Test Of Null Hypothesis Research Articles

Instructional Humour and Cognitive Affective Learning with Multimedia (IHCALM)

While background music and interesting yet irrelevant to the topic adjuncts were found to harm learning (and were classified as seductive detail) in the Cognitive-Affective Theory of Learning with Media model (CATLM), emotionally appealing shapes and color were found to foster learning (and were classified as multimedia with emotional design). However, although humour is used in education during class and has both psychological and physiological benefits, there is no published research about instructional humour (IH) in CATLM to date. The purpose of the current research was to clarify whether IH in CATLM fosters learning, or if it is yet another type of seductive detail. Total of 96 young undergraduate student participants were randomly assigned to watch a stimuli depicting 3D animations of brain cells either with IH (named as IHCALM) or without it (named as NH). All student data regarding mirth duration were measured with cameras, while how funny they found the stimuli, as well as their cognitive load, emotions, motivation, knowledge, and metacognition were all measured with Open Sesame. To test if the IHCALM harms learning, similarity between conditions was analyzed with both Bayesian Factor analysis and null hypothesis testing, which jointly reveal 3 outcomes. Outcome results show that IHCALM does not harms learning, due to being similar with the non-humorous condition. Implications of these findings for education are considered.

Bayesian Inference on the Bimodality of the Generalized von Mises Distribution

This article introduces Bayesian inference on the bimodality of the generalized von Mises (GvM) distribution for planar directions (Gatto and Jammalamadaka in Stat Methodol 4(3):341–353, 2007). The GvM distribution is a flexible model that can be axial symmetric or asymmetric, unimodal or bimodal. Two inferential approaches are analysed. The first is the test of null hypothesis of bimodality and Bayes factors are obtained. The second approach provides a two-dimensional highest posterior density (HPD) credible set for two parameters relevant to bimodality. Based on the identification of the two-dimensional parametric region associated with bimodality, the inclusion of the HPD credible set in that region allows us to infer on the bimodality of the underlying GvM distribution. A particular implementation of the Metropolis–Hastings algorithm allows for the computation of the Bayes factors and the HPD credible sets. A Monte Carlo study reveals that, whenever the samples are generated under a bimodal GvM, the Bayes factors and the HPD credible sets do clearly confirm the underlying bimodality.

Think like a Bayesian and avoid pitfalls from our frequentist past

Bayesian inference is a powerful tool that is increasingly being used by ecologists. This is largely due to the flexibility in model specification and improvements in software that makes this tool easier to use. However, with increasing ease of use comes a risk of misuse or abuse. We review four major issues we have identified in the use of Bayesian methods and offer reminders and suggestions that will improve the application and reporting of Bayesian inference while at the same time, hopefully, avoiding the pitfalls that have plagued null hypothesis statistical testing (NHST). These issues include; 1) understanding software and model specification; 2) use of prior probability distributions; 3) maximizing utility of posterior probability distributions; and 4) avoiding dichotomous thinking (i.e., the NHST pitfall). We suggest ecologists should strive for openness in their use of statistical software by understanding their model and providing the full computer code used, develop reasonable and informative priors, and make full use of posterior information that Bayesian methods provide. At the same time, ecologists should avoid dichotomizing results into significant/ non-significant boxes, eliminate null hypothesis tests (including probability intervals for hypothesis testing), and use clear language when describing results. Finally, quantitative training should be expanded in undergraduate curricula to provide students with a larger suite of foundational core concepts that extend beyond NHST.

Equivalence tests for ratio of means in bioequivalence studies under crossover design

There are several problems concerning the statistical definition of average bioequivalence provided by the U.S. Food and Drug Administration. We proposed a ratio of means based on the original bioavailability measure as the definition for average bioequivalence. Under the log-normal distribution assumption, we proposed a hypothesis testing-based method and a confidence interval-based method to answer the question of whether the ratio of means falls into a predetermined interval. For the hypothesis testing-based method, we decomposed the null two-sided hypothesis of the ratio of means into two one-sided hypotheses. With the intersection-union theorem for asymptotic tests, we constructed two asymptotic size- tests for the original null hypothesis. The method of variance estimation recovery was adopted to develop the confidence interval-based method. Simulation studies showed that the proposed methods can maintain the empirical type I error rate closely at the nominal level and is as powerful as the two one-sided -test for testing the ratio of means under different settings. The application of the proposed methods was illustrated through six datasets in real-world examples.

Without Commitment to an Ontology, There Could Be No Causal Inference.

Without Commitment to an Ontology, There Could Be No Causal Inference.

Characterization of anisotropic Gaussian random fields by Minkowski tensors

Gaussian random fields are among the most important models of amorphous spatial structures and appear across length scales in a variety of physical, biological, and geological applications, from composite materials to geospatial data. Anisotropy in such systems can be sensitively and comprehensively characterized by the so-called Minkowski tensors (MTs) from integral geometry. Here, we analytically calculate expected MTs of arbitrary rank for the level sets of Gaussian random fields. The explicit expressions for interfacial MTs are confirmed in detailed simulations. We demonstrate how the MTs detect and characterize the anisotropy of the level sets, and we clarify which shape information is contained in the MTs of different rank. Using an irreducible representation of the MTs in the Euclidean plane, we show that higher-rank tensors indeed contain additional anisotropy information compared to a rank two tensor. Surprisingly, we can nevertheless predict this information from the second-rank tensor if we assume that the random field is Gaussian. This relation between tensors of different rank is independent of the details of the model. It is, therefore, useful for a null hypothesis test that detects non-Gaussianities in anisotropic random fields.

The Importance of Random Slopes in Mixed Models for Bayesian Hypothesis Testing

Mixed models are gaining popularity in psychology. For frequentist mixed models, previous research showed that excluding random slopes—differences between individuals in the direction and size of an effect—from a model when they are in the data can lead to a substantial increase in false-positive conclusions in null-hypothesis tests. Here, I demonstrated through five simulations that the same is true for Bayesian hypothesis testing with mixed models, which often yield Bayes factors reflecting very strong evidence for a mean effect on the population level even if there was no such effect. Including random slopes in the model largely eliminates the risk of strong false positives but reduces the chance of obtaining strong evidence for true effects. I recommend starting analysis by testing the support for random slopes in the data and removing them from the models only if there is clear evidence against them.

Lightweight ProteinUnet2 network for protein secondary structure prediction: a step towards proper evaluation

BackgroundThe prediction of protein secondary structures is a crucial and significant step for ab initio tertiary structure prediction which delivers the information about proteins activity and functions. As the experimental methods are expensive and sometimes impossible, many SS predictors, mainly based on different machine learning methods have been proposed for many years. Currently, most of the top methods use evolutionary-based input features produced by PSSM and HHblits software, although quite recently the embeddings—the new description of protein sequences generated by language models (LM) have appeared that could be leveraged as input features. Apart from input features calculation, the top models usually need extensive computational resources for training and prediction and are barely possible to run on a regular PC. SS prediction as the imbalanced classification problem should not be judged by the commonly used Q3/Q8 metrics. Moreover, as the benchmark datasets are not random samples, the classical statistical null hypothesis testing based on the Neyman–Pearson approach is not appropriate.ResultsWe present a lightweight deep network ProteinUnet2 for SS prediction which is based on U-Net convolutional architecture and evolutionary-based input features (from PSSM and HHblits) as well as SPOT-Contact features. Through an extensive evaluation study, we report the performance of ProteinUnet2 in comparison with top SS prediction methods based on evolutionary information (SAINT and SPOT-1D). We also propose a new statistical methodology for prediction performance assessment based on the significance from Fisher–Pitman permutation tests accompanied by practical significance measured by Cohen’s effect size.ConclusionsOur results suggest that ProteinUnet2 architecture has much shorter training and inference times while maintaining results similar to SAINT and SPOT-1D predictors. Taking into account the relatively long times of calculating evolutionary-based features (from PSSM in particular), it would be worth conducting the predictive ability tests on embeddings as input features in the future. We strongly believe that our proposed here statistical methodology for the evaluation of SS prediction results will be adopted and used (and even expanded) by the research community.

Effect sizes for contrasts of estimated marginal effects

The statistical literature is replete with calls to report standardized measures of effect size alongside traditional p-values and null hypothesis tests. While effect-size measures such as Cohen’s d and Hedges’s g are straightforward to calculate for t tests, this is not the case for parameters in more complex linear models, where traditional effect-size measures such as η 2 and ω 2 face limitations. After a review of effect sizes and their implementation in Stata, I introduce the community-contributed command mces. This postestimation command reports standardized effect-size statistics for dichotomous comparisons of marginal-effect contrasts obtained from margins and mimrgns, including with complex samples, for continuous outcome variables. mces provides Stata users the ability to report straightforward estimates of effect size in many modeling applications.

Spatio-temporal distribution of Vachellia stuhlmannii (Taub.) in Venetia-Limpopo nature reserve – SPOT-imagery

The impact of African elephant (Loxodonta africana) on selected woody plants varies from local extirpation to persistence amid continuous utilisation. Species generally selected by elephants and the way they are foraged makes them vulnerable to high utilisation levels. This paper studied the spatial and temporal distribution of Vachellia stuhlmannii in Venetia-Limpopo Nature Reserve using SPOT ( S ytéme Pour l’Observation de la Terre) imagery. The spatial pattern distribution of the species was computed using the Spatial Point Pattern Analysis (SPPA) and the Complete Spatial Randomness (CSR) of the species was tested using the Nearest Neighbour Ripley’s K(r) function. Results indicate a significant decrease in the area covered by V. stuhlmannii between 2013 and 2017, with a subsequent decrease in the density of the species. This result was positively correlated (Pearson correlation r 3 = 0.796, P = 0.634) to a decrease in annual rainfall. V. stuhlmannii follows an aggregated spatial pattern, which led to the rejection of the null hypothesis (H o) test of complete spatial randomness. Limitation of this study is that the results are based on five-year data between 2013 and 2017. This was influenced by the SPOT data used that is made available commercially. It is recommended that further studies evaluate the species distribution and density prior to 2013 to determine whether the decrease is because of other browsers, diseases, elephant impact, decreasing annual rainfall or combination of factors.

Predictability of Exfoliation Time of Deciduous Teeth

Background: The predictability of the eruption time of primary teeth has an important place in clinical applications, forensic medicine, planning of orthodontic treatments and age estimation. Aim: In this study, we tried to contribute to dental literature by trying to prove the predictability of the falling time of primary teeth. Design: From the periapical radiographs taken from the patients who applied to the clinic with the complaint of primary second molars, primary first molars were detected without caries and with 1 to 3 months to fall. The teeth of the patients for whom follow-up was recommended, which were due to fall out and adhered only to the gingiva, were extracted with the help of cotton, according to the patient’s complaint. Fall times were recorded. Results: Null Hypothesis test was applied to estimate the falling time of the teeth followed in the study, and the success rate in the 90-day exfoliation period was statistically significant, but no statistically significant difference was found in the 30-day samples. Conclusion: We think that, besides estimating the falling time of primary teeth in stem cell research, orthodontic treatment, clinical treatment planning and age determination, examining of primary teeth with correct without causing ethical problems will make a significant contribution to the literature. Clinical Benefit: Naturally and physiologically shedding deciduous teeth are unique ex vivo human tissue resources with abundance of multiple cells, including stem cells. The predictability of shedding time of deciduous teeth would be very important while planning sensible stem cell experiments.

Knowledge, Attitude and Practice of Care of Elderly Among Health Workers in Tertiary Health Facilities in Uyo Metropolis

Elderly care emphasizes the social and personal requirements of senior citizens who need some assistance with daily activities and health care, but who desire to age with dignity. With increasing rate of elderly in the society, there have been insufficient healthcare services for the elderly. Most health workers have very poor knowledge of health conditions which are common with the elderly and this lack of adequate knowledge triggers negative attitude towards the care of the elderly likely to result in serious problems in our society in the near future. This study was conducted to examine the knowledge, attitude and practice of care of elderly among health workers in tertiary health facilities in Uyo Metropolis. Three specific objectives, three research questions and three null hypotheses guided the study. The study employed a cross-sectional descriptive research design. The population of study consisted of 394 health workers. The sample of 241 health workers comprising of 27 doctors, 190 nurses, 10dentists and 14 Physiotherapist was determined using Taro Yamane and selected for the study using stratified sampling technique. The instrument used for data collection was questionnaire titled “Knowledge, Attitude and Practice of Care of Elderly Questionnaire (KAPCEQ)”. The instrument yielded the reliability coefficient of 0.84. The data were collected by the researcher and four trained research assistants. Frequency and percentage were used for the answering of the research questions while chi-square was used for testing of null hypotheses at .05 level of significance. Findings of the study indicated that health workers had adequate knowledge of care of elderly, exhibit good attitude toward care of elderly and sometimes practice care of the elderly in tertiary health facilities in Uyo Metropolis. Based on the findings of the study, the researcher recommended among others that Hospital Management Board should develop a system of periodical health workers evaluation to determine strategies of upgrading health workers knowledge and enhancing practice of care of the elderly. Furthermore, Professional bodies in the health sectors should develop a follow up courses and in-service training programmes to maintain efficient performance of individuals previously trained in the care of the elderly.

From explanation of the past to prediction of the future: A comparative and predictive research design in the Social Sciences

Business and Psychology research (and the Social Sciences, in general) is heavily biased toward explaining the past. The holy grail in such explanation-oriented research is to develop causal theory, and to test this theory with historical data against a null no-effect benchmark. We seek to expand the methodological toolkit by adding a comparative and predictive research design. First, by organizing an inter-theory battle, we move away from classic null hypothesis testing. Second, by predicting the future, we add prediction as a complement to the traditional explanation of the past. By way of illustration, we select a case in the Entrepreneurship field and theorize about the ranked predictions as to the relative growth performance of a sample of Small and Medium-sized Enterprises (SMEs). For this, we adopt two widely acknowledged theories in the literatures of Business and Psychology: The Competitive Strategy theory and the Motive Disposition theory. We use Gamblers' Ruin or Random Walk theory, arguing that company growth cannot be predicted, as the null benchmark. After identifying key explanatory predictive variables of our basic pair of theories, with Gamblers' Ruin or Random Walk theory's non-predictability as our benchmark, we produce ranked predictions as to the relative growth performance of 294 Belgian entrepreneurs and their SMEs. Later in 2023, we will test the predictive accuracy of these two selected theories and their predictive variables by comparing the predictive rankings with realized growth, as well as vis-à-vis randomness.

A comprehensive gene-centric pleiotropic association analysis for 14 psychiatric disorders with GWAS summary statistics.

BackgroundRecent genome-wide association studies (GWASs) have revealed the polygenic nature of psychiatric disorders and discovered a few of single-nucleotide polymorphisms (SNPs) associated with multiple psychiatric disorders. However, the extent and pattern of pleiotropy among distinct psychiatric disorders remain not completely clear.MethodsWe analyzed 14 psychiatric disorders using summary statistics available from the largest GWASs by far. We first applied the cross-trait linkage disequilibrium score regression (LDSC) to estimate genetic correlation between disorders. Then, we performed a gene-based pleiotropy analysis by first aggregating a set of SNP-level associations into a single gene-level association signal using MAGMA. From a methodological perspective, we viewed the identification of pleiotropic associations across the entire genome as a high-dimensional problem of composite null hypothesis testing and utilized a novel method called PLACO for pleiotropy mapping. We ultimately implemented functional analysis for identified pleiotropic genes and used Mendelian randomization for detecting causal association between these disorders.ResultsWe confirmed extensive genetic correlation among psychiatric disorders, based on which these disorders can be grouped into three diverse categories. We detected a large number of pleiotropic genes including 5884 associations and 2424 unique genes and found that differentially expressed pleiotropic genes were significantly enriched in pancreas, liver, heart, and brain, and that the biological process of these genes was remarkably enriched in regulating neurodevelopment, neurogenesis, and neuron differentiation, offering substantial evidence supporting the validity of identified pleiotropic loci. We further demonstrated that among all the identified pleiotropic genes there were 342 unique ones linked with 6353 drugs with drug-gene interaction which can be classified into distinct types including inhibitor, agonist, blocker, antagonist, and modulator. We also revealed causal associations among psychiatric disorders, indicating that genetic overlap and causality commonly drove the observed co-existence of these disorders.ConclusionsOur study is among the first large-scale effort to characterize gene-level pleiotropy among a greatly expanded set of psychiatric disorders and provides important insight into shared genetic etiology underlying these disorders. The findings would inform psychiatric nosology, identify potential neurobiological mechanisms predisposing to specific clinical presentations, and pave the way to effective drug targets for clinical treatment.

Reverse Bayesian Implications of p-Values Reported in Critical Care Randomized Trials.

BackgroundMisinterpretations of the p-value in null-hypothesis statistical testing are common. We aimed to determine the implications of observed p-values in critical care randomized controlled trials (RCTs).MethodsWe included three cohorts of published RCTs: Adult-RCTs reporting a mortality outcome, Pediatric-RCTs reporting a mortality outcome, and recent Consecutive-RCTs reporting p-value ≤.10 in six higher-impact journals. We recorded descriptive information from RCTs. Reverse Bayesian implications of obtained p-values were calculated, reported as percentages with inter-quartile ranges.ResultsObtained p-value was ≤.005 in 11/216 (5.1%) Adult-RCTs, 2/120 (1.7%) Pediatric-RCTs, and 37/90 (41.1%) Consecutive-RCTs. An obtained p-value .05–.0051 had high False Positive Rates; in Adult-RCTs, minimum (assuming prior probability of the alternative hypothesis was 50%) and realistic (assuming prior probability of the alternative hypothesis was 10%) False Positive Rates were 16.7% [11.2, 21.8] and 64.3% [53.2, 71.4]. An obtained p-value ≤.005 had lower False Positive Rates; in Adult-RCTs the realistic False Positive Rate was 7.7% [7.7, 16.0]. The realistic probability of the alternative hypothesis for obtained p-value .05–.0051 (ie, Positive Predictive Value) was 28.0% [24.1, 34.8], 30.6% [27.7, 48.5], 29.3% [24.3, 41.0], and 32.7% [24.1, 43.5] for Adult-RCTs, Pediatric-RCTs, Consecutive-RCTs primary and secondary outcome, respectively. The maximum Positive Predictive Value for p-value category .05–.0051 was median 77.8%, 79.8%, 78.8%, and 81.4% respectively. To have maximum or realistic Positive Predictive Value >90% or >80%, RCTs needed to have obtained p-value ≤.005. The credibility of p-value .05–.0051 findings were easy to challenge, and the credibility to rule-out an effect with p-value >.05 to .10 was low. The probability that a replication study would obtain p-value ≤.05 did not approach 90% unless the obtained p-value was ≤.005.ConclusionsUnless the obtained p-value was ≤.005, the False Positive Rate was high, and the Positive Predictive Value and probability of replication of “statistically significant” findings were low.

FORENSIC AUDIT SKILLS ON CONTEMPORARY AUDIT REPORTS AND ORGANIZATIONAL PRODUCTIVITY

<p>This study sought to determine the forensic audit skills on contemporary audit reports and organizational productivity. Two research questions and two null hypotheses were formulated and tested at 0.05 level of significance. The descriptive survey design was adopted and it was carried out in Akwa Ibom State using two organizations namely; Champion Breweries Plc and Akwa Ibom Water Company. The population consists of 2836 staff while the sample size was 351 staff determined statistically using Taro Yamane formula. The purposive sampling technique was used. The instrument used for data collection was Forensic Audit Skills on Contemporary Audit Report and Organizational Productivity Questionnaire (FASCAROPQ). It was validated by three experts from the University of Uyo. The Cronbach Alpha reliability technique was used in which 0.79 index was obtained. The regression analysis was used in answering the research questions as well as testing of null hypotheses at 0.05 level of significance. The findings reveal that data mining skill and computer assisted audit techniques significantly predict organizational productivity to a high level. Based on the findings of the study, it was concluded that forensic audit skills are very paramount to operators of any organization in order to correct the irregularities of contemporary audit report. Therefore, it is recommended among others that owners and managers of organization should ensure that their auditors adhere to data mining skill for effective audit report that would lead to optimal productivity of the organization.</p><p> </p><p><strong> Article visualizations:</strong></p><p><img src="/-counters-/edu_01/0976/a.php" alt="Hit counter" /></p>

Permutation tests for hypothesis testing with animal social network data: Problems and potential solutions.

Permutation tests are widely used to test null hypotheses with animal social network data, but suffer from high rates of type I and II error when the permutations do not properly simulate the intended null hypothesis.Two common types of permutations each have limitations. Pre‐network (or datastream) permutations can be used to control ‘nuisance effects’ like spatial, temporal or sampling biases, but only when the null hypothesis assumes random social structure. Node (or node‐label) permutation tests can test null hypotheses that include nonrandom social structure, but only when nuisance effects do not shape the observed network.We demonstrate one possible solution addressing these limitations: using pre‐network permutations to adjust the values for each node or edge before conducting a node permutation test. We conduct a range of simulations to estimate error rates caused by confounding effects of social or non‐social structure in the raw data.Regressions on simulated datasets suggest that this ‘double permutation’ approach is less likely to produce elevated error rates relative to using only node permutations, pre‐network permutations or node permutations with simple covariates, which all exhibit elevated type I errors under at least one set of simulated conditions. For example, in scenarios where type I error rates from pre‐network permutation tests exceed 30%, the error rates from double permutation remain at 5%.The double permutation procedure provides one potential solution to issues arising from elevated type I and type II error rates when testing null hypotheses with social network data. We also discuss alternative approaches that can provide robust inference, including fitting mixed effects models, restricted node permutations, testing multiple null hypotheses and splitting large datasets to generate replicated networks. Finally, we highlight ways that uncertainty can be explicitly considered and carried through the analysis.

Improved group sequential Holm procedures for testing multiple correlated hypotheses over time

ABSTRACT Clinical trials can typically feature two different types of multiple inference: testing of more than one null hypothesis and testing at multiple time points. These modes of multiplicity are closely related mathematically but distinct statistically and philosophically. Regulatory agencies require strong control of the family-wise error rate (FWER), the risk of falsely rejecting any null hypothesis at any analysis. The correlations between test statistics at interim analyses and the final analysis are therefore routinely used in group sequential designs to achieve less conservative critical values. However, the same type of correlations between different comparisons, endpoints or sub-populations are less commonly used. As a result, FWER is in practice often controlled conservatively for commonly applied procedures. Repeated testing of the same null hypothesis may give changing results, when the hypothesis is rejected at an interim but accepted at the final analysis. The mathematically correct overall rejection is at odds with an inference theoretic approach and with common sense. We discuss these two issues, of incorporating correlations and how to interpret time-changing conclusions, and provide case studies where power can be increased while adhering to sound statistical principles.

Six-Month Outcomes from the NEXit Junior Trial of a Text Messaging Smoking Cessation Intervention for High School Students: Randomized Controlled Trial With Bayesian Analysis.

BackgroundThe prevalence of daily or occasional smoking among high school students in Sweden was approximately 20% in 2019, which is problematic since lifestyle behaviors are established in adolescence and track into adulthood. The Nicotine Exit (NEXit) Junior trial was conducted in response to a lack of evidence for the effects of text message smoking cessation interventions among high school students in Sweden.ObjectiveThe aim of this study was to estimate the 3- and 6-month effects of a text messaging intervention among high school students in Sweden on smoking cessation outcomes.MethodsA 2-arm, single-blind randomized controlled trial was employed to estimate the effects of the intervention on smoking cessation in comparison to treatment as usual. Participants were recruited from high schools in Sweden using advertising and promotion by school staff from January 10, 2018, to January 10, 2019. Weekly or daily smokers who were willing to make a quit attempt were eligible for inclusion. Prolonged abstinence and point prevalence of smoking cessation were measured at 3 and 6 months after randomization.ResultsComplete case analysis was possible on 57.9% (310/535) of the participants at 6 months, with no observed statistically significant effect on 5-month prolonged abstinence (odds ratio [OR] 1.27, 95% CI 0.73-2.20; P=.39) or 4-week smoking cessation (OR 1.42; 95% CI 0.83-2.46; P=.20). Sensitivity analyses using imputation yielded similar findings. Unplanned Bayesian analyses showed that the effects of the intervention were in the anticipated direction. The findings were limited by the risk of bias induced by high attrition (42.1%). The trial recruited high school students in a pragmatic setting and included both weekly and daily smokers; thus, generalization to the target population is more direct compared with findings obtained under more strict study procedures.ConclusionsHigher than expected attrition rates to follow-up 6 months after randomization led to null hypothesis tests being underpowered; however, unplanned Bayesian analyses found that the effects of the intervention were in the anticipated direction. Future trials of smoking cessation interventions targeting high school students should aim to prepare strategies for increasing retention to mid- and long-term follow-up.Trial RegistrationIRCTN Registry ISRCTN15396225; https://www.isrctn.com/ISRCTN15396225International Registered Report Identifier (IRRID)RR2-10.1186/s13063-018-3028-2

Relating the Friedman test adjusted for ties, the Cochran–Mantel–Haenszel mean score test and the ANOVA F test

The Friedman test is used to nonparametrically test the null hypothesis of equality of the treatment distributions in the randomized block design. The simple form of the test statistic is for data that is untied within blocks. When ties occur and mid-ranks are used, an adjustment to the simple form of the test statistic is needed. Here such adjustments are given, and it is shown that the Friedman tests, both with untied rank data and with tied data using mid-ranks, are Cochran–Mantel–Haenszel mean score tests. Additionally, for the randomized block design, the Cochran–Mantel–Haenszel mean score statistic is shown to be a simple function of the ANOVA F statistic. Using this relationship for the Friedman tests is shown to give more accurate p-values close to the nominal significance level. Moreover, since the ANOVA F test null hypothesis specifies equality of mean treatment ranks, so does the Friedman test. Therefore the null hypothesis of the Friedman test is sharper than equality of the treatment distributions.