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49-524: One difficulty with discussing the Behrens–Fisher problem and proposed solutions, is that there are many different interpretations of what is meant by "the Behrens–Fisher problem". These differences involve not only what is counted as being a relevant solution, but even the basic statement of the context being considered. Let X 1 , ...,  X n and Y 1 , ...,  Y m be i.i.d. samples from two populations which both come from

98-419: A Bayesian inference point of view and either solution would be notionally invalid judged from the other point of view. If consideration is restricted to classical statistical inference only, it is possible to seek solutions to the inference problem that are simple to apply in a practical sense, giving preference to this simplicity over any inaccuracy in the corresponding probability statements. Where exactness of

147-513: A location parameter and a non-negative scale parameter . For any random variable X {\displaystyle X} whose probability distribution function belongs to such a family, the distribution function of Y = d a + b X {\displaystyle Y{\stackrel {d}{=}}a+bX} also belongs to the family (where = d {\displaystyle {\stackrel {d}{=}}} means " equal in distribution "—that is, "has

196-482: A one-tailed test , or partitioned to both sides of the distribution, as in a two-tailed test , with each tail (or rejection region) containing 2.5% of the distribution. The use of a one-tailed test is dependent on whether the research question or alternative hypothesis specifies a direction such as whether a group of objects is heavier or the performance of students on an assessment is better . A two-tailed test may still be used but it will be less powerful than

245-408: A one-tailed test, because the rejection region for a one-tailed test is concentrated on one end of the null distribution and is twice the size (5% vs. 2.5%) of each rejection region for a two-tailed test. As a result, the null hypothesis can be rejected with a less extreme result if a one-tailed test was used. The one-tailed test is only more powerful than a two-tailed test if the specified direction of

294-415: A significance level, Fisher did not intend this cutoff value to be fixed. In his 1956 publication Statistical Methods and Scientific Inference, he recommended that significance levels be set according to specific circumstances. The significance level α {\displaystyle \alpha } is the threshold for p {\displaystyle p} below which the null hypothesis

343-447: A study is chosen before data collection, and is typically set to 5% or much lower—depending on the field of study. In any experiment or observation that involves drawing a sample from a population , there is always the possibility that an observed effect would have occurred due to sampling error alone. But if the p -value of an observed effect is less than (or equal to) the significance level, an investigator may conclude that

392-585: A symptom of the problem. There is nothing wrong with hypothesis testing and p -values per se as long as authors, reviewers, and action editors use them correctly." Some statisticians prefer to use alternative measures of evidence, such as likelihood ratios or Bayes factors . Using Bayesian statistics can avoid confidence levels, but also requires making additional assumptions, and may not necessarily improve practice regarding statistical testing. The widespread abuse of statistical significance represents an important topic of research in metascience . In 2016,

441-400: Is a random variable. A t distribution with a random number of degrees of freedom does not exist. Nevertheless, the Behrens–Fisher T can be compared with a corresponding quantile of Student's t distribution with these estimated numbers of degrees of freedom, ν ^ {\displaystyle {\hat {\nu }}} , which is generally non-integer. In this way,

490-479: Is also a difference between statistical significance and practical significance. A study that is found to be statistically significant may not necessarily be practically significant. Effect size is a measure of a study's practical significance. A statistically significant result may have a weak effect. To gauge the research significance of their result, researchers are encouraged to always report an effect size along with p -values. An effect size measure quantifies

539-490: Is designed for R but should generalize to any language and library. The example here is of the Student's t -distribution , which is normally provided in R only in its standard form, with a single degrees of freedom parameter df . The versions below with _ls appended show how to generalize this to a generalized Student's t-distribution with an arbitrary location parameter m and scale parameter s . Note that

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588-522: Is known as the multivariate Behrens–Fisher problem . The nonparametric Behrens–Fisher problem does not assume that the distributions are normal. Tests include the Cucconi test of 1968 and the Lepage test of 1971. Location%E2%80%93scale family In probability theory , especially in mathematical statistics , a location–scale family is a family of probability distributions parametrized by

637-400: Is made. While Lehmann discusses a number of approaches to the more general problem, mainly based on nonparametrics, most other sources appear to use "the Behrens–Fisher problem" to refer only to the case where the distribution is assumed to be normal: most of this article makes this assumption. Solutions to the Behrens–Fisher problem have been presented that make use of either a classical or

686-458: Is more accurate to use Student's t-test . A number of different approaches to the general problem have been proposed, some of which claim to "solve" some version of the problem. Among these are, In Dudewicz’s comparison of selected methods, it was found that the Dudewicz–Ahmed procedure is recommended for practical use. For several decades, it was commonly believed that no exact solution to

735-408: Is rejected even though by assumption it were true, and something else is going on. This means that α {\displaystyle \alpha } is also the probability of mistakenly rejecting the null hypothesis, if the null hypothesis is true. This is also called false positive and type I error . Sometimes researchers talk about the confidence level γ = (1 − α ) instead. This

784-411: Is set to 5%, the conditional probability of a type I error , given that the null hypothesis is true , is 5%, and a statistically significant result is one where the observed p -value is less than (or equal to) 5%. When drawing data from a sample, this means that the rejection region comprises 5% of the sampling distribution . These 5% can be allocated to one side of the sampling distribution, as in

833-426: Is the probability of not rejecting the null hypothesis given that it is true. Confidence levels and confidence intervals were introduced by Neyman in 1937. Statistical significance plays a pivotal role in statistical hypothesis testing. It is used to determine whether the null hypothesis should be rejected or retained. The null hypothesis is the hypothesis that no effect exists in the phenomenon being studied. For

882-486: Is the probability of the study rejecting the null hypothesis, given that the null hypothesis is true; and the p -value of a result, p {\displaystyle p} , is the probability of obtaining a result at least as extreme, given that the null hypothesis is true. The result is statistically significant, by the standards of the study, when p ≤ α {\displaystyle p\leq \alpha } . The significance level for

931-491: The American Statistical Association (ASA) published a statement on p -values, saying that "the widespread use of 'statistical significance' (generally interpreted as ' p  ≤ 0.05') as a license for making a claim of a scientific finding (or implied truth) leads to considerable distortion of the scientific process". In 2017, a group of 72 authors proposed to enhance reproducibility by changing

980-541: The Higgs boson particle's existence was based on the 5 σ criterion, which corresponds to a p -value of about 1 in 3.5 million. In other fields of scientific research such as genome-wide association studies , significance levels as low as 5 × 10 are not uncommon —as the number of tests performed is extremely large. Researchers focusing solely on whether their results are statistically significant might report findings that are not substantive and not replicable. There

1029-441: The p -value threshold for statistical significance from 0.05 to 0.005. Other researchers responded that imposing a more stringent significance threshold would aggravate problems such as data dredging ; alternative propositions are thus to select and justify flexible p -value thresholds before collecting data, or to interpret p -values as continuous indices, thereby discarding thresholds and statistical significance. Additionally,

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1078-441: The Behrens–Fisher statistic T , which also depends on the variance ratio σ 1 / σ 2 , could now be approximated by Student's t distribution with these ν degrees of freedom. But this ν contains the population variances σ i , and these are unknown. The following estimate only replaces the population variances by the sample variances: This ν ^ {\displaystyle {\hat {\nu }}}

1127-483: The Type III Pearson distribution (a scaled chi-squared distribution ) whose first two moments agree with that of s d ¯ 2 {\displaystyle s_{\bar {d}}^{2}} . This applies to the following number of degrees of freedom (d.f.), which is generally non-integer: Under the null hypothesis of equal expectations, μ 1 = μ 2 , the distribution of

1176-410: The alternative hypothesis is correct. If it is wrong, however, then the one-tailed test has no power. In specific fields such as particle physics and manufacturing , statistical significance is often expressed in multiples of the standard deviation or sigma ( σ ) of a normal distribution , with significance thresholds set at a much stricter level (for example 5 σ ). For instance, the certainty of

1225-411: The appendix. A minor variant of the Behrens–Fisher problem has been studied. In this instance the problem is, assuming that the two population-means are in fact the same, to make inferences about the common mean: for example, one could require a confidence interval for the common mean. One generalisation of the problem involves multivariate normal distributions with unknown covariance matrices, and

1274-430: The boundary between acceptance and rejection region of the test statistic T is calculated based on the empirical variances s i , in a way that is a smooth function of these. This method also does not give exactly the nominal rate, but is generally not too far off. However, if the population variances are equal, or if the samples are rather small and the population variances can be assumed to be approximately equal, it

1323-451: The change to 0.005 would increase the likelihood of false negatives, whereby the effect being studied is real, but the test fails to show it. In 2019, over 800 statisticians and scientists signed a message calling for the abandonment of the term "statistical significance" in science, and the ASA published a further official statement declaring (page 2): We conclude, based on our review of

1372-404: The classic paired t -test is a central Behrens–Fisher problem with a non-zero population correlation coefficient and derived its corresponding probability density function by solving its associated non-central Behrens–Fisher problem with a nonzero population correlation coefficient. It also solved a more general non-central Behrens–Fisher problem with a non-zero population correlation coefficient in

1421-443: The common Behrens–Fisher problem existed. However, it was proved in 1966 that it has an exact solution. In 2018 the probability density function of a generalized Behrens–Fisher distribution of m means and m distinct standard errors from m samples of distinct sizes from independent normal distributions with distinct means and variances was proved and the paper also examined its asymptotic approximations. A follow-up paper showed that

1470-436: The distribution function G ( x ) = F ( a + b x ) {\displaystyle G(x)=F(a+bx)} is also a member of Ω {\displaystyle \Omega } . Moreover, if X {\displaystyle X} and Y {\displaystyle Y} are two random variables whose distribution functions are members of the family, and assuming existence of

1519-423: The effect reflects the characteristics of the whole population, thereby rejecting the null hypothesis. This technique for testing the statistical significance of results was developed in the early 20th century. The term significance does not imply importance here, and the term statistical significance is not the same as research significance, theoretical significance, or practical significance. For example,

Behrens–Fisher problem - Misplaced Pages Continue

1568-518: The first two moments and X {\displaystyle X} has zero mean and unit variance, then Y {\displaystyle Y} can be written as Y = d μ Y + σ Y X {\displaystyle Y{\stackrel {d}{=}}\mu _{Y}+\sigma _{Y}X} , where μ Y {\displaystyle \mu _{Y}} and σ Y {\displaystyle \sigma _{Y}} are

1617-472: The generalized functions do not have standard deviation s since the standard t distribution does not have standard deviation of 1. Significance level In statistical hypothesis testing , a result has statistical significance when a result at least as "extreme" would be very infrequent if the null hypothesis were true. More precisely, a study's defined significance level , denoted by α {\displaystyle \alpha } ,

1666-622: The idea of statistical hypothesis testing, which he called "tests of significance", in his publication Statistical Methods for Research Workers . Fisher suggested a probability of one in twenty (0.05) as a convenient cutoff level to reject the null hypothesis. In a 1933 paper, Jerzy Neyman and Egon Pearson called this cutoff the significance level , which they named α {\displaystyle \alpha } . They recommended that α {\displaystyle \alpha } be set ahead of time, prior to any data collection. Despite his initial suggestion of 0.05 as

1715-412: The journal Basic and Applied Social Psychology banned the use of significance testing altogether from papers it published, requiring authors to use other measures to evaluate hypotheses and impact. Other editors, commenting on this ban have noted: "Banning the reporting of p -values, as Basic and Applied Social Psychology recently did, is not going to solve the problem because it is merely treating

1764-400: The likelihood that the result was a false positive. Starting in the 2010s, some journals began questioning whether significance testing, and particularly using a threshold of α =5%, was being relied on too heavily as the primary measure of validity of a hypothesis. Some journals encouraged authors to do more detailed analysis than just a statistical significance test. In social psychology,

1813-475: The mean and standard deviation of Y {\displaystyle Y} . In decision theory , if all alternative distributions available to a decision-maker are in the same location–scale family, and the first two moments are finite, then a two-moment decision model can apply, and decision-making can be framed in terms of the means and the variances of the distributions. Often, location–scale families are restricted to those where all members have

1862-494: The mean-difference being zero: clearly this would not be "optimal" in any sense. The task of specifying interval estimates for this problem is one where a frequentist approach fails to provide an exact solution, although some approximations are available. Standard Bayesian approaches also fail to provide an answer that can be expressed as straightforward simple formulae, but modern computational methods of Bayesian analysis do allow essentially exact solutions to be found. Thus study of

1911-532: The means were in fact equal. Many other methods of treating the problem have been proposed since, and the effect on the resulting confidence intervals have been investigated. A widely used method is that of B. L. Welch , who, like Fisher, was at University College London . The variance of the mean difference results in Welch (1938) approximated the distribution of s d ¯ 2 {\displaystyle s_{\bar {d}}^{2}} by

1960-496: The null hypothesis is true. The null hypothesis is rejected if the p -value is less than (or equal to) a predetermined level, α {\displaystyle \alpha } . α {\displaystyle \alpha } is also called the significance level , and is the probability of rejecting the null hypothesis given that it is true (a type I error ). It is usually set at or below 5%. For example, when α {\displaystyle \alpha }

2009-426: The null hypothesis to be rejected, an observed result has to be statistically significant, i.e. the observed p -value is less than the pre-specified significance level α {\displaystyle \alpha } . To determine whether a result is statistically significant, a researcher calculates a p -value, which is the probability of observing an effect of the same magnitude or more extreme given that

Behrens–Fisher problem - Misplaced Pages Continue

2058-570: The problem can be used to elucidate the differences between the frequentist and Bayesian approaches to interval estimation. Ronald Fisher in 1935 introduced fiducial inference in order to apply it to this problem. He referred to an earlier paper by Walter-Ulrich Behrens from 1929. Behrens and Fisher proposed to find the probability distribution of where x ¯ 1 {\displaystyle {\bar {x}}_{1}} and x ¯ 2 {\displaystyle {\bar {x}}_{2}} are

2107-418: The same location–scale family of distributions. The scale parameters are assumed to be unknown and not necessarily equal, and the problem is to assess whether the location parameters can reasonably be treated as equal. Lehmann states that "the Behrens–Fisher problem" is used both for this general form of model when the family of distributions is arbitrary, and for when the restriction to a normal distribution

2156-474: The same distribution as"). In other words, a class Ω {\displaystyle \Omega } of probability distributions is a location–scale family if for all cumulative distribution functions F ∈ Ω {\displaystyle F\in \Omega } and any real numbers a ∈ R {\displaystyle a\in \mathbb {R} } and b > 0 {\displaystyle b>0} ,

2205-415: The same functional form. Most location–scale families are univariate , though not all. Well-known families in which the functional form of the distribution is consistent throughout the family include the following: The following shows how to implement a location–scale family in a statistical package or programming environment where only functions for the "standard" version of a distribution are available. It

2254-419: The significance levels of statistical tests is required, there may be an additional requirement that the procedure should make maximum use of the statistical information in the dataset. It is well known that an exact test can be gained by randomly discarding data from the larger dataset until the sample sizes are equal, assembling data in pairs and taking differences, and then using an ordinary t-test to test for

2303-412: The strength of an effect, such as the distance between two means in units of standard deviation (cf. Cohen's d ), the correlation coefficient between two variables or its square , and other measures. A statistically significant result may not be easy to reproduce. In particular, some statistically significant results will in fact be false positives. Each failed attempt to reproduce a result increases

2352-432: The term clinical significance refers to the practical importance of a treatment effect. Statistical significance dates to the 18th century, in the work of John Arbuthnot and Pierre-Simon Laplace , who computed the p -value for the human sex ratio at birth, assuming a null hypothesis of equal probability of male and female births; see p -value § History for details. In 1925, Ronald Fisher advanced

2401-402: The two sample means , and s 1 and s 2 are their standard deviations . See Behrens–Fisher distribution . Fisher approximated the distribution of this by ignoring the random variation of the relative sizes of the standard deviations, Fisher's solution provoked controversy because it did not have the property that the hypothesis of equal means would be rejected with probability α if

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