Bartlett's test

From Wikipedia, the free encyclopedia

In statistics, Bartlett's test (see Snedecor and Cochran, 1989) is used to test if k samples are from populations with equal variances. Equal variances across samples is called homoscedasticity or homogeneity of variances. Some statistical tests, for example the analysis of variance, assume that variances are equal across groups or samples. The Bartlett test can be used to verify that assumption.

Bartlett's test is sensitive to departures from normality. That is, if the samples come from non-normal distributions, then Bartlett's test may simply be testing for non-normality. Levene's test and the Brown–Forsythe test are alternatives to the Bartlett test that are less sensitive to departures from normality.^[1]

The test is named after Maurice Stevenson Bartlett.

Specification

Bartlett's test is used to test the null hypothesis, H₀ that all k population variances are equal against the alternative that at least two are different.

If there are k samples with size $n_{i}$ and sample variances $S_{i}^{2}$ then Bartlett's test statistic is

$X^{2}={\frac {(N-k)\ln(S_{p}^{2})-\sum _{{i=1}}^{k}(n_{i}-1)\ln(S_{i}^{2})}{1+{\frac {1}{3(k-1)}}\left(\sum _{{i=1}}^{k}({\frac {1}{n_{i}-1}})-{\frac {1}{N-k}}\right)}}$

where $N=\sum _{{i=1}}^{k}n_{i}$ and $S_{p}^{2}={\frac {1}{N-k}}\sum _{i}(n_{i}-1)S_{i}^{2}$ is the pooled estimate for the variance.

The test statistic has approximately a $\chi _{{k-1}}^{2}$ distribution. Thus the null hypothesis is rejected if $X^{2}>\chi _{{k-1,\alpha }}^{2}$ (where $\chi _{{k-1,\alpha }}^{2}$ is the upper tail critical value for the $\chi _{{k-1}}^{2}$ distribution).

Bartlett's test is a modification of the corresponding likelihood ratio test designed to make the approximation to the $\chi _{{k-1}}^{2}$ distribution better (Bartlett, 1937).

References

↑ NIST/SEMATECH e-Handbook of Statistical Methods. Available online, URL: http://www.itl.nist.gov/div898/handbook/eda/section3/eda357.htm. Retrieved December 31, 2013.

Bartlett, M. S. (1937). "Properties of sufficiency and statistical tests". Proceedings of the Royal Statistical Society, Series A 160, 268–282 JSTOR 96803
Snedecor, George W. and Cochran, William G. (1989), Statistical Methods, Eighth Edition, Iowa State University Press. ISBN 978-0-8138-1561-9

External links

NIST page on Bartlett's test

Statistics

Descriptive statistics

Continuous data

Location	Mean (Arithmetic, Geometric, Harmonic) Median Mode

Dispersion	Range Standard deviation Coefficient of variation Percentile Interquartile range

Shape	Variance Skewness Kurtosis Moments L-moments

Count data

Index of dispersion

Summary tables

Dependence

Statistical graphics

Data collection

Designing studies	Effect size Standard error Statistical power Sample size determination

Survey methodology	Sampling Stratified sampling Cluster sampling Opinion poll Questionnaire

Controlled experiment	Design of experiments Randomized experiment Random assignment Replication Blocking Factorial experiment Optimal design

Uncontrolled studies	Natural experiment Quasi-experiment Observational study

Statistical inference

Statistical theory	Sampling distribution Order statistic Scan statistic Record value Sufficiency Completeness Exponential family Permutation test (Randomization test) Empirical distribution Bootstrap U statistic Efficiency Asymptotics Robustness

Frequentist inference	Unbiased estimator (Mean unbiased minimum variance, Median unbiased) Biased estimators (Maximum likelihood, Method of moments, Minimum distance, Density estimation) Confidence interval Testing hypotheses Power Parametric tests (Likelihood-ratio, Wald, Score)

Specific tests	Z (normal) Student's t-test F Goodness of fit (Chi-squared, G, Sample source, sample normality, Skewness & kurtosis Normality, Model comparison, Model quality) Signed-rank (1-sample, 2-sample, 1-way anova) Shapiro–Wilk Kolmogorov–Smirnov

Bayesian inference	Bayesian probability Prior Posterior Credible interval Bayes factor Bayesian estimator Maximum posterior estimator

Correlation and regression analysis

Correlation	Pearson product–moment correlation Partial correlation Confounding variable Coefficient of determination

Regression analysis	Errors and residuals Regression model validation Mixed effects models Simultaneous equations models MARS

Linear regression	Simple linear regression Ordinary least squares General linear model Bayesian regression

Non-standard predictors	Nonlinear regression Nonparametric Semiparametric Isotonic Robust Heteroscedasticity Homoscedasticity

Generalized linear model	Exponential families Logistic (Bernoulli) Binomial Poisson

Partition of variance	Analysis of variance (ANOVA) Analysis of covariance Multivariate ANOVA Degrees of freedom

Categorical, multivariate, time-series, or survival analysis

Categorical data

Multivariate statistics

Time series analysis

General	Decomposition Trend Stationarity Seasonal adjustment Exponential smoothing Cointegration

Specific tests	Granger causality Q-Statistic Durbin–Watson

Time domain	ACF PACF XCF ARMA model ARIMA model ARCH Vector autoregression

Frequency domain	Spectral density estimation Fourier analysis

Survival analysis

Applications

Biostatistics	Bioinformatics Clinical trials & studies Epidemiology Medical statistics

Engineering statistics	Chemometrics Methods engineering Probabilistic design Process & Quality control Reliability System identification

Social statistics	Actuarial science Census Crime statistics Demography Econometrics National accounts Official statistics Population Psychometrics

Spatial statistics	Cartography Environmental statistics Geographic information system Geostatistics Kriging

Category
Portal
Outline
Index

This article is issued from Wikipedia. The text is available under the Creative Commons Attribution/Share Alike; additional terms may apply for the media files.