Bias and Precision of the Squared Canonical Correlation Coefficient under Nonnormal Data Conditions

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This dissertation: (a) investigated the degree to which the squared canonical correlation coefficient is biased in multivariate nonnormal distributions and (b) identified formulae that adjust the squared canonical correlation coefficient (Rc2) such that it most closely approximates the true population effect under normal and nonnormal data conditions. Five conditions were manipulated in a fully-crossed design to determine the degree of bias associated with Rc2: distribution shape, variable sets, sample size to variable ratios, and within- and between-set correlations. Very few of the condition combinations produced acceptable amounts of bias in Rc2, but those that did were all found with first ... continued below

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Leach, Lesley Ann Freeny August 2006.

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  • Leach, Lesley Ann Freeny

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Description

This dissertation: (a) investigated the degree to which the squared canonical correlation coefficient is biased in multivariate nonnormal distributions and (b) identified formulae that adjust the squared canonical correlation coefficient (Rc2) such that it most closely approximates the true population effect under normal and nonnormal data conditions. Five conditions were manipulated in a fully-crossed design to determine the degree of bias associated with Rc2:
distribution shape, variable sets, sample size to variable ratios, and within- and between-set correlations.
Very few of the condition combinations produced acceptable amounts of bias in Rc2, but those that did were all found with first function results. The sample size to variable ratio (n:v)was determined to have the greatest impact on the bias associated with the Rc2 for the first, second, and third functions. The variable set condition also affected the accuracy of Rc2, but for the second and third functions only. The kurtosis levels of the marginal distributions (b2), and the
between- and within-set correlations demonstrated little or no impact on the bias associated with Rc2. Therefore, it is recommended that researchers use n:v ratios of at least 10:1 in canonical analyses, although greater n:v ratios have the potential to produce even less bias. Furthermore,because it was determined that b2 did not impact the accuracy of Rc2, one can be somewhat confident that, with marginal distributions possessing homogenous kurtosis levels ranging
anywhere from -1 to 8, Rc2 will likely be as accurate as that resulting from a normal distribution.
Because the majority of Rc2 estimates were extremely biased, it is recommended that all Rc2 effects, regardless of which function from which they result, be adjusted using an appropriate adjustment formula. If no rationale exists for the use of another formula, the Rozeboom-2 would likely be a safe choice given that it produced the greatest number of unbiased Rc2 estimates for the greatest number of condition combinations in this study.

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  • August 2006

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  • May 5, 2008, 2:46 p.m.

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  • Dec. 12, 2013, 12:49 p.m.

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Leach, Lesley Ann Freeny. Bias and Precision of the Squared Canonical Correlation Coefficient under Nonnormal Data Conditions, dissertation, August 2006; Denton, Texas. (digital.library.unt.edu/ark:/67531/metadc5361/: accessed October 18, 2017), University of North Texas Libraries, Digital Library, digital.library.unt.edu; .