Determining the Correlation Structure of Random Variables

You can determine the correlation structure of the random variables, which are initially generated without correlations. Variables are replaced by linear combinations to yield the specified correlation structure. This will change the probability distribution in most cases, although Normal variables will still follow that distribution.

About this task

Use the dialog to determine the correlation structure of the random variables, which are initially generated without correlations.

Procedure

  1. Rationalization: The variables that are generated in the first stage will not be perfectly uncorrelated due to randomness. Click the radio button for Exact correlations, to orthogonalize the data before transformation. The resulting variables will have zero correlation with each other. If a correlation structure is, then specified, the final result will be variables that have exactly that structure. Otherwise, the correlation structure varies due to randomness.
  2. Correlation Structure: Choose the desired correlation structure and enter the required parameters in the edit field below. The Help in the Analyze>Mixed Models>Linear dialog gives the definitions of these structures.
    1. None: The variables will be approximately or exactly uncorrelated. Leave the edit field empty.
    2. Equal: Each pair of variables will have the same correlation. Enter one correlation coefficient.
    3. Toeplitz: The correlations will be the same down each diagonal of the correlation matrix. Enter as many correlation coefficients as there are variables starting with 1.
    4. Factor Analytic: The correlation for each pair of variables will be the product lambda(i) * lambda(j) plus a constant, d, on the diagonal. Enter d followed by one lambda for each variable.
    5. Arbitrary: The correlation for each pair will be a specified value. Enter the lower triangle of the correlation matrix, ending each row with 1.
    6. Random: The correlation for each pair will be a random value drawn from a uniform distribution. Enter the minimum and maximum correlations.
  3. Display requested correlation matrix: Check this box to display the specified correlation matrix.
  4. Correlation Parameters: Enter the parameters appropriate for the chosen correlation structure separated by blanks. Always use the period as the decimal point.