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  • Copyright 2010 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458

    All rights reserved.

    Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit

    Statistics and Data Analysisfor Nursing Research

    Second Edition

    CHAPTER

    Analysis of Covariance, Multivariate ANOVA, and Related Multivariate Analyses

    11

  • Copyright 2010 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458

    All rights reserved.

    Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit

    General Linear Model General linear model (GLM):

    encompasses a broad class of techniques (e.g., ANOVA, multiple regression)

    Flexible, complex model, used to fit data to straight-line solutions

    Involves partitioning variance, different methods possible Type III sums of squares is standard

  • Copyright 2010 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458

    All rights reserved.

    Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit

    Analysis of Covariance A useful analytic tool within the GLM:

    Analysis of covariance Often abbreviated as ANCOVA Used to compare group means after

    removing the effect of a variable called a covariate

    A covariate is typically: A preintervention measure of an outcome A measure of a confounding variable

  • Copyright 2010 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458

    All rights reserved.

    Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit

    ANCOVA Principles Conceptualize as a two-step process First step, similar to hierarchical multiple

    regressionEffects of covariates are removed from variance of the dependent variable

    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

  • Copyright 2010 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458

    All rights reserved.

    Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit

    Venn Diagram of ANCOVA Two-step process:

  • Copyright 2010 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458

    All rights reserved.

    Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit

    Adjusted Means In ANCOVA, the means on the

    dependent variable (DV) can be adjusted by removing the covariates effect

    Adjusted means indicate net effectsGroup differences on the DV net of the effect of the covariate

  • Copyright 2010 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458

    All rights reserved.

    Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit

    ANCOVA Hypotheses Similar to ANOVA, tested via the F

    statistic The null hypothesis: The adjusted group

    means are equal The alternative hypothesis: At least some

    of the adjusted group means are not equal

  • Copyright 2010 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458

    All rights reserved.

    Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit

    Multiple Comparisons Like ANOVA, if there is a significant

    overall group difference, pairwise multiple comparisons must be done when there are three or more groups

    Purpose of these post hoc tests is to identify which pairs of adjusted group means are significantly different from one another

  • Copyright 2010 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458

    All rights reserved.

    Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit

    Effect Size in ANCOVA

    In ANCOVA, the effect size is similar to that in ANOVA (eta-squared)

    In ANCOVA, eta-squared is adjusted for the covariate

    Partial eta-squared is variability accounted for in the DV by the IV, after adjusting for the covariate

  • Copyright 2010 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458

    All rights reserved.

    Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit

    Measurement in ANCOVA Dependent variable:

    Interval or ratio level (e.g., heart rate)

    Independent variable: Nominal level (e.g., experimental versus control)

    Covariate: Interval or ratio level (or dichotomous with near 50-50

    split) Dummy-variable covariates can be included as

    additional independent variables

  • Copyright 2010 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458

    All rights reserved.

    Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit

    Selection of Covariates Multiple covariates can be included More than three to four covariates is

    usually unnecessary and potentially unwise Mutlicollinearity is likely to occur with too many

    covariates Good covariates: A pretest measure of

    outcome variable; key demographic or clinical variables

    It is important that covariates be reliably measured

  • Copyright 2010 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458

    All rights reserved.

    Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit

    Assumptions for ANCOVA Basic assumptionslike ANOVA:

    Random sampling from the populations Dependent variable normally distributed Variances in the populations are equal

    Can be tested with Levenes statistic

    Robustness: With reasonably large sample and groups of approximately equal size, statistical tests are robust to violations of latter two

  • Copyright 2010 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458

    All rights reserved.

    Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit

    Assumptions for ANCOVA (contd)

    Additional assumption in ANCOVA: Homogeneity of regression across groups The slope of the dependent variable on the

    covariate should be the same for every group

    When this assumption is violated, there is a heightened risk of a Type II error

  • Copyright 2010 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458

    All rights reserved.

    Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit

    Homogeneity of Regression Illustration

  • Copyright 2010 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458

    All rights reserved.

    Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit

    Homogeneity of Regression: Evaluation

    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

  • Copyright 2010 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458

    All rights reserved.

    Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit

    Testing Homogeneity of Regression Assumption

    Here, the interaction is not significant, and the assumption holds

  • Copyright 2010 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458

    All rights reserved.

    Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit

    ANCOVA Applications Appropriate primarily in randomized

    controlled trials (RCTs) Often used in nonequivalent control group

    (quasiexperimental) and case-control (nonexperimental) designs Intent: Enhance internal validity by reducing

    selection threat (enhancing group comparability)

    Use in non-RCTs is controversial but common; great care is needed in interpreting ANCOVA results in non-RCTs

  • Copyright 2010 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458

    All rights reserved.

    Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit

    Alternatives to ANCOVA When the covariate is the same measure

    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

    constrained by floor effects or ceiling effects (limits on amount of change), and tend to be unreliable

    RM-ANOVA is another option (i.e., comparing two or more groups at two points in time)

  • Copyright 2010 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458

    All rights reserved.

    Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit

    SPSS and ANCOVA Analyze

    General Linear Model Univariate

    Select: Dependent

    variable Independents

    (Fixed Factor) Covariates

  • Copyright 2010 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458

    All rights reserved.

    Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit

    SPSS and ANCOVA (contd) GLM Options include

    options for descriptive statistics such as adjusted marginal means

    Many other options, such as: Effect size (partial

    eta-squared) Levenes test Post hoc power

  • Copyright 2010 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458

    All rights reserved.

    Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit

    Multivariate ANOVA Multivariate analysis of variance

    (MANOVA) is another extension of ANOVA

    Used to test mean group differences on two or more dependent variables

  • Copyright 2010 by Pearson Education, Inc.Upper Saddle River, New Jersey 0