Question

A researcher is interested in establishing if there is a difference between cricketers and hockey players as measured by three dimensions of athletic ability: 1. Strength 2. Agility 3. Balance A MANOVA is undertaken on data collected from 115 Cricketers and 73 Hockey players, the output from which is given below a) Why is the MANOVA procedure appropriate in this instance? b) Discuss the checks that the researcher should have made to ensure no violation 2 marks of MANOVA assumptions 10 marks 18 marks TOTAL 30 marks c) Interpret the MANOVA output produced by the researcher: Between-Subjects Factors cricket or Hockey 1 Cricket Hockey 115 Descriptive Statistics Deviation Mean 11.3739 11.5068 11.4255 3.0437 3.3830 3.1755 9.9304 0.9589 cricket or Hocke Strength Cricket 2.83295 2.92557 2.86222 37606 41320 42360 1681 3.29325 115 73 188 115 73 188 Total Agility Cricket Hockey Total Balance Cricket Hockey 73 188 Box's Test of Equality of Covariance Box's M 12.356 2.021 dfl df2 157011.737 059 Sig hypothesis that the observed covariance matrices of the dependent variables are equal across groups a. Design: Intercept + group note: output continues overleaf]

Answer 1

a)
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.
Multivariate analysis of variance (MANOVA) is simply an ANOVA with several dependent variables. That is to say, ANOVA tests for the difference in means between two or more groups, while MANOVA tests for the difference in two or more "vectors" of means.

Here, vectors would be
vector-1: means of Strength, Agility & Balance of Cricket
vector-2: corresponding means of Strength, Agility & Balance of Hockey

[Cricket -- measured at three levels : Strength, Agility, Balance

Hockey -- measured at three levels : Strength, Agility, Balance]

Thus, MANOVA is appropriate here.

b)
Checks that the researcher should have made to ensure no violation of MANOVA assumptions:

One way to check this is :

Use the residual plots to determine whether the model is adequate and meets the assumptions of the analysis. If the assumptions are not met, the model may not fit the data well and caution is needed when interpret the results.

Assumptions:
--the residuals are randomly distributed and have constant variance.[In Residuals versus fits plot, the points should fall randomly on both sides of 0, with no recognizable patterns in the points. The patterns like curvilinear, points far away from zero etc may indicate that the model does not meet the model assumptions.]

--the residuals are independent from one another.[In Residuals versus order plot,ideally, the residuals on the plot should fall randomly around the center line.Independent residuals show no trends or patterns when displayed in time order. Patterns in the points may indicate that residuals near each other may be correlated, and thus, not independent.]

--the residuals are normally distributed.[Points should fall along the straight line. If in the normal probability plot, the points generally follow a straight line, then there is no evidence of nonnormality, outliers, or unidentified variables.]
  
c)
Note: If p-value is less than alpha=0.05, the the variable is significant.

The Box’s Test of Equality of Covariance Matrices checks the assumption of homogeneity using covariance across the groups using p-value as a criterion. For this example, the Box’s M (12.356) is not significant, p (0.059) > alpha (0.05) – indicating that there are no significant differences between the covariance matrices. Therefore, the assumption is not violated and Wilk’s Lambda is an appropriate test to use.

MANOVA using the Wilk’s Lambda test:
Using an alpha level of .05, we see that this test is significant, Wilk’s lambda (for group) = .835, F(3,184) = 12.092, p < .001, multivariate partial eta-squared =0.165.
This significant F indicates that there are significant differences among the groups on a linear combination of the dependent variables. [If we had violated the assumption of homogeneity of variance-covariance, we would use the Pillai’s Trace test.]

The Levene’s Test of Equality of Error Variances tests the assumption of MANOVA and ANOVA that the variances of each variable are equal across the groups. Here,all p-values (0.901,0.630,0.091) are greater than alpha=0.05, the Levene’s test is insignificant, this means that the assumption is met. Variances are equal across groups.

Tests of Between-Subjects Effects:
As we conclude that MANOVA is significant, further univariate ANOVA is carried out. Univariate ANOVAs indicated that,
For Strength: F(1,186) = 0.096, p-value =0.757,
For Agility: F(1,186) = 33.646, p-value=0.000,
For Balance: F(1,186) = 5.415, p-value=0.021

Here, for Agility and Balance, p-value is less than alpha, therefore, dependent variable Agility and Balance are significant.