Polit ln ch11

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<ul><li><p> Copyright 2010 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458</p><p>All rights reserved.</p><p>Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit</p><p>Statistics and Data Analysisfor Nursing Research</p><p>Second Edition</p><p>CHAPTER</p><p>Analysis of Covariance, Multivariate ANOVA, and Related Multivariate Analyses</p><p>11</p></li><li><p> Copyright 2010 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458</p><p>All rights reserved.</p><p>Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit</p><p>General Linear Model General linear model (GLM): </p><p>encompasses a broad class of techniques (e.g., ANOVA, multiple regression) </p><p> Flexible, complex model, used to fit data to straight-line solutions</p><p> Involves partitioning variance, different methods possible Type III sums of squares is standard</p></li><li><p> Copyright 2010 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458</p><p>All rights reserved.</p><p>Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit</p><p>Analysis of Covariance A useful analytic tool within the GLM: </p><p>Analysis of covariance Often abbreviated as ANCOVA Used to compare group means after </p><p>removing the effect of a variable called a covariate </p><p> A covariate is typically: A preintervention measure of an outcome A measure of a confounding variable</p></li><li><p> Copyright 2010 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458</p><p>All rights reserved.</p><p>Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit</p><p>ANCOVA Principles Conceptualize as a two-step process First step, similar to hierarchical multiple </p><p>regressionEffects of covariates are removed from variance of the dependent variable</p><p> Second step, ANOVA-like partitioning of variance, with sum of squares for the independent variable contrasted against what remains of sum of squares due to error</p></li><li><p> Copyright 2010 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458</p><p>All rights reserved.</p><p>Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit</p><p>Venn Diagram of ANCOVA Two-step process:</p></li><li><p> Copyright 2010 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458</p><p>All rights reserved.</p><p>Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit</p><p>Adjusted Means In ANCOVA, the means on the </p><p>dependent variable (DV) can be adjusted by removing the covariates effect</p><p> Adjusted means indicate net effectsGroup differences on the DV net of the effect of the covariate</p></li><li><p> Copyright 2010 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458</p><p>All rights reserved.</p><p>Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit</p><p>ANCOVA Hypotheses Similar to ANOVA, tested via the F </p><p>statistic The null hypothesis: The adjusted group </p><p>means are equal The alternative hypothesis: At least some </p><p>of the adjusted group means are not equal</p></li><li><p> Copyright 2010 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458</p><p>All rights reserved.</p><p>Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit</p><p>Multiple Comparisons Like ANOVA, if there is a significant </p><p>overall group difference, pairwise multiple comparisons must be done when there are three or more groups</p><p> Purpose of these post hoc tests is to identify which pairs of adjusted group means are significantly different from one another</p></li><li><p> Copyright 2010 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458</p><p>All rights reserved.</p><p>Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit</p><p>Effect Size in ANCOVA</p><p> In ANCOVA, the effect size is similar to that in ANOVA (eta-squared)</p><p> In ANCOVA, eta-squared is adjusted for the covariate</p><p> Partial eta-squared is variability accounted for in the DV by the IV, after adjusting for the covariate</p></li><li><p> Copyright 2010 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458</p><p>All rights reserved.</p><p>Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit</p><p>Measurement in ANCOVA Dependent variable:</p><p> Interval or ratio level (e.g., heart rate)</p><p> Independent variable: Nominal level (e.g., experimental versus control) </p><p> Covariate: Interval or ratio level (or dichotomous with near 50-50 </p><p>split) Dummy-variable covariates can be included as </p><p>additional independent variables </p></li><li><p> Copyright 2010 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458</p><p>All rights reserved.</p><p>Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit</p><p>Selection of Covariates Multiple covariates can be included More than three to four covariates is </p><p>usually unnecessary and potentially unwise Mutlicollinearity is likely to occur with too many </p><p>covariates Good covariates: A pretest measure of </p><p>outcome variable; key demographic or clinical variables</p><p> It is important that covariates be reliably measured</p></li><li><p> Copyright 2010 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458</p><p>All rights reserved.</p><p>Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit</p><p>Assumptions for ANCOVA Basic assumptionslike ANOVA:</p><p> Random sampling from the populations Dependent variable normally distributed Variances in the populations are equal</p><p> Can be tested with Levenes statistic</p><p> Robustness: With reasonably large sample and groups of approximately equal size, statistical tests are robust to violations of latter two</p></li><li><p> Copyright 2010 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458</p><p>All rights reserved.</p><p>Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit</p><p>Assumptions for ANCOVA (contd) </p><p> Additional assumption in ANCOVA: Homogeneity of regression across groups The slope of the dependent variable on the </p><p>covariate should be the same for every group</p><p> When this assumption is violated, there is a heightened risk of a Type II error</p></li><li><p> Copyright 2010 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458</p><p>All rights reserved.</p><p>Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit</p><p>Homogeneity of Regression Illustration</p></li><li><p> Copyright 2010 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458</p><p>All rights reserved.</p><p>Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit</p><p>Homogeneity of Regression: Evaluation</p><p> This assumption is evaluated by testing whether the interaction of the covariate and the independent (group) variable (group*covariate) is statistically significant in the model for the dependent variable</p></li><li><p> Copyright 2010 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458</p><p>All rights reserved.</p><p>Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit</p><p>Testing Homogeneity of Regression Assumption </p><p> Here, the interaction is not significant, and the assumption holds</p></li><li><p> Copyright 2010 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458</p><p>All rights reserved.</p><p>Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit</p><p>ANCOVA Applications Appropriate primarily in randomized </p><p>controlled trials (RCTs) Often used in nonequivalent control group </p><p>(quasiexperimental) and case-control (nonexperimental) designs Intent: Enhance internal validity by reducing </p><p>selection threat (enhancing group comparability)</p><p> Use in non-RCTs is controversial but common; great care is needed in interpreting ANCOVA results in non-RCTs</p></li><li><p> Copyright 2010 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458</p><p>All rights reserved.</p><p>Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit</p><p>Alternatives to ANCOVA When the covariate is the same measure </p><p>as the DV, some people use change scores (DV at posttest minus DV at pretest) as the DV in an ANOVA Not advisable: Change scores can be </p><p>constrained by floor effects or ceiling effects (limits on amount of change), and tend to be unreliable</p><p> RM-ANOVA is another option (i.e., comparing two or more groups at two points in time)</p></li><li><p> Copyright 2010 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458</p><p>All rights reserved.</p><p>Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit</p><p>SPSS and ANCOVA Analyze </p><p>General Linear Model Univariate</p><p> Select: Dependent </p><p>variable Independents </p><p>(Fixed Factor) Covariates</p></li><li><p> Copyright 2010 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458</p><p>All rights reserved.</p><p>Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit</p><p>SPSS and ANCOVA (contd) GLM Options include </p><p>options for descriptive statistics such as adjusted marginal means </p><p> Many other options, such as: Effect size (partial </p><p>eta-squared) Levenes test Post hoc power</p></li><li><p> Copyright 2010 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458</p><p>All rights reserved.</p><p>Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit</p><p>Multivariate ANOVA Multivariate analysis of variance </p><p>(MANOVA) is another extension of ANOVA</p><p> Used to test mean group differences on two or more dependent variables</p></li><li><p> Copyright 2010 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458</p><p>All rights reserved.</p><p>Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit</p><p>MANOVA Concepts MANOVA involves the creation of a new </p><p>dependent variable that is a linear combination (composite) of the original DVs</p><p> Like ANOVA, MANOVA involves partitioning variance (of the composite DV) into different sources</p></li><li><p> Copyright 2010 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458</p><p>All rights reserved.</p><p>Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit</p><p>MANOVA Composite DV</p><p> A composite of two DVs (Y1 and Y2) for three groups (A, B, C)</p></li><li><p> Copyright 2010 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458</p><p>All rights reserved.</p><p>Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit</p><p>MANOVA Applications MANOVA is most often used in RCTs </p><p>with multiple outcome measures MANOVA works best when DVs are </p><p>highly negatively correlated or moderately positively correlated</p><p> Not necessary if DVs are uncorrelated</p></li><li><p> Copyright 2010 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458</p><p>All rights reserved.</p><p>Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit</p><p>Extensions of MANOVA MANOVA can be extended to have </p><p>multiple independent variablese.g., a two-way MANOVA (more generally, a multi-factor MANOVA)</p><p> MANOVA can also include covariate adjustments, in which case it is a multivariate analysis of covariance (MANCOVA)</p></li><li><p> Copyright 2010 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458</p><p>All rights reserved.</p><p>Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit</p><p>Statistical Tests for MANOVA For testing the significance of group </p><p>differences in MANOVA, there are four tests: Pillais trace criterion Wilks Lambda () Hotellings trace criterion Roys largest root criterion</p><p> These statistics are typically transformed to an approximate F distribution</p></li><li><p> Copyright 2010 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458</p><p>All rights reserved.</p><p>Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit</p><p>Evaluation of Tests The most frequently used statistic for </p><p>MANOVA is probably Wilks lambda Lambda represents the pooled ratio of error </p><p>variance in composite DV to error + treatment variance</p><p> Thus, conceptually, = 1 R2</p><p> However, Pillais criterion is the most robust It is preferred to Wilks when sample size is </p><p>small, group ns are unequal, or assumptions are violated</p></li><li><p> Copyright 2010 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458</p><p>All rights reserved.</p><p>Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit</p><p>MANOVA Tests in SPSS </p><p> MANOVA test, for two DVs, three groups: No significant group differencesfor any of the four tests</p></li><li><p> Copyright 2010 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458</p><p>All rights reserved.</p><p>Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit</p><p>Evaluation of Dependent Variables in MANOVA</p><p> When the MANOVA test is significant, which DV contributed to group differences? </p><p> One approachInspect individual ANOVA results for each DV, but make a Bonferroni adjustment</p><p> Use a stepdown analysis, which involves stepping DVs into a series of ANCOVAs, using previously entered DVs as covariates</p></li><li><p> Copyright 2010 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458</p><p>All rights reserved.</p><p>Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit</p><p>MANOVA/MANCOVA Assumptions</p><p> Multivariate normality of the DVs is assumed, but tests are fairly robust to violations if group ns are &gt; 20</p><p> Multivariate analogue of homogeneous variances: Homogeneity of the variance-covariance matrix Assumption is evaluated via Box M test (use p </p><p>&lt; .001 as criterion) Tests are robust to violation if group sizes are </p><p>roughly equal</p></li><li><p> Copyright 2010 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458</p><p>All rights reserved.</p><p>Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit</p><p>Mixed Design RM-ANOVA A mixed-design repeated-measures </p><p>ANOVA (RM-ANOVA) is appropriate when two or more groups are measured at multiple points in time</p><p> Mixed designs have a: Within-subjects factor (measurements of the </p><p>same people over time) Between-subjects factor (different people in </p><p>the groups being compared, such as experimental versus control )</p></li><li><p> Copyright 2010 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458</p><p>All rights reserved.</p><p>Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit</p><p>Mixed Design RM-ANOVA: Assumptions</p><p> In addition to the usual (normality, homogeneous between-group variances), a mixed design RM-ANOVA has a complex set of assumptions:</p><p> Compound symmetry assumption has two parts: (1) Homogeneity of within-group variances across all measurements; (2) homogeneity of correlations between different pairs of time measurements</p><p> Sphericity: Variance of the difference between estimated means for any pair of groups is the same as for any other pair</p></li><li><p> Copyright 2010 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458</p><p>All rights reserved.</p><p>Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit</p><p>Sphericity Assumption If compound symmetry assumption holds, </p><p>so does sphericity assumption RM-ANOVA not robust to violations of </p><p>sphericity Sphericity is most often evaluated using </p><p>Mauchlys test If Mauchlys test is significant (i.e., </p><p>sphericity assumption is violated), there are two alternatives</p></li><li><p> Copyright 2010 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458</p><p>All rights reserved.</p><p>Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit</p><p>Violations of Sphericity</p><p> Two main alternative routes:1. Proceed with the RM-ANOVA, BUT make </p><p>adjustments to compensate for the inflated risk of a Type I error</p><p>2. Use an alternative statistical test, such as MANOVA, using each measurement of the DV as a separate DV in a MANOVA analysis First alternative is preferred</p></li><li><p> Copyright 2010 b...</p></li></ul>