covariance matrix for unequal sample sizes. Cov is between two random vectors

Rasika -

This might explain this fairly well:

 http://www.itl.nist.gov/div898/handbook/pmc/section5/pmc541.htm

As I think Subrata indicated, you may be confused about which matrix you are examining, and I think that is the difference between the first and last matrices in the short attachment from a longer handbook shown by using this URL.  (I think there are some good parts, and at least one not-so-good part to the handbook from which this was taken.  I think the attached seems to be very helpful here for your question.) 

Note also, that for individual off-diagonal members of the covariance matrix, that is, the actual covariances, not variances, you can only include data points that are paired between the two variables on the column and row matches, for any one element of the matrix. So the sample data considered are of equal size, if that is part of your question. For example, if you have a variable on a periodically collected survey (say, for Official Statistics), and you look at a covariance between two periods for that variable, if the sample has changed between those periods, you can only use data to estimate covariance from those who responded to both surveys.  -  Perhaps that is not a good example here.  Consider then, two variables on the same survey.  If there is an item nonresponse for one, then the corresponding response for the other can be used in variance estimation for that one variable (item), but not for the covariance estimate between variables.  The covariance matrix would be a 2 x 2 matrix, with variances on the diagonal and the covariance repeated off-diagonal.  Sample sizes used for the covariance would be the same as the lesser of the sample sizes for the two variance estimations. 

Cheers - Jim 

PS - To be clear, it is the last matrix in the handbook section found through the URL here that is an example of a covariance matrix. The elements on the diagonal are variances. 

Thank you both. But actually real question is this. I have one data set which measures service quality (in social science ) measured with 24 question items (Likert scale) administered to customers. Totally about 1500. In addition to that we have administered 10 question items (Likert Scale) to 25 respondents (Organizational members) measuring market orientation. Now i want to calculate the relationship between market orientation and service quality. but here rows are not equal. one is having 1500 and other is having 25. i cant increase 25 into 1500 due to limitation of employees in the organization. I would be great full if you can advice on this  

Rasika -

So you have one set of questions for customers and another set of questions for organizations, and likert scales.  I don't see anything pairing specific observations that would be appropriate for obtaining covariances.

You wrote that "Now I want to calculate the relationship between market orientation and service quality..." but that seems to me to be a bit vague, not based on specifically paired, appropriate data for a covariance matrix. 

It sounds like you need to match customer responses with the organizations, and that would not be a one-to-one match.  I'm not familiar with likert scale data, but perhaps there is some standard technique for looking into this, but as your data currently are configured, if I understand this question, covariance matrices would not be an option.

Best wishes - Jim 

sampath, look at it like this. what you have is two groups of data of unequal sizes. what you are trying to do is to establish association between these two groups. in statistics group statistics are mainly dealt through t statistics, anova etc. you can look for weighted techniques, ancova etc. 

another concern: service quality causes mo or mo causes service quality?

Thank you shamil sir. Yes. This is one of MBA student's issue. I tried to help. As you said i took weighted values and made equal the rows. Problem solved. Thank you agian for the comment since it assure my decision. 

Dear Rasika,

As the others I understand:

1500 customer responses on one scale:'SQ  ( of 24 items)

25 Organizational member responses on another scale:  MO (of 10 items).

You seem to want to know if there is a relation between the  SQ and  the MO.

As Jim said before:"you need to match customers with the organizations,".

Like Jim , I do not see where a covariance matrix could help or fit in.

And: either  I do not understand your question either weighting is wrong as (in this case) it increases artificially the reliabiltiy of the variable MO.

Take Care,

Pierre

Thank you sir. Actually i may have worded the questions wrongly.

But when we calculate correlations i believe co-variance matrix also calculates and then finally get the correlation coefficient. I just wanted to test the correlation to help my student. What do you think. How should assist this student to calculate correlation.   

I have the following suggestion:

   You have qualitative variables.with m categories in one, and n categories in the other. 

Use the local ccvarianve between the random events {X=i} and {Y=j}. It is defined by the difference

P{X=i, Y=j} - P{X=i}.P{Y=j}.

This function is called "connection between the two events". Its values are between [-1/4, 1/4}, and actually measures the covariance between indicators of the two random events I_{X=i} and I_{Y=j}, i=1,...,m; j=1,...,n.

.

Dear Rasika,

You want to calculate correlations. Indeed, a covariance matrix gives you the unstandardized correlations (that is what covariances are).

The possibilities to calculate correlations in your data are:

you have:1500 subjects, customers, who responded on 24 items.

You can calculate interitem correlations (or Spearmans Rho because the data are likely to be ordinal), that would be 288 correlations, about 15 will be significant, by chance allone. You  should rather look at the size of the correlations and ignore the significance (besides: whith 1500 cases anything will be significant). These correlations give information about the coherence of the items (or 'internal consistency' of the scale/test/instrument).

The other dataset is 25 employees who have responded to 10 items.

Here you can do the same; that will give you 45 correlations, about 2 will be significant by chance alone. Again, focus on size. Here also these correlations give information about the coherence of the items.

As long as there is no relation defined between the customers and the employees (like: this (ot this group of) customer was  client of this (or this group of) employee) you can not calculate a correlation between the (sum)scores on the two different tests.

I hope this helps,

Cheers,

Pierre

Dear Rasika,

It is possible, but you have to make a homogeneity test previously. If the, e.g. Levene's test is NOT significant, then you can perform an ANOVA test.

If you make an multivariate analysis of variance (MANOVA) you have to mae an Box's M test and it also should NOT be significant (p>.005). Remeber, that .006 can already be interpreted as non significant based on Huberty and Petoskey's guideline!

Huberty, C. J., & Petoskey, M. D. (2000). Multivariate analysis of variance and covariance. Handbook of Applied Multivariate Statistics and Mathematical Modeling, 183–208.

Best,

Eugene