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Wilks Lambda

Wilks' lambda (Λ) is a statistic used in MANOVA and discriminant analysis to test differences between group means on multiple dependent variables. In MANOVA, a lower lambda value closer to 0 indicates more variance is explained by group differences. In discriminant analysis, lambda ranges from 0 to 1, with 0 indicating perfect discrimination between groups and 1 no discrimination. Software like SPSS can perform stepwise discriminant analysis using lambda to determine which independent variables best discriminate between groups. Lambda output includes the statistic value, significance level, and associated F-statistic and degrees of freedom.

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2K views4 pages

Wilks Lambda

Wilks' lambda (Λ) is a statistic used in MANOVA and discriminant analysis to test differences between group means on multiple dependent variables. In MANOVA, a lower lambda value closer to 0 indicates more variance is explained by group differences. In discriminant analysis, lambda ranges from 0 to 1, with 0 indicating perfect discrimination between groups and 1 no discrimination. Software like SPSS can perform stepwise discriminant analysis using lambda to determine which independent variables best discriminate between groups. Lambda output includes the statistic value, significance level, and associated F-statistic and degrees of freedom.

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Wilks’ Lambda: Simple Definition

Multivariate Analysis > Wilks’ Lambda

What is Wilks’ Lambda?

Wilks’ lamdba (Λ) is a test statistic that’s reported in


results from MANOVA , discriminant analysis, and other
multivariate procedures. Other similar test statistics
include Pillai’s trace criterion and Roy’s ger criterion.

• In MANOVA, Λ tests if there are differences between


group means for a particular combination
of dependent variables. It is similar to the F-test
statistic in ANOVA. Lambda is a measure of the
percent variance in dependent variables not explained
by differences in levels of the independent variable. A
value of zero means that there isn’t any variance not
explained by the independent variable (which is ideal).
In other words, the closer to zero the statistic is, the
more the variable in question contributes to the
model. You would reject the null hypothesis when
Wilk’s lambda is close to zero, although this should be
done in combination with a small p-value.
•
• In discriminant analysis, Wilk’s lambda tests how well
each level of independent variablecontributes to the
model. The scale ranges from 0 to 1, where 0 means
total discrimination, and 1 means no discrimination.
Each independent variable is tested by putting it into
the model and then taking it out — generating a Λ
statistic. The significance of the change in Λ is
measured with an F-test; if the F-value is greater than
the critical value, the variable is kept in the model.
This stepwise procedure is usually performed using
software like Minitab, R, or SPSS. The following SPSS
output shows which variables (from a list of a dozen or
more) were kept in using this procedure.
SPSS Wilk’s Lambda output. Image: Bournemouth
University.

Formula

1 – λ in the denominator is the proportion of variance in


dependent variables explained by the model’s effect.
Caution should be used in interpreting results as this
statistic tends to be biased, especially for small samples.

Output Components

Wilks’ lambda output has several components, including:


• “Sig” or significance (p-value). If this is small, (i.e.
under .05) reject the null hypothesis.
• “Value” column in the output: the value of Wilk’s
Lambda.
• “Statistic” is the F-statistic associated with the listed
degrees of freedom. It would be reported in APA
format as F(df1,df2) = value. For example, if you had
an f-value of 36.612 with 1 and 2 degrees of freedom
you would report that as F(1,2) = 36.612.

References

Nath, R. and Pavur, R. (1985) A new statistic in the one


way multivariate analysis of variance, Computational
Statistics and Data Analysis, 2, 297–315.
Todorov, V. and Filzmoser, P. (2007) Robust statistic for
the one-way MANOVA, submitted to the Journal of
Environmetrics

https://www.statisticshowto.datasciencecentral.com/wil
ks-lambda/

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