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MLR_PRFTest
Calculates the pvalue and related statistics of the partial ftest (used for testing the inclusion/exclusion variables).
Syntax
X
is the independent (explanatory) variables data matrix, such that each column represents one variable.
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.
Mask1
is the boolean array for the explanatory variables in the first model. If missing, all variables in X are included.
Mask2
is the boolean array for the explanatory variables in the second model. If missing, all variables in X are included.
Return_type
is a switch to select the return output (1 = PValue (default), 2 = Test Stats, 3 = Critical Value.)
Method  Description 

1  PValue 
2  Test Statistics (e.g. Zscore) 
3  Critical Value 
Alpha
is the statistical significance of the test (i.e. alpha). If missing or omitted, an alpha value of 5% is assumed.
Remarks
 The underlying model is described here.
 Model 1 must be a submodel of Model 2. In other words, all variables included in Model 1 must be included in Model 2.
 The coefficient of determination (i.e. ) increases in value as we add variables to the regression model, but we often wish to test whether the improvement in R square by adding those variables is statistically significant.

To do so, we developed an inclusion/exclusion test for those variables. First, let's start with a regression model with variables:
Now, let's add a few more variables :
 The test of the hypothesis is as follows:
 Using the change in the coefficient of determination (i.e. ) as we add new variables, we can calculate the test statistics:
Where:
 is the of the full model (with added variables).
 is the of the reduced model (without the added variables).
 is the number of variables in the reduced model.
 is the number of variables in the full model.
 is the number of observations in the sample data.
 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_ANOVA function is available starting with version 1.60 APACHE.
Examples
References
 Hamilton, J .D.; Time Series Analysis , Princeton University Press (1994), ISBN 0691042896
 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. 252285