From a6e7a2226130013e1a5c1e571dfe59b89a9f8180 Mon Sep 17 00:00:00 2001 From: MarcoRianiUNIPR Date: Tue, 27 Aug 2024 08:31:44 +0200 Subject: [PATCH] Funcion boxtest regenerated and typo corrected --- toolbox/helpfiles/FSDA/boxtest.html | 217 ++++++++++++++-------------- toolbox/multivariate/boxtest.m | 6 +- 2 files changed, 112 insertions(+), 111 deletions(-) diff --git a/toolbox/helpfiles/FSDA/boxtest.html b/toolbox/helpfiles/FSDA/boxtest.html index ff1952a38..587d076dd 100644 --- a/toolbox/helpfiles/FSDA/boxtest.html +++ b/toolbox/helpfiles/FSDA/boxtest.html @@ -1,41 +1,41 @@ - boxtest

boxtest

boxtest performs Box test of equality of covariance matrices

expand all in page

Syntax

Description

Box test is used to determine whether two or more covariance matrices - are equal. Note that Box test is sensitive to departures from - normality. If the samples come from non-normal distributions, then Box - test may simply be testing for non-normality.

example

out =boxtest(Y, group) Box test for Iris data.

example

out =boxtest(Y, group, Name, Value) Box test for Iris data displaying results.

Examples

expand all

  • Box test for Iris data.

    • - load iris data 

    load fisheriris
-% Compute Box test of equality of covariance matrices
-out=boxtest(meas,species);

  • Box test for Iris data displaying results.

    • - load iris data 

    load fisheriris
-% Compute Box test of equality of covariance matrices
+ 



boxtest








































  
    

    

    

    

    



    


    

    
 

    
 

    


    

    


    

    

    


    

    

    

    

    

    


    

    

    

    

    

    

    

    

    boxtest
    

    

    

    
boxtest performs Box test of equality of covariance matrices

    

    

expand all in page
    

    

    

    

    

    Syntax
    

    

    


    

    

    
	
    
							
    Description
    
							
    
								
    

    Box test is used to determine whether two or more covariance matrices
+ are equal. Note that Box test is sensitive to departures from
+ normality. If the samples come from non-normal distributions, then Box
+ test may simply be testing for non-normality.
    
	
    
	
	example
    
	
    
out
=boxtest(Y,
group)
Box test for Iris data.
    

    
	
    
	
	example
    
	
    
out
=boxtest(Y,
group,
Name, Value)
Box test for Iris data displaying results.
    

    								
    
							
    
							
    
							
    
						
    

    

    Examples
    

    

    expand all
    

    

    

    

 
  • Box test for Iris data.
    • 

    

    
+     load iris data

    

    load fisheriris
+% Compute Box test of equality of covariance matrices
+out=boxtest(meas,species);
    

    

    

    

    

    

    

    

    

 
  •  Box test for Iris data displaying results.
    • 

    

    
+     load iris data

    

    load fisheriris
+% Compute Box test of equality of covariance matrices
 out=boxtest(meas,species,'dispresults',true);
    

    *****************************************************************
 Test of homogeneity of covariances (without correction factor)
-  146.6632
+     1.466632492125116e+02
 
 LRchi2approx = Chi2 approximation of Box test of homogeneity of covariances
-  140.9430
+     1.409430499234975e+02
 
 p value of the Chi2 approximation of Box test
-   3.3520e-20
+     3.352034178317562e-20
 
 *****************************************************************
-
    

    

    

    

    

     
    

    

    Related Examples
    

    

    expand all
    

    

    

    

    


  •  Box test for Iris with option Fapprox.
    • 

    

    
-     load iris data

    

    load fisheriris
-% Compute Box test of equality of covariance matrices
+
    

    

    

    

    

     
    

    

    Related Examples
    

    

    expand all
    

    

    

    

    


  •  Box test for Iris with option Fapprox.
    • 

    

    
+     load iris data

    

    load fisheriris
+% Compute Box test of equality of covariance matrices
 out=boxtest(meas,species,'dispresults',true,'Fapprox',true)
    

    *****************************************************************
 Test of homogeneity of covariances (without correction factor)
-  146.6632
+     1.466632492125116e+02
 
 LRchi2approx = Chi2 approximation of Box test of homogeneity of covariances
-  140.9430
+     1.409430499234975e+02
 
 p value of the Chi2 approximation of Box test
-   3.3520e-20
+     3.352034178317562e-20
 
 LRFapprox = F approximation of Box test of homogeneity of covariances
-    7.0453
+   7.045261695711880
 
 p value of the F approximation of Box test
-   3.5781e-20
+     3.578106019823562e-20
 
 *****************************************************************
 
@@ -43,95 +43,96 @@
 
   struct with fields:
 
-                   LR: 146.6632
-         LRchi2approx: 140.9430
-    LRchi2approx_pval: 3.3520e-20
-            LRFapprox: 7.0453
-       LRFapprox_pval: 3.5781e-20
+                   LR: 1.466632492125116e+02
+         LRchi2approx: 1.409430499234975e+02
+    LRchi2approx_pval: 3.352034178317562e-20
+            LRFapprox: 7.045261695711880
+       LRFapprox_pval: 3.578106019823562e-20
                   Spl: [4×4 double]
 
-
    

    

    

    

    

    

    

    Input Arguments
    

    

    

    

    

expand all
    
 
    
 
    
 
    
 
 
 Y — Input data.  
 Matrix.
    
 
    
 
    n x v data matrix. n observations and v
-               variables. Rows of Y represent observations, and columns represent
-               variables. Missing values (NaN's) and infinite values
-               (Inf's) are allowed, since observations (rows) with missing
-               or infinite values will automatically be excluded from the
-               computations.
     
    
-               
    
 
    
 Data Types: single | double
    
 
    
 
    

    
 
    
 
    
 
    
 
 
 group — Grouping variable.  
 Numeric vector | logical vector | character array | cell
-               array of character vectors.
    
 
    
 
    group is a grouping variable defined as a categorical
-               variable, vector, character array, or cell array of
-               character vectors. Two observations are in the same group
-               if they have the same value in the group array. The
-               observations in each group represent a sample from a
-               population.
    
 
    
 Data Types: single| double
    
 
    
 
    

    

    

    
Name-Value Pair Arguments
    

    

    Specify optional comma-separated pairs of Name,Value arguments.
 Name is the argument name and Value
 is the corresponding value. Name must appear 
 inside single quotes (' '). 
 You can specify several name and value pair arguments in any order as  
 Name1,Value1,...,NameN,ValueN.
     
 Example:
 'Fapprox',true
-, 'dispresults',true
-

    

    

    

    


Fapprox 
—Test based on F approximation.boolean.
    

    
	
    If Fapprox is
-               true, the asymptotic F distribution of the value of Box
-               test is also computed. On the other hand is
-               Fapprox is false (default) just the chi2 approximation is
-               computed.
     
    
-               
    
	
    
       Example:  'Fapprox',true
-
    
	
    Data Types: logical
    

    

    

    

    

    


dispresults 
—Display results.boolean.
    

    
	
    If dispresults is
-               true, the value of the test and the associated p-value
-               will be shown on the screen. On the other hand is
-               dispresults is false (default) nothing is shown on the
-               screen.
     
    
-               
    
	
    
       Example:  'dispresults',true
-
    
	
    Data Types: logical
    

    

    

    

    

    

    

    Output Arguments
    

    

    

    

    
expand all
    

    

    

    


out — description
Structure
    

    

    Structure which contains the following fields
-   
    





    
Value
Description

    

    
LR


    scalar which contains Box test (uncorrected)
-                       for equality of covariances.
     
    
-                       This is -2ln M (see 'More About section' for the
-                       definition of M).
    
    		

    
LRchi2approx


    scalar which contains Box test (corrected)
-                       for equality of covariances. This version is
-                       called $\chi^2$ approximation of the Box test.
     
    
-                       This is $-2(1-c_1)\ln M$ (see further details for the
-                       definition of $c_1$ and $M$). This value must
-                       be compared with a $\chi^2$ with $0.5v(v+1)(g-1)$
-                       degrees of freedom.
    
    		

    
LRchi2approx_pval


    scalar which contains the p-value of
-                       $\chi^2$ approximation of Box test of homogeneity of
-                       covariances.
    
    		

    
LRFapprox


    scalar which contains the $F$ approximation of Box
-                       test of homogeneity of covariances. This field is
-                       given just if input option Fapprox is true.
    
    		

    
LRFapprox_pval


    scalar which contains the p-value of
-                       $F$ approximation of Box test of homogeneity of
-                       covariances. This field is given just if input
+

    





    

    Input Arguments
    

    

    

    

    

expand all
    
 
    
 
    
 
    
 
 
 Y — Input data.  
 Matrix.
    
 
    
 
    n x v data matrix. n observations and v
+               variables. Rows of Y represent observations, and columns represent
+               variables. Missing values (NaN's) and infinite values
+               (Inf's) are allowed, since observations (rows) with missing
+               or infinite values will automatically be excluded from the
+               computations.
     
    
+               
    
 
    
 Data Types: single | double
    
 
    
 
    

    
 
    
 
    
 
    
 
 
 group — Grouping variable.  
 Grouping variable defined as a categorical
+               variable, vector, character array, or cell array of
+               character vectors.
    
 
    
 
    Two observations are in the same group
+               if they have the same value in the group array. The
+               observations in each group represent a sample from a
+               population.
     
    
+               
    
 
    
 Data Types: Numeric vector | logical vector | character array | cell
+               array of character vectors | string array.
    
 
    
 
    

    

    

    
Name-Value Pair Arguments
    

    

    Specify optional comma-separated pairs of Name,Value arguments.
 Name is the argument name and Value
 is the corresponding value. Name must appear 
 inside single quotes (' '). 
 You can specify several name and value pair arguments in any order as  
 Name1,Value1,...,NameN,ValueN.
     
 Example:
 'Fapprox',true
+, 'dispresults',true
+

    

    

    

    


Fapprox 
—Test based on F approximation.boolean.
    

    
	
    If Fapprox is
+               true, the asymptotic F distribution of the value of Box
+               test is also computed. On the other hand is
+               Fapprox is false (default) just the chi2 approximation is
+               computed.
     
    
+               
    
	
    
       Example:  'Fapprox',true
+
    
	
    Data Types: logical
    

    

    

    

    

    


dispresults 
—Display results.boolean.
    

    
	
    If dispresults is
+               true, the value of the test and the associated p-value
+               will be shown on the screen. On the other hand is
+               dispresults is false (default) nothing is shown on the
+               screen.
     
    
+               
    
	
    
       Example:  'dispresults',true
+
    
	
    Data Types: logical
    

    

    

    

    

    

    

    Output Arguments
    

    

    

    

    
expand all
    

    

    

    


out — description
Structure
    

    

    Structure which contains the following fields
+   
    





    
Value
Description

    

    
LR


    scalar which contains Box test (uncorrected)
+                       for equality of covariances.
     
    
+                       This is -2ln M (see 'More About section' for the
+                       definition of M).
    
    		

    
LRchi2approx


    scalar which contains Box test (corrected)
+                       for equality of covariances. This version is
+                       called $\chi^2$ approximation of the Box test.
     
    
+                       This is $-2(1-c_1)\ln M$ (see further details for the
+                       definition of $c_1$ and $M$). This value must
+                       be compared with a $\chi^2$ with $0.5v(v+1)(g-1)$
+                       degrees of freedom.
    
    		

    
LRchi2approx_pval


    scalar which contains the p-value of
+                       $\chi^2$ approximation of Box test of homogeneity of
+                       covariances.
    
    		

    
LRFapprox


    scalar which contains the $F$ approximation of Box
+                       test of homogeneity of covariances. This field is
+                       given just if input option Fapprox is true.
    
    		

    
LRFapprox_pval


    scalar which contains the p-value of
+                       $F$ approximation of Box test of homogeneity of
+                       covariances. This field is given just if input
                        option Fapprox is true.
    
    		

    
Spl


    pooled variance covariance matrix.
    
    		

    

    

    

    

    

    

    More About
    

    

    

expand all
    

    

    

Additional Details
    

    

    
- We assume independent samples of size $n_1$, $n_2$, $\ldots$,
- $n_g$ from a $v$-multivariate normal distribution. The hypothesis of
- equality of covariances is:
- 
- \[
- H_0= \Sigma_1 = \Sigma_2 = \ldots = \Sigma_g
- \]
-
- To make the test we calculate:
- 
- \[
- M=\frac{ |S_1|^{n_1-1} |S_2|^{n_2-1} \ldots |S_g|^{n_g-1} }{|S_{pl}|^{\sum_{i=1}^g n_i-1} }
- \]
    
-
- where $S_i$ is the covariance matrix of group $i$ and $S_{pl}$ is the
- pooled sample covariance matrix. It is clear that we must have $n_i-1>v$
- otherwise $|S_i|=0$ for some $i$ and $M$ would be zero. The statistic $M$
- is a modification of the likelihood ratio test and varies between 0 and 1
- with values near 1 favouring $H_0$ and values near 0 leading to the
- rejection of $H_0$ (see Rencher (2002) p. 256 for further details). The
- quantity $-2 \ln M$ is approximately distributed as a $\chi^2$
- distribution and is given in $\mbox{out.LR}$.  The quantity $-2(1-c_1) \ln  M$
- (where $c_1$ is a small sample correction factor) is usually called
- correct Box test and is approximately distributed as a $\chi^2$ with $0.5
- (g-1) v(v+1)$ degrees of freedom. We reject $H_0$ if 
- $$-2(1-c_1) \ln M  >\chi^2_{1-\alpha}$$ where $\chi^2_{1-\alpha}$ is the 
- $1-\alpha$ quantile. The value of this test is contained in
- $\mbox{out.LRchi2approx}$  and the corresponding p-value of this test is
- given in $\mbox{out.LRchi2approx_pval}$.
     
    
- Box also derived and F approximation to the test. This test is computed
- just if input option Fapprox is true. The value of this last test and the
- corresponding p-values are given in $\mbox{out.LRFapprox}$ and
- $\mbox{LRFapprox_pval}$.
     
    
- WARNING: if the absolute value of the determinant of the
- covariance matrix of any group is less than 1e-40,
- a missing value for LR test is reported.
    
-
-
-
    

    

    

    

    

    

    References
     

    Mardia, K.V., J.T. Kent, and J.M. Bibby (1979), "Multivariate Analysis,"
- Academic Press, London, pp. 140. [MKB].
    

    Rencher A.C. (2002), "Methods of Multivariate Analysis", 2nd edition,
+ We assume independent samples of size $n_1$, $n_2$, $\ldots$,
+ $n_g$ from a $v$-multivariate normal distribution. The hypothesis of
+ equality of covariances is:
+ 
+ \[
+ H_0= \Sigma_1 = \Sigma_2 = \ldots = \Sigma_g
+ \]
+
+ To make the test we calculate:
+ 
+ \[
+ M=\frac{ |S_1|^{n_1-1} |S_2|^{n_2-1} \ldots |S_g|^{n_g-1} }{|S_{pl}|^{\sum_{i=1}^g n_i-1} }
+ \]
    
+
+ where $S_i$ is the covariance matrix of group $i$ and $S_{pl}$ is the
+ pooled sample covariance matrix. It is clear that we must have $n_i-1>v$
+ otherwise $|S_i|=0$ for some $i$ and $M$ would be zero. The statistic $M$
+ is a modification of the likelihood ratio test and varies between 0 and 1
+ with values near 1 favouring $H_0$ and values near 0 leading to the
+ rejection of $H_0$ (see Rencher (2002) p. 256 for further details). The
+ quantity $-2 \ln M$ is approximately distributed as a $\chi^2$
+ distribution and is given in $\mbox{out.LR}$.  The quantity $-2(1-c_1) \ln  M$
+ (where $c_1$ is a small sample correction factor) is usually called
+ correct Box test and is approximately distributed as a $\chi^2$ with $0.5
+ (g-1) v(v+1)$ degrees of freedom. We reject $H_0$ if 
+ $$-2(1-c_1) \ln M  >\chi^2_{1-\alpha}$$ where $\chi^2_{1-\alpha}$ is the 
+ $1-\alpha$ quantile. The value of this test is contained in
+ $\mbox{out.LRchi2approx}$  and the corresponding p-value of this test is
+ given in $\mbox{out.LRchi2approx_pval}$.
     
    
+ Box also derived and F approximation to the test. This test is computed
+ just if input option Fapprox is true. The value of this last test and the
+ corresponding p-values are given in $\mbox{out.LRFapprox}$ and
+ $\mbox{LRFapprox_pval}$.
     
    
+ WARNING: if the absolute value of the determinant of the
+ covariance matrix of any group is less than 1e-40,
+ a missing value for LR test is reported.
    
+
+
+
    

    

    

    

    

    

    References
     

    Mardia, K.V., J.T. Kent, and J.M. Bibby (1979), "Multivariate Analysis,"
+ Academic Press, London, pp. 140. [MKB].
    

    Rencher A.C. (2002), "Methods of Multivariate Analysis", 2nd edition,
  Wiley, New York, pp. 280-284.
    

    

    

    See Also
    

    


 | 


    
This page has been automatically generated by our routine publishFS







    

\ No newline at end of file
diff --git a/toolbox/multivariate/boxtest.m b/toolbox/multivariate/boxtest.m
index 02443c02d..0545a7046 100644
--- a/toolbox/multivariate/boxtest.m
+++ b/toolbox/multivariate/boxtest.m
@@ -19,14 +19,14 @@
 %               Data Types - single | double
 %
 % group :       Grouping variable.
-%               Numeric vector | logical vector | character array | cell
-%               array of character vectors.
-%               group is a grouping variable defined as a categorical
+%               Grouping variable defined as a categorical
 %               variable, vector, character array, or cell array of
 %               character vectors. Two observations are in the same group
 %               if they have the same value in the group array. The
 %               observations in each group represent a sample from a
 %               population.
+%               Data Types - Numeric vector | logical vector | character array | cell
+%               array of character vectors | string array.
 %
 %
 % Optional input arguments: