Y học - Chapter 14: Inferential data analysis

Chapter 14 Inferential Data AnalysisInferential Statistics Techniques that allow us to study samples and then make generalizations about the population. Inferential statistics are a very crucial part of scientific research in that these techniques are used to test hypothesesUses for Inferential StatisticsStatistics for determining differences between experimental and control groups in experimental researchStatistics used in descriptive research when comparisons are made between different groupsThese statistics enable the researcher to evaluate the effects of an independent variable on a dependent variableSampling ErrorDifferences between a sample statistic and a population parameter because the sample is not perfectly representative of the populationHypothesis TestingThe purpose of the statistical test is to evaluate the null hypothesis (H0) at a specified level of significance (e.g., p < .05)In other words, do the treatment effects differ significantly so that these differences would be attributable to chance occurrence less than 5 times in 100?Hypothesis Testing Procedures State the hypothesis (H0)Select the probability level (alpha)Determine the value needed for significanceCalculate the test statisticAccept or reject H0Statistical SignificanceA statement in the research literature that the statistical test was significant indicates that the value of the calculated statistic warranted rejection of the null hypothesisFor a difference question, this suggests a real difference and not one due to sampling errorParametric StatisticsTechniques which require basic assumptions about the data, for example:normality of distributionhomogeneity of variancerequirement of interval or ratio dataMost prevalent in HHPMany statistical techniques are considered robust to violations of the assumptions, meaning that the outcome of the statistical test should still be considered validt-testsCharacteristics of t-testsrequires interval or ratio level scoresused to compare two mean scoreseasy to computepretty good small sample statisticTypes of t-testOne-Group t-testt-test between a sample and population meanIndependent Groups t-testcompares mean scores on two independent samplesDependent Groups (Correlated) t-testcompares two mean scores from a repeated measures or matched pairs designmost common situation is for comparison of pretest with posttest scores from the same sampleHypothesis Testing ErrorsHypothesis testing decisions are made without direct knowledge of the true circumstance in the population. As a result, the researcher’s decision may or may not be correctType I ErrorType II ErrorType I Error. . . is made when the researcher rejects the null hypothesis when in fact the null hypothesis is trueprobability of committing Type I error is equal to the significance (alpha) level set by the researcherthus, the smaller the alpha level the lower the chance of committing a Type I errorType II Error. . . occurs when the researcher accepts the null hypothesis, when in fact it should have been rejectedprobability is equal to beta (B) which is influenced by several factorsinversely related to alpha levelincreasing sample size will reduce BStatistical Power – the probability of rejecting a false null hypothesisPower = 1 – betaDecreasing probability of making a Type II error increases statistical powerHypothesis Truth TableCORRECT DECISIONCORRECT DECISIONTYPE IIERRORTYPE IERRORNULL HYPOTHESISTRUEFALSEACCEPTREJECTDECISIONANOVA - Analysis of VarianceA commonly used family of statistical tests that may be considered a logical extension of the t-testrequires interval or ratio level scoresused for comparing 2 or more mean scoresmaintains designated alpha level as compared to experimentwise inflation of alpha level with multiple t-testsmay also test more than 1 independent variable as well as interaction effectOne-way ANOVAExtension of independent groups t-test, but may be used for evaluating differences among 2 or more groupsRepeated Measures ANOVAExtension of dependent groups t-test, where each subject is measured on 2 or more occasionsa.k.a “within subjects design”Test of sphericity assumption is recommendedRandom Blocks ANOVAThis is an extension of the matched pairs t-test when there are three or more groups or the same as the matched pairs t-test when there are two groupsParticipants similar in terms of a variable are placed together in a block and then randomly assigned to treatment groupsFactorial ANOVAThis is an extension of the one-way ANOVA for testing the effects of 2 or more independent variables as well as interaction effectsTwo-way ANOVA (e.g., 3 X 2 ANOVA)Three-way ANOVA (e.g., 3 X 3 X 2 ANOVA)Assumptions of Statistical TestsParametric tests are based on a variety of assumptions, such asInterval or ratio level scoresRandom sampling of participantsScores are normally distributed N = 30 considered minimum by someHomogeneity of varianceGroups are independent of each otherOthersResearchers should try to satisfy assumptions underlying the statistical test being usedImproving the Probability of Meeting AssumptionsUtilize a sample that is truly representative of the population of interestUtilize large sample sizesUtilize comparison groups that have about the same number of participantsTwo-Group Comparison Testsa.k.a. Multiple Comparison or Post Hoc TestsThe various ANOVA tests are often referred to as “omnibus” tests because they are used to determine if the means are different but they do not specify the location of the differenceif the null hypothesis is rejected, meaning that there is a difference among the mean scores, then the researcher needs to perform additional tests in order to determine which means (groups) are actually differentCommon Post Hoc TestsMultiple comparison (post hoc) tests are used to make specific comparisons following a significant finding from ANOVA in order to determine the location of the differenceDuncanTukeyBonferroniScheffeNote that post hoc tests are only necessary if there are more than two levels of the independent variableAnalysis of CovarianceANOVAANOVA design which statistically adjusts the difference among group means to allow for the fact that the groups differ on some other variablefrequently used to adjust for inequality of groups at the start of a research studyNonparametric StatisticsConsidered assumption free statisticsAppropriate for nominal and ordinal data or in situations where very small sample sizes (n < 10) would probably not yield a normal distribution of scoresLess statistical power than parametric statisticsChi SquareA nonparametric test used with nominally scaled data which are common with survey research The statistic is used when the researcher is interested in the number of responses, objects, or people that fall in two or more categoriesSingle Sample Chi-Squarea.k.a one-way chi-square or goodness of fit chi-squareUsed to test the hypothesis that the collected data (observed scores) fits an expected distributioni.e. are the observed frequencies and expected frequencies for a questionnaire item in agreement with each other?Independent Groups Chi-Squarea.k.a. two-way chi-square or contingency table chi-squareUsed to test if there is a significant relationship (association) between two nominally scaled variablesIn this test we are comparing two or more patterns of frequencies to see if they are independent from each otherOverview of Multivariate TestsUnivariate statistic – used in situations where each participant contributed one score to the data analysis, or in the case of a repeated measures design, one score per cellMultivariate statistic – used in situations where each participant contributes multiple scoresExample Multivariate TestsMANOVACanonical correlationDiscriminant analysisFactor analysisMultiple Analysis of Variance MANOVAAnalogous to ANOVA except that there are multiple dependent variablesRepresents a type of multivariate testPrediction and Regression AnalysisCorrelational techniqueSimple predictionPredicting an unknown score (Y) based on a single predictor variable (X)Y’ = bX + cMultiple predictionInvolves more than one predictor variableY’ = b1X1 + b2X2 + cMultiple Regression/Predictiona.k.a multiple correlationDetermines the relationship between one dependent variable and 2 or more predictor variablesUsed to predict performance on one variable from anotherY’ = b1X1 + b2X2 + cStandard error of prediction is an index of accuracy of the predictionStatistical PowerThe probability that the statistical test will correctly reject a false null hypothesis. . . it is effectively the probability of finding significance, that the experimental treatment actually does have an effecta researcher would like to have a high level of powerStatistical Power alpha = probability of a Type I errorrejecting a true null hypothesisthis is your significance levelbeta = probability of a Type II errorfailing to reject a false null hypothesisStatistical power = 1 - betaFactors Affecting PowerAlpha levelSample sizeEffect sizeOne-tailed or two-tailed testAlpha levelReducing the alpha level (moving from .05 to .01) will reduce the power of a statistical test. This makes it harder to reject the null hypothesisSample size In general, the larger the sample size the greater the power. This is because the standard error of the mean decreases as the sample size increasesOne-tailed versus two-tailed testsIt is easier to reject the null hypothesis using a one-tailed test than a two-tailed test because the critical region is largerEffect size This is an indication of the size of the treatment effect, its meaningfulnessWith a large effect size, it will be easy to detect differences and statistical power will be highBut, if the treatment effect is small, it will be difficult to detect differences and power will be lowEffect SizeNumerous authors have indicated the need to estimate the magnitude of differences between groups as well as to report the significance of the effectsOne way to describe the strength of a treatment effect, or meaningfulness of the findings, is the computation of “effect size” (ES)ES = M1 - M2SDNote: SD represents the standard deviation of the control group or the pooled standard deviation if there is no control groupEffect SizeInterpretation of ES by Cohen (1988)0.2 represents a small ES0.5 represents a moderate ES0.8 represents a large ESResearchers using experimental designs are advised to provide post hoc estimates of ES for any significant findings as a way to evaluate the meaningfulnessA Priori ProceduresCalculate the power for each of the statistical procedures to be appliedrequires three indices - alpha, sample size, effect sizeEstimate the sample size needed to detect a certain effect (ES) given a specific alpha and powermay require an estimation of ES from previous published studies or from a pilot study