MANCOVA
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Multivariate analysis of covariance (MANCOVA) is an extension of
analysis of covariance Analysis of covariance (ANCOVA) is a general linear model that blends ANOVA and regression analysis, regression. ANCOVA evaluates whether the means of a dependent variable (DV) are equal across levels of one or more Categorical variable, categori ...
( ANCOVA) methods to cover cases where there is more than one dependent variable and where the control of concomitant continuous independent variables – covariates – is required. The most prominent benefit of the MANCOVA design over the simple
MANOVA In statistics, multivariate analysis of variance (MANOVA) is a procedure for comparing multivariate sample means. As a multivariate procedure, it is used when there are two or more dependent variables, and is often followed by significance tests ...
is the 'factoring out' of
noise Noise is sound, chiefly unwanted, unintentional, or harmful sound considered unpleasant, loud, or disruptive to mental or hearing faculties. From a physics standpoint, there is no distinction between noise and desired sound, as both are vibrat ...
or error that has been introduced by the covariant. A commonly used multivariate version of the
ANOVA Analysis of variance (ANOVA) is a family of statistical methods used to compare the means of two or more groups by analyzing variance. Specifically, ANOVA compares the amount of variation ''between'' the group means to the amount of variation ''w ...
F-statistic is Wilks' Lambda (Λ), which represents the ratio between the error variance (or covariance) and the effect variance (or covariance).
Statsoft Textbook, ANOVA/MANOVA.


Goals

Similarly to all tests in the
ANOVA Analysis of variance (ANOVA) is a family of statistical methods used to compare the means of two or more groups by analyzing variance. Specifically, ANOVA compares the amount of variation ''between'' the group means to the amount of variation ''w ...
family, the primary aim of the MANCOVA is to test for significant differences between group means. The process of characterising a covariate in a data source allows the reduction of the magnitude of the error term, represented in the MANCOVA design as ''MSerror''. Subsequently, the overall Wilks' Lambda will become larger and more likely to be characterised as significant. This grants the researcher more
statistical power In frequentist statistics, power is the probability of detecting a given effect (if that effect actually exists) using a given test in a given context. In typical use, it is a function of the specific test that is used (including the choice of tes ...
to detect differences within the data. The multivariate aspect of the MANCOVA allows the characterisation of differences in group means in regards to a linear combination of multiple dependent variables, while simultaneously controlling for covariates. ''Example situation where MANCOVA is appropriate:'' Suppose a scientist is interested in testing two new drugs for their effects on depression and anxiety scores. Also suppose that the scientist has information pertaining to the overall responsivity to drugs for each patient; accounting for this covariate will grant the test higher sensitivity in determining the effects of each drug on both dependent variables.


Assumptions

Certain assumptions must be met for the MANCOVA to be used appropriately: # Normality: For each group, each dependent variable must represent a
normal distribution In probability theory and statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is f(x) = \frac ...
of scores. Furthermore, any linear combination of dependent variables must be normally distributed. Transformation or removal of outliers can help ensure this assumption is met.
French, A. et al., 2010. Multivariate analysis of variance (MANOVA).
Violation of this assumption may lead to an increase in
Type I error Type I error, or a false positive, is the erroneous rejection of a true null hypothesis in statistical hypothesis testing. A type II error, or a false negative, is the erroneous failure in bringing about appropriate rejection of a false null hy ...
rates.
Davis, K., 2003. Multiple analysis of variance (MANOVA) or multiple analysis of covariance (MANCOVA). Louisiana State University.
# Independence of observations: Each observation must be independent of all other observations; this assumption can be met by employing random sample, random sampling techniques. Violation of this assumption may lead to an increase in Type I error rates. # Homogeneity of variances: Each dependent variable must demonstrate similar levels of variance across each independent variable. Violation of this assumption can be conceptualised as a correlation existing between the variances and the means of dependent variables. This violation is often called '
heteroscedasticity In statistics, a sequence of random variables is homoscedastic () if all its random variables have the same finite variance; this is also known as homogeneity of variance. The complementary notion is called heteroscedasticity, also known as hete ...
'
Bors, D. A. University of Toronto at Scarborough.
and can be tested for using
Levene's test In statistics, Levene's test is an inferential statistic used to assess the equality of variances for a variable calculated for two or more groups. This test is used because some common statistical procedures assume that variances of the population ...
.
McLaughlin, M., 2009. University of Southern Carolina.
# Homogeneity of covariances: The intercorrelation matrix between dependent variables must be equal across all levels of the independent variable. Violation of this assumption may lead to an increase in Type I error rates as well as decreased
statistical power In frequentist statistics, power is the probability of detecting a given effect (if that effect actually exists) using a given test in a given context. In typical use, it is a function of the specific test that is used (including the choice of tes ...
.


Logic of MANOVA

Inspired by
ANOVA Analysis of variance (ANOVA) is a family of statistical methods used to compare the means of two or more groups by analyzing variance. Specifically, ANOVA compares the amount of variation ''between'' the group means to the amount of variation ''w ...
, MANOVA is based on a generalization of sum of squares explained by the model S_\text and the inverse of the sum of squares unexplained by the model S_\text^. The most common statistics are summaries based on the roots (or
eigenvalues In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by a ...
) \lambda_p of the matrix A:= S_S_^. *
Samuel Stanley Wilks Samuel Stanley Wilks (June 17, 1906 – March 7, 1964) was an American mathematician and academic who played an important role in the development of mathematical statistics, especially in regard to practical applications. Early life and educat ...
' \Lambda_\text = \prod_(1/(1 + \lambda_)) = \det(I + A)^ = \det(S_\text)/\det(S_\text + S_\text) distributed as lambda (Λ) * the K. C. Sreedharan PillaiM. S. Bartlett
trace Trace may refer to: Arts and entertainment Music * ''Trace'' (Son Volt album), 1995 * ''Trace'' (Died Pretty album), 1993 * Trace (band), a Dutch progressive rock band * ''The Trace'' (album), by Nell Other uses in arts and entertainment * ...
, \Lambda_\text = \sum_(\lambda_p/(1 + \lambda_p)) = \operatorname(A(I + A)^) * the Lawley– Hotelling trace, \Lambda_\text = \sum_(\lambda_) = \operatorname(A) * Roy's greatest root (also called ''Roy's largest root''), \Lambda_\text = \max_p(\lambda_p)


Covariates

In statistics, a covariate represents a source of variation that has not been controlled in the experiment and is believed to affect the dependent variable. The aim of such techniques as ANCOVA is to remove the effects of such uncontrolled variation, in order to increase statistical power and to ensure an accurate measurement of the true relationship between independent and dependent variables. An example is provided by the analysis of trend in sea-level by Woodworth (1987). Here the
dependent variable A variable is considered dependent if it depends on (or is hypothesized to depend on) an independent variable. Dependent variables are studied under the supposition or demand that they depend, by some law or rule (e.g., by a mathematical functio ...
(and variable of most interest) was the annual mean sea level at a given location for which a series of yearly values were available. The primary independent variable was "time". Use was made of a "covariate" consisting of yearly values of annual mean atmospheric pressure at sea level. The results showed that inclusion of the covariate allowed improved estimates of the trend against time to be obtained, compared to analyses which omitted the covariate.


See also

*
Discriminant function analysis Linear discriminant analysis (LDA), normal discriminant analysis (NDA), canonical variates analysis (CVA), or discriminant function analysis is a generalization of Fisher's linear discriminant, a method used in statistics and other fields, to fi ...
* ANCOVA *
MANOVA In statistics, multivariate analysis of variance (MANOVA) is a procedure for comparing multivariate sample means. As a multivariate procedure, it is used when there are two or more dependent variables, and is often followed by significance tests ...


References

{{DEFAULTSORT:Mancova Analysis of variance