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##### Significance and Meaningfulness

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**Significance and Meaningfulness**Effect Sizes**Significance vs. meaningfulness**• Is your significant difference a real difference?**Significance vs. meaningfulness**• Is your significant difference a real difference?**Significance vs. meaningfulness**• Statistical Power**Significance vs. meaningfulness**• Statistical Power • Smaller difference between means reduces power • Larger SEM reduces power**Significance vs. meaningfulness**• Statistical Power • Smaller reduces power**Significance vs. meaningfulness**• As sample size increases, likelihood of significant difference increases The fact that this sample size is buried down here in the denominator of the test statistic means that as n , p 0. So if your sample is big enough, it will generate significant results**Significance vs. meaningfulness**• As sample size increases, likelihood of significant difference increases • So statistical difference does not always mean important difference • What to do about this? • Calculate a measure of the difference that is standardized to be expressed in terms of the variability in the 2 samples • = EFFECT SIZE**Significance vs. meaningfulness**• EFFECT SIZE - FORMULA**Significance vs. meaningfulness**• EFFECT SIZE – from SPSS • Using appendix B data set 2, and submitting DV salary to test of difference across gender, gives the following output (squashed here to fit): T-Test**Significance vs. meaningfulness**• EFFECT SIZE – from SPSS Mean difference to use T-Test SD’s to pool**Significance vs. meaningfulness**• EFFECT SIZE – from SPSS So…**Significance vs. meaningfulness**• EFFECT SIZE – from SPSS Substituting…**Significance vs. meaningfulness**• EFFECT SIZE – from SPSS Calculating…**Significance vs. meaningfulness**• From Cohen, 1988: • d = .20 is small • d = .50 is moderate • d = .80 is large • So our effect size of .25 is small, and concurs on this occasion with the insignificant result • The finding is both insignificant and small • (a pathetic, measly, piddling little difference of no consequence whatsoever – trivial and beneath us)**Statistical Power**Maximizing the likelihood of significance**Statistical Power**• The likelihood of getting a significant relationship when you should (i.e. when there is a relationship in reality) • Recall from truth table, power = 1 - ( = type II error)**Factors Affecting Statistical Power**The big ones: • Effect size (bit obvious) • Select samples such that difference between them is maximized • Sample size • Most important: as n increases, SEM decreases, and test statistic then increases**Factors Affecting Statistical Power**The others: • Level of significance • Smaller , less power • Larger , more power • 1-tailed vs. 2-tailed tests • With good a priori info (i.e. research literature), selecting 1-tailed test increases power • Dependent samples • Correlation between samples reduces standard error, and thus increases test statistic**Calculating sample size a priori**• Specify effect size • Set desired level of power • Enter values for effect size and power in appropriate table, and generate desired sample size: • Applet for calculating sample size based on above: http://www.stat.uiowa.edu/~rlenth/Power/ • Applets for seeing power acting (and interacting) with sample size, effect size, etc… http://statman.stat.sc.edu/~west/applets/power.html http://acad.cgu.edu/wise/power/powerapplet1.html http://www.stat.sc.edu/%7Eogden/javahtml/power/power.html