MANOVA vs ANOVA: Multivariate Analysis of Variance

MANOVA (Multivariate Analysis of Variance)

• MANOVA is a statistical method that can be used to compare the means of two or more independent groups based on two or more dependent variables.
• It is the multivariate version of ANOVA and is used when there are at least two dependent variables.

Difference between ANOVA and MANOVA

• ANOVA is based on one dependent variable, while MANOVA is based on at least two dependent variables.
• ANOVA is used to compare the means of two or more independent groups based on one dependent variable.

Null Hypothesis of MANOVA

• The null hypothesis of MANOVA states that the mean of the dependent variables is equal for all treatment groups.

Effective Data for MANOVA

• The data used for MANOVA represents measurements of some clinical variables on 12 patients.
• The data includes the concentration of the C-reactive protein in blood and the body temperature of the same patients.

Testing the Null Hypothesis

• To test the null hypothesis, we can use a p-value which is less than the general significance level of 0.05.
• If the p-value is less than 0.05, we can reject the null hypothesis and conclude that there is a significant difference in the mean body temperature and CRP concentration between the two groups.

Post-Hoc Test

• When the null hypothesis is rejected, we might want to continue with a post-hoc test.
• A post-hoc test can be used to check if there is a difference in the mean therapy concentration and mean body temperature between the two groups.
• Running two separate ANOVAs or two separate t-tests for each dependent variable is an option.
• However, when we study the two variables separately, it is possible that we may no longer find a significant difference between the groups.

Assumptions of MANOVA

• The first assumption is that the groups being compared should be independent.
• The second assumption is multivariate normality.
• The third assumption is homogeneity of the covariance matrices.
• The fourth assumption is that there should be no multicollinearity between the dependent variables.
• The fifth assumption is that there should be a linear relationship between the dependent variables for each group.

Multivariate Normality

• Multivariate normality can be tested using tests such as the generalized Shapiro-Wilks test.
• In the example, the data is normally distributed and fulfills the assumption of multivariate normality.

Homogeneity of Covariance Matrices

• The assumption of homogeneity of covariance matrices can be tested using Box's M test.
• In the example, the two covariance matrices are fairly similar, which indicates that the assumption is fulfilled.

No Multicollinearity

• There should not be a strong correlation between the dependent variables.
• In the example, there is no indication of multicollinearity.

Linear Relationship

• There should be a linear relationship between the dependent variables for each group.
• In the example, the two variables show a fairly linear pattern in the two groups.

Basic Math behind MANOVA

• MANOVA uses sums of squares and cross-product matrices, which are similar to the covariance matrix used in LDA.
• The sums of squares and cross-product matrix can be calculated using the following formula:
  • d represents the data set with centered values
  • d^t represents the transpose of the matrix d
  • The product of the two matrices results in the sums of squares and cross-product matrix of the total variation

Calculating the Between Groups Sums of Squares and Cross-Product Matrix

• The between groups sums of squares and cross-product matrix can be calculated by subtracting the within-group sums of squares and cross-product matrix from the total sums of squares and cross-product matrix.

Calculating the Test Statistic

• The test statistic can be calculated using the eigenvalues of the matrix.
• There are four different methods to calculate the test statistic: Pillai's trace, Hotelling's trace, Wilks' lambda, and Roy's largest root.

Selecting the Test Statistic

• Pillai's test is generally considered to be the most robust test against violations of the assumptions behind MANOVA and is therefore a common test to select.

Calculating the F-statistic and p-value

• Once the test statistic has been selected, the F-statistic and p-value can be calculated using statistical software tools.

MANOVA (Multivariate Analysis of Variance)

• MANOVA is a statistical method used to compare the means of two or more independent groups based on two or more dependent variables.
• It is the multivariate version of ANOVA and is used when there are at least two dependent variables.

Difference between ANOVA and MANOVA

• ANOVA is based on one dependent variable, while MANOVA is based on at least two dependent variables.
• ANOVA compares the means of two or more independent groups based on one dependent variable.

Null Hypothesis of MANOVA

• The null hypothesis of MANOVA states that the mean of the dependent variables is equal for all treatment groups.

Effective Data for MANOVA

• MANOVA can be applied to data that represents measurements of clinical variables, such as the concentration of C-reactive protein in blood and the body temperature of patients.

Testing the Null Hypothesis

• The null hypothesis can be tested using a p-value, which should be less than 0.05 to indicate significance.
• If the p-value is less than 0.05, the null hypothesis can be rejected, and a significant difference in the mean body temperature and CRP concentration between the two groups can be concluded.

Post-Hoc Test

• If the null hypothesis is rejected, a post-hoc test can be used to check for differences in the mean therapy concentration and mean body temperature between the two groups.

Assumptions of MANOVA

• The assumptions of MANOVA include:
  • Independent groups
  • Multivariate normality
  • Homogeneity of the covariance matrices
  • No multicollinearity between the dependent variables
  • Linear relationship between the dependent variables for each group

Multivariate Normality

• Multivariate normality can be tested using tests such as the generalized Shapiro-Wilks test.

Homogeneity of Covariance Matrices

• The assumption of homogeneity of covariance matrices can be tested using Box's M test.

No Multicollinearity

• There should not be a strong correlation between the dependent variables.

Linear Relationship

• There should be a linear relationship between the dependent variables for each group.

Basic Math behind MANOVA

• MANOVA uses sums of squares and cross-product matrices, similar to the covariance matrix used in LDA.
• The sums of squares and cross-product matrix can be calculated using the formula: d^t * d, where d represents the data set with centered values.

Calculating the Between Groups Sums of Squares and Cross-Product Matrix

• The between groups sums of squares and cross-product matrix can be calculated by subtracting the within-group sums of squares and cross-product matrix from the total sums of squares and cross-product matrix.

Calculating the Test Statistic

• The test statistic can be calculated using the eigenvalues of the matrix, and there are four different methods to calculate the test statistic: Pillai's trace, Hotelling's trace, Wilks' lambda, and Roy's largest root.

Selecting the Test Statistic

• Pillai's test is generally considered to be the most robust test against violations of the assumptions behind MANOVA.

Calculating the F-statistic and p-value

• Once the test statistic has been selected, the F-statistic and p-value can be calculated using statistical software tools.