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Questions and Answers
What is the primary difference between ANOVA and MANOVA?
What is the primary difference between ANOVA and MANOVA?
What does the null hypothesis of MANOVA state?
What does the null hypothesis of MANOVA state?
What is the purpose of a post-hoc test in MANOVA?
What is the purpose of a post-hoc test in MANOVA?
What is an assumption of MANOVA?
What is an assumption of MANOVA?
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What is the purpose of MANOVA?
What is the purpose of MANOVA?
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What is the significance of a p-value less than 0.05 in MANOVA?
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?
What is an alternative to running a post-hoc test in MANOVA?
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What type of data is used for MANOVA?
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?
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?
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?
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?
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?
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?
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?
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?
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.
