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Text of the page (random words):
tandard contents of a contingency table edit multiple columns historically they were designed to use up all the white space of a printed page where each row refers to a specific sub group in the population in this case men or women the columns are sometimes referred to as banner points or cuts and the rows are sometimes referred to as stubs significance tests typically either column comparisons which test for differences between columns and display these results using letters or cell comparisons which use color or arrows to identify a cell in a table that stands out in some way nets or netts which are sub totals one or more of percentages row percentages column percentages indexes or averages unweighted sample sizes counts measures of association edit the degree of association between the two variables can be assessed by a number of coefficients the following subsections describe a few of them for a more complete discussion of their uses see the main articles linked under each subsection heading odds ratio edit main article odds ratio the simplest measure of association for a 2 2 contingency table is the odds ratio given two events a and b the odds ratio is defined as the ratio of the odds of a in the presence of b and the odds of a in the absence of b or equivalently due to symmetry the ratio of the odds of b in the presence of a and the odds of b in the absence of a two events are independent if and only if the odds ratio is 1 if the odds ratio is greater than 1 the events are positively associated if the odds ratio is less than 1 the events are negatively associated the odds ratio has a simple expression in terms of probabilities given the joint probability distribution b 1 b 0 a 1 p 11 p 10 a 0 p 01 p 00 displaystyle begin array c cc b 1 b 0 hline a 1 p_ 11 p_ 10 a 0 p_ 01 p_ 00 end array the odds ratio is o r p 11 p 00 p 10 p 01 displaystyle or frac p_ 11 p_ 00 p_ 10 p_ 01 phi coefficient edit main article phi coefficient a simple measure applicable only to the case of 2 2 contingency tables is the phi coefficient φ defined by ϕ χ 2 n displaystyle phi pm sqrt frac chi 2 n where χ 2 is computed as in pearson s chi squared test and n is the grand total of observations φ varies from 0 corresponding to no association between the variables to 1 or 1 complete association or complete inverse association provided it is based on frequency data represented in 2 2 tables then its sign equals the sign of the product of the main diagonal elements of the table minus the product of the off diagonal elements φ takes on the minimum value 1 0 or the maximum value of 1 0 if and only if every marginal proportion is equal to 0 5 and two diagonal cells are empty 2 cramér s v and the contingency coefficient c edit main article cramér s v two alternatives are the contingency coefficient c and cramér s v the formulae for the c and v coefficients are c χ 2 n χ 2 displaystyle c sqrt frac chi 2 n chi 2 and v χ 2 n k 1 displaystyle v sqrt frac chi 2 n k 1 k being the number of rows or the number of columns whichever is less c suffers from the disadvantage that it does not reach a maximum of 1 0 notably the highest it can reach in a 2 2 table is 0 707 it can reach values closer to 1 0 in contingency tables with more categories for example it can reach a maximum of 0 870 in a 4 4 table it should therefore not be used to compare associations in different tables if they have different numbers of categories 3 c can be adjusted so it reaches a maximum of 1 0 when there is complete association in a table of any number of rows and columns by dividing c by k 1 k displaystyle sqrt frac k 1 k where k is the number of rows or columns when the table is square citation needed or by r 1 r c 1 c 4 displaystyle sqrt scriptstyle 4 r 1 over r times c 1 over c where r is the number of rows and c is the number of columns 4 tetrachoric correlation coefficient edit main article polychoric correlation another choice is the tetrachoric correlation coefficient but it is only applicable to 2 2 tables polychoric correlation is an extension of the tetrachoric correlation to tables involving variables with more than two levels tetrachoric correlation assumes that the variable underlying each dichotomous measure is normally distributed 5 the coefficient provides a convenient measure of the pearson product moment correlation when graduated measurements have been reduced to two categories 6 the tetrachoric correlation coefficient should not be confused with the pearson correlation coefficient computed by assigning say values 0 0 and 1 0 to represent the two levels of each variable which is mathematically equivalent to the φ coefficient lambda coefficient edit main article goodman and kruskal s lambda the lambda coefficient is a measure of the strength of association of the cross tabulations when the variables are measured at the nominal level values range from 0 0 no association to 1 0 the maximum possible association asymmetric lambda measures the percentage improvement in predicting the dependent variable symmetric lambda measures the percentage improvement when prediction is done in both directions uncertainty coefficient edit main article uncertainty coefficient the uncertainty coefficient or theil s u is another measure for variables at the nominal level its values range from 1 0 100 negative association or perfect inversion to 1 0 100 positive association or perfect agreement a value of 0 0 indicates the absence of association also the uncertainty coefficient is conditional and an asymmetrical measure of association which can be expressed as u x y u y x displaystyle u x y neq u y x this asymmetrical property can lead to insights not as evident in symmetrical measures of association 7 others edit main articles goodman and kruskal s gamma and kendall rank correlation coefficient gamma tau b and tau c are used when the categories or levels of both variables have a natural order gamma test no adjustment for either table size or ties kendall s tau adjustment for ties tau b used for square tables tau c used for rectangular tables see also edit confusion matrix pivot table in spreadsheet software cross tabulates sampling data with counts contingency table and or sums tpl tables is a tool for generating and printing crosstabs the iterative proportional fitting procedure essentially manipulates contingency tables to match altered joint distributions or marginal sums the multivariate statistics in special multivariate discrete probability distributions some procedures used in this context can be used in dealing with contingency tables olap cube a modern multidimensional computing form of contingency tables panel data multidimensional data over time references edit karl pearson f r s 1904 mathematical contributions to the theory of evolution dulau and co ferguson g a 1966 statistical analysis in psychology and education new york mcgraw hill smith s c albaum g s 2004 fundamentals of marketing research sage thousand oaks ca p 631 blaikie n 2003 analyzing quantitative data sage thousand oaks ca p 100 ferguson full citation needed ferguson 1966 p 244 the search for categorical correlation 26 december 2019 further reading edit andersen erling b 1980 discrete statistical models with social science applications north holland 1980 bishop y m m fienberg s e holland p w 1975 discrete multivariate analysis theory and practice mit press isbn 978 0 262 02113 5 mr 0381130 christensen ronald 1997 log linear models and logistic regression springer texts in statistics second ed new york springer verlag pp xvi 483 isbn 0 387 98247 7 mr 1633357 lauritzen steffen l 1979 lectures on contingency tables aalborg university pdf 4th edition first electronic edition 2002 ed gokhale d v kullback solomon 1978 the information in contingency tables marcel dekker isbn 0 824 76698 9 external links edit wikimedia commons has media related to contingency tables on line analysis of contingency tables calculator with examples interactive cross tabulation chi squared independent test and tutorial fisher and chi squared calculator of 2 2 contingency table more correlation coefficients nominal association phi contingency coefficient tschuprow s t cramer s v lambda uncertainty coefficient march 24 2008 g david garson north carolina state university custominsight com cross tabulation the powermutt project iv displaying categorical data stats steves attempt to teach statistics odds ratio versus relative risk january 9 2001 epi info community health assessment tutorial lesson 5 analysis creating statistics v t e statistics outline index descriptive statistics continuous data center mean arithmetic arithmetic geometric contraharmonic cubic generalized power geometric harmonic heronian heinz lehmer median mode dispersion average absolute deviation coefficient of variation interquartile range percentile range standard deviation variance shape central limit theorem moments kurtosis l moments skewness count data index of dispersion summary tables contingency table frequency distribution grouped data dependence partial correlation pearson product moment correlation rank correlation kendall s τ spearman s ρ scatter plot graphics bar chart biplot box plot control chart correlogram fan chart forest plot histogram pie chart q q plot radar chart run chart scatter plot stem and leaf display violin plot heatmap scatter plot matrix ecdf plot line chart statistical data processing transformations data transformation log transformation power transform box cox transformation yeo johnson transformation variance stabilizing transformation anscombe transform fisher transformation scaling and normalization feature scaling normalization standardization z score min max normalization unit vector normalization data cleaning data cleaning outlier winsorizing truncation missing data data reduction dimensionality reduction principal component analysis factor analysis time series preprocessing differencing detrending seasonal adjustment stationarity transformation data collection study design effect size missing data optimal design population replication sample size determination statistic statistical power survey methodology sampling cluster stratified opinion poll questionnaire standard error controlled experiments blocking factorial experiment interaction random assignment randomized controlled trial randomized experiment scientific control adaptive designs adaptive clinical trial stochastic approximation up and down designs observational studies cohort study cross sectional study natural experiment quasi experiment statistical inference statistical theory population statistic probability distribution sampling distribution order statistic empirical distribution density estimation statistical model model specification l p space parameter location scale shape parametric family likelihood monotone location scale family exponential family completeness sufficiency statistical functional bootstrap u v optimal decision loss function efficiency statistical distance divergence asymptotics robustness sensitivity analysis frequentist inference point estimation estimating equations maximum likelihood method of moments m estimator minimum distance unbiased estimators mean unbiased minimum variance rao blackwellization lehmann scheffé theorem median unbiased plug in interval estimation confidence interval pivot likelihood interval prediction interval tolerance interval resampling bootstrap jackknife testing hypotheses 1 2 tails power uniformly most powerful test permutation test randomization test multiple comparisons parametric tests likelihood ratio g test score lagrange multiplier wald z test normal specific tests parametric student s t test f test goodness of fit chi squared kolmogorov smirnov anderson darling lilliefors jarque bera normality shapiro wilk model selection cross validation aic bic rank statistics sign sample median signed rank wilcoxon hodges lehmann estimator rank sum mann whitney nonparametric anova 1 way kruskal wallis 2 way friedman ordered alternative jonckheere terpstra van der waerden test bayesian inference bayesian probability prior posterior credible interval bayes factor bayesian estimator maximum posterior estimator correlation regression analysis correlation pearson product moment partial correlation confounding variable coefficient of determination regression analysis errors and residuals regression validation mixed effects models simultaneous equations models multivariate adaptive regression splines mars template least squares and regression analysis linear regression simple linear regression ordinary least squares general linear model bayesian regression non standard predictors nonlinear regression nonparametric semiparametric isotonic robust homoscedasticity and heteroscedasticity generalized linear model exponential families logistic bernoulli binomial poisson regressions partition of variance analysis of variance anova analysis of variance ancova manova degrees of freedom categorical multivariate time series survival analysis categorical cohen s kappa contingency table graphical model log linear model mcnemar s test cochran mantel haenszel statistics multivariate regression manova principal components canonical correlation discriminant analysis cluster analysis classification structural equation model factor analysis multivariate distributions elliptical distributions normal time series general decomposition trend stationarity seasonal adjustment exponential smoothing cointegration structural break granger causality specific tests dickey fuller johansen q statistic ljung box durbin watson breusch godfrey time domain autocorrelation acf partial pacf cross correlation xcf arma model arima model box jenkins autoregressive conditional heteroskedasticity arch vector autoregression var autoregressive model ar frequency domain spectral density estimation fourier analysis least squares spectral analysis wavelet whittle likelihood survival survival function kaplan meier estimator product limit proportional hazards models accelerated failure time aft model first hitting time hazard function nelson aalen estimator test log rank test 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 jurimetrics national accounts official statistics population statistics psychometrics spatial statistics cartography environmental statistics geographic information system geostatistics kriging category mathematics portal commons wikiproject authority control databases national united states france bnf data spain israel other yale lux retrieved from https en wikipedia org w index php title contingency_table oldid 1366698527 categories contingency table tables information frequency distrib...