NationMaster - Encyclopedia: ANCOVA

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 statistical technique of controlling extraneous variables in correlational studies. In probability theory and statistics, the covariance between two real-valued random variables X and Y, with expected values and is defined as: where E is the expected value. ... Extraneous variables are variables other than the independent variable that may bear any effect on the behavior of the person being studied. ...

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. In statistics, analysis of variance (ANOVA) is a collection of statistical models and their associated procedures which compare means by splitting the overall observed variance into different parts. ... The power of a statistical test is the probability that the test will reject a false null hypothesis, or in other words that it will not make a Type II error. ...

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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): $S S T y$ , $S S T r y$ , and $S S E y$ 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 $S S T x$ and $S S E x$ . Least squares or ordinary least squares (OLS) is a mathematical optimization technique which, when given a series of measured data, attempts to find a function which closely approximates the data (a best fit). It attempts to minimize the sum of the squares of the ordinate differences (called residuals) between points...

The total sum of squares determines the variability of all the samples. $n T$ represents the total number of samples:

$SST_y=sum_{i=1}^nsum_{j=1}^kY_{ij}^2-frac{left(sum_{i=1}^nsum_{j=1}^kY_{ij}right)^2}{n_T}$

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

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:

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

$SST_y=SSTr_y+SSE_y.,$

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:

Adjusting SST_y

The correlation between X and Y is $r_T^2$ . Linear correlations between 1000 pairs of numbers. ...

$r_T^2=frac{SCT^2}{SST_xSST_y}$

$r_n^2=frac{SCE^2}{SSE_xSSE_y}$

The proportion of covariance is subtracted from the dependent, $S S y$ values:

$SST_{yadj}=SST_y-r_T^2,$

S S T r y a d j = S S T y a d j - S S E y a d j

Adjusting the means of each population k

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

$M_{y_iadj}=M_{y_i}-frac{SCE_y}{SCE_x}(M_{x_i}-M_{x_T})$

Analysis using adjusted sum of squares values

Mean squares for treatments where $d f T r$ is equal to $N T - k - 1$ . $d f T r$ is one less than in ANOVA to account for the covariance and $d f E = k - 1$ :

$MSTr=frac{SSTr}{df_{Tr}}$

$MSE=frac{SSE}{df_E}$

The F statistic is In statistics and probability, the F-distribution is a continuous probability distribution. ...

$F_{df_E,df_mathrm{Tr}}=frac{mathrm{MSTr}}{mathrm{MSE}}.$

External links

One-Way Analysis of Covariance for Independent Samples
THE ANALYSIS OF COVARIANCE

Categories: Statistics stubs | Analysis of variance | Covariance and correlation

Results from FactBites:

Multivariate Statistics - ANCOVA Applications (779 words)
	Because ANCOVA estimates an additional population parameter (the pooled population correlation), another degree of freedom is lost from the error term.
	A very controversial use of ANCOVA is to correct for initial group differences (prior to assigned to x) that exists on y among several intact, state variable groups.
	Remember that ANCOVA is simply a statistical method of accounting for a third variable; in experimental studies, the randomized blocks design is used to accomplish the same thing.

NationMaster - Encyclopedia: ANCOVA (764 words)
	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.
	ANCOVA is a statistical technique of controlling extraneous variables in correlational studies.
	ANCOVA is a merger of ANOVA and regression for continuous variables.

More results at FactBites »

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