yawiki.org entry for ANCOVA

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ANCOVA, or analysis of ''cova''riance is an old-fashioned name for a linear regression model with one continuous explanatory variable and one or more factors. The name exists for historical reasons, but there is no particular reason to distinguish the method from the general purpose linear model. ANCOVA is a statistical technique of controlling extraneous variables in correlational studies.
ANCOVA is a merger of ANOVA and regression for continuous variables. ANCOVA tests whether certain factors have an effect after controlling for quantitative predictors. The inclusion of covariates increases statistical power because it accounts for some of the variability.

    ANCOVA
                One-factor ANCOVA analysis
                Calculating the sum of squared deviates for the independent variable X and the dependent variable Y
                Calculating the covariance of X and Y
                Adjusting SST
                Adjusting the means of each population k
                Analysis using adjusted sum of squares values

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One-factor ANCOVA analysis
One factor analysis is appropriate when dealing with more than 3 populations; k populations. The single factor has k levels equal to the k populations. n samples from each population are chosen random from their respective population.

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Calculating the sum of squared deviates for the independent variable X and the dependent variable Y
The sum of squared deviates (SS):

SST_y

SSTr_y

, and

SSE_y

must be calculated using the following equations for the dependent variable, Y. The SS for the covariate must also be calculated, the two necessary values are

SST_x

and

SSE_x

.

The total sum of squares determines the variability of all the samples.

n_T

represents the total number of samples:

$SST_y=sum_^nsum_^kY_^2-rac$

The sum of squares for treatments determines the variability between populations or factors.

n_k

represents the number of factors:

$SSTr_y=sum_^nleft(sum_^kY_^2-rac$

ight)

The sum of squares for error determines the variability within each population or factor.

n_n

represents the number of samples with a given population:

$SSE_y=sum_^kleft(sum_^nY_^2-rac$

ight).

The total sum of squares is equal to the sum of the sum of squares for treatments and the sum of squares for error:

$SST_y=SSTr_y+SSE_y.,$

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Calculating the covariance of X and Y
The total sum of square covariates determines the covariance of X and Y within the all the data samples:

$SCT=sum_^nsum_^kX_^2Y_^2-rac$

SCE=sum_^kleft(sum_^nX_^2Y_^2-rac
ight)

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Adjusting SST
The correlation between X and Y is

r_T^2

$r_T^2=rac$

$r_n^2=rac$

The proportion of covariance is subtracted from the dependent,

SS_y

values:

$SST_=SST_y-r_T^2,$

$SSE_=SSE_y-r_n^2$

$SSTr_=SST_y adj -SSE_y adj$

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Adjusting the means of each population k
The mean of each population is adjusted in the following manner:

$M_=M_-rac(M_-M_)$

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Analysis using adjusted sum of squares values
Mean squares for treatments where

df_

is equal to

N_T-k-1

df_

is one less than in ANOVA to account for the covariance and

df_E=k-1

$MSTr=rac$

$MSE=rac$

The F statistic is

$F_=rac.$

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