ANCOVA Class

ANCOVA Class

Analyzes a one-way classification model with covariates.

Inheritance Hierarchy

System.Object
Imsl.Stat.ANCOVA

Namespace: Imsl.Stat
Assembly: ImslCS (in ImslCS.dll) Version: 6.5.2.0

Syntax

C++

Copy

[SerializableAttribute]
public class ANCOVA

<SerializableAttribute>
Public Class ANCOVA

[SerializableAttribute]
public ref class ANCOVA

[<SerializableAttribute>]
type ANCOVA =  class end

The ANCOVA type exposes the following members.

Constructors

	Name	Description
	ANCOVA	Constructs a one-way classification model with covariates.

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Methods

	Name	Description
	Compute	Performs one-way analysis of covariance assuming parallelism and returns an array containing the parallelism tests for the one-way analysis of covariance.
	Equals	Determines whether the specified object is equal to the current object. (Inherited from Object.)
	Finalize	Allows an object to try to free resources and perform other cleanup operations before it is reclaimed by garbage collection. (Inherited from Object.)
	GetAdjustedANOVA	Returns the partial sum of squares for the one-way analysis of covariance.
	GetANCOVA	Returns an array containing the one-way analysis of covariance assuming parallelism.
	GetANOVATables	Returns a matrix of size ngroup by 15 containing the analysis of variance tables for each linear regression model fitted separately to each treatment group.
	GetCoefficientTable	Returns a matrix of size ncov + 1 by 4 containing statistics for a linear regression model fitted separately for each of the ngroup treatment groups.
	GetCoefficientTables	Returns an array containing statistics for a linear regression model fitted separately for all ngroup treatments.
	GetHashCode	Serves as a hash function for a particular type. (Inherited from Object.)
	GetMeans	Returns a matrix containing the unadjusted means for the covariates and the response variate and the means for the response variate adjusted for the covariates.
	GetModelCoefficients	Returns a matrix containing statistics for the regression coefficients for the model assuming parallelism.
	GetR	Returns the R matrix from the QR decomposition.
	GetType	Gets the Type of the current instance. (Inherited from Object.)
	GetVarCovAdjustedMeans	Returns a matrix containing the estimated variances and covariances for the adjusted means assuming parallelism.
	GetVarCovCoefficients	Returns a matrix containing the estimated variances and covariances for the coefficients returned using GetModelCoefficients.
	MemberwiseClone	Creates a shallow copy of the current Object. (Inherited from Object.)
	ToString	Returns a string that represents the current object. (Inherited from Object.)

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Properties

	Name	Description
	NumberOfMissing	The number of cases with missing values in covariates or responses. Cases with any missing values are not used in the analysis.

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Remarks

Class ANCOVA performs analyses for models that combine the features of a one-way analysis of variance model with that of a multiple linear regression model. The basic one-way analysis of covariance model is

$y_{ij}=\beta_{0i}+\beta_{1}x_{ij1}+\beta_{2} x_{ij2}+\ldots+\beta_mx_{ijm}+\varepsilon_{ij}$

$i=1,2,\ldots,ngroup$

$j=1,2,\ldots,n_i$

where, ngroup is the number of treatment groups, the observed value of $y_{ij}$ constitutes the j-th response in the i-th group, $\beta_{0i}$ denotes the y intercept for the regression function for the i-th group, $\beta_{1}$ , $\beta_{2}$ , ..., $\beta_{m}$ are the regression coefficients for the covariates, and the $\varepsilon_{ij}$ 's are independently distributed normal errors with mean zero and variance $\sigma^2$ . This model allows the regression function for each group to have different intercepts. However, the remaining m regression coefficients are the same for each group, i.e., the regression functions are parallel.

In practice, sometimes the regression functions are not parallel. In addition to estimates for the model assuming parallelism (parallel regression planes), ANCOVA computes estimates and summary statistics for the separate regressions of each group. These estimates can be examined using the methods GetCoefficientTables and GetANOVATables.

Estimates for the $\beta_{0i}$ 's and $\beta_{1}$ , $\beta_{2}$ , ..., $\beta_{m}$ in the model assuming parallelism are returned using the method GetModelCoefficients. Summary statistics are also computed for this model and returned by the Compute method.

The adjusted group means, stored in the last column of xymean, are computed using the formula:

$\hat{\beta}_{0i}+\hat{\beta}_{1} \overline{x}_{1}+\hat{\beta}_{2}\overline{x}_{2}+\ldots+ \hat{\beta}_{m}\overline{x}_{m}$

where xymean is the matrix returned by GetMeans and ncov is the number of covariates.

The estimated covariance between the $i_{1}$ -th and $i_{2}$ -th adjusted group mean is given by

$\nu_{i_{1}i_{2}}+\sum_{r=1}^{m}\sum_{s=1}^{m} \overline{x}_{r}\nu_{k+r, k+s}\overline{x}_{s}+\sum_{r=1}^{m} \overline{x}_{r}\nu_{i_{1}, k+r}+\sum_{r=1}^{m}\overline{x}_{r} \nu_{i_{2}, k+r}$

where $\nu_{pq}$ is the entry in covb[p-1][q-1], where covb is returned by GetVarCovCoefficients and is the estimated covariance between the p-th and q-th estimated coefficients in the regression function.

Reference

Imsl.Stat Namespace

Other Resources

Example 1

Example 2