Psy 524More ReviewHypothesis Testing and Inferential StatisticsZ-test Where all your misery began!!Z-testSlide 6Making a DecisionHypothesis TestingPowerSlide 10t-tests“Why is it called analysis of variance anyway?”Factorial between-subjects ANOVAsRepeated MeasuresMixed designsSpecific ComparisonsOrthogonality revisitedSlide 18ComparisonsMeasuring strength of associationEta Square (η2 )Slide 22Slide 23Partial Eta SquarePartial Eta SquareBivariate StatisticsChi SquarePsy 524Lecture 2Andrew AinsworthMore ReviewHypothesis Testing and Inferential Statistics•Making decisions about uncertain events•The use of samples to represent populations•Comparing samples to given values or to other samples based on probability distributions set up by the null and alternative hypothesesZ-test Where all your misery began!!•Assumes that the population mean and standard deviation are known (therefore not realistic for application purposes)•Used as a theoretical exercise to establish tests that followZ-test•Sampling distributions are established; either by rote or by estimation (hypotheses deal with means so distributions of means are what we use) compared to y yyyNs sss =Z-test•Decision axes established so we leave little chance for errorMaking a Decision•Type 1 error – rejecting null hypothesis by mistake (Alpha)•Type 2 error – keeping the null hypothesis by mistake (Beta)Hypothesis TestingPower•Power is established by the probability of rejecting the null given that the alternative is true.•Three ways to increase it–Increase the effect size–Use less stringent alpha level–Reduce your variability in scores (narrow the width of the distributions) •more control or more subjectsPower•“You can never have too much power!!” – –this is not true –too much power (e.g. too many subjects) hypothesis testing becomes meaningless (really should look at effects size only)t-tests•realistic application of z-tests because the population standard deviation is not known (need multiple distributions instead of just one)“Why is it called analysis of variance anyway?”/Total wg bgTotal S A ASS SS SSSS SS SS= += +Factorial between-subjects ANOVAs•really just one-way ANOVAs for each effect and an additional test for the interaction. •What’s an interaction?1 21 1 11 22 12 2DV IV IVdv g gg gg gdvN g gMMRepeated Measures•Error broken into error due (S) and (S * T)•carryover effects, subject effects, subject fatigue etc…1 2 31 11 12 131 2 3Subject Trial Trial Trials r r rsn rn rn rnM M M MMixed designs1 2 31 1 11 12 131 12 1 13 1 2Group Subject Trial Trial Trials r r rsnsnsn n++M M M M MM M MM M MM M M M MM M MSpecific Comparisons•Use specific a priori comparisons in place of doing any type of ANOVA•Any number of planned comparisons can be done but if the number of comparisons surpasses the number of DFs than a correction is preferable (e.g. Bonferoni)•Comparisons are done by assigning each group a weight given that the weights sum to zero10kiiw==�Orthogonality revisited•If the weights are also orthogonal than the comparisons also have desirable properties in that it covers all of the shared variance•Orthogonal contrast must sum to zero and the sum of the cross products must also be orthogonal•If you use polynomial contrasts they are by definition orthogonal, but may not be interesting substantively1 2 1*22 0 01 1 11 1 10 0 0Constrast Constrast- -- -Comparisons•where nc is the number of scores used to get the mean for the group and MSerror comes from the omnibus ANOVA•These tests are compared to critical F’s with 1 degree of freedom•If post hoc than an adjustment needs to be made in the critical F (critical F is inflated in order to compensate for lack of hypothesis; e.g. Scheffé adjustment is (k-1)Fcritical)( )22/c j j jerrorn w Y wFMS=� �Measuring strength of association•It’s not the size of your effect that matters!!! (yes it is)Eta Square (η2 )•ratio of between subjects variation to total variance, it is the same as squared correlationA CBEta Square (η2 )•For one way analysis = B/A+BA CBEta Square (η2 )•For factorial D + F/ A + D + E + FAEBFGDCPartial Eta Square •Ratio of between subjects variance to between variance plus error•For one way analysis eta squared and partial are the samePartial Eta Square•For factorial designs D + F /D + F + A–Because A is the unexplained variance in the DV or errorAEBFGDCBivariate Statistics•Correlation•Regression ( ) ( )2 22 2( )( )N XY X YrN X X N Y Y-=� �� �- -� �� �� �� �� � �� � � �( )22( )( )N XY X YBN X X-=� �-� �� �� � �� �Chi Square( )2/( )( ) /o e ee sum sumf f ff row column