# Manova hypothesis example

MANOVA is used to model two or more dependent variables that are continuous with one or more categorical predictor variables.

Please note: The purpose of this page is to show how to use various data analysis commands. It does not cover all aspects of the research process which researchers are expected to do. In particular, it does not cover data cleaning and checking, verification of assumptions, model diagnostics or potential follow-up analyses. A researcher randomly assigns 33 subjects to one of three groups.

The first group receives technical dietary information interactively from an on-line website. Group 2 receives the same information from a nurse practitioner, while group 3 receives the information from a video tape made by the same nurse practitioner. The researcher looks at three different ratings of the presentation, difficulty, usefulness and importance, to determine if there is a difference in the modes of presentation. In particular, the researcher is interested in whether the interactive website is superior because that is the most cost-effective way of delivering the information.

Example 2. A clinical psychologist recruits people who suffer from panic disorder into his study. Each subject receives one of four types of treatment for eight weeks. The clinical psychologist wants to know which type of treatment most reduces the symptoms of the panic disorder as measured on the physiological, emotional and cognitive scales. This example was adapted from Grimm and Yarnold,page We have a data file, manova. The response variables are ratings called usefuldifficulty and importance.

Level 1 of the group variable is the treatment group, level 2 is control group 1 and level 3 is control group 2. Below is a list of some analysis methods you may have encountered. Some of the methods listed are quite reasonable, while others have either fallen out of favor or have limitations. We will start by running the manova command. After the categorical predictor variable groupwe need to specify the minimum and maximum values of that variable in parentheses.

We will begin by comparing the treatment group group 1 to an average of the control groups groups 2 and 3. This tests the hypothesis that the mean of the control groups equals the treatment group. We will also compare control group 1 group 2 to control group 2 group 3. The first hypothesis is given on the second line of the contrast subcommand, and the second hypothesis is given on the third line of the contrast subcommand.

We can use the pmeans subcommand to obtain adjusted predicted values for each of the groups. In the first table below, we get the predicted means for the dependent variable difficulty. In the next two tables, we get the predicted means for the dependent variables useful and importance.

These values can be helpful in seeing where differences between levels of the predictor variable are and describing the model. In each of the three tables above, we see that the predicted means for groups 2 and 3 are very similar; the predicted mean for group 1 is higher than those for groups 2 and 3.The MANOVA multivariate analysis of variance is a type of multivariate analysis used to analyze data that involves more than one dependent variable at a time.

MANOVA allows us to test hypotheses regarding the effect of one or more independent variables on two or more dependent variables. See Statistical Data Analysis for more information. We may want to look at the effect of teaching style independent variable on the average values of several dependent variables such as student satisfaction, number of student absences and math scores.

Interestingly, in addition to detecting differences in the average values, a MANOVA test can also detect differences in correlations among the dependent variables between the different levels of the independent variable. Multiple Linear Regression is another type of multivariate analysis, which is described in its own tutorial topic.

Get the Statistics Help you need When you hire me to do the data analysis for your dissertation Results ChapterI will determine which statistical methods are appropriate for your hypotheses.

I also provide ongoing statistical help as needed to ensure that you fully understand all of the statistics that I used for your dissertation. Simply contact me by phone or email to get started.For example, we may conduct a study where we try two different textbooks, and we are interested in the students' improvements in math and physics. In that case, improvements in math and physics are the two dependent variables, and our hypothesis is that both together are affected by the difference in textbooks.

The "covariance" here is included because the two measures are probably correlated and we must take this correlation into account when performing the significance test. Testing the multiple dependent variables is accomplished by creating new dependent variables that maximize group differences. These artificial dependent variables are linear combinations of the measured dependent variables. How may they be utilized? If the overall multivariate test is significant, we conclude that the respective effect e.

However, our next question would of course be whether only math skills improved, only physics skills improved, or both. In fact, after obtaining a significant multivariate test for a particular main effect or interaction, customarily one would examine the univariate F tests for each variable to interpret the respective effect.

In other words, one would identify the specific dependent variables that contributed to the significant overall effect. MANOVA is useful in experimental situations where at least some of the independent variables are manipulated. First, by measuring several dependent variables in a single experiment, there is a better chance of discovering which factor is truly important.

However, there are several cautions as well. It is a substantially more complicated design than ANOVA, and therefore there can be some ambiguity about which independent variable affects each dependent variable.

Thus, the observer must make many potentially subjective assumptions. Moreover, one degree of freedom is lost for each dependent variable that is added. The gain of power obtained from decreased SS error may be offset by the loss in these degrees of freedom. Finally, the dependent variables should be largely uncorrelated. If the dependent variables are highly correlated, there is little advantage in including more than one in the test given the resultant loss in degrees of freedom.

Normal Distribution : - The dependent variable should be normally distributed within groups. Overall, the F test is robust to non-normality, if the non-normality is caused by skewness rather than by outliers. Therefore, when the relationship deviates from linearity, the power of the analysis will be compromised. Homogeneity of Variances : - Homogeneity of variances assumes that the dependent variables exhibit equal levels of variance across the range of predictor variables.

Remember that the error variance is computed SS error by adding up the sums of squares within each group. If the variances in the two groups are different from each other, then adding the two together is not appropriate, and will not yield an estimate of the common within-group variance. Homoscedasticity can be examined graphically or by means of a number of statistical tests.

Homogeneity of Variances and Covariances : - In multivariate designs, with multiple dependent measures, the homogeneity of variances assumption described earlier also applies.

However, since there are multiple dependent variables, it is also required that their intercorrelations covariances are homogeneous across the cells of the design. There are various specific tests of this assumption. Two special cases arise in MANOVA, the inclusion of within-subjects independent variables and unequal sample sizes in cells. This causes tests of main effects and interactions to be correlated. Within-subjects design - Problems arise if the researcher measures several different dependent variables on different occasions.

This situation can be viewed as a within-subject independent variable with as many levels as occasions, or it can be viewed as separate dependent variables for each occasion. Tabachnick and Fidell provide examples and solutions for each situation.

This situation often lends itself to the use of profile analysis, which is explained below. Outliers may produce either a Type I or Type II error and give no indication as to which type of error is occurring in the analysis. There are several programs available to test for univariate and multivariate outliers.The one-way multivariate analysis of variance one-way MANOVA is used to determine whether there are any differences between independent groups on more than one continuous dependent variable.

For example, you could use a one-way MANOVA to understand whether there were differences in the perceptions of attractiveness and intelligence of drug users in movies i. Alternatively, you could use a one-way MANOVA to understand whether there were differences in students' short-term and long-term recall of facts based on three different lengths of lecture i. In addition, if your independent variable consists of repeated measures, you can use the one-way repeated measures MANOVA.

## One-way Manova | SPSS Data Analysis Examples

It is important to realize that the one-way MANOVA is an omnibus test statistic and cannot tell you which specific groups were significantly different from each other; it only tells you that at least two groups were different.

Since you may have three, four, five or more groups in your study design, determining which of these groups differ from each other is important.

You can do this using a post-hoc test N. However, before we introduce you to this procedure, you need to understand the different assumptions that your data must meet in order for a one-way MANOVA to give you a valid result. We discuss these assumptions next.

Do not be surprised if, when analysing your own data using SPSS Statistics, one or more of these assumptions is violated i. This is not uncommon when working with real-world data. However, even when your data fails certain assumptions, there is often a solution to overcome this.

## Conduct and Interpret a One-Way MANOVA

In practice, checking for these nine assumptions adds some more time to your analysis, requiring you to work through additional procedures in SPSS Statistics when performing your analysis, as well as thinking a little bit more about your data.

These nine assumptions are presented below:. Before doing this, you should make sure that your data meets assumptions 1, 2, 3 and 4, although you don't need SPSS Statistics to do this.

Just remember that if you do not run the statistical tests on these assumptions correctly, the results you get when running a one-way MANOVA might not be valid. You can find out about our enhanced content as a whole on our Features: Overview page, or more specifically, learn how we help with testing assumptions on our Features: Assumptions page. The pupils at a high school come from three different primary schools. The head teacher wanted to know whether there were academic differences between the pupils from the three different primary schools.

As such, she randomly selected 20 pupils from School A, 20 pupils from School B and 20 pupils from School C, and measured their academic performance as assessed by the marks they received for their end-of-year English and Maths exams. Therefore, the two dependent variables were "English score" and "Maths score", whilst the independent variable was "School", which consisted of three categories: "School A", "School B" and "School C". This latter variable is required to test whether there are any multivariate outliers i.

We do not include it in the test procedure in the next section because we do not show you how to test for the assumptions of the one-way MANOVA in this "quick start" guide. You can learn about our enhanced data setup content on our Features: Data Setup.By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service. Cross Validated is a question and answer site for people interested in statistics, machine learning, data analysis, data mining, and data visualization.

It only takes a minute to sign up. For each task, I have accuracy score and reaction times. Mainly, my doubts arise from the fact that I feel I don't understand when from a theoretical point of view a series of dependent variables can be considered a group.

I have recently answered a very similar question, maybe you want to take a look: Assessing group differences on multiple outcomes. However, as the questions have not been marked as duplicates and I am too new here to attempt itlet me add here the following.

Therefore I would start with separate t-tests, and if you can show that your groups differ according to several dependent variables, then perfect. Note that they should better differ consistentlye. See my linked answer about multiple comparisons. If you don't get convincing and consistent differences with individual t-tests, well, then you can try MANOVA -- again, see my answer with some further tips. I guess the answer is whenever you want. If you have several dependent variables, then whatever they are you can ask if your groups differ with regard to them.

Many different statistical tests of significance can be applied in research studies. Factors such as the scale of measurement represented by the data, method of participant selection, number of groups being compared and number of independent variables determine which test of significance should be used in a given study. The t test is used to determine whether two groups of scores are significantly different at a selected probability level.

Simple ANOVA is a test of significance used to determine whether scores from two or more groups are significantly different at a selected probability level. Whereas, the t test is appropriate test of difference between the means of two groups at a time e.

It is also possible to compute a series of t tests, one for each pair of means. If a research study uses a factorial design to investigate two or more independent variables and the interactions between them, the appropriate statistical analysis is factorial, or multifactor analysis of variance. This analysis yields a separate F ratio for each independent variable and one for each interaction. MANOVA will allow us to consider both independent variables gender and economic status and multiple dependent variables e.

Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. Asked 7 years, 2 months ago. Active 4 years, 7 months ago. Viewed 8k times. Active Oldest Votes. The "flag" link is right below the tags; "It is a duplicate You just have to paste the URL of the original question into the field that appears after you select that option.

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Surprised, you plot the raw data for both response variables using individual value plots.

Introduction to Two-Way Multivariate Analysis of Variance (Two-Way MANOVA)

This time the results are significant with p-values less than 0. You create a scatterplot to better understand the results. The individual value plots show, from a univariate perspective, that the alloys do not significantly affect either strength or flexibility. However, the scatterplot of the same data shows that the different alloys change the relationship between the two response variables.

That is, for a specified flexibility score, Alloy 3 usually has a higher strength score than Alloys 1 and 2. Usually, you should graph the data before conducting any analyses because it will help you decide what approach is appropriate. You can request to have these eigenvalues displayed.

If the eigenvalues are repeated, corresponding eigenvectors are not unique and in this case, the eigenvectors Minitab displays and those in books or other software may not agree.

MANOVA is a test that analyzes the relationship between several response variables and a common set of predictors at the same time. Increased power You can use the covariance structure of the data between the response variables to test the equality of means at the same time. If the response variables are correlated, then this additional information can help detect differences too small to be detected through individual ANOVAs.

Detects multivariate response patterns The factors may affect the relationship between responses instead of affecting a single response.

ANOVAs will not detect these multivariate patterns as the following figures show. Controls the family error rate Your chance of incorrectly rejecting the null hypothesis increases with each successive ANOVA. Doing one MANOVA to test all response variables at the same time keeps the family error rate equal to your alpha level. Note Usually, you should graph the data before conducting any analyses because it will help you decide what approach is appropriate.

Minitab automatically does four multivariate tests for each term in the model and for specially requested terms: Wilk's test Lawley-Hotelling test Pillai's test Roy's largest root test. All four tests are based on two SSCP sums of squares and cross products matrices: An H hypothesis matrix associated with each term; also called between sample sums of squares An E error matrix associated with the error for the test; also called within sample sums of squares.

By using this site you agree to the use of cookies for analytics and personalized content. Read our policy.Group j is said to have n j subjects in its sample. We also define. We use the following definitions for the total Tbetween groups B and within groups W sum of squares SSdegrees of freedom df and mean square MS :. In this case, you treat the repeated levels as dependent variables.

The total or grand mean vector is the column vector. The sample group mean vector for group j is a column vector. Example 1 : A new type of corn seed has been developed and a team of agronomists wanted to determine whether there was a significant difference between the types of soils that they are planted in loam, sandy, salty, clay based on the yield of the crop, amount of water required and amount of herbicide needed. Eight fields of each type were chosen for the analysis.

Based on the data in Figure 1, determine whether there is a significant difference between the results for each type of soil condition. Figure 1 — Data for Example 1 in standard form. We also calculate the total mean vector and group vectors expressed as row vectors in Figure Figure 2 — Total mean and group mean vectors.

The other total mean values are calculated by highlighting range GI10 and pressing Ctrl-R. It can be useful to create a chart with the group means shown in Figure 2. The result is shown on the left side of Figure 3. Figure 3 — Chart of group mean vectors. The group mean vectors all look fairly similar although as we will soon see there are significant differences.

It seems that the loam and sandy mean group vectors are very similar and a bit different from the salty and clay group mean vectors which are also very similar. These distinctions are even more evident when we look at the group means minus the total mean shown in Figure 4 below. The result is shown on the right side of Figure 3.

### Factorial Anova

Definition 2 : Using the terminology from Definition 1, we define the following total cross products for p and q. The multivariate equivalent of the total sum of square is the total sum of squares and cross productsi. Note that the diagonal terms are SS 1…, SS k. An alternative way of expressing T is as follows:. If the sample data is expressed as a range R1 in the format shown in Figure 1 what we will henceforth call the standard format and R2 is the total mean row vector, then T can be calculated by.

Definition 3 : We define the hypothesis cross products for p and q as follows:.