## Podcast Beta

## Questions and Answers

What is the primary difference between ANOVA and MANOVA?

What does the null hypothesis of MANOVA state?

What is the purpose of a post-hoc test in MANOVA?

What is an assumption of MANOVA?

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What is the purpose of MANOVA?

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What is the significance of a p-value less than 0.05 in MANOVA?

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What is an alternative to running a post-hoc test in MANOVA?

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What type of data is used for MANOVA?

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What is the assumption that states there should be no strong correlation between the dependent variables?

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What is the test used to determine if the assumption of homogeneity of covariance matrices is fulfilled?

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What is the purpose of calculating the test statistic in MANOVA?

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What is the formula used to calculate the sums of squares and cross-product matrix in MANOVA?

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Which of the following tests is generally considered to be the most robust test against violations of the assumptions behind MANOVA?

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What is the assumption that states there should be a linear relationship between the dependent variables for each group?

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What is the purpose of calculating the between groups sums of squares and cross-product matrix in MANOVA?

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What is the name of the test used to determine if the assumption of multivariate normality is fulfilled?

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## Study Notes

### 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.

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## Description

Learn about MANOVA, a statistical method to compare means of multiple groups based on multiple dependent variables, and how it differs from ANOVA.