MLR_GOF

Calculates different measures for the regression goodness of fit (e.g. R^2).

Syntax

MLRⁱ_GOF(X, Mask, Y, Intercept, Return_type)

X
is the independent (explanatory) variables data matrix, such that each column represents one variable.

Mask
is the boolean array to choose the explanatory variables in the model. If missing, all variables in X are included.

Y
is the response or the dependent variable data array (one dimensional array of cells (e.g. rows or columns)).

Intercept
is the constant or the intercept value to fix (e.g. zero). If missing, an intercept will not be fixed and is computed normally.

Return_type
is a switch to select a fitness measure (1 = R-Square (default), 2 = Adjusted R Square, 3 = RMSEⁱ, 4 = LLFⁱ, 5 = AICⁱ, 6 = BICⁱ/SICⁱ).

Method	Description
1	R-Square (Coefficient of determination)
2	Adjusted R Square
3	Regression Error (RMSE)
4	log-likelyhood (LLF)
5	Akaike-information criterion (AIC)
6	Schwartz/Bayesian Information criterion (SBIC)

Remarks

The underlying model is described here.
The coefficient of determination, denoted $R^2$ provides a measure of how well observed outcomes are replicated by the model.

$R^2 = \frac{\mathrm{SSR}} {\mathrm{SST}} = 1 - \frac{\mathrm{SSE}} {\mathrm{SST}}$
The adjusted R-square (denoted ) is an attempt to take account of the phenomenon of the automatically and spuriously increasing when extra explanatory variables are added to the model. The adjusts for the number of explanatory terms in a model relative to the number of data points.

$\bar R^2 = {1-(1-R^{2}){N-1 \over N-p-1}} = {R^{2}-(1-R^{2}){p \over N-p-1}} = 1 - \frac{\mathrm{SSE}/(N-p-1)}{\mathrm{SST}/(N-1)}$

Where:
- $p$ is the number of explanatory variables in the model.
- $N$ is the number of observations in the sample.
The regression error is defined as the square root for the mean square error (RMSE):

$\mathrm{RMSE} = \sqrt{\frac{SSE}{N-p-1}}$
The log likelihood of the regression is given as:

$\mathrm{LLF}=-\frac{N}{2}\left(1+\ln(2\pi)+\ln\left(\frac{\mathrm{SSR}}{N} \right ) \right )$

The Akaike and Schwarz/Bayesian information criterion are given as:

$\mathrm{AIC}=-\frac{2\mathrm{LLF}}{N}+\frac{2(p+1)}{N}$

$\mathrm{BIC} = \mathrm{SIC}=-\frac{2\mathrm{LLF}}{N}+\frac{(p+1)\times\ln(p+1)}{N}$
The sample data may include missing values.
Each column in the input matrix corresponds to a separate variable.
Each row in the input matrix corresponds to an observation.
Observations (i.e. row) with missing values in X or Y are removed.
The number of rows of the response variable (Y) must be equal to the number of rows of the explanatory variables (X).
The MLR_GOF function is available starting with version 1.60 APACHE.

Examples

References

Hamilton, J .D.; Time Series Analysis , Princeton University Press (1994), ISBN 0-691-04289-6
Kenney, J. F. and Keeping, E. S. (1962) "Linear Regression and Correlation." Ch. 15 in Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 252-285

MLR_GOF

Syntax

Remarks

Examples

References

External Links

Follow Us