STA 431 Assignment Eight:

Quiz in Tutorial on Friday March 11th

A farm co-operative (co-op) is an association of farmers. The co-op can buy fertilizer and other suppies in large quantities for a lower price, it often provides a common storage location for harvested crops, and it arranges sale of farm products in large quantities to grocery store chains and other food suppliers. Farm co-ops usually have professional managers, and some do a better job than others.

The file manager.data has data from a study of farm co-op managers. The dependent variable of interest is job performance, a latent variable. The independent variables are

The data file has these observable variables in addition to an identification code for the managers.

  1. w11: Knowledge measurement 1
  2. w12: Knowledge measurement 2
  3. w21: Profit-Loss Orientation 1
  4. w22: Profit-Loss Orientation 2
  5. w31: Job Satisfaction 1
  6. w32: Job Satisfaction 2
  7. x4:    Formal education, assumed measured without error
  8. v1:    Job Performance 1
  9. v2:    Job Performance 2

In this study, the double measurements are obtained by just splitting questionnaires in two, as in split half reliability. Furthermore, all the measurement errors are assumed independent of one another. This is consistent with mainstream psychometric theory, though maybe not with common sense. For this assignment, please assume that the measurement errors are independent of one another. Of course the errors (including εi) are independent of the independent variables, and εi is independent of the measurement errors.

This data set almost fits the double measurement design, but not quite. The problem is the presence of a variable x4 (education) that is measured without error. Later, we will prove a rule saying that if the parameters of a model are identifiable, adding an observable variable to the model results in a new model whose parameters are also identifiable. For right now please just take the rule for granted. Otherwise, you would have to prove identifiability by solving the covariance structure equations manually. It's not hard mathematically, but it's a big job. Anyway, we know that the parametersin the covariance matrix are identifiable, and as usual you'll confine your attention to those parameters.

  1. Using proc calis, fit the appropriate model. Using your list file when necessary, be ready to answer questions like the following on the quiz.
    1. There is one manifest exogenous variable. What is it?
    2. There is one latent endogenous variable. What is it?
    3. Based on the number of covariance structure equations and the number of unknown paramters, how many equality restrictions should the model impose on the covariance matrix? The answer is a single number; you need not say what they all are.
    4. Does your model fit the data adequately? Answer Yes or No and give three numbers: a chisquare statistic, the degrees of freedom, and a p-value.
    5. Controlling for knowledge, profit-loss orientation and job satisfaction, is there evidence that formal education is related to job performance? Answer Yes or No and give the value of a test statistic (actually it's a Z) that supports your conclusion. Of course in all these questions you are using the α=0.05 significance level and a 2-sided test.
    6. Controlling for formal education, knowledge and profit-loss orientation, is there evidence that job satisfaction is related to job performance? Answer Yes or No and give the value of a test statistic (actually it's a Z) that supports your conclusion. If the answer is Yes, say whether satisfaction is positively related to performance, or negatively related.
    7. Controlling for job satisfaction, formal education and knowledge, is there evidence that profit-loss orientation is related to job performance? Answer Yes or No and give the value of a test statistic (actually it's a Z) that supports your conclusion. If the answer is Yes, say whether profit-loss orientation is positively related to performance, or negatively related.
    8. Estimate the reliability of Knowledge Measurement 1. Your answer is a number.
    9. Estimate the reliability of Knowledge Measurement 2. Your answer is a number.
    10. Show that the reliabilities of Knowledge Measurements 1 and 2 are equal if and only if the variances of the two measurement error terms are equal. This is the basis of the next (and last) you are asked to carry out.
  2. Carry out a likelihood ratio test of the null hypothesis that the variances of the two measurement error terms for Knowledge Measurements 1 and 2 are equal. By the last calculation you did, this is equivalent to testing whether the two reliabilities are equal.
    1. What is the value of the chisquare statistic? The answer is a number.
    2. What are the degrees of freedom? The answer is a number.
    3. What is the p-value? The answer is a number. It would be best to use proc iml for this, so the number will be on the printout.
    4. Do you reject the null hypothesis at α=0.05? Answer Yes or No.
    5. What do you conclude about the reliabilties of the two measurements? Do you have sufficient evidence to conclude that they are different? Answer Yes or No. If the answer is Yes, say which one is more reliable. (Of course the answer may not be Yes).

    If you use the lincon statement to carry out the likelihood ratio test for the last item (recommended), you will get a loud warning that there is "one active constraint at the solution," and there is a lot of stuff about altering the degrees of freeedom. All that is completely inappropriate and beside the point. The active constraint is ω1 = ω2, which is your null hypothesis. You can always ignore warnings about active constraints when they come from null hypotheses that you have imposed with lincon. On the other hand, active constraints that come from the bounds statement are a different matter, and should be taken seriously.