13 MANOVA: Multiple Analysis of Variance

13.1 Outliers

To determine whether the data set has outliers let’s use a graphical representation. The primary plot represents the ordered squared robust Mahalanobis distances against the empirical distribution function \(MD^2_i\). Of the three additional graphs the first shows the data, the second attempts to highlight potential outliers as detected using the \(chisq_p\) distribution, and the third presenting the outliers by the adjusted quantile (Filzmoser, Garret, and Reimann ⊕2005Filzmoser, P., R.G. Garret, and C. Reimann. 2005. “Multivariate Outlier Detection in Exploration Geochemistry.” Computers and Geosciences, no. 31: 579–87.).

Projection to the first and second robust principal components.
Proportion of total variation (explained variance): 0.9381

Figure 13.1: Outliers in the dataset

Two of plots in Figure 13.1 show two cases, 42 and 99, as potential outliers while the third, the outliers based on adjusted quantiles, suggests that the dataset has no outliers. In this case, the decision is left to the researcher. Each outlier should be analyzed individually and a decision should be made if it is to be removed or not. The criteria used for such decisions can range from theoretical necessity of including the possible outliers in the analysis to the point highlighted as outlier to be a valid data point based on an in-depth analysis of the data. In this example, because the analysis does not conclusively highlight the two cases as outliers, they will not be removed9494 A more extensive analysis, outside the scope of this chapter, focused on the two possible outliers conducted on the dataset led to the same conclusion..

13.2 Assumptions

Before conducting the analysis, assumptions of normality, homogeneity of covariances, homogeneity of variances, and independence of observations need to be tested.

Summary statistics including skewness and kurtosis9595 In the output, X1 refers to the dependent variable being analyzed: punish, repeats, dispos..

DV: punish

 Descriptive statistics by group 
group: African-American
   vars  n  mean    sd median trimmed   mad min max range skew kurtosis
X1    1 24 49.33 11.09   50.5   49.25 13.34  30  71    41 0.08    -1.13
     se
X1 2.26
-------------------------------------------------------- 
group: Hispanic
   vars  n  mean    sd median trimmed   mad min max range  skew kurtosis
X1    1 26 57.77 14.71   60.5   58.64 16.31  28  79    51 -0.55    -0.68
     se
X1 2.89
-------------------------------------------------------- 
group: Caucasian
   vars  n  mean    sd median trimmed   mad min max range  skew kurtosis
X1    1 55 65.89 10.54     67   65.91 11.86  46  85    39 -0.04    -1.13
     se
X1 1.42

The output shows various basic statistics, such as the mean of the variable, its median, mean, and so forth. Of interest here is the reported swkewness and kurtosis. The values suggest a normally distributed dataset for the punish variable for each of the three ethnic groups. A similar analysis is performed below for the two other variables in the analysis, repeats and dispos. In each case, the skewness and kurtosis are well within the limits.

DV: repeats

 Descriptive statistics by group 
group: African-American
   vars  n  mean   sd median trimmed  mad min max range skew kurtosis   se
X1    1 24 24.12 2.17     24   24.05 2.97  21  28     7 0.26    -1.24 0.44
-------------------------------------------------------- 
group: Hispanic
   vars  n  mean   sd median trimmed  mad min max range skew kurtosis   se
X1    1 26 24.23 2.25     24   24.32 2.97  20  28     8 -0.3       -1 0.44
-------------------------------------------------------- 
group: Caucasian
   vars  n  mean   sd median trimmed  mad min max range  skew kurtosis
X1    1 55 25.53 1.99     26   25.58 1.48  21  29     8 -0.21    -0.62
     se
X1 0.27

DV: dispos

 Descriptive statistics by group 
group: African-American
   vars  n mean   sd median trimmed  mad min max range skew kurtosis   se
X1    1 24   23 2.27     23    22.9 2.97  20  28     8 0.26    -1.03 0.46
-------------------------------------------------------- 
group: Hispanic
   vars  n  mean   sd median trimmed  mad min max range  skew kurtosis  se
X1    1 26 23.46 2.04     24   23.45 1.48  20  27     7 -0.15     -1.1 0.4
-------------------------------------------------------- 
group: Caucasian
   vars  n  mean   sd median trimmed  mad min max range  skew kurtosis
X1    1 55 24.91 2.47     25   25.07 2.97  20  29     9 -0.44    -0.84
     se
X1 0.33

13.2.1 Visual Evaluation of Data Normality

For analysis we produce the appropriate qq-plots (Fig. 13.2) and histograms (Fig. 13.3). A cursory review of the plots seem to suggest that departure from normality may exist, at least for some of the groups. Observe the top-left plot in Fig. 13.2 (level of punishment for African Americans). The points representing the data align neatly along the diagonal line, which suggests that data is normally distributed. Other plots show an S-shaped distribution of the data point alongside the diagonal, which suggests departures from normality. The histograms in Fig. 13.3 seem to suggest similar possible departures from normality. In this situation further analysis is necessary to determine if the assumption of normality is met.

Figure 13.2: qq-plots

Figure 13.3: Histograms

    Box's M-test for Homogeneity of Covariance Matrices

data:  variables
Chi-Sq (approx.) = 17, df = 12, p-value = 0.2

Level of Puhishment

Levene's Test for Homogeneity of Variance (center = median)
       Df F value Pr(>F)
group   2    1.14   0.32
      102               

Potential to Repeat

Levene's Test for Homogeneity of Variance (center = median)
       Df F value Pr(>F)
group   2    0.55   0.58
      102               

Natural Disposition

Levene's Test for Homogeneity of Variance (center = median)
       Df F value Pr(>F)
group   2    0.65   0.52
      102               

13.3 The MANOVA Test

Now that the assumptions have been tested and the results show that the data set is appropriate for analysis, run the MANOVA test.

           Df Wilks approx F num Df den Df  Pr(>F)    
ethnicity   2 0.731     5.66      6    200 1.9e-05 ***
Residuals 102                                         
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

To better understand what the results tell us let’s look at the effect size (\(\eta^{2}\)). The threshold values for evaluating the effect size (partial \(\eta^{2}\)) are .01 for small effect size, .08 for moderate effect size, and .14 for large effect size. Any value below .01 indicates no effect.

           eta^2
ethnicity 0.1452

The write-up may look like below:

A significance value p < .001, together with F(6,200) = 5.66, p < .05 and a large effect size, suggested by a partial \(\eta^{2}= .145 (> .14)\) for the multivariate tests, indicate a significant multivariate effect of ethnicity on the three dependent variables. Therefore, the null hypothesis (H₀) can be rejected.

 Response 1 :
             Df Sum Sq Mean Sq F value Pr(>F)    
ethnicity     2   4768    2384    17.1  4e-07 ***
Residuals   102  14243     140                   
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

 Response 2 :
             Df Sum Sq Mean Sq F value Pr(>F)   
ethnicity     2     48    23.8    5.42 0.0058 **
Residuals   102    449     4.4                  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

 Response 3 :
             Df Sum Sq Mean Sq F value Pr(>F)   
ethnicity     2     76    37.8    6.97 0.0014 **
Residuals   102    553     5.4                  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Level of Punishment

          Partial eta^2
ethnicity        0.2508
Residuals            NA

Potential to Repeat

          Partial eta^2
ethnicity       0.09601
Residuals            NA

Natural Disposition

          Partial eta^2
ethnicity        0.1203
Residuals            NA