Anonymous:

Specific question: We discussed the test statistic based which was derived from SSM and SSE in the context of univariate ANOVA models. If a certain set of assumptions hold then the distributional assumption F­_g-1,n-g  for the statistic is plausible. Assumptions in a quick recap are: data from population (group) i is sampled from mean value m_i, homogeneity of variance in each group, normality and independent sampling from groups. My first question is how these set of assumptions change when we have more than one variable. We delved into the computations related to Wilks’ lambda but did not discuss the assumptions based on which we can contend that the statistic is F distributed. Does the usefulness of Wilks’ lambda necessitate assuming “multivariate normal distribution of order p” for variables?             Returning back to univariate two sample t test and Texas Tech dataset, I find that the variance homogeneity assumption to be less likely when considering the ariable FacTeaching. The histogram for BA class has more weights in extremes compared to class of non BA sample. Using the Levene’s test for difference in variance and deriving the p value equal to 0.0125, I am convinced that the t statistic needs to be adjusted for inequality in variance. In ISQS5347 we learned to replace the pooled variance with squared root of (S₁² /n₁+ S₂²/n₂).  Firstly how can we identify the heteroscedsticity in MANOVA model where we have multiple variables? For example in the Texas Tech dataset, for all variables except FacTeaching, I did not observe significant variance differences between BA and non BA observations, does that imply that the eventual computed Wilks’ statistic will be almost reliable? Is there any multivariate test for differences in variances among groups? Secondly, how can we extend the heteroscedasticity adjustment discussed in two sample t test to ANOVA and multivariate cases and derive better F and Wilks’ lambda statistics? 

Please work on writing.  You wrote

"We discussed the test statistic based which was derived from SSM and SSE in the context of univariate ANOVA models. If a certain set of assumptions hold then the distributional assumption F­_g-1,n-g  for the statistic is plausible."

Lots of problems. Better:

"We discussed the F statistic based on SSM and SSE in the context of univariate ANOVA models. If the model assumptions hold then the F statistic is distributed as F­_g-1,n-g  under the null hypothesis."  

Your next question has to do with extending univariate assumptions to multivariate. You stated the univariate assumptions as

"Assumptions in a quick recap are: data from population (group) i is sampled from mean value m_i, homogeneity of variance in each group, normality and independent sampling from groups. "

 To extend these to the multivariate case, here are the changes, along with fixing a couple minor problems:

data vectors in group i are sampled from a process with mean vector m_i, with common covariance matrices across groups, with multivariate normality within groups, and with all data vectors  independently sampled.

Then you said,

" In ISQS5347 we learned to replace the pooled variance with squared root of (S₁² /n₁+ S₂²/n₂).  ..."

Sure, we can use the same kind of combined covariance matrix in the multivariate case - (S1/n1 + S2/n2) where S1 and S2 are the within-group covariance matrices.  It's not as standard to do this, though.  

There are good reasons to use the pooled variance test: (1) the results have known distributions (F related).  When you use the (S1/n1 + S2/n2) types of poolings, nothing is exact anymore, it is only approximate. (2) Often the hypotheses of interest refer to equality of distributions, rather than just means (ie, a hypothesis might be Ho:  no difference between groups).  So you might as well make the distributions completely the same under your null.  Of course in this case you have to interpret the significance a little more carefully as a difference between distributions, as not specifically as a difference between means.  (3)  The problems with non-constant variance are extremely minor even when the null of interest is the mean comparison, especially when the sample sizes are roughly equal.  Even when the sample sizes differ greatly, it is quite unusual that the differences between variances is so  large that it hurts the mean test.   Too, in those cases where the variance difference is large, it is quite unlikely that the mean hypothesis would be of any interest at all, because it would be well understood that the groups are so different that the null hypothesis of difference between means is not even tenable, and therefore not worth testing.

If you need a test for equality of covariance matrices, let me know and I'll get you one.  It's real easy.  I just don't think it is worth spending time talking about it.   The covariance matrix test is extremely non-robust to non-normality, unlike the means test in addition.  So again, I'll get you a covariance test if you want it, or just google it.  It's called "Bartlett's test".  But I really would like to discourage you from going down that road, because it is usually silly to do so.

I know that econmetricians and finance researchers are getting giddy over heteroscedastic adjustments, and in some cases they are warranted, but in many others the use of heteroscedastic "robust" standard errors is like throwing a rug over the dog poo on the carpet rather than cleaning it up.  See  http://econpapers.repec.org/article/besamstat/v_3A60_3Ay_3A2006_3Am_3Anovember_3Ap_3A299-302.htm

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  General questions: This question concerns the problem when we observe several layers of clustering (grouping) rather than just one. For instance in time series sequences we expect to see temporally correlated observations. To illustrate the point consider a sample of companies with a class variable identifying five major industries. After some market wide major event and in specific points of time the price changes are measured for each firm. Therefore in this example we have variations caused within industries, within all time points in a specific company and between price changes of all firms in each point of time. We learned about mixed models in regression class, where each effect is implemented with the associated mean and interaction effects. Using mixed models we can model our problem as follows: R_ijk = m + a­_i + b_j(i) + t­_k +  (at)_ik + e_ijk             Where mu is the overall mean, alpha is the industry effect mean, beta is the effect associated to individual companies, tau is related to time points and interaction between time and industry is embedded in (at). Here we assume that the effect of time does not depend on company and no interaction is defined in this regard.             I think a better approach would be to follow from multivariate MANOVA model to tackle these settings. Treating measured price changes in time points as distinct random variables seems to result in a more parsimonious model. The vector of random returns for three time points would be as follows and in the right hand side we have only to include px1 vectors of overall, industry effect and company effect means. Are there any differences in inferences made from these two models? Which model is more constrained? Which model is advantageous?           [ R_ij1]      R_ij = | R_ij2]            _|| R_ij3]            

Note - without further qualifications, what you call a "mixed model" would be understood to be an "ANOVA model."  The qualifications that make it "Mixed" would be a statement that, eg, industry effects are random vaiables.  "Mixed" means specifically, "mixed fixed and random effects."   So without a statement of which effects are random, it could not be called a "Mixed model." 

When you want to emphasize that you are *not* using a mixed effects model, you would call it a "fixed effects ANOVA model".

Mixed models can indeed be used for Multivariate data.  Their advantages are more parsimonious covariance structures, and allowance of unbalanced data and certain missing data patterns.  Advantages of the multivariate approach are fewer assumptions regarding covariance and distributional structures.  It's not  slam dunk either way.  See

http://jfec.oxfordjournals.org/cgi/content/abstract/2/3/451  for an example of the use of the multivariate approach with non-normally distributed data for multivariate event studies.

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