Score Statistics Research Articles

Genome-wide screening for selection signatures in native and cosmopolitan cattle breeds reared in Türkiye.

Via long-term natural and artificial selection pressure, homozygosity may extend across the genome, leaving genomic patterns called selection signatures. This study is the first attempt to assess genome-wide selection signatures in six native Turkish and two cosmopolitan cattle breeds by 211.119 bi-allelic SNPs recovered using the double digest restriction associated DNA sequencing method. The integrated haplotype score (iHS) statistic was utilised to reveal selection signatures within populations, whereas the cross-population extended haplotype homozygosity (XP-EHH) and fixation index (FST ) approaches were preferred to reveal differently fixed genomic regions between native Turkish and cosmopolitan cattle breeds. Selection signatures in 142 genomic regions containing 305 genes were detected within eight cattle breeds by iHS statistics. The XP-EHH and FST approaches revealed that 197 and 114 SNPs were under selection pressure, respectively, which overlapped with 144 and 190 genes, respectively. A total of 18 genes were detected by at least two approaches. Six genes related to disease resistance (TTP2), meat yield (DIAPH3 and METTL21C), meat quality (ZNF24 and ZNF397) and first calving interval (ZSCAN30) turned out to be differently fixed between native Turkish and cosmopolitan cattle breeds, as they were identified by both XP-EHH and FST approaches. In addition, the iHS approach revealed that eight genes associated with visual modality (LSGN), olfaction (MOXD2, OR4C1F and OR4C1F), and immune response (TRBV3-1 and CLDN10) were under selection pressure in both native and cosmopolitan cattle breeds. Owing to their being significantly related to survival traits, these regions may have played a key role in cattle genome evolution. Future studies utilising denser genetic data are required to obtain deeper knowledge on effects of natural and artificial selection in Anatolian cattle breeds. © 2023 Stichting International Foundation for Animal Genetics.

Diagnosis of single and multiple-source faults of chiller sensors using EWEEMD-ICKNN by time sequence denoising and non-Gaussian distribution feature extraction

Currently, fault detection of chiller sensors primarily focuses on single-source fault (SSF) sensors. These methods are typically ineffective in multiple source faults (MSF) cases. These methods assume a Gaussian distribution of the chiller data and ignore the coupling between the sensors. Therefore, we propose temporal feature denoising and non-Gaussian distribution-based methods in this study. First, the chiller sensor signals are denoised based on exponential weighted moving average ensemble empirical mode decomposition (EWEEMD). Subsequently, an independent component analysis k-nearest neighbour (ICKNN) classifier is proposed based on the non-Gaussian distribution features of the chiller sensor data. A residual vector combined with independent component analysis was used for feature extraction to propose the ICKNN classifier. Finally, according to their characteristics, different SSF and MSF were applied to the chiller sensors. Experimental comparisons with other methods were performed for four single-source faults and six multisource faults. The EWEEMD-ICKNN was experimentally confirmed to have excellent fault detection performance in the SSF and MSF chiller sensors. The experimental results demonstrate that the fault diagnosis rate of EWEEMD-ICKNN ranges from 99.90% to 86.80%. Additionally, the fault diagnosis efficiency of EWEEMD-ICKNN remains consistently stable across various fault states, exhibiting an average F1 score of 0.9589 with F1 score statistics of 0.038.

Using item scores and response times in person-fit assessment.

The use of joint models for item scores and response times is becoming increasingly popular in educational and psychological testing. In this paper, we propose two new person-fit statistics for such models in order to detect aberrant behaviour. The first statistic is computed by combining two existing person-fit statistics: one for the item scores, and one for the item response times. The second statistic is computed directly using the likelihood function of the joint model. Using detailed simulations, we show that the empirical null distributions of the new statistics are very close to the theoretical null distributions, and that the new statistics tend to be more powerful than several existing statistics for item scores and/or response times. A real data example is also provided using data from a licensure examination.

Research on adaptive state prediction method for the metering error of capacitor voltage transformer

A capacitor voltage transformer (CVT) is widely used in high voltage power systems because of its good insulation performance. However, the structure of CVT is more complex and the stability of its metering error is poor, which easily causes the loss of power metering. The conventional evaluation method for the metering error of CVT is to compare and calibrate with a standard transformer under off-line condition in a fixed period. Because of the long evaluation period, it is impossible to accurately predict the state change of CVT metering error, which is of more significance practically. To solve this problem, this paper proposes an adaptive state prediction method: Analyze the measurement data of CVT using principal component analysis method under the constraint of electrical physical relationship, the metering of CVT is mapped to residual and score (CRS) statistic. For this way, the self-evaluation of CVT metering error in real-time is realized without a standard transformer to get the high frequency time series of error data. According to the measurement data of CVT in process, the CRS statistics are batch processed adaptively, and the prediction model of CRS statistics is established based on the time series analysis. Experiments show that the method can accurately predict the state change of CVT metering error, and the prediction error is better than 15%. It is helpful to promote the development of CVT metering error detection into on-demand detection.

Science Subject Specific Pedagogy to Support Disaster Risk Reduction in Education: Its Feasibility and Influence

As a country with a high level of vulnerability to volcanic disasters, Indonesia must have good preparedness in dealing with these disasters. One of the efforts to improve preparedness is through learning in schools so that the community has been introduced to disaster mitigation from an early age. This study aims to (1) produce a science subject-specific pedagogy that is integrated with volcanic eruption disasters and (2) find out the effectiveness of the use of integrated science subject-specific pedagogy of volcanic eruption disasters to students’ disaster concept mastery and preparedness in disaster mitigation. This research is R&D research with a 4-D model consisting of define, design, develop, and disseminate. The subject of this study was 7th-grade students whose school is located in a disaster-prone area for the eruption of Mount Merapi, Yogyakarta. Data were obtained from tests and non-tests, including concept mastery tests, preparedness tests, questionnaires, and performance observations. The data were analyzed descriptively using MANOVA's gain score and inferential statistics using the SPSS 22.0 program. The results of this study show that (1) the science subject-specific pedagogy product developed is included in the very good category based on validation results from expert and learning practitioners, and (2) science subject-specific pedagogy is effective for improving students’ disaster concept mastery and student preparedness in facing disasters. Overall, this science subject-specific pedagogy product is feasible for science learning and has been proven to improve students' preparedness and mastery of concepts regarding volcanic eruption disasters.

Combination fairness with scores in outlier detection ensembles

We revisit score combinations for outlier detection ensembles (ODEs). Different detectors' scores vary in ranges and scales. Normalization transforms these into a common range. We assert that ODE score combinations may remain biased despite normalization. Determining a suitable normalization for an outlier model is challenging, which captures original score statistics apt for different parameters across different datasets. Thus, treating higher normalized scores of one detector as superior outlying probabilities to other detectors may be inappropriate. With differences between top rankers' scores, detectors with higher normalized scores dominate during combination. This limits the effective utilization of diversity, undesired for ensembles. We propose a framework FairComb for providing unbiased combination treatment to outputs of all detectors. It stands for combination fairness in ODEs. We define combination parity, which quantifies differences between detectors' participation in ensemble results. Higher combination parity indicates higher combination unfairness. This fairness work differs from others, which aim for fair treatment of protected data groups. The proposed FairComb checks for significant score variations across an ensemble. A reduction approach is used to mitigate these variations. We use a baseline approach WtSR to prove that addressing this issue promotes fairer combinations. Experiments on benchmark datasets demonstrate the effectiveness of FairComb.

Extent of Accessibility and Utilization of Reference Materials by Nigerian University Undergraduates

The reference section of every library remains an important arm of such library. It contains materials needed for teaching, learning, community development and research. However, studies have shown that most of these materials are rarely consulted by library users, hence, the rationale of the present study which investigated the extent to which reference materials are accessed and utilized by the undergraduates of library and information science (LIS), Taraba State University (TSU), Jalingo. Six research objectives with corresponding research questions were considered and answered. The descriptive survey method was adopted using a total population of 133 final year undergraduates of LIS, TSU, Jalingo. Furthermore, the complete census technique was employed to study the entire population. A fifty-eight (58) items questionnaire was used for data collection. 133 copies were distributed, but 127 (95.5%) copies were returned and found suitable for data analysis. The data analysis was done using descriptive statistics of mean scores and frequency counts. Findings revealed a high extent of accessibility of a good number of reference materials. Few of these materials were reported to be utilized to a high extent leading to a relationship between the extent of accessibility and utilization. However, numerous challenges were identified to be facing the accessibility and utilization of reference materials by the undergraduates. The study concludes that the extent of accessibility determines the extent of utilization which all proved to be high in the library studied. It was, however, among other things, recommended that the management of university libraries should intensify efforts to improve the number and quality of reference materials provided and make them accessible to their users.

Genetic architecture of plasma Alzheimer disease biomarkers.

Genome-wide association studies (GWAS) of cerebrospinal fluid (CSF) Alzheimer's Disease (AD) biomarker levels have identified novel genes implicated in disease risk, onset and progression. However, lumbar punctures have limited availability and may be perceived as invasive. Blood collection is readily available and well accepted, but it is not clear whether plasma biomarkers will be informative for genetic studies. Here we perform genetic analyses on concentrations of plasma amyloid-β peptides Aβ40 (n = 1,467) and Aβ42 (n = 1,484), Aβ42/40 (n = 1467) total tau (n = 504), tau phosphorylated (p-tau181; n = 1079) and neurofilament light (NfL; n = 2,058). GWAS and gene-based analysis was used to identify single variant and genes associated with plasma levels. Finally, polygenic risk score and summary statistics were used to investigate overlapping genetic architecture between plasma biomarkers, CSF biomarkers and AD risk. We found a total of six genome-wide significant signals. APOE was associated with plasma Aβ42, Aβ42/40, tau, p-tau181 and NfL. We proposed 10 candidate functional genes on the basis of 12 single nucleotide polymorphism-biomarker pairs and brain differential gene expression analysis. We found a significant genetic overlap between CSF and plasma biomarkers. We also demonstrate that it is possible to improve the specificity and sensitivity of these biomarkers, when genetic variants regulating protein levels are included in the model. This current study using plasma biomarker levels as quantitative traits can be critical to identification of novel genes that impact AD and more accurate interpretation of plasma biomarker levels.

Health literacy in women with chronic pelvic pain.

Limited health literacy (HL) is a risk factor for poor patient outcomes, including pain. Chronic pelvic pain (CPP) is a prevalent disorder affecting up to 25% of women and coexists with multiple overlapping conditions. This study aimed to describe health literacy in women with CPP, primarily correlate HL to pain intensity and pain duration, and secondarily correlate HL to mood symptoms and pain catastrophizing. We hypothesized that women with CPP with higher HL would report lower levels of pain intensity and duration. This was a prospective, cross-sectional study. Forty-five women with CPP were recruited from outpatient Physical Medicine & Rehabilitation and Female Pelvic Medicine & Reconstructive Surgery clinics. Validated questionnaires were administered to evaluate pain intensity and duration, pain disability, psychological symptoms, pain catastrophizing, and health literacy. Statistical analyses included descriptive statistics of patient characteristics and summary scores, as well as Spearman's rank correlation coefficients (rho) to assess the strength of associations between summary scores and health literacy. Forty-five women with CPP were enrolled with mean age of 49years, majority non-Hispanic White, and median chronic pelvic pain duration of 7years. Possible or high likelihood of limited health literacy was identified in 20% women with CPP (11.1% and 8.9%, respectively). Limited health literacy was moderately correlated with pain intensity, depressive symptoms, and pain catastrophizing. Pain duration was not significantly correlated with health literacy. The remaining 80% of women with CPP were likely to have adequate health literacy. A majority of women with CPP in this single center study were likely to have adequate health literacy. Limited health literacy was seen in a minority of women with CPP but was moderately correlated with greater pain intensity, more depressive symptoms, and higher pain catastrophizing. This study identified that women with CPP were likely to have adequate HL, but underscores the importance of considering HL screening and interventions in those with higher pain intensity, depression, and pain catastrophizing.

A score test for detecting extreme values in a vector autoregressive model

In this paper, we propose a score test to study a vector autoregressive model and its detection of extreme values. We take a likelihood approach to derive the corresponding maximum likelihood estimators and information matrix. We establish the score statistic for the vector autoregressive model under two perturbation schemes for identifying possible influential cases or outliers. The effectiveness of the proposed diagnostics is examined by a simulation study. To make an application, a data analysis is performed using the model to fit monthly log-returns of International Business Machines Corporation stock and the Standard & Poor's 500 index. Lastly, comparisons between the results by the score test and the local influence method are made. We establish two important findings that the score test is more effective while the local influence analysis can be used to diagnose more influential cases.

Norms and T-scores for screeners of alcohol use, depression and anxiety in the population of Suriname.

There is a considerable gap between care provision and the demand for care for common mental disorders in low-and-middle-income countries. Screening for these disorders, e.g., in primary care, will help to close this gap. However, appropriate norms and threshold values for screeners of common mental disorders are lacking. In a survey study, we gathered data on frequently used screeners for alcohol use disorders, (AUDIT), depression, (CES-D), and anxiety disorders (GAD-7, ACQ, and BSQ) in a representative sample from Suriname, a non-Latin American Caribbean country. A stratified sampling method was used by random selection of 2,863 respondents from 5 rural and 12 urban resorts. We established descriptive statistics of all scale scores and investigated unidimensionality. Furthermore, we compared scores by gender, age-group, and education level with t-test and Mann-Whitney U tests, using a significance level of p < 0.05. Norms and crosswalk tables were established for the conversion of raw scores into a common metric: T-scores. Furthermore, recommended cut-off values on the T-score metric for severity levels were compared with international cut-off values for raw scores on these screeners. The appropriateness of these cut-offs and the value of converting raw scores into T-scores are discussed. Cut-off values help with screening and early detection of those who are likely to have a common mental health disorder and may require treatment. Conversion of raw scores to a common metric in this study facilitates the interpretation of questionnaire results for clinicians and can improve health care provision through measurement-based care.

Fast and accurate recurrent event analysis for genome-wide association studies.

Many diseases recur after recovery, for example, recurrences in cancer and infections. However, research is often focused on analysing only time-to-first recurrence, thereby ignoring any subsequent recurrences that may occur after the first. Statistical models for the analysis of recurrent events are available, of which the extended Cox proportional hazards frailty model is the current state-of-the-art. However, this model is too statistically complex for computationally efficient application in high-dimensional data sets, including genome-wide association studies (GWAS). Here, we develop an application for fast and accurate recurrent event analysis in GWAS, called SPARE (SaddlePoint Approximation for Recurrent Event analysis). In SPARE, every DNA variant is tested for association with recurrence risk using a modified score statistic. A saddlepoint approximation is implemented to achieve statistical accuracy. SPARE controls the Type I error, and its statistical power is similar to existing recurrent event models, yet SPARE is significantly faster. An application of SPARE in a recurrent event GWAS on bladder cancer for 6.2 million DNA variants in 1,443 individuals required less than 15 min, whereas existing recurrent event methods would require several weeks.

460 Integrating pharmacist independent prescribing in community pharmacy: investigating NHS Pharmacy First Plus using Normalization Process Theory

Abstract Introduction Limited evidence exists for independent prescribing in community pharmacy (CP) (1). In 2020 NHS Scotland launched NHS Pharmacy First Plus (PFP), aiming to facilitate access to pharmacist independent prescribers (PIPs) in CP, focusing on common clinical conditions. Aim To investigate, using Normalization Process Theory (NPT), the integration of PIPs in CP in the context of NHS PFP. Methods A cross-sectional online survey of all 120 community pharmacists offering PFP in 13 Scottish NHS Health Boards. A ‘Think Aloud’ tested, pre-piloted questionnaire was developed using the NPT derived NoMAD tool (2) and including demographic and practice characteristic questions for contextualisation. This was disseminated via email with 3 reminders. Data exported from Online Surveys (JISC) to SPSS was analysed descriptively. Scale score statistics for each of the four NPT constructs (Table 1) were calculated through summation of item scores (5 = Strongly agree to 1 = Strongly disagree) for each respondent with scale item reliability testing using Cronbach’s alpha. Results A 73% (88/120) response rate showed the majority were female (63%, 55/88), &amp;lt;40 years old (51%, 45/88), had worked in CP for &amp;gt;15 years (56%, 49/88) and qualified as a PIP &amp;lt;5 years ago (59%, 52/88). 76% (67/88) indicated on average they consulted with patients under PFP &amp;gt;6 times a week. 38% (33/88) indicated they had &amp;lt;3 staff when offering PFP and 47% (41/88) indicated that they provided the service while working as the only pharmacist; 35% (25/71) had one other pharmacist. Generally, there were high levels of NoMAD item agreement with &amp;gt;90% ‘strongly agreeing’ or ‘agreeing’ in items relating to ‘coherence’, ‘cognitive participation’ and ‘reflexive monitoring’. Responses to ‘collective action’ items showed diversity with more answering ‘neither agree nor disagree’ or ‘disagree’. Conclusion This theory-based work with an excellent response rate offers a robust and unique perspective on PIP integration within CP. The positivity could perhaps be expected from trailblazer respondents but it also highlights challenges within the ‘collective action’ construct and a need to focus on training, staff resource, working relationships and management. More work is planned on scale score inferential analysis and qualitative exploration of key findings.

Тест пищевых установок (ЕАТ-26): оценка психометрических характеристик и факторной структуры на неклинической выборке 876 студенток

&lt;p style="text-align: justify;"&gt;The Eating Attitudes Test (EAT-26) is widely used all over the world, including Russia, for screening eating disorders, however, the psychometric characteristics and factorial structure of the questionnaire have not yet been studied on Russian non-clinical samples. This study involved 876 female students aged 17&amp;ndash;26 (M=20.11&amp;plusmn;1.79) from three cities (Moscow, Ryazan, and Arkhangelsk). All respondents were asked to fill out an adapted Russian version of EAT-26. To assess the retest reliability, 48 students from Ryazan filled out the questionnaire twice with an interval of one month. To assess construct validity, 293 female students from Ryazan filled out the Body Appreciation Scale (BAS) to assess a positive attitude towards the body, 281 students from Arkhangelsk filled out the Hospital Anxiety and Depression Scale (HADS). As a result, descriptive statistics of the total score and responses to individual items of EAT-26 were obtained. In accordance with the critical value of EAT-26 (&amp;ge;20), the risk group for eating disorders included 13.5% of the respondents. The high internal consistency (Cronbach's alpha &amp;mdash; 0.883) and retest reliability (r=0.865) of the EAT-26 total were shown. The confirmatory factor analyses failed to confirm the three-factor structure originally proposed by the authors of the questionnaire. Confirmatory factor analysis confirmed a five-factor model (&amp;chi;&lt;sup&gt;2&lt;/sup&gt;=226.584, df=81, p&amp;lt;10&lt;sup&gt;-6&lt;/sup&gt;, CFI=0.970, RMSEA=0.046, 90% CI [0.038; 0.052] with 15 items included (EAT-15), which has high rates of internal consistency and retest reliability (Cronbach's alpha &amp;mdash; 0.840; r=0.861). Psychometric characteristics and indicators of reliability and validity of EAT-26, EAT-15, and the five selected scales are given.&lt;/p&gt;

Pengajaran Tarannum Dalam Kalangan Guru Pendidikan Islam Di Sekolah Rendah

The Quran is a miracle from Allah SWT that is very valuable to humans in terms of language, history, and civilization, as well as its values. There is an ongoing effort that is carried out informally through tarannum learning classes run by Islamic education teachers. Therefore, the purpose of this study is to evaluate the tarannum teaching practice among Islamic education teachers in primary schools. This quantitative study was conducted using a questionare and involving 62 Islamic education teachers from 24 primary schools in the Perak Tengah District. This study focuses on the practice of teaching tarannum knowledge based on the teaching model proposed by al-Qabisi. The level of teaching practice of tarannum among Islamic education teachers was analyzed using the mean score and variance inference statistics (ANOVA). Overall, the data analysis shows that the practice of teaching tarannum among Islamic education teachers is at a high level. All factors of the practice of teaching tarannum knowledge among Islamic education teachers in primary schools, which include teaching goals, teaching objectives, teaching methods, teaching techniques, teaching aids, and teaching evaluation, recorded a high level. The findings of the study also show that there is no significant difference in the level of teaching practice among Islamic education teachers in primary schools based on their teaching experience. This study also confirms the practice model of al-Quran teaching introduced by al-Qabisi as a basis for strengthening the practice of teaching tarannum knowledge for Islamic education teachers in primary schools.

Modeling Lower-Order Statistics to Enable Decoy-Free FDR Estimation in Proteomics.

One of the chief objectives in mass spectrometry-based peptide identification in proteomics is the statistical validation of top-scoring peptide-spectrum matches (PSMs) in the form of false discovery rate (FDR) estimation. Existing methods construct a null model that captures the characteristics of incorrect target PSMs to estimate the FDR, most often with the help of decoys. Decoy-based methods, however, increase the computational cost and rely on the difficult-to-verify assumption that decoy PSMs constitute a sufficient and representative sample of the population of possible incorrect target PSMs. On the other hand, the possibility of FDR estimation assisted by the plentiful non-top-scoring PSMs, which are almost always incorrect, has been scarcely explored. In this work, we propose a novel decoy-free procedure for developing null models for top-scoring PSMs using the transformed e-value (TEV) score and the distributions of non-top-scoring target PSMs. The method relies on a theoretically derivable relationship between the parameters of the distributions of lower-order statistics of the TEV score and a necessary empirical optimization to fit a single parameter to actual data. The framework was tested on multiple different data sets and two search engines. We present evidence that our method is comparable to and occasionally outperforms popular decoy-free and decoy-based methods in FDR estimation.

Association of Biochemical &amp; Anthropometric parameters with depressive disorder among adults in a Health care centre

Background: Depression may be a common psychiatric disorder. In spite of the fact that numerous risk factors for sadness have been reported, the affiliations of biochemical and anthropometric records with depressive disorder remain unclear. Depressive disorder is a leading cause of illness around the world. Aim: To access the association between biochemical and anthropometric parameters with depressive disorders among adults Methods: This quantitative cross sectional study was conducted at B.V. Hospital, Bahawalpur from July 1, 2022 to December 31, 2022. A pre validated questionnaire was used. Study was carried out at outdoor patient department of Bahawal Victoria Hospital Bahawalpur. The total numbers of patients were 250 who participated after taking the informed consent. The study was done by simple random sampling technique. Detailed information was given to patient about the current research. Results: The descriptive statistics for anthropometric measures, demographic, baseline variables and CES-D scores for all the subjects are summarized in Table 1. Total number of participants in the study was 250 in which 115 were males and 135 were females. Baseline characteristics were measured include weight, waist circumference and body mass index and others parameters include systolic and diastolic BP, blood sugar fasting, glycemic control and lipid profile was measured both in depressed and not depressed group. Conclusion: Depression and obesity are major wellbeing issues worldwide and may increase metabolic disorders. Obesity, which is related with numerous wellbeing issues related to changes in way of life, can take after depression that happened earlier in life or in early adulthood, a vital time for implementation of health-promoting interventions. Keywords: Depression, Biochemical parameters, Anxiety, Adults, Health Care

Wald, QLR, and score tests when parameters are subject to linear inequality constraints

This paper develops Wald-type, QLR, and score-type tests for linear equality constraints in a general class of extremum estimation problems where the parameter space is characterized by a finite number of linear equality and inequality constraints. We show that the asymptotic null distributions of the Wald and QLR statistics are discontinuous in an implicit nuisance parameter and propose an algorithm to identify it. In contrast, the asymptotic null distribution of the score statistic is not discontinuous in any model parameter but depends on a polytope projection. We present an algorithm based on the Fourier–Motzkin elimination to compute such a projection. We study the consistency and local power properties of the three tests. Finally, we present numerical results of our tests’ finite sample performance from a Monte Carlo study and conduct an empirical illustration of a Mincer earnings regression.

Simple methods for comparing two predictive values with incomplete data

ABSTRACT Statistical methods have been well developed for comparing the predictive values of two binary diagnostic tests under a paired design. However, existing methods do not make allowance for incomplete data. Although maximum likelihood based method can be used to deal with incomplete data, it requires iterative algorithm for implementation. A simple and easily implemented statistical method is therefore needed. Simple methods exist for comparing two sensitivities or specificities with incomplete data but such simple methods are not available for comparing two predictive values with incomplete data. In this paper, we propose two simple methods for comparing two predictive values with incomplete data. The test statistics derived by these two methods are simple to compute, only involving some minor modification of the existing weighted generalized score statistics with complete data. Simulation results demonstrate that the proposed methods are more efficient than the ad-hoc method that only uses the subjects wit complete data. As an illustration, the proposed methods are applied to an observational study comparing two non-invasive methods in detecting endometriosis.

Homogeneity Test of the First-Order Agreement Coefficient in a Stratified Design

Gwet's first-order agreement coefficient (AC1) is widely used to assess the agreement between raters. This paper proposes several asymptotic statistics for a homogeneity test of stratified AC1 in large sample sizes. These statistics may have unsatisfactory performance, especially for small samples and a high value of AC1. Furthermore, we propose three exact methods for small pieces. A likelihood ratio statistic is recommended in large sample sizes based on the numerical results. The exact E approaches under likelihood ratio and score statistics are more robust in the case of small sample scenarios. Moreover, the exact E method is effective to a high value of AC1. We apply two real examples to illustrate the proposed methods.