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Computations using analysis of covariance

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Abstract Although analysis of covariance (ANCOVA) was introduced long ago, it is not well understood by many researchers and is frequently misused. ANCOVA is an extension of analysis of variance (ANOVA) with the inclusion of one or more covariates. Its benefits compared to ANOVA include (1) increased power and (2) a reduction in biases caused by differences in experimental units [the covariate(s)] among groups. ANCOVA can be presented using an adjusted means procedure (testing for differences in means adjusted for the covariate), but the procedure can be easily understood using a multiple linear regression method using indicator variables to represent groups. Statistical software packages use general linear models to perform ANCOVA which are identical to the multiple linear regression models. Failure to meet assumptions of ANCOVA can lead to misinterpretation of results. Failing to meet the assumption of parallel group regression slopes is common in many data sets and methods are available to analyze these data sets (e.g., the Johnson–Neyman technique). Although ANCOVA is robust to violations of some assumptions (e.g., normality and equality of variances) when sample sizes are equal, many nonparametric tests based on ranks are available as nonparametric alternatives to ANCOVA. WIREs Comp Stat 2011 3 260–268 DOI: 10.1002/wics.165 This article is categorized under: Statistical and Graphical Methods of Data Analysis > Nonparametric Methods

(a) Scatterplot of Y versus X fit to least squares regression lines with a common group regression slope; group 1 (blue circles) and group 2 (red squares). (b) Scatterplot of Y versus X for group 1 (blue circles) and group 2 (red squares) fit to least squares regression lines with a common slope among groups. Observations are adjusted to the grand mean of the covariate (45.63) by ‘sliding’ parallel along their group regression lines as an illustration of the adjustment procedure. The adjusted values (pink circles, group 1; green squares, group 2) align at the grand mean of the covariate. (c) Individual value plot of the adjusted Y values, adjusted to the grand mean of the covariate for group 1 (pink circles) and group 2 (green squares).

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(a) Scatterplot of a response (y) versus covariate (x1) with regressions with unique slopes and intercepts (full regression model); y = b0 + b1x1 for group 1 (blue circles), and y = (b0 + b2) + (b1 + b3)x1 for group 2 (red squares). (b) Scatterplot of a response (y) versus covariate (x1) with regressions with equal slopes, but unique intercepts (reduced regression model); y = b0 + b1x1 for group 1 (blue circles), and y = (b0 + b2) + b1x1 for group 2 (red squares). (c) Scatterplot of a response (y) versus covariate (x1) with a common regression line (i.e., equal slopes and intercepts); y = b0 + b1x1 for group 1 (blue circles) and group 2 (red squares).

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(a) Illustration of parallel regression lines of the regression of a response Y on a covariate X for two groups. When the regression lines are parallel, differences in the response among the two groups are equal at each value of the covariate. (b) Illustration of nonparallel regression lines of the regression of a response Y on a covariate X for two groups. When regression lines are not parallel, differences in the response among the two groups vary depending on which value of the covariate is examined.

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Statistical and Graphical Methods of Data Analysis > Nonparametric Methods

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