barttest

Bartlett's test for multivariate analysis of variance (MANOVA)

Since R2023b

Syntax

d = barttest(maov)

d = barttest(maov,factor)

d = barttest(___,Alpha=alpha)

[d,tbl] = barttest(___)

Description

d = barttest(maov) returns the result of a Bartlett's test to determine the minimum number of dimensions needed by a linear space to contain the mean response vectors of the manova object maov. In other words, barttest calculates the number of linearly independent mean response vectors. This syntax is supported only when maov is a one-way manova object.

d = barttest(maov,factor) specifies the factor barttest uses to group the response data. Use this syntax when maov is a two- or N-way manova object.

d = barttest(___,Alpha=alpha) specifies the confidence level barttest uses to return the minimum dimension, using any of the input argument combinations in previous syntaxes. The confidence level is (1 – alpha)*100.

[d,tbl] = barttest(___) additionally returns a table containing the Bartlett's test statistics.

Examples

Perform Bartlett's Test for Two-Way MANOVA

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 of factor values from the data in Model_Year and Cylinders.

tbl = table(Model_Year,Cylinders,VariableNames=["Year" "Cylinders"]);

Create a matrix of response variables from Acceleration and Displacement.

y = [Acceleration Displacement];

Perform a two-way MANOVA using the factor values in tbl and the response variables in y.

maov = manova(tbl,y)

maov = 
2-way manova

Y1,Y2 ~ 1 + Year + Cylinders

     Source      DF    TestStatistic     Value        F       DFNumerator    DFDenominator      pValue  
    _________    __    _____________    ________    ______    ___________    _____________    __________

    Year          2       pillai        0.084893    2.1056          4             190           0.081708
    Cylinders     2       pillai         0.94174     42.27          4             190         2.5049e-25
    Error        95                                                                                     
    Total        99                                                                                     


  Properties, Methods

maov is a two-way manova object that contains the results of the two-way MANOVA. The output displays the formula for the MANOVA model and a MANOVA table. In the formula, the car acceleration and displacement are represented by the variables Y1 and Y2, respectively. The MANOVA table contains a small p-value corresponding to the Cylinders term in the MANOVA model. The small p-value indicates that, at the 95% confidence level, enough evidence exists to conclude that Cylinders has a statistically significant effect on the mean response vector. Year has a p-value larger than 0.05, which indicates that not enough evidence exists to conclude that Year has a statistically significant effect on the mean response vector at the 95% confidence level.

Use the barttest function to determine the dimension of the space spanned by the mean response vectors corresponding to the factor Year.

barttest(maov,"Year")

ans = 
0

The output shows that the mean response vectors corresponding to Year span a point, indicating that they are not statistically different from each other. This result is consistent with the large p-value for Year.

Get Statistics for Bartlett's Test

Load the fisheriris data set.

load fisheriris

The column vector species contains iris flowers of three different species: setosa, versicolor, and virginica. The matrix meas contains four types of measurements for the flower: the length and width of sepals and petals in centimeters.

Perform a one-way MANOVA with species as the factor and the measurements in meas as the response variables.

maov = manova(species,meas);

maov is a one-way manova object that contains the results of the one-way MANOVA.

Perform a Bartlett's test to determine the minimum number of dimensions needed by a linear space to contain the mean response vectors. maov has three mean response vectors corresponding to the three iris species.

[d,tbl] = barttest(maov)

d = 
2

tbl=2×5 table
    Dimension    WilksLambda    ChiSquared    DF      pValue   
    _________    ___________    __________    __    ___________

        0         0.023439        546.12      8     8.8708e-113
        1          0.77797         36.53      3      5.7861e-08

Each row of the table output corresponds to a dimension checked by the Bartlett's test. The small p-values in the table indicate that not enough evidence exists to conclude that the mean response vectors are elements of a zero- or one-dimensional space. Three points are guaranteed to be elements of a two-dimensional space, so the Bartlett's test returns 2 as the number of dimensions.

Input Arguments

maov — MANOVA results

manova object

MANOVA results, specified as a manova object. The properties of maov contain the response data and factor values used by barttest to perform the Bartlett's test.

factor — Factor used to group response data

string scalar | character array

Factor used to group the response data, specified as a string scalar or character array. factor must be a name in maov.FactorNames.

Example: "Factor2"

Data Types: char | string

alpha — Significance level

0.05 (default) | scalar value in the range (0,1)

Significance level barttest uses to return d, specified as a scalar value in the range (0,1). The confidence level for d is (1 – alpha)*100.

Example: Alpha=0.01

Data Types: single | double

Output Arguments

d — Dimension

nonnegative integer

Dimension of the lowest dimensional linear space containing the response vectors of maov, returned as a nonnegative integer. d is the number of linearly independent mean response vectors for maov.

tbl — Bartlett's test statistics

table

Bartlett's test statistics, returned as a table. Each row of tbl corresponds to a dimension checked by barttest using the Wilks' lambda test statistic. tbl has the same number of rows as the minimum of the number of response variables and the number of values in factor minus 1. tbl has the following columns:

• Dimension — Dimension checked by the Bartlett's test

• WilksLambda — Value of the Wilks' lambda test statistic

• ChiSquared — Value of the chi-square test statistic corresponding to the Wilks' lambda test statistic

• DF — Degrees of freedom of the chi-square test statistic

• pValue — p-value for the chi-square test statistic

Data Types: table

Alternative Functionality

The manova1 function returns the output of the barttest object function, and a subset of the manova object properties. manova1 is limited to one-way MANOVA.

Version History

Introduced in R2023b