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PubMed1995

MedCalc: a new computer program for medical statistics.

In recent years, the use and abuse of statistics in the medical literature has extensively been reviewed. Amongst others, the importance of the P-value has been challenged and the use of misleading graphics, including 3-dimensional displays, has been criticized. The ease of access to more complex statistical procedures, since the introduction of several statistical software packages for personal computers, has been identified as one of the factors involved in the misuse of statistics. Therefore, we have developed a new computer program that includes those statistical procedures commonly encountered in the medical literature and in statistical textbooks for medical researchers. More complex statistical analyses are not implemented in the software. If researchers with limited statistical training require more sophisticated statistical analyses, they should refer to a statistician, not to a more complete statistical software package.

Computer Graphics↗
PubMed2002

[Possibilities of utilising official statistics for health services research].

This report deals with the experiences and problems with using official statistics on a limited area level of Germany's 440 administrative districts. The data comprised health-related issues and were used for an ecological analysis of the determinants of hospital use. We obtained health statistics data from homepages of the German health report, the Federal statistics office, and the 16 statistics offices of the Federal states. We also made use of the data base of the joint regional statistics published by the Federal and regional statistics offices. Finally, there was a co-ordinated inquiry to the regional statistics offices about the supply of data on mortality, hospital discharges, structure of hospital care and several socio-economic indicators. The process of data collecting was mostly unproblematic. On the other hand, not all of the data were available on a limited area level in a satisfying form and quality. For reasons of data protection some information of small districts was transmitted incompletely. Additionally, there are procedural differences between regional statistics offices regarding the differentiation between some variables and in the exchange of data between the federal states. Finally, some problems were due to varying age classifications for different health indicators. For health services research the use of official statistics becomes increasingly comfortable via health reports and Internet. However, for specific questions of research some problems still remain, rising from missing standardised health indicators and deficient availability of limited area health data. Data collection is exacerbated by restrictive interpretation of data protection acts and the different handling of data inquiries in the regional statistics offices. Finally, there is a lack of data on morbidity and use of ambulatory care.

Data Collection↗
PubMed1994

The statistical analysis of single-subject data: a comparative examination.

BACKGROUND AND PURPOSE: The purposes of this study were to examine whether the use of three different statistical methods for analyzing single-subject data led to similar results and to identify components of graphed data that influence agreement (or disagreement) among the statistical procedures. METHODS: Forty-two graphs containing single-subject data were examined. Twenty-one were AB charts of hypothetical data. The other 21 graphs appeared in Journal of Applied Behavioral Analysis, Physical Therapy, Journal of the Association for Persons With Severe Handicaps, and Journal of Behavior Therapy and Experimental Psychiatry. Three different statistical tests--the C statistic, the two-standard deviation band method, and the split-middle method of trend estimation--were used to analyze the 42 graphs. RESULTS: A relatively low degree of agreement (38%) was found among the three statistical tests. The highest rate of agreement for any two statistical procedures (71%) was found for the two-standard deviation band method and the C statistic. A logistic regression analysis revealed that overlap in single-subject graphed data was the best predictor of disagreement among the three statistical tests (beta = .49, P < .03). CONCLUSION AND DISCUSSION: The results indicate that interpretation of data from single-subject research designs is directly influenced by the method of data analysis selected. Variation exists across both visual and statistical methods of data reduction. The advantages and disadvantages of statistical and visual analysis are described.

Data Interpretation, Statistical↗
PubMed1995

Microcomputer statistics packages for biomedical scientists.

1. There are hundreds of commercially available microcomputer statistics packages, ranging from the very cheap and elementary to the very expensive and complex, and from the very general to the very specialized. This review covers only those that appear to be popular with biomedical investigators who deal with fairly small sets of data but may wish to use relatively complex analytical techniques. 2. It is highly desirable, if not essential, that biomedical investigators who use microcomputer statistics packages have access to a spreadsheet program. These provide sample statistics and simple statistical analyses but, more importantly, they are often the best way of entering data into the files of the statistics packages proper. 3. A vital component of any statistics package is its manual. This should be easy to follow, but at the same time it must provide full documentation of, and references to, precisely how the various statistical tests are performed. 4. Some packages are elementary and offer only a narrow range of test procedures (mini-packages). Some are designed to be used as statistical libraries and programming tools for professional statisticians. Between these extremes are the general purpose packages (mid-range, maxi- and supermaxi-packages) that constitute the main body of this review. 5. All the packages reviewed have some shortcomings or flaws. It is argued that the ideal package for biomedical investigators should have the following features: (i) it should provide a wide range of test procedures for analysing continuous, rank-ordered, and categorical data; (ii) the way in which these tests are carried out should be clearly stated in the manual; and (iii) lastly, although not unimportantly, the package should be easy to use. 6. It is recommended that biomedical investigators purchase a package that provides many more statistical routines than they use in their everyday practice. Provided the manual is a good one and the package itself has no serious flaws, this is an excellent stimulus to continuing education in statistical techniques.

Humans↗
PubMed2004

Powerful allele sharing statistics for nonparametric linkage analysis.

Nonparametric linkage analysis is widely used to map susceptibility genes for complex diseases. This paper introduces six nonparametric statistics for measuring marker allele sharing among the affected members of a pedigree. We compare the power of these new statistics and three previous statistics to detect linkage with Mendelian diseases having recessive, additive, and dominant modes of inheritance. The nine statistics represent all possible combinations of three different IBD scoring functions and three different schemes for sampling genes among affecteds. Our results strongly suggest that the statistic T(rec)(blocks) is best for recessive traits, while the two statistics T(kin)(pairs) and T(all)(kin) vie for best for an additive trait. The best statistic for a dominant trait is less clear. The statistics T(kin)(pairs) and T(all)(kin) are equally promising for small sibships, but in extended pedigrees the statistics T(dom)(blocks) and T(dom)(pairs) appear best. For a complex trait, we advocate computing several of these statistics.

Alleles↗
PubMed2004

Statistics in the pharmacy literature.

BACKGROUND: Research in statistical methods is essential for maintenance of high quality of the published literature. OBJECTIVE: To update previous reports of the types and frequencies of statistical terms and procedures in research studies of selected professional pharmacy journals. METHODS: We obtained all research articles published in 2001 in 6 journals: American Journal of Health-System Pharmacy, The Annals of Pharmacotherapy, Canadian Journal of Hospital Pharmacy, Formulary, Hospital Pharmacy, and Journal of the American Pharmaceutical Association. Two independent reviewers identified and recorded descriptive and inferential statistical terms/procedures found in the methods, results, and discussion sections of each article. Results were determined by tallying the total number of times, as well as the percentage, that each statistical term or procedure appeared in the articles. RESULTS: One hundred forty-four articles were included. Ninety-eight percent employed descriptive statistics; of these, 28% used only descriptive statistics. The most common descriptive statistical terms were percentage (90%), mean (74%), standard deviation (58%), and range (46%). Sixty-nine percent of the articles used inferential statistics, the most frequent being chi(2) (33%), Student's t-test (26%), Pearson's correlation coefficient r (18%), ANOVA (14%), and logistic regression (11%). CONCLUSIONS: Statistical terms and procedures were found in nearly all of the research articles published in pharmacy journals. Thus, pharmacy education should aim to provide current and future pharmacists with an understanding of the common statistical terms and procedures identified to facilitate the appropriate appraisal and consequential utilization of the information available in research articles.

Education, Pharmacy↗
PubMed1989

Review of use of statistics in The American Journal of Tropical Medicine and Hygiene for January-December 1988.

A statistical review of all 201 scientific articles published during the calendar year 1988 in The American Journal of Tropical Medicine and Hygiene was made to determine the type of study published by the journal, the level of statistics that was employed by the published articles, and the extent of common statistical errors. Sixty-five (32%) of the articles were classified as experimental, with most of the remaining observational studies classified as cross-sectional in design. A reader with the knowledge of only simple descriptive statistics would be able to understand 60% of the statistical procedures used in the published papers. Knowledge of the usual components of a single semester course in introductory statistics increases understanding to 91% of the techniques used. It was determined that 148 (73.5%) of the 201 articles had at least 1 detectable statistical error; most of these errors involved improper documentation or application of statistical hypothesis testing. The most common descriptive statistical error was the misuse of the concepts of "standard deviation" and "standard error" which was found in greater than 20% of the articles. Examples of each of the common statistical errors seen are given with suggestions for improvement.

Analysis of Variance↗
PubMed2002

Understanding statistics. Putting p-values into perspective.

Understanding statistics can be one of the more difficult and daunting tasks facing the clinician attempting to understand and use the research literature. While the fundamental aim of including statistics in literature is to justify the conclusions and to enhance the information presented, all too often, statistics serve only to intimidate and alienate clinicians. There are two sources for this barrier to understanding: poor comprehension of statistics by clinicians, and inappropriate presentation of statistics by researchers. In the third of our papers on evidence-based practice and the clinician, some basic principles of statistics are introduced. Aspects such as the concepts of clinical and statistical significance are discussed, and an outline of the appropriate use of specific statistics is provided. The hope is that this will enable the clinician to better judge the applicability of statistical techniques as applied in the published literature. Some of the newer, more 'clinician friendly' statistical approaches are also presented. Finally some of the cheats and shortcuts commonly encountered in the presentation of research are discussed, and the methods for dealing with them are outlined.

Humans↗
PubMed1998

[Current use of statistical methods in five core Chinese medical journals].

OBJECTIVE: To determine the frequency of statistical techniques used in Chinese medical journals, types of statistical errors present, and trends over a 10-year period. METHODS: We reviewed the statistical methods used in all original articles published in 5 leading journals in 1985 (n = 640) and 1995 (n = 954). RESULTS: Compared with 1985, significant improvement was seen in 1995. The proportion of papers using statistical tests increased from 40% to 60% (chi 2 = 60.9, P < 0.01), and more sophisticated statistical methods were used. Of those articles using statistics, the proportion using appropriate methods increased from 22% to 46% (chi 2 = 45.3, P < 0.01). In both years, the most commonly used statistical methods were t tests and contingency tables. The most common errors were presentation of P values without specifying the test used, use of multiple t tests instead of ANOVAs, and use of unpaired t tests when paired tests were required. CONCLUSION: The use of statistical methods in Chinese medical journals is improving, and by 1995, the frequency of using statistical methods in published articles was similar to that in other journals around the world. However, the lack of or inappropriate sue of statistics remains a serious problem.

China↗
PubMed1988

Applications of statistical analysis in diagnostic histopathology and cytopathology.

Corresponding to the rapid increase in the amount of data available for use in clinical diagnoses, there is an increased need for procedures that can provide the diagnostician with meaningful statistical summaries of data and with statements concerning the statistical significance associated with a diagnostic evaluation. It has been demonstrated that multivariate statistical assessment of clinical material can provide consistent, reliable and highly sensitive diagnostic clues, even in instances in which trained personnel are unable to see any change. Several examples of applications of statistical analyses in diagnostic cytology and histopathology are given in this paper. The examples were chosen to be illustrative of the different types of problems for which statistical analyses have been found useful. These problems differ with respect to the extent of the statistical methods thus far developed and the difficulty involved in developing further analyses. For many problems, appropriate statistical analyses are readily available; other problems require definition of custom-made test statistics and, in some cases, also definition of new statistical distributions. The problems discussed here are only a small sample of the existing problems, but they provide at least an indication of the scope of the role that statistics plays in cytopathologic and histopathologic diagnosis.

Cytological Techniques↗
PubMed1997

[Statistical analyses in the Revista Clínica Española: their actual use and accessibility for the reader].

BACKGROUND: In the original articles published in Revista Clínica Española in 1994 and 1995, the use frequency of different statistical analyses is reported, the statistical repertoire necessary for the reader to interpret statistical methods determined, and results are compared with those found in similar studies. METHODS: Ninety-one original articles were reviewed. Statistical procedures were classified into 18 hierarchical categories and the statistical accessibility to the reader was quantitated assuming the knowledge up to a determined category (basic course in biostatistics) as reference level. RESULTS: The five categories used most frequently were Bivariate contingency tables (56%), t and z tests (42%), at the most descriptive statistics (23%), Epidemiological statistics (22%) and Non parametric tests (20%). Sixty-eight percent of original articles in Revista Clínica Española were statistically accessible to the reader with a basic knowledge on statistics. CONCLUSIONS: To have access to a considerable percentage of originals in this and other similar medical journals, the reader of Revista Clínica Española does no need to manage a statistical repertoire higher than a basic course in biostatistics. On the other hand, the reader is advised to deepen his/her knowledge on the design of an investigational study.

Clinical Medicine↗
PubMed1999

Statistical methods for the Ames Salmonella assay: a review.

The Ames Salmonella assay remains the most widely used in vitro genotoxicity assay. Several statistical methods have been proposed for its analysis [B.H. Margolin, N. Kaplan, E. Zeiger, Statistical analysis of the Ames Salmonella/microsome test, Proc. Natl. Acad. Sci., 78 (1981) 3779-3783; L.E. Myers, N.H. Saxton, L.I. Southerland, T.J. Wolff, Regression analysis of Ames test data, Environ. Mol. Mutagen., 3 (1981) 575-586; A.G. Stead, V. Hasselblad, J.P. Creason, L. Claxton, Modelling the Ames test, Mutation Res., 85 (1981) 13-27; L. Bernstein, J. Kaldor, J. McCaan, M.C. Pike, An empirical approach to the statistical analysis of mutagenesis data from the Salmonella test, Mutation Res., 97 (1982) 267-281; N.E. Breslow, Extra-Poisson variation in log-linear models, Appl. Stat., 33 (1984) 38-44; J. Wahrendorf, G.A.T. Mahon, M. Schumacher, A nonparametric approach to the statistical analysis of mutagenicity data, Mutation Res., 147 (1985) 5-13; D.G. Simpson, B.H. Margolin, Recursive nonparametric testing for dose-response relationships subject to downturns at high doses, Biometrika, 73 (1986) 589-596; D.G. Simpson, B.H. Margolin, Nonparametric testing for dose-response curves subject to downturns: Asymptotic power considerations, Annals Stat., 18 (1990) 373-390.]. In this paper we review recent literature to see what statistical methods are in fact employed for the analysis of the Ames assay. We then note that these methods can be classified into a common category in the framework of Haynes and Eckardt's mutation induction kinetics model [R.H. Haynes, F. Eckardt, Mathematical analysis of mutation induction kinetics, in: F.J. de Serres, A. Hollaender (Eds. ), Chemical Mutagens, Principles and Methods for Their Detection, Vol. 6, Plenum, New York, 1980, pp. 271-307]. The value in knowing this is that most methods of analysis considered here will likely exhibit common statistical behavior. These analyses are computationally intensive, e.g., [B.H. Margolin, N. Kaplan, E. Zeiger, Statistical analysis of the Ames Salmonella/microsome test, Proc. Nat. Acad. Sci., 78 (1981) 3779-3783], hence the ready availability of computer programs is essential if biologists are to use these methods. We briefly review two statistical software programs that are available in the public domain, and describe in detail a third program, Salm, [B.H. Margolin, N. Kaplan, E. Zeiger, Statistical analysis of the Ames Salmonella/microsome test, Proc. Nat. Acad. Sci., 78 (1981) 3779-3783; B.H. Margolin, B.S. Kim, K. Risko, The Ames Salmonella/microsome assay: Issues of inference and validation, J. Amer. Stat. Assoc., 84 (1989) 651-661]. The Salm program is obtainable through the file transfer protocol (ftp) or using a WWW browser. Finally, we discuss two statistical consequences of naively applying the two-fold rule, a method of analysis employed by a number of researchers.

Dose-Response Relationship, Drug↗
PubMed2000

Composite statistics for QTL mapping with moderately discordant sibling pairs.

Extreme discordant sibling-pair (EDSP) designs have been shown in theory to be very powerful for mapping quantitative-trait loci (QTLs) in humans. However, their practical applicability has been somewhat limited by the need to phenotype very large populations to find enough pairs that are extremely discordant. In this paper, we demonstrate that there is also substantial power in pairs that are only moderately discordant, and that designs using moderately discordant pairs can yield a more practical balance between phenotyping and genotyping efforts. The power we demonstrate for moderately discordant pairs stems from a new statistical result. Statistical analysis in discordant-pair studies is generally done by testing for reduced identity by descent (IBD) sharing in the pairs. By contrast, the most commonly-used statistical methods for more standard QTL mapping are Haseman-Elston regression and variance-components analysis. Both of these use statistics that are functions of the trait values given IBD information for the pedigree. We show that IBD sharing statistics and "trait value given IBD" statistics contribute complementary rather than redundant information, and thus that statistics of the two types can be combined to form more powerful tests of linkage. We propose a simple composite statistic, and test it with simulation studies. The simulation results show that our composite statistic increases power only minimally for extremely discordant pairs. However, it boosts the power of moderately discordant pairs substantially and makes them a very practical alternative. Our composite statistic is straightforward to calculate with existing software; we give a practical example of its use by applying it to a Genetic Analysis Workshop (GAW) data set.

Chromosome Mapping↗
PubMed2006

Composition-based statistics and translated nucleotide searches: improving the TBLASTN module of BLAST.

BACKGROUND: TBLASTN is a mode of operation for BLAST that aligns protein sequences to a nucleotide database translated in all six frames. We present the first description of the modern implementation of TBLASTN, focusing on new techniques that were used to implement composition-based statistics for translated nucleotide searches. Composition-based statistics use the composition of the sequences being aligned to generate more accurate E-values, which allows for a more accurate distinction between true and false matches. Until recently, composition-based statistics were available only for protein-protein searches. They are now available as a command line option for recent versions of TBLASTN and as an option for TBLASTN on the NCBI BLAST web server. RESULTS: We evaluate the statistical and retrieval accuracy of the E-values reported by a baseline version of TBLASTN and by two variants that use different types of composition-based statistics. To test the statistical accuracy of TBLASTN, we ran 1000 searches using scrambled proteins from the mouse genome and a database of human chromosomes. To test retrieval accuracy, we modernize and adapt to translated searches a test set previously used to evaluate the retrieval accuracy of protein-protein searches. We show that composition-based statistics greatly improve the statistical accuracy of TBLASTN, at a small cost to the retrieval accuracy. CONCLUSION: TBLASTN is widely used, as it is common to wish to compare proteins to chromosomes or to libraries of mRNAs. Composition-based statistics improve the statistical accuracy, and therefore the reliability, of TBLASTN results. The algorithms used by TBLASTN are not widely known, and some of the most important are reported here. The data used to test TBLASTN are available for download and may be useful in other studies of translated search algorithms.

Algorithms↗
PubMed1990

R.A. Fisher's contributions to genetical statistics.

R. A. Fisher (1890-1962) was a professor of genetics, and many of his statistical innovations found expression in the development of methodology in statistical genetics. However, whereas his contributions in mathematical statistics are easily identified, in population genetics he shares his preeminence with Sewall Wright (1889-1988) and J. B. S. Haldane (1892-1965). This paper traces some of Fisher's major contributions to the foundations of statistical genetics, and his interactions with Wright and with Haldane which contributed to the development of the subject. With modern technology, both statistical methodology and genetic data are changing. Nonetheless much of Fisher's work remains relevant, and may even serve as a foundation for future research in the statistical analysis of DNA data. For Fisher's work reflects his view of the role of statistics in scientific inference, expressed in 1949: There is no wide or urgent demand for people who will define methods of proof in set theory in the name of improving mathematical statistics. There is a widespread and urgent demand for mathematicians who understand that branch of mathematics known as theoretical statistics, but who are capable also of recognising situations in the real world to which such mathematics is applicable. In recognising features of the real world to which his models and analyses should be applicable, Fisher laid a lasting foundation for statistical inference in genetic analyses.

Animals↗
PubMedDate not supplied

The original calibre of the lower limbs arteries as a possible risk factor for complications of atherosclerosis: a statistical investigation in 90 subjects by echocolor-doppler.

Ninety subjects with emodynamically significant atherosclerotic disease of the lower limbs were examined. They had no history of diabetes mellitus or hypertension. Each subject underwent a color Doppler ultrasonographic study of the common iliac, superficial femoral, and popliteal arteries. In each arterial segment, diameter and blood flow velocity were measured. In evaluating the hemodynamic significance of the stenoses, we used the Windsor method. In a comparison of the calibers of the arteries significant relationship emerged in each given subject in two sexes. Males: a statistically significant difference was found only in the iliac artery, in which the calibre and Windsor indices were greater in the right as compared to the left; right: 1) There was a statistically significant relationship between mean caliber and Windsor indices, (p < 0.01); 2) there was a statistically significant correlation between mean calibre and age (p < 0.05); Left: 1) There was a statistically significant correlation between mean calibre and Windsor indices calculated at the levels of both the posterior tibialis (p < 0.05) and dorsalis pedis arteries (p < 0.01); 2) there was a statistically significant correlation between mean calibre and age (p < 0.05). Females: Student's t test for paired samples of the three arteries did not reveal a statistically significant predominance of one side over the other. With respect to the coefficients of correlation between mean calibre and Windsor indices, the results were as follows. Right: 1) There was a statistically significant correlation between mean calibre and Windsor indices calculated at the levels of both the posterior tibialis and dorsalis pedis arteries (p < 0.01); 2) there was a statistically significant correlation between age and mean calibre (p < 0.05); Left: 1) There was a statistically significant correlation between mean calibre and Windsor indices calculated at the levels of both the posterior tibialis (p < 0.05) and dorsalis pedis arteries (p < 0.01); 2) there was a statistically significant correlation between mean calibre and age (p < 0.05).

Aged↗
PubMedDate not supplied

The measurement of the calibers of the branches of the aortic arch: a statistical investigation of 430 living subjects using ultrasonic tomography.

Using Doppler ultrasound, we have measured the calibers of the branch arteries of the aortic arch in 430 subjects (230 men and 200 women; age range: 18 to 84 years). These included: the brachiocephalic trunk, the subclavian arteries, the common carotid arteries, the internal and external carotid arteries, and the vertebral arteries. Statistical analysis of the values obtained yielded a number of interesting results. The average caliber of the arteries of the women examined was lower than that of the men. Nonetheless, the statistical difference in the arterial diameters of the two sexes was not significant, with the exception of the left subclavian artery. In the case of this artery, its caliber was larger among the men than the women. However, this relationship was not confirmed in the right subclavian artery. The left vertebral artery was larger than the right one. This difference was statistically significant among the women (p < 0.05), but not among the men (p = ns). The right subclavian artery was larger than the left one by a statistically significant margin (p < 0.05), but the statistical significance may be accounted for by the values in the women alone (p < 0.02). When the men were considered separately, the difference between right and left did not achieve statistical significance (p = ns). There was no statistically significant correlation between arterial caliber and either height or body weight. Furthermore, the caliber of the arteries examined increased with age, but not in a statistically significant manner. The mean values of the single arteries examined were lower than those described by the principal investigators of the cadaveric studies to which we have referred. However, with three exceptions, the differences were not statistically significant. In the case of the subclavian and internal and external carotid arteries, our values were lower by a statistically significant margin.

Adolescent↗
PubMed1995

Optimal experimental design and sample size for the statistical evaluation of data from somatic mutation and recombination tests (SMART) in Drosophila.

In genetic toxicology it is important to know whether chemicals should be regarded as clearly hazardous or whether they can be considered sufficiently safe, which latter would be the case from the genotoxicologist's view if their genotoxic effects are nil or at least significantly below a predefined minimal effect level. A previously presented statistical decision procedure which allows one to make precisely this distinction is now extended to the question of how optimal experimental sample size can be determined in advance for genotoxicity experiments using the somatic mutation and recombination tests (SMART) of Drosophila. Optimally, the statistical tests should have high power to minimise the chance for statistically inconclusive results. Based on the normal test, the statistical principles are explained, and in an application to the wing spot assay, it is shown how the practitioner can proceed to optimise sample size to achieve numerically satisfactory conditions for statistical testing. The somatic genotoxicity assays of Drosophila are in principle based on somatic spots (mutant clones) that are recovered in variable numbers on individual flies. The underlying frequency distributions are expected to be of the Poisson type. However, some care seems indicated with respect to this latter assumption, because pooling of data over individuals, sexes, and experiments, for sample, can (but need not) lead to data which are overdispersed, i.e., the data may show more variability than theoretically expected. It is an undesired effect of overdispersion that in comparisons of pooled totals it can lead to statistical testing which is too liberal, because overall it yields too many seemingly significant results. If individual variability considered alone is not in contradiction with Poisson expectation, however, experimental planning can help to minimise the undesired effects of overdispersion on statistical testing of pooled totals. The rule for the practice is to avoid disproportionate sampling. It is recalled that for optimal power in statistical testing, it is preferable to use equal total numbers of flies in the control and treated series. Statistical tests which are based on Poisson expectations are too liberal if there is overdispersion in the data due to excess individual variability. In this case we propose to use the U test as a non-parametric two-sample test and to adjust the estimated optimal sample size according to (i) the overdispersion observed in a large historical control and (ii) the relative efficiency of the U test in comparison to the t test and related parametric tests.

Animals↗