PsyPsy524524Lecture 2Lecture 2Andrew AinsworthAndrew AinsworthMore ReviewMore ReviewHypothesis Testing and Inferential Hypothesis Testing and Inferential StatisticsStatisticsMaking decisions about uncertain eventsMaking decisions about uncertain eventsThe use of samples to represent populationsThe use of samples to represent populationsComparing samples to given values or to other Comparing samples to given values or to other samples based on probability distributions set up samples based on probability distributions set up by the null and alternative hypothesesby the null and alternative hypothesesZZ--test test Where all your misery began!!Where all your misery began!!Assumes that the population mean and Assumes that the population mean and standard deviation are known (therefore standard deviation are known (therefore not realistic for application purposes)not realistic for application purposes)Used as a theoretical exercise to establish Used as a theoretical exercise to establish tests that followtests that followZZ--testtestSampling distributions are established; Sampling distributions are established; either by rote or by estimation either by rote or by estimation (hypotheses deal with means so (hypotheses deal with means so distributions of means are what we use)distributions of means are what we use)compared to yyyyNσσσσ=ZZ--testtestDecision axes established so we leave little Decision axes established so we leave little chance for errorchance for error Reality Reality H0 HA H0 HA “H0” 1 - α β “H0” .95 .16 Your Decision “HA” α 1 - β Your Decision “HA” .05 .84 1.00 1.00 1.00 1.00Making a DecisionMaking a DecisionType 1 error Type 1 error ––rejecting null hypothesis by rejecting null hypothesis by mistake (Alpha)mistake (Alpha)Type 2 error Type 2 error ––keeping the null hypothesis keeping the null hypothesis by mistake (Beta)by mistake (Beta)Hypothesis TestingHypothesis TestingPowerPowerPower is established by the probability of rejecting the Power is established by the probability of rejecting the null given that the alternative is true.null given that the alternative is true.Three ways to increase itThree ways to increase it––Increase the effect sizeIncrease the effect size––Use less stringent alpha levelUse less stringent alpha level––Reduce your variability in scores (narrow the width of the Reduce your variability in scores (narrow the width of the distributions) distributions) more control or more subjectsmore control or more subjectsPowerPower““You can never have too much You can never have too much power!!” power!!” ––––this is not true this is not true ––too much power (e.g. too many too much power (e.g. too many subjects) hypothesis testing becomes subjects) hypothesis testing becomes meaningless (really should look at meaningless (really should look at effects size only)effects size only)tt--teststestsrealistic application of zrealistic application of z--tests because the tests because the population standard deviation is not population standard deviation is not known (need multiple distributions instead known (need multiple distributions instead of just one)of just one)““Why is it called analysis of Why is it called analysis of variance anyway?”variance anyway?”/Total wgbgTotal S A ASS SS SSSS SS SS=+=+Factorial betweenFactorial between--subjects ANOVAssubjects ANOVAsreally just onereally just one--way way ANOVAs for each ANOVAs for each effect and an effect and an additional test for additional test for the interaction. the interaction. What’s an What’s an interaction?interaction?1211 1122122DVIV IVdv g gggggdvN g g##Repeated MeasuresRepeated MeasuresError broken into error due (S) and (S * T)Error broken into error due (S) and (S * T)carryover effects, subject effects, subject carryover effects, subject effects, subject fatigue etc…fatigue etc…1231111213123Subject Trial Trial Trialsrrrsn rn rn rn####Mixed designsMixed designs1231 1 11 12 1311211312Group Subject Trial Trial Trialsrrrsnsnsn n++###################Specific ComparisonsSpecific ComparisonsUse specific a priori comparisons in Use specific a priori comparisons in place of doing any type of ANOVAplace of doing any type of ANOVAAny number of planned comparisons Any number of planned comparisons can be done but if the number of can be done but if the number of comparisons surpasses the number comparisons surpasses the number of of DFsDFsthan a correction is than a correction is preferable (e.g. preferable (e.g. BonferoniBonferoni))Comparisons are done by assigning Comparisons are done by assigning each group a weight given that the each group a weight given that the weights sum to zeroweights sum to zero10kiiw==∑OrthogonalityOrthogonalityrevisitedrevisitedIf the weights are also orthogonal than the If the weights are also orthogonal than the comparisons also have desirable properties in comparisons also have desirable properties in that it covers all of the shared variancethat it covers all of the shared varianceOrthogonal contrast must sum to zero and the Orthogonal contrast must sum to zero and the sum of the cross products must also be sum of the cross products must also be orthogonalorthogonalIf you use polynomial contrasts they are by If you use polynomial contrasts they are by definition orthogonal, but may not be interesting definition orthogonal, but may not be interesting substantivelysubstantively121*2200111111000Constrast Constrast−−−−ComparisonsComparisonswhere where nnccis the number of scores used to get the mean is the number of scores used to get the mean for the group and for the group and MSerrorMSerrorcomes from the omnibus comes from the omnibus ANOVAANOVAThese tests are compared to critical F’s with 1 degree of These tests are compared to critical F’s with 1 degree of freedomfreedomIf post hoc than an adjustment needs to be made in the If post hoc than an adjustment needs to be made in the critical F (critical F is inflated in order to compensate for critical F (critical F is inflated in order to compensate for lack of hypothesis; e.g. lack of hypothesis; e.g. SchefféSchefféadjustment is (kadjustment is (k--1)Fcritical)1)Fcritical)()22/cjj jerrornwY wFMS=∑∑Measuring strength of associationMeasuring strength
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