benefits from selective genotyping in the same manner as in a backcross. The benefit remains in the presence of a second unlinked QTL with small effect (explaining <20% of the variance), but diminishes if the second QTL has a large effect. Software for performing power calculations for backcross and F(2) intercross incorporating selective genotyping and marker spacing is available from http://www.biostat.ucsf.edu/sen.

Animals↗

PubMed2005

Nonstationary noise estimation in functional MRI.

An important issue in functional MRI analysis is accurate characterisation of the noise processes present in the data. Whilst conventional fMRI noise representations often assume stationarity (or time-invariance) in the noise generating sources, such approaches may serve to suppress important dynamic information about brain function. As an alternative to these fixed temporal assumptions, we present in this paper two time-varying procedures for examining nonstationary noise structure in fMRI data. In the first procedure, we approximate nonstationary behaviour by means of a collection of simple but numerous time-varying parametric models. This is accomplished through the derivation of a locally parametric AutoRegressive (AR) plus drift model which tracks temporal covariance by allowing the model parameters to evolve over time. Before exploring time variation in these parameters, window-widths (bandwidths) that are well suited to the latent time-varying noise structure must be determined. To do this, we employ a bandwidth selection mechanism based on Stein's Unbiased Risk Estimator (SURE) criterion. In the second procedure, we describe the fMRI noise using a nonparametric method based on Functional Data Analysis (FDA). This process generates well-conditioned nonstationary covariance estimates that reflect temporal continuity in the underlying data structure whilst penalizing effective model dimension. We demonstrate both methods on simulated data and investigate the presence of nonstationary noise in resting fMRI data using the whitening capabilities of the locally parametric procedure. We evaluate the comparative behaviour of the stationary and nonstationary AR-based methods on data acquired at 1.5, 3 and 7 T magnetic field strengths and show that incorporation of time variation in the AR parameters leads to an overall decrease in the level of residual structure in the data. The FDA noise modelling technique is formulated within an activation mapping procedure and compared to the SPM (Statistical Parametric Mapping) toolbox on a cognitive face recognition task. Both the SPM and FDA methods show good sensitivity on this task, but we find that inclusion of the nonstationary FDA noise model seems to improve detection power in important task-related medial temporal regions.

Algorithms↗

PubMed2001

Large upward bias in estimation of locus-specific effects from genomewide scans.

The primary goal of a genomewide scan is to estimate the genomic locations of genes influencing a trait of interest. It is sometimes said that a secondary goal is to estimate the phenotypic effects of each identified locus. Here, it is shown that these two objectives cannot be met reliably by use of a single data set of a currently realistic size. Simulation and analytical results, based on variance-components linkage analysis as an example, demonstrate that estimates of locus-specific effect size at genomewide LOD score peaks tend to be grossly inflated and can even be virtually independent of the true effect size, even for studies on large samples when the true effect size is small. However, the bias diminishes asymptotically. The explanation for the bias is that the LOD score is a function of the locus-specific effect-size estimate, such that there is a high correlation between the observed statistical significance and the effect-size estimate. When the LOD score is maximized over the many pointwise tests being conducted throughout the genome, the locus-specific effect-size estimate is therefore effectively maximized as well. We argue that attempts at bias correction give unsatisfactory results, and that pointwise estimation in an independent data set may be the only way of obtaining reliable estimates of locus-specific effect-and then only if one does not condition on statistical significance being obtained. We further show that the same factors causing this bias are responsible for frequent failures to replicate initial claims of linkage or association for complex traits, even when the initial localization is, in fact, correct. The findings of this study have wide-ranging implications, as they apply to all statistical methods of gene localization. It is hoped that, by keeping this bias in mind, we will more realistically interpret and extrapolate from the results of genomewide scans.

Bias↗

PubMed2000

New world monkey pepsinogens A and C, and prochymosins. Purification, characterization of enzymatic properties, cDNA cloning, and molecular evolution.

Pepsinogens A and C, and prochymosin were purified from four species of adult New World monkeys, namely, common marmoset (Callithrix jacchus), cotton-top tamarin (Saguinus oedipus), squirrel monkey (Saimiri sciureus), and capuchin monkey (Cebus apella). The occurrence of prochymosin was quite unique since this zymogen is known to be neonate-specific and, in primates, it has been thought that the prochymosin gene is not functional. No multiple form has been detected for any type of pepsinogen except that two pepsinogen-A isozymogens were identified in capuchin monkey. Pepsins A and C, and chymosin hydrolyzed hemoglobin optimally at pH 2-2.5 with maximal activities of about 20, 30, and 15 units/mg protein. Pepsins A were inhibited in the presence of an equimolar amount of pepstatin, and chymosins and pepsins C needed 5- and 100-fold molar excesses of pepstatin for complete inhibition, respectively. Hydrolysis of insulin B chain occurred first at the Leu15-Tyr16 bond in the case of pepsins A and chymosins, and at either the Leu15-Tyr16 or Tyr16-Leu17 bond in the case of pepsins C. The presence of different types of pepsins might be advantageous to New World monkeys for the efficient digestion of a variety of foods. Molecular cloning of cDNAs for three types of pepsinogens from common marmoset was achieved. A phylogenetic tree of pepsinogens based on the nucleotide sequence showed that common marmoset diverged from the ancestral primate about 40 million years ago.

Amino Acid Sequence↗

PubMed2005

Partly conditional survival models for longitudinal data.

It is common in longitudinal studies to collect information on the time until a key clinical event, such as death, and to measure markers of patient health at multiple follow-up times. One approach to the joint analysis of survival and repeated measures data adopts a time-varying covariate regression model for the event time hazard. Using this standard approach, the instantaneous risk of death at time t is specified as a possibly semi-parametric function of covariate information that has accrued through time t. In this manuscript, we decouple the time scale for modeling the hazard from the time scale for accrual of available longitudinal covariate information. Specifically, we propose a class of models that condition on the covariate information through time s and then specifies the conditional hazard for times t, where t > s. Our approach parallels the "partly conditional" models proposed by Pepe and Couper (1997, Journal of the American Statistical Association 92, 991-998) for pure repeated measures applications. Estimation is based on the use of estimating equations applied to clusters of data formed through the creation of derived survival times that measure the time from measurement of covariates to the end of follow-up. Patient follow-up may be terminated either by the occurrence of the event or by censoring. The proposed methods allow a flexible characterization of the association between a longitudinal covariate process and a survival time, and facilitate the direct prediction of survival probabilities in the time-varying covariate setting.

Acquired Immunodeficiency Syndrome↗

PubMed2005

The development and psychometric analyses of ADEPT: an instrument for assessing the interactions between doctors and their elderly patients.

BACKGROUND: The lack of instruments and methodologies designed specifically for assessing doctor-elderly patient interactions has constricted research on effective communication in the medical care of older adults. PURPOSE: This article reports on the development, qualitative analyses, and psychometric testing of the Assessment of Doctor-Elderly Patient Transactions (ADEPT), an instrument for assessing interactions between doctors and their elderly patients. METHODS: The ADEPT was based on the recommendations of an expert panel and designed around the three-function model of the medical interview. The ADEPT is meant to operationalize the research findings of interactional analysis studies of doctor-patient interaction. Following preliminary testing with standardized patients, the ADEPT was applied to videotaped visits of 433 patients 65 years of age and older to the doctor (n = 40) identified as their primary source of care. RESULTS: Four final scales derived from exploratory and confirmatory factor analyses were scored: Supporting, comprised of the 12 items from the first factor; Eliciting Needs, containing the 5 items from the second factor; and Informing, based on the final 6-item factor. Individual Cronbach's alphas across raters for this sample ranged from .71 to .79 for the first scale, from .83 to .88 for the second scale, and from .64 to .81 for the third scale. The reliability estimates for the total scale (23 items) ranged from .80 to .86 across raters. A fifth summed index composed of 46 binary checklist items also was computed. CONCLUSIONS: The findings indicate that credible scales can be developed for assessing communication behaviors.

Adult↗

PubMed2005

Estimating treatment efficacy over time: a logistic regression model for binary longitudinal outcomes.

This paper presents a case study in longitudinal data analysis where the goal is to estimate the efficacy of a new drug for treatment of a severe chronic constipation. Data consist of long sequences of binary outcomes (relief/no relief) on each of a large number of patients randomized to treatment (low and high dose) or placebo. Data characteristics indicate: (1) the treatment effects vary non-linearly with time; (2) there is substantial heterogeneity across subjects in their responses to treatment; and (3) there is a high proportion of subjects who never experience any relief (the non-responders). To overcome these challenges, we develop a hierarchical model for binary longitudinal data with a mixture distribution on the probability of response to account for the high frequency of non-responders. While the model is specified conditionally on subject-specific latent variables, we also draw inferences on key population-average parameters for the assessment of the treatments' efficacy in a population. In addition we employ a model-checking method to compare the goodness-of-fit for our model against simpler modelling approaches for aggregated counts, such as the zero-inflated Poisson and zero-inflated negative binomial models. We estimate subject-specific and population-average rate ratios of relief for the treatment with respect to the placebo as functions of time (RR(t)), and compare them with the rate ratios estimated from the models for aggregated counts. We find that: (1) the treatment is effective with respect to the placebo with higher efficacy at the beginning of the study; (2) the estimated rate ratios from the models for aggregated counts appear to be similar to the average across time of the population-average rate ratios estimated under our model; and (3) model-checking suggests that the hierarchical and zero-inflated negative binomial model fit the data best. If we are mainly interested to establish the overall efficacy (or safety) of a new drug, it is appropriate to aggregate the longitudinal data over time and analyse the count data by use of standard statistical methods. However, the models for aggregated counts cannot capture time trend of treatment such as the initial treatment benefit or the development of tolerance during the early stage of the treatment which may be important information to physicians to predict the treatment effects for their patients.

Biometry↗