During my graduate work at San Diego State University (MA, Psychology ’11) I spent the bulk of my time developing behavioral models to study healthy aging; more specifically, I used existing/published paradigms using hippocampal lesioned rodents to test some of the plausible sites of neural deterioration in a strain of rat known to have fairly low incidence of ageassociated disorders.
At its most basic AN(C)OVA, and its multivariate counterpart MAN(C)OVA, tests for meanvector differences between groups of discrete variables (i.e., data that is nominal, ordinal, bivariate, etc.). For example, if we are simply comparing two groupmeans, a oneway ANOVA will produce the same result as an independent samples ttest, or a ttest for dependent samples if we are comparing two variables in one set of observations (e.g., how a group of students function on one test that contains multiple memory components). At the very heart of the underlying theory of ANOVA is, you guessed it, the analysis of variance. That is, the functionality of this type of analysis comes from the fact that we can partition variance (s).
(i.e., ())
I will illustrate the partitioning of variance using a very simple example:
Table 1 
Group 1 
Group 2 
Observation 1 
2 
6 
Observation 2 
3 
7 
Observation 3 
1 
5 
Mean 
2 
6 
SS 
2 
2 
Overall Mean 
4 

Total SS 
28 
As we can see the means for the two groups (Group 1, Group 2) look to be quite different from one another (M_{Group }_{1} = 2; M_{Group 2} = 6), but the sum of squared deviations from the mean (SS within groups in this case) happens to be the same (SS_{Group }_{1} = 2; SS_{Group 2} = 2). If we totally ignore group membership (i.e., Group 1 v. Group 2), we can see that our SS value increases to 28 (SS Total). Ultimately, a ratio is made of the variance between the groups compared to the variance within the groups (s_{Between}/s_{Within}) and a value is obtained (Fvalue). As previously mentioned, ANOVA allows us to test if it is mathematically beneficial to group things together based on the partitioning of individual/group variance.
Lets look at a more in depth version of the previous example:
Table 2 
Group 1 
Group 2 
2 
6 

Males 
3 
5 
1 
7 

Mean 
2 
6 
4 
8 

Females 
5 
9 
3 
7 

Mean 
4 
8 
Controlling for factors: We can partition the variance from the Table 2, above, into a couple of distinct sources, including: (1) error withingroups (i.e., the male participants in Group 1 have different scores); and (2) variability betweengroups due to group membership (i.e., Males v. Females; Group 1 v. Group 2).
MANOVA
A multivariate analysis of variance (MANOVA) is an extension of ANOVA. When using MANOVA we have the ability to analyze two or more correlated dependent variables (DV’s) together. This can be advantageous for a number of reasons: (1) we are going to be mathematically accounting for the partial redundancy between the correlated DV’s; and (2) MANOVA can detect when groups differ on a system of variables (taken individually, the DV’s may not show significant group differences, but taken together¾as a system defining one or more theoretical constructs–differences caused by the independent variables may be revealed). This is accomplished by finding a linear composite of the dependent measures that maximizes the separation between groups defined by the independent variables, resulting in the most statistically significant value of Wilk’s lambda.
The logic and nature of the computations do not change when there is more than one DV at a time. For example, we may conduct a study where we try two different textbooks for both math and physics. In this case, we have two different dependent variables (i.e., math skills and physics skills), and instead of using a univariate F, we have a multivariate F (Wilk’s l) based on a comparison of the error variance/covariance matrix and the effect variance/covariance matrix. The sampling covariance(s) can be defined as follows:
At this point the covariance is included because the two dependent measures are, in all likelihood, correlated, and we have to take this correlation into account. The problem with MANOVA is that if the overall multivariate test is significant, we conclude that the respective effect (i.e., Textbook) is significant; however, we do not know if math skills improved, physics skill improved, or both.
Neural activation induced by a stimulus, such as an odor, image, or sound, can be recorded with multielectrode arrays (MEA; from Wu, Kendrick, & Feng, 2007). Considering that we are recording from hundreds or even thousands of local field potentials, the data sets from these types of experiments can be hundreds of Gigabytes. The question that is address by Wu et al. (2007) is ‘how do we differentiate neuronal network activity with MANOVA when we know that a traditional MANOVA is designed to test the changes in mean activities’; more specifically, the mean activity of a neuronal network could remain, on average, unchanged when a stimulus is presented, but it could reorganize itself to show more or less coherent network activity (e.g., Consider two neurons firing independently, but synchronizing themselves when a stimulus is applied. The mean firing rates before the stimulus, and during the stimulus are identical). As mentioned by Wu et al. (2006; 2007), a prerequisite for the applicability of [adaptive] MANOVA is that the covariance matrices among the variables are equal. The problem with this is that for the local field potentials and spike trains data, the correlations between variables are likely to be highly…variable. A Fisher’sZ transformation and Edgeworth expansion must be applied to transform the distribution of covariance, which is not ‘normal’, into Gaussian distribution (which is one of the assumption of MANOVA).
Let X_{1} = (x_{11},…,x_{1m}) (prestimulus Local Field Potentials; LFPs), and X_{2} = (x_{21},…, x_{2m}) (duringstimulus LFPs) be two random matrices; the spatial and temporal pattern of the recordings can be represented by the rows and columns of the two matrices (X_{1} and X_{2}). For example, consider x_{1i} is the one prestimulus LFP variable recorded during time period [0, Nh], where h is the sampling resolution and N is the total sample size. Correspondingly, we would represent the random covariance matrix of X_{1} and X_{2} as follows:
The main MANOVA test is designed to find all variables between X_{1} and X_{2}, which are ‘significantly’ different (based on the assumption that ). In other words, we have to know if the aforementioned assumption is true or not.
In order to evaluate Wilk’s l, we need the covariance matrices of both populations. The covariance matrice is assumed to be equal and generated randomly. As previously mention, MANOVA is defined by Wilk’s l, the significance of which is given by the following:
where is the upper (100a)th percentile of a Chisquare distribution with m degrees. The element of set A is the index of the variable (i.e., the ith element of A stands for the ith variable for both populations ().
Further review of this novel approach to using MANOVA in MEA research can be found in the article written by Wu et al., 2007. The novel conclusion found is that by applying correlational MANOVA to simulated and biological data sets we can detect variables which statistically responsive to an external stimulus. Additionally, due to the interactions between correlated dependent variables, the proper analysis of MEA requires a tool that takes those interactions and redundancies into account.
References
Wu, J.H., Kendrick, K., & Feng, J.F. (2006). Detecting hotspot in spatiotemporal patterns of complex biological data. Online Source: http://www.dcs.warwick.ac.uk/~jianhua/doc/hottor.html.
Wu, J.H., Kendrick, K., & Feng, J.F. (2007). Detecting correlational changes in electrophysiological data. Journal of Neuroscience Methods, 161, 155165.
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