MONOVA is a statistical technique used to examine data with multiple dependent variables. It enables researchers to determine whether the means of two or more groups differ significantly while controlling for other variables that may influence the outcome. MONOVA is a potent tool for obtaining insights from complex datasets, but it can be difficult to interpret the results. In this blog post, we will demonstrate how to interpret MONOVA results for your research or project.

## Step 1: Understand the MONOVA Output

Before interpreting the output of a MONOVA analysis, it is necessary to have a firm grasp of the various components that comprise the output. Typical output sections include the following:

This section contains descriptive statistics for the dependent and independent variables employed in the analysis. This comprises the mean, standard deviation, and sample size for each condition or group.

**Multivariate Tests:**This section contains the results of the multivariate tests, which examine the significance of the differences between the groups or conditions in general. Wilks' Lambda is the most frequently used multivariate test. It ranges from 0 to 1, with 0 indicating perfect discrimination between groups and 1 indicating no discrimination.**Univariate Tests:**This section contains the results of the univariate tests, which examine the significance of the differences between the groups or conditions for each dependent variable separately. The F-test, which contrasts the mean scores of each group or condition on the dependent variable, is the most frequently used univariate test.**Tests of Between-Subjects Effects:**This section contains the results of the tests of between-subjects effects, which examine the significance of differences between groups or conditions on the dependent variable(s) while controlling for any covariates.**Tests of Within-Subjects Effects:**This section contains the results of the tests of within-subjects effects, which examine the significance of the differences between the levels of the independent variable(s) and dependent variable(s) in a repeated measures design.**Multivariate Tests of Within-Subjects Effects:**This section contains the results of the multivariate tests of within-subjects effects, which examine the significance of the differences between the levels of the independent variable(s) and the dependent variable(s) in a repeated measures design.

This section contains the results of Mauchly's test of sphericity, which evaluates the assumption of sphericity in repeated measures designs. If the assumption of sphericity is violated, the degrees of freedom should be adjusted using a correction factor such as Greenhouse-Geisser or Huynh-Feldt.

**Tests of Within-Subjects Contrasts:**This section contains the results of tests of within-subjects contrasts, which examine the significance of linear, quadratic, or higher-order trends in the data.**Tests of Simple Effects:**This section contains the results of the tests of simple effects, which examine the significance of the differences between the levels of the independent variable(s) and the dependent variable(s) at each level of any covariables.

To interpret the output of a MONOVA analysis, it is necessary to thoroughly examine each of these sections and understand the results in the context of the research question and design.

## Step 2: Check Assumptions

Prior to interpreting the output, it is essential to confirm that the fundamental assumptions of the analysis have been satisfied. Any violation of the assumptions could result in inaccurate results and interpretations.

**The key assumptions of a one-way ANOVA are as follows:
**

**Normality:**The dependent variable's distribution within each group should be normal.**Homogeneity of variance:**The variance of the dependent variable must be the same for all categories.**Independence:**The observations must be distinct from one another.

Create a histogram or Q-Q plot of the dependent variable for each group to test for normality. A normal distribution has a bell-shaped curve, whereas an atypical distribution has asymmetrical tails or an irregular shape. Normality can also be confirmed using statistical analyses such as the Shapiro-Wilk and Kolmogorov-Smirnov tests.

Statistical measures, such as Levene's test or the Bartlett test, can be utilized to examine variance homogeneity. These analyses examine whether the variances of the dependent variable across all groups are statistically comparable.

Lastly, to ensure independence, you must ensure that the observations are completely unrelated. For instance, if you measure the height of members of the same family, the observations may not be independent because the heights may be genetically determined.

Once you have confirmed that the assumptions are met, you can interpret the MONOVA analysis's results.

## Step 3: Interpret the Multivariate Tests

After confirming the validity of the assumptions and results, the next stage is to interpret the multivariate tests. This is a crucial stage because it allows you to comprehend the overall relationship between the dependent and independent variables.

The multivariate analyses determine if there is a statistically significant difference between groups or conditions on the dependent variables. Several multivariate tests, including Pillai's Trace, Wilks' Lambda, Hotelling's Trace, and Roy's Largest Root, are frequently employed in MONOVA.

It is essential to choose the appropriate test for your data and research query, as each has its own assumptions and limitations. A significant result on any of the tests typically indicates that there is a difference between the groups or conditions on the dependent variables.

Nevertheless, it is also crucial to consider the effect size, which quantifies the magnitude of the difference between the categories. Even if a modest effect size is statistically significant, it may not be practically significant. A large effect size, however, may be practically significant even if it is not statistically significant.

The pattern of results is also an essential consideration. For instance, if there is a significant multivariate effect but no significant univariate effect, it suggests that the groups differ on a combination of the dependent variables, as opposed to a single variable in particular.

Overall, interpreting multivariate tests requires careful consideration of statistical significance, effect magnitude, and pattern of results, as well as knowledge of the research question and the context of the study.

## Step 4: Examine the Univariate Tests

The next stage is to examine the univariate tests after examining the multivariate tests. These tests investigate the influence of independent variables on the dependent variable. The univariate analyses can provide more specific information regarding the study's variable relationships.

**Among the univariate assays are:
**

**ANOVA:**ANOVA examines the differences in means between three or more groups. It describes the influence of each independent variable on the dependent variable.**T-Tests:**T-tests compare the means of two groups to determine whether they differ significantly. They are employed to investigate the effects of a single independent variable on the dependent variable.**Chi-Square Tests:**Chi-square tests are used to determine whether or not two categorical variables are related. They provide information regarding the relationship between variables.

The correlation analysis investigates the relationship between two continuous variables. It provides information regarding the direction and intensity of the relationship.

When interpreting univariate tests, it is crucial to consider the p-values, which indicate the likelihood of obtaining results by coincidence. A p-value less than 0.05 is typically considered statistically significant, indicating that the results are unlikely to have occurred by accident.

The effect sizes, which provide information about the practical significance of the results, must also be considered. Sizes of effect indicate the intensity of the relationship between variables. Cohen's d is a frequently employed measure of effect size, with values of.2,.5, and.8 corresponding to minor, medium, and large effects, respectively.

Overall, the univariate analyses provide greater insight into the relationships between the study's variables. By examining the effects of each independent variable on the dependent variable, researchers can obtain a deeper understanding of the factors that contribute to their study's outcomes.

## Step 5: Look at the Descriptive Statistics

Statistics that describe the central tendency, variability, and shape of the data are descriptive statistics. This information is useful for comprehending the characteristics of the data and interpreting the MONOVA results.

In the MONOVA output, descriptive statistics for each variable can be discovered. The means, standard deviations, sample sizes, and skewness and kurtosis coefficients are provided for each variable. Skewness measures the degree of asymmetry in the variable's distribution, whereas kurtosis measures the degree of peakedness or flatness.

The means and standard deviations, in general, provide information about the central tendency and variability of the data, respectively. A high degree of variability indicates that the data may not be normally distributed, which could compromise the reliability of the MONOVA results.

The skewness and kurtosis coefficients can also shed light on the data's distribution. A skewness coefficient of zero implies a symmetric distribution, whereas a positive or negative value indicates a right- or left-skewed distribution, respectively. Similarly, a kurtosis coefficient of zero indicates a normal distribution, whereas a positive or negative kurtosis coefficient indicates a distribution that is more or less peaked than normal.

It is essential to observe that deviations from normality can compromise the validity of the MONOVA results, particularly for smaller sample sizes. If the data has a high degree of skewness or kurtosis, it may be necessary to transform the data or use a non-parametric test instead.

In addition to descriptive statistics, it is essential to investigate any outliers or influential cases in the data. Outliers are extreme values that fall outside the normal range of values for the variable, whereas influential cases are observations that have a significant impact on the analysis's results.

Boxplots and scatterplots can be used to identify outliers, which should be investigated to determine whether they represent valid data points or errors. Using leverage statistics or Cook's distance, influential cases can be identified and should be examined to determine if they are driving the results of the analysis.

## Step 6: Post-hoc Tests (if necessary)

Typically, post-hoc tests are conducted when multivariate tests reveal statistical significance. The purpose of these tests is to identify the distinctions between the groups that contributed to the significant results. The choice of post-hoc test should be determined by the research question and sample size.

**There are several post-hoc tests available for MONOVA, including:
**

**Bonferroni correction:**This test adjusts the significance level to control for multiple comparisons. It is a conservative approach and is known for having low statistical power.**Tukey's HSD (Honestly Significant Difference) test:**This test compares all possible pairs of means to determine which pairs are significantly different from each other.**Scheffe's test:**This test is a more conservative approach that can be used when there are unequal sample sizes or when the assumption of homogeneity of variance is violated.**Dunnett's test:**This test is used to compare each treatment group with a control group.**Games-Howell test:**This test is a non-parametric alternative to Tukey's HSD test

## Step 7: Interpret the Results

After completing all preceding stages, it is now time to interpret the results. You should begin by summarizing and connecting your findings to your research question or hypothesis.

**When interpreting your results, it is essential to bear the following in mind:
**

**Effect Size:**The effect size is one means to evaluate the practical significance of your results. The effect size quantifies the significance of the relationship between variables. A larger effect magnitude generally indicates a stronger relationship.**Level of Significance:**Another crucial factor is the significance level of the analyses. Often denoted as alpha, the significance level indicates the probability of obtaining the observed results by coincidence alone. A common level of significance is.05. This indicates that there is a 5% possibility that the results were obtained by chance. The results are regarded statistically significant if the p-value is less than.05.**Additionally, it is essential to consider the confidence interval:**The confidence interval is a range of values that contains the true population parameter with a high degree of probability. The estimation is less precise the wider the confidence interval.**Validity of Assumptions:**Prior to interpreting your results, you should verify the validity of the MONOVA analysis's underlying assumptions. Inconsistencies with the assumptions can affect the precision of the results.**Significance on a Substantive Level:**Finally, you should evaluate the findings' significance on a substantive level. The results' substantive significance refers to their practical or real-world implications. For instance, if the effect size is very small, a statistically significant result may not be practically significant.

Overall, interpreting MONOVA results requires a combination of statistical knowledge, critical thinking skills, and a comprehensive comprehension of the research query.

## Conclusion

It can be difficult to interpret the results of a MONOVA analysis, but doing so is essential for deriving valid conclusions from your study or project. By adhering to the seven steps outlined in this blog, you can interpret the results of your MONOVA analysis and obtain a deeper understanding of the relationships between your variables. Remember to evaluate assumptions, multivariate and univariate tests, descriptive statistics, and, if necessary, post-hoc tests. Use your interpretation of the results to guide your conclusions and future directions of research. With these guidelines in mind, you will soon be an expert at interpreting MONOVA results.