coeftest
Syntax
Description
tests the null hypothesis that tbl
= coeftest(maov
,A
)A
*B
= 0, where B
is the matrix of coefficients in
maov.Coefficients
. A
is an
a-by-p transform matrix with rank a ≤ p, and p is the number of terms in the MANOVA model for
maov
. Use this syntax to test for statistically significant
differences in model coefficients between factor values.
Examples
Test Significance of MANOVA Model Coefficients
Load the carsmall
data set.
load carsmall
The variable Model_Year
contains data for the year a car was manufactured, and the variable Cylinders
contains data for the number of engine cylinders in the car. The Acceleration
and Displacement
variables contain data for car acceleration and displacement.
Use the table
function to create a table from the data in Model_Year
, Cylinders
, Acceleration
, and Displacement
.
data = table(Model_Year,Cylinders,Acceleration,Displacement, ... VariableNames=["Year" "Cylinders" "Acceleration" "Displacement"]);
Perform a two-way MANOVA, using Model_Year
and Cylinders
as factors, and Acceleration
and Displacement
as response variables. Display the test statistic used to perform the MANOVA.
maov = manova(data,["Acceleration" "Displacement"]); maov.TestStatistic
ans = "pillai"
The output shows that the function uses Pillai's to compute the MANOVA statistics for maov
.
Test the null hypothesis that the model coefficients for maov
are not statistically different from zero. By default, coeftest
uses the statistic in maov.TestStatistic
to perform the test.
tbl = coeftest(maov)
tbl=1×6 table
TestStatistic Value F DFNumerator DFDenominator pValue
_____________ ______ ______ ___________ _____________ ___________
"pillai" 1.8636 259.68 10 190 4.4695e-105
The p-value in the table output is very small, indicating that enough evidence exists to conclude that at least one of the model coefficients is statistically significant.
Test Significance of MANOVA Model Term
Load the carsmall
data set.
load carsmall
The variable Model_Year
contains data for the year a car was manufactured, and the variable Cylinders
contains data for the number of engine cylinders in the car. The Acceleration
and Displacement
variables contain data for car acceleration and displacement.
Use the table
function to create a table from the data in Model_Year
, Cylinders
, Acceleration
, and Displacement
.
data = table(Model_Year,Cylinders,Acceleration,Displacement, ... VariableNames=["Year" "Cylinders" "Acceleration" "Displacement"]);
Perform a two-way MANOVA using Model_Year
and Cylinders
as factors, and Acceleration
and Displacement
as response variables.
maov = manova(data,["Acceleration" "Displacement"]);
maov
is a manova
object that contains the results of the two-way MANOVA.
Display the fitted MANOVA model coefficients for maov
.
coefs = maov.Coefficients
coefs = 5×2
14.9360 228.5164
-0.8342 4.5054
0.6874 -10.0817
1.5827 -115.6528
1.3065 -7.8655
The first and second columns of the matrix coefs
correspond to the car acceleration and car displacement response variables, respectively. Each row corresponds to a term in the MANOVA model, with the first row containing intercept terms.
Display the names of the terms for the fitted coefficients.
maov.ExpandedFactorNames
ans = 1x5 string
"(Intercept)" "Year_70" "Year_76" "Cylinders_4" "Cylinders_6"
The output shows that the last two rows of coefs
correspond to the terms for number of engine cylinders.
Test the null hypothesis that, for both response variables, the sum of the coefficients corresponding to the number of engine cylinders is zero.
A = [0 0 0 1 1]; tbl = coeftest(maov,A)
tbl=1×6 table
TestStatistic Value F DFNumerator DFDenominator pValue
_____________ _______ ______ ___________ _____________ __________
"pillai" 0.81715 210.04 2 94 2.0833e-35
The small p-value in the table output indicates that enough evidence exists to conclude that the sum of the engine cylinders coefficients is statistically different from zero.
Test Significance of Coefficients for Each Response Variable
Load the carsmall
data set.
load carsmall
The variable Model_Year
contains data for the year a car was manufactured, and the variable Cylinders
contains data for the number of engine cylinders in the car. The Acceleration
and Displacement
variables contain data for car acceleration and displacement.
Use the table
function to create a table from the data in Model_Year
, Cylinders
, Acceleration
, and Displacement
.
data = table(Model_Year,Cylinders,Acceleration,Displacement, ... VariableNames=["Year" "Cylinders" "Acceleration" "Displacement"]);
Perform a two-way MANOVA using Model_Year
and Cylinders
as factors, and Acceleration
and Displacement
as response variables.
maov = manova(data,["Acceleration" "Displacement"]);
maov
is a manova
object that contains the results of the two-way MANOVA.
Display the fitted MANOVA model coefficients for maov
.
coefs = maov.Coefficients
coefs = 5×2
14.9360 228.5164
-0.8342 4.5054
0.6874 -10.0817
1.5827 -115.6528
1.3065 -7.8655
The first and second columns of the matrix coefs
correspond to the car acceleration and car displacement response variables, respectively. Each row corresponds to a term in the MANOVA model, with the first row containing intercept terms.
Test the null hypothesis that the coefficients corresponding to the car acceleration sum to zero for each response variable.
A = [0 0 0 1 1]; C = [1;0]; tbl = coeftest(maov,A,C)
tbl=1×6 table
TestStatistic Value F DFNumerator DFDenominator pValue
_____________ _______ ______ ___________ _____________ __________
"pillai" 0.33182 47.176 1 95 6.6905e-10
The small p-value in the table output indicates that enough evidence exists to conclude that at least one of the sums is statistically different from zero.
Input Arguments
maov
— MANOVA results
manova
object
MANOVA results, specified as a manova
object.
The properties of maov
contain the coefficient estimates and MANOVA
statistics used by coeftest
to perform the
F-test.
A
— Transform matrix
p-by-p identity matrix (default) | a-by-p numeric matrix
Transform matrix, specified as a p-by-p
identity matrix or an a-by-p numeric matrix, where
p is the number of terms in the MANOVA model for
maov
. A
has rank a ≤ p.
coeftest
uses A
to perform an
F-test with the null hypothesis A
*B
*C
=
D
. B
is the matrix of coefficients in
maov.Coefficients
, C
is a contrast matrix,
and D
is a matrix of hypothesized values. Specify
A
to test for statistically significant differences in model
coefficients between factor values. For more information, see Multivariate Analysis of Variance for Repeated Measures.
Example: [1 1 3 0;0 0 2 1]
Data Types: single
| double
C
— Contrast matrix
r-by-r identity matrix (default) | r-by-c numeric matrix
Contrast matrix, specified as an r-by-r
identity matrix or an r-by-c numeric matrix, where
r is the number of response variables in the MANOVA model for
maov
.
coeftest
uses C
to perform an
F-test with the null hypothesis A
*B
*C
=
D
. B
is the matrix of coefficients in
maov.Coefficients
, A
is a transform matrix,
and D
is a matrix of hypothesized values. Specify
C
to test for statistically significant differences in model
coefficients between response variables. For more information, see Multivariate Analysis of Variance for Repeated Measures.
Example: [0.25 0.4]
Data Types: single
| double
D
— Hypothesized values
0
(default) | numeric scalar | a-by-c numeric matrix
Hypothesized values, specified as 0
, a numeric scalar, or an
a-by-c numeric matrix. a is
the number of rows in the transform matrix A
, and
c is the number of columns in the contrast matrix
C
. If D
is a scalar, the function expands it
to be an a-by-c matrix.
coeftest
uses D
to perform an
F-test with the null hypothesis A
*B
*C
=
D
. B
is the matrix of coefficients in
maov.Coefficients
, A
is a transform matrix,
and C
is a contrast matrix. Specify D
to
determine whether linear combinations of the coefficients estimated for
maov
are statistically equal to certain values. For more
information, see Multivariate Analysis of Variance for Repeated Measures.
Example: [0 0 0 0;1 1 1 1;2 2 2 2]
Data Types: single
| double
testStat
— MANOVA test statistics
maov.TestStatistic
(default) | "all"
| "pillai"
| "hotelling"
| "wilks"
| "roy"
MANOVA test statistics, specified as maov.TestStatistic
,
"all"
, or one or more of the following values.
Value | Test Name | Equation |
---|---|---|
"pillai" (default) | Pillai's trace | where θi values are the solutions of the characteristic equation Qh – θ(Qh + Qe) = 0. Qh and Qe are, respectively, the hypotheses and the residual sum of squares product matrices. |
"hotelling" | Hotelling-Lawley trace | where λi are the solutions of the characteristic equation |Qh – λQe| = 0. |
"wilks" | Wilk's lambda |
|
"roy" | Roy's maximum root statistic |
|
If you specify testStat
as "all"
,
coeftest
calculates all the test statistics in the table
above.
Example: TestStatistic
="hotelling"
Data Types: char
| string
| cell
Output Arguments
tbl
— Hypothesis test results
table
Hypothesis test results, returned as a table with the following variables:
TestStatistic
— Test statistic used bycoeftest
to perform the F-test.Value
— Value of the test statistic.F
— F-statistic value. To calculate the F-statistic,coeftest
transformsValue
so that it follows an F-distribution under the null hypothesis.DFNumerator
— Degrees of freedom for the numerator of the F-statistic.DFDenominator
— Degrees of freedom for the denominator of the F-statistic.pValue
— p-value for the F-statistic.
H
— Hypothesis matrix
numeric matrix
Hypothesis matrix, returned as a numeric matrix. coeftest
uses H
to calculate the test statistic. For more information about
H
, see Qh in Multivariate Analysis of Variance for Repeated Measures.
Data Types: single
| double
E
— Error matrix
numeric matrix
Error matrix, returned as a numeric matrix. coeftest
uses
E
to calculate the test statistic. For more information about
E
, see Qe in Multivariate Analysis of Variance for Repeated Measures.
Data Types: single
| double
Version History
Introduced in R2023b
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