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Bartlett's Test
In statistics, Bartlett's test, named after Maurice Stevenson Bartlett, is used to test homoscedasticity, that is, if multiple samples are from populations with equal variances. Some statistical tests, such as the analysis of variance, assume that variances are equal across groups or samples, which can be verified with Bartlett's test. In a Bartlett test, we construct the null and alternative hypothesis. For this purpose several test procedures have been devised. The test procedure due to M.S.E (Mean Square Error/Estimator) Bartlett test is represented here. This test procedure is based on the statistic whose sampling distribution is approximately a Chi-Square distribution with (''k'' − 1) degrees of freedom, where ''k'' is the number of random samples, which may vary in size and are each drawn from independent normal distributions. Bartlett's test is sensitive to departures from normality. That is, if the samples come from non-normal distributions, then Bartlett's test may simp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Statistics
Statistics (from German: '' Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with a statistical population or a statistical model to be studied. Populations can be diverse groups of people or objects such as "all people living in a country" or "every atom composing a crystal". Statistics deals with every aspect of data, including the planning of data collection in terms of the design of surveys and experiments.Dodge, Y. (2006) ''The Oxford Dictionary of Statistical Terms'', Oxford University Press. When census data cannot be collected, statisticians collect data by developing specific experiment designs and survey samples. Representative sampling assures that inferences and conclusions can reasonably extend from the sample to the population as a whole. An ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homoscedasticity
In statistics, a sequence (or a vector) of random variables is homoscedastic () if all its random variables have the same finite variance. This is also known as homogeneity of variance. The complementary notion is called heteroscedasticity. The spellings ''homoskedasticity'' and ''heteroskedasticity'' are also frequently used. Assuming a variable is homoscedastic when in reality it is heteroscedastic () results in unbiased but inefficient point estimates and in biased estimates of standard errors, and may result in overestimating the goodness of fit as measured by the Pearson coefficient. The existence of heteroscedasticity is a major concern in regression analysis and the analysis of variance, as it invalidates statistical tests of significance that assume that the modelling errors all have the same variance. While the ordinary least squares estimator is still unbiased in the presence of heteroscedasticity, it is inefficient and generalized least squares should b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Variance
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers is spread out from their average value. Variance has a central role in statistics, where some ideas that use it include descriptive statistics, statistical inference, hypothesis testing, goodness of fit, and Monte Carlo sampling. Variance is an important tool in the sciences, where statistical analysis of data is common. The variance is the square of the standard deviation, the second central moment of a distribution, and the covariance of the random variable with itself, and it is often represented by \sigma^2, s^2, \operatorname(X), V(X), or \mathbb(X). An advantage of variance as a measure of dispersion is that it is more amenable to algebraic manipulation than other measures of dispersion such as the expected absolute deviatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
George W
George Walker Bush (born July 6, 1946) is an American politician who served as the 43rd president of the United States from 2001 to 2009. A member of the Republican Party, Bush family, and son of the 41st president George H. W. Bush, he previously served as the 46th governor of Texas from 1995 to 2000. While in his twenties, Bush flew warplanes in the Texas Air National Guard. After graduating from Harvard Business School in 1975, he worked in the oil industry. In 1978, Bush unsuccessfully ran for the House of Representatives. He later co-owned the Texas Rangers of Major League Baseball before he was elected governor of Texas in 1994. As governor, Bush successfully sponsored legislation for tort reform, increased education funding, set higher standards for schools, and reformed the criminal justice system. He also helped make Texas the leading producer of wind powered electricity in the nation. In the 2000 presidential election, Bush defeated Democratic incum ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
William Gemmell Cochran
William Gemmell Cochran (15 July 1909 – 29 March 1980) was a prominent statistician. He was born in Scotland but spent most of his life in the United States. Cochran studied mathematics at the University of Glasgow and the University of Cambridge. He worked at Rothamsted Experimental Station from 1934 to 1939, when he moved to the United States. There he helped establish several departments of statistics. His longest spell in any one university was at Harvard, which he joined in 1957 and from which he retired in 1976. Writings Cochran wrote many articles and books. His books became standard texts: * ''Experimental Designs'' (with Gertrude Mary Cox) 1950 * * ''Statistical Methods Applied to Experiments in Agriculture and Biology'' by George W. Snedecor George Waddel Snedecor (October 20, 1881 – February 15, 1974) was an American mathematician and statistician. He contributed to the foundations of analysis of variance, data analysis, experimental design, and statist ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Analysis Of Variance
Analysis of variance (ANOVA) is a collection of statistical models and their associated estimation procedures (such as the "variation" among and between groups) used to analyze the differences among means. ANOVA was developed by the statistician Ronald Fisher. ANOVA is based on the law of total variance, where the observed variance in a particular variable is partitioned into components attributable to different sources of variation. In its simplest form, ANOVA provides a statistical test of whether two or more population means are equal, and therefore generalizes the ''t''-test beyond two means. In other words, the ANOVA is used to test the difference between two or more means. History While the analysis of variance reached fruition in the 20th century, antecedents extend centuries into the past according to Stigler. These include hypothesis testing, the partitioning of sums of squares, experimental techniques and the additive model. Laplace was performing hypothesis test ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Levene's Test
In statistics, Levene's test is an inferential statistic used to assess the equality of variances for a variable calculated for two or more groups. Some common statistical procedures assume that variances of the populations from which different samples are drawn are equal. Levene's test assesses this assumption. It tests the null hypothesis that the population variances are equal (called ''homogeneity of variance'' or ''homoscedasticity''). If the resulting ''p''-value of Levene's test is less than some significance level (typically 0.05), the obtained differences in sample variances are unlikely to have occurred based on random sampling from a population with equal variances. Thus, the null hypothesis of equal variances is rejected and it is concluded that there is a difference between the variances in the population. Some of the procedures typically assuming homoscedasticity, for which one can use Levene's tests, include analysis of variance and t-tests. Levene's test is s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Brown–Forsythe Test
The Brown–Forsythe test is a statistical test for the equality of group variances based on performing an Analysis of Variance (ANOVA) on a transformation of the response variable. When a one-way ANOVA is performed, samples are assumed to have been drawn from distributions with equal variance. If this assumption is not valid, the resulting ''F''-test is invalid. The Brown–Forsythe test statistic is the F statistic resulting from an ordinary one-way analysis of variance on the absolute deviations of the groups or treatments data from their individual medians. Transformation The transformed response variable is constructed to measure the spread in each group. Let : z_=\left\vert y_ - \tilde_j \right\vert where \tilde_j is the median of group ''j''. The Brown–Forsythe test statistic is the model ''F'' statistic from a one way ANOVA on ''zij'': : F = \frac \frac where ''p'' is the number of groups, ''nj'' is the number of observations in group ''j'', and ''N'' is the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sample Variance
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers is spread out from their average value. Variance has a central role in statistics, where some ideas that use it include descriptive statistics, statistical inference, hypothesis testing, goodness of fit, and Monte Carlo sampling. Variance is an important tool in the sciences, where statistical analysis of data is common. The variance is the square of the standard deviation, the second central moment of a distribution, and the covariance of the random variable with itself, and it is often represented by \sigma^2, s^2, \operatorname(X), V(X), or \mathbb(X). An advantage of variance as a measure of dispersion is that it is more amenable to algebraic manipulation than other measures of dispersion such as the expected absolute deviatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Likelihood Ratio Test
In statistics, the likelihood-ratio test assesses the goodness of fit of two competing statistical models based on the ratio of their likelihoods, specifically one found by maximization over the entire parameter space and another found after imposing some constraint. If the constraint (i.e., the null hypothesis) is supported by the observed data, the two likelihoods should not differ by more than sampling error. Thus the likelihood-ratio test tests whether this ratio is significantly different from one, or equivalently whether its natural logarithm is significantly different from zero. The likelihood-ratio test, also known as Wilks test, is the oldest of the three classical approaches to hypothesis testing, together with the Lagrange multiplier test and the Wald test. In fact, the latter two can be conceptualized as approximations to the likelihood-ratio test, and are asymptotically equivalent. In the case of comparing two models each of which has no unknown parameters, us ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Box's M Test
Box's ''M'' test is a multivariate statistical test used to check the equality of multiple variance-covariance matrices. The test is commonly used to test the assumption of homogeneity of variances and covariances in MANOVA and linear discriminant analysis. It is named after George E. P. Box, who first discussed the test in 1949. The test uses a chi-squared approximation. Box's ''M'' test is susceptible to errors if the data does not meet model assumptions or if the sample size is too large or small. Box's ''M'' test is especially prone to error if the data does not meet the assumption of multivariate normality. See also * Bartlett's test * Levene's test In statistics, Levene's test is an inferential statistic used to assess the equality of variances for a variable calculated for two or more groups. Some common statistical procedures assume that variances of the populations from which different sam ... References Multivariate statistics Statistical hypothesis testing ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kaiser–Meyer–Olkin Test
The Kaiser–Meyer–Olkin (KMO) test is a statistical measure to determine how suited data is for factor analysis. The test measures sampling adequacy for each variable in the model and the complete model. The statistic is a measure of the proportion of variance among variables that might be common variance. The higher the proportion, the higher the KMO-value, the more suited the data is to factor analysis. History Henry Kaiser introduced a Measure of Sampling Adequacy (MSA) of factor analytic data matrices in 1970. Kaiser and Rice then modified it in 1974. Measure of sampling adequacy The measure of sampling adequacy is calculated for each indicator as :MSA_j = \frac and indicates to what extent an indicator is suitable for a factor analysis. Kaiser–Meyer–Olkin criterion : The Kaiser–Meyer–Olkin criterion is calculated and returns values between 0 and 1. : :KMO = \frac Here r_ is the correlation between the variable in question and another, and p_ is the pa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
