MANOVA test table for General MANOVA

Find definitions and interpretations for every statistic in the MANOVA test table.

In This Topic

s
m
n
Criterion
Test Statistic
F-value
DF Num
DF Denom
P-Value

s

Minitab uses s to calculate the F-statistics for the Wilk's, Lawley-Hotelling, and Pillai's tests. The F-statistic is exact if s=1 or 2. If s ≠ 1 or 2, the F-statistic is approximate. For more information about how Minitab calculates s, go to Methods and formulas for MANOVA tests.

Interpretation

Minitab uses s to calculate the F-value and the p-value. Usually, you assess the p-value because it is easier to interpret.

m

Minitab uses m to calculate the F-statistics for the Wilk's, Lawley-Hotelling, and Pillai's tests. For more information about how Minitab calculates m, go to Methods and formulas for MANOVA tests.

Interpretation

Minitab uses m to calculate the F-value and the p-value. Usually, you assess the p-value because it is easier to interpret.

n

Minitab uses n to calculate the F-statistics for the Wilk's, Lawley-Hotelling, and Pillai's tests. For more information about how Minitab calculates n, go to Methods and formulas for MANOVA tests.

Interpretation

Minitab uses n to calculate the F-value and then the p-value. Usually, you assess the p-value because it is easier to interpret.

Criterion

By default, Minitab displays a table of four multivariate tests for each term in the model:

Wilk's test is the most commonly used test because it was the first test to be derived and has a well-known F approximation.
Lawley-Hotelling is also known as Hotelling's generalized T² statistic.
Pillai's test is the best test to use in most situations. Pillai's gives similar results to the Wilks' and Lawley-Hotelling's tests.
Roy's largest root test is the best when the mean vectors are collinear. Roy's test does not have a satisfactory F approximation.

Interpretation

Examine the p-values for the Wilk's, Lawley-Hotelling, and Pillai's test statistic to judge whether significant evidence exists for model effects. If the p-value is less than your significance level, the effect is statistically significant. Generally, you will draw the same conclusion using any of the tests. In cases when conclusions differ, base your decision on the test that is best for your data.

Test Statistic

Minitab provides a test statistic for each multivariate test. The name of the test statistic for each test is as follows:

Wilk's lambda
Lawley-Hotelling trace
Pillai's trace
Largest eigenvalue, λ1

For more information about how Minitab calculates each test statistic, go to Methods and Formulas.

Interpretation

Minitab uses the test statistic to calculate the F-value and the p-value. Usually, you assess the p-value because it is easier to interpret.

F-value

An F-value appears for each term in the Analysis of Variance table:

F-value for the model or the terms: The F-value is the test statistic used to determine whether the term is associated with the response.
F-value for the lack-of-fit test: The F-value is the test statistic used to determine whether the model is missing higher-order terms that include the predictors in the current model.

Interpretation

Minitab uses the F-value to calculate the p-value, which you use to make a decision about the statistical significance of the terms and model. The p-value is a probability that measures the evidence against the null hypothesis. Lower probabilities provide stronger evidence against the null hypothesis.

A sufficiently large F-value indicates that the term or model is significant.

If you want to use the F-value to determine whether to reject the null hypothesis, compare the F-value to your critical value. You can calculate the critical value in Minitab or find the critical value from an F-distribution table in most statistics books. For more information on using Minitab to calculate the critical value, go to Using the inverse cumulative distribution function (ICDF) and click "Use the ICDF to calculate critical values".

DF Num

The DF Num is the degrees of freedom for the numerator that Minitab uses to calculate F.

Interpretation

Minitab uses the F-value to calculate the p-value. Usually, you assess the p-value because it is easier to interpret.

DF Denom

The DF Denom is degrees of freedom for the denominator that Minitab uses to calculate F.

Interpretation

Minitab uses the F-value to calculate the p-value. Usually, you assess the p-value because it is easier to interpret.

P-Value

The p-value is a probability that measures the evidence against the null hypothesis. Lower probabilities provide stronger evidence against the null hypothesis.

Interpretation

To simultaneously test the equality of means from all the responses, compare the p-values in the MANOVA test tables for each term to your significance level. Usually, a significance level (denoted as α or alpha) of 0.05 works well. A significance level of 0.05 indicates a 5% risk of concluding that an association exists when there is no actual association.

P-value ≤ α: The differences between the means are statistically significant: If the p-value is less than or equal to the significance level, you can conclude that the differences between the means are statistically significant.
P-value > α: The differences between the means are not statistically significant: If the p-value is greater than the significance level, you cannot conclude that the differences between the means are statistically significant. You may want to refit the model without the term.; If there are multiple predictors without a statistically significant association with the response, you can reduce the model by removing terms one at a time. For more information on removing terms from the model, go to Model reduction.

If a model term is statistically significant, the interpretation depends on the type of term. The interpretations are as follows:

If a main effect is significant, the level means for the factor are significantly different from each other across all responses in your model.
If an interaction term is significant, the effects of each factor are different at each level of the other factors across all responses in your model. For this reason, you should not analyze the individual effects of terms involved in significant higher-order interactions.