ANCOVA, or analysis of covariance 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 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.

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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.

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:

$SST\_y=\backslash sum\_\{i=1\}^n\backslash sum\_\{j=1\}^kY\_\{ij\}^2-\backslash frac\{\backslash left(\backslash sum\_\{i=1\}^n\backslash sum\_\{j=1\}^kY\_\{ij\}\backslash right)^2\}\{n\_T\}$

The sum of squares for treatments determines the variability between populations or factors. $n\_k$ represents the number of factors:

$SSTr\_y=\backslash sum\_\{i=1\}^n\backslash left(\backslash sum\_\{j=1\}^kY\_\{ij\}-\backslash frac\{\backslash sum\_\{j=1\}^k(Y\_\{ij\})^2\}\{n\_k\}\backslash right)$

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:

$SSE\_y=\backslash sum\_\{j=1\}^k\backslash left(\backslash sum\_\{i=1\}^nY\_\{ij\}^2-\backslash frac\{\backslash sum\_\{i=1\}^k(Y\_\{ij\})^2\}\{n\_n\}\backslash right).$

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

$SST\_y=SSTr\_y+SSE\_y.\backslash ,$

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:

$SCT=\backslash sum\_\{i=1\}^n\backslash sum\_\{j=1\}^kX\_\{ij\}^2Y\_\{ij\}^2-\backslash frac\{\backslash left(\backslash sum\_\{i=1\}^n\backslash sum\_\{j=1\}^kX\_\{ij\}\backslash right)^2Y\_\{ij\}^2\}\{n\_T\}$

$SCE=\backslash sum\_\{j=1\}^k\backslash left(\backslash sum\_\{i=1\}^nX\_\{ij\}^2Y\_\{ij\}^2-\backslash frac\{\backslash sum\_\{i=1\}^k(X\_\{ij\}Y\_\{ij\})^2\}\{n\_n\}\backslash right)$

Adjusting SST_y

The correlation between X and Y is $r\_T^2$.

$r\_T^2=\backslash frac\{SCT^2\}\{SST\_xSST\_y\}$

$r\_n^2=\backslash frac\{SCE^2\}\{SSE\_xSSE\_y\}$

The proportion of covariance is subtracted from the dependent, $SS\_y$ values:

$SST\_\{yadj\}=SST\_y-r\_T^2\backslash ,$

$SSE\_\{yadj\}=SSE\_y-r\_n^2$

$SSTr\_\{yadj\}=SST\_y*-SSE\_y*$

Adjusting the means of each population k

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

$M\_\{y\_iadj\}=M\_\{y\_i\}-\backslash frac\{SCE\_y\}\{SCE\_x\}(M\_\{x\_i\}-M\_\{x\_T\})$

Analysis using adjusted sum of squares values

Mean squares for treatments where $df\_\{Tr\}$ is equal to $N\_T-k-1$. $df\_\{Tr\}$ is one less than in ANOVA to account for the covariance and $df\_E=k-1$:

$MSTr=\backslash frac\{SSTr\}\{df\_\{Tr\}\}$

$MSE=\backslash frac\{SSE\}\{df\_E\}$

The F statistic is

$F\_\{df\_E,df\_\backslash mathrm\{Tr\}\}=\backslash frac\{\backslash mathrm\{MSTr\}\}\{\backslash mathrm\{MSE\}\}.$

External links

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• One-Way Analysis of Covariance for Independent Samples

Statistics