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No Use replication, i.e. get several values for each level of the factor Yes Are the samples from populations with equal variance? Are there any “nuisance” factors? Testing whether all means are equal Yes No More advanced techniques needed (e.g. transformations or General Linear Model). Beyond the scope of this poster Yes One One Might the “nuisance” factors interact with each other an d/or the factor of interest? How many factors of key interest are there? No No How many “nuisance” factors are there? Two More than two Use each level of the “nuisance” factor as a block Is it possible to inclu de each level of the factor of interest in each block? Are the number of levels the same for all three factors? Yes No No Latin square design Randomised block design. Possibly replication Are the samples fro m populations with equal variance, which are at least approximately Nor mal? Testing whether all means are equal, for each factor Yes Possibly balanced incomplete blocks or partially balanced incomplete blocks For each combination of factors, does the population have the same variance and is it at least approximately Normal? Specialised design beyond the scope of this poster Yes How many “nuisance” factors are there? Two or more Analysis beyond the scope of this poster Yes Testing whether all means are equal, for each factor No More than two Are there any “nuisance” factors? Beyond the scope of this poster This poster is one of a series of three, designed by Stella Dudzic. The series includes: Hypothesis tests for one sample, Hypothesis tests for two samples, and Experimental Design and Hypothesis tests for several samples: ANOVA (Analysis of Variance) To view the other two posters and to place orders for these and for extra sets of all three posters, please visit the MEI website at www.mei.org.uk The set of three posters is also available in a simplified format (on CD) to print in A4 size for student folders or for use on a whiteboard. Included on the CD are test statistics for one and two samples, and worked examples for analysis of variance (ANOVA). Yes No Yes Use a two way factorial design with randomisation and, possibly, replication Are the samples from populations with equal variance and at least approximately Normal? Use a two way factorial design with randomisation and, possibly, replication Beyond the scope of this poster No suitable common test Analysis similar to that for two factors of key interest Possibly factorial design Have yo u used replication? Yes No No Analysis of variance for Latin square Analysis beyond the scope of this poster No Two Analysis of variance for randomised blocks Beyond the scope of this poster, possibly Friedman's two-way analysis of variance by rank No Specialised design (possibly Graeco Latin square) One One-way analysis of variance (one between subjects factor) Kruskal-Wallis one-way analysis of variance No Yes See www.winterolympics .external.bbc.co.uk/ event-results-sched ules/index.html for results fro m the Winter Olympics Are the populations Normal (at least approximately)? Might the factors interact with each other ? Are you prepared to assume that the factors do not interact? Beyond the scope of this poster Yes Analysis beyond the scope of this poster Testing whether all means are equal, for each factor Testing whether all means are equal, for each factor, and whether interactions between factors exist Yes No Two-way analysis of variance (no interaction) Two-way analysis of variance (two between subjects factors) Two-way analysis of variance, with interaction interpreted as residual No simple general procedure - beyond the scope of this poster Yes Test on mean/ me dian Do you have a large sample? No Yes Do you know the variance? Are the data from a Normal distribution? No Single variable Test on variance See www.lotter y.co.uk/st atistics/ for data Symmetrical Distribution Other For Normal population This poster is one of a series of three, designed by Stella Dudzic. The series includes: Hypothesis tests for one sample, Hypothesis tests for two samples, and Experimental Design and Hypothesis tests for several samples: ANOVA (Analysis of Variance) To view the other two posters and to place orders for these and for extra sets of all three posters, please visit the MEI website at www.mei.org.uk The set of three posters is also available in a simplified format (on CD) to print in A4 size for student folders or for use on a whiteboard. Included on the CD are test statistics for one and two samples, and worked examples for analysis of variance (ANOVA). Wilcoxon single sample test Sign test test for variance Binomial test or Normal approximation Go odness of fit test test or Kolmogorov-Smirnov (see fig 3) Number pairs fig 1 test Yes Are the data fro m a bivariate Normal distribution? (see fig 2) fig 2 No fig 3 1 female right handed 32 28 left handed 7 5 Contingency table Poisson test Test of proportion Are the variables categories or numbers? Male Estimate variance as s² and use t test Poisson What distribution are the data from? cumulative probability 38 has co me up 213 times to en d March 2010 but 20 has only co me up 148 times Yo u co uld u se a go o dness o f fit test to che ck if there is ev idence that the lo tter y is not fair Normal test No Categories in a contingency table (see fig 1) Bivariate data Estimate variance as s² and use Normal test Yes Do you know the variance? Yes No Are the data single variable or bivariate? Normal test 0.75 observed D 0.5 EXPECTED 0.25 0 x With a large set of data, the scatter diagram for a bivariate Normal distribution is approximately elliptical Test statistic for Kolmogorov-Smirnov test Pearson’s product moment correlation test Spearman’s rank correlation test or Kendall’s rank correlation test Test on difference of means/medians Yes Do yo u know the variance of the differences? No Do you know the variance of the differences? Do you have large samples? Matched (paired) samples Yes Are the differences Normally distribute d? To do a test on paired samples, first find the differences between paired data values and then proceed as for a single sample test Are your samples matched? Unpaired samples No Do yo u have large samples? Yes Are the data from Normal distributions? No A survey of TV watching habits is conducted with the following results Women Men Sample size 50 60 Number of hours of TV watched per week Sample mean 11.2 9.6 Sample variance 135.2 66.9 Does this provide evidence that there is a difference in the mean number of hours of TV watched by men and women? This poster is one of a series of three, designed by Stella Dudzic. The series includes: Hypothesis tests for one sample, Hypothesis tests for two samples, and Experimental Design and Hypothesis tests for several samples: ANOVA (Analysis of Variance) To view the other two posters and to place orders for these and for extra sets of all three posters, please visit the MEI website at www.mei.org.uk The set of three posters is also available in a simplified format (on CD) to print in A4 size for student folders or for use on a whiteboard. Included on the CD are test statistics for one and two samples, and worked examples for analysis of variance (ANOVA). Test on difference of variances Are the differences symmetrically distribute d? Do you know the variances? Yes No Normal test No Estimate variance of differences using s² and use Normal test Yes Normal test No Estimate variance of differences using s² and use t test Yes Testing whether they are from the same distribution Test on difference of means/medians Yes Do you know the variances? Are the data from distribut ions w ith the sam e shape? Wilcoxon paired sample test No Sign test Kolmogorov-Smirnov 2-sample test Yes Normal test No Estimate variances using s²,s² and use Normal test Yes Yes No Yes No Are the variances equal? No Normal test t test with pooled estimate of variance No suitable simple test Wilcoxon rank sum test or Mann Whitney U test No suitable simple test F test