# Population Structures for Monte Carlo Experiments

In Monte Carlo analysis, the program generates data that simulate samples from a particular population structure.  The option you check in this group determines how that population is obtained.  These options are available on the Advanced tab of the Monte Carlo Analysis dialog.

Model. If you select this option, the program generates a population structure corresponding to the current model. Whatever model is active in the current window, with the starting values specified for the parameters, is used to generate a population covariance matrix. You can specify the numerical value you want for various free parameters by specifying these values as starting values within braces after the free parameter number. So, for example, if you wanted your population to be based on a factor model where the first factor loads .5 on the first variable, you would have a line like this in your model description:

(F1)-1{.5}->[X1]

Data. If you select this option, the program will take your data, calculate the appropriate matrix (correlation, covariance, augmented moment) and generate simulated data from it based on the currently selected distributional characteristics.

It is important that you understand the fine points of how this option works. If your input data file is a correlation or covariance matrix, the program will treat that as the population matrix, or convert it to the appropriate matrix type if necessary. If your input file contains raw data, the program will calculate the appropriate matrix type, and sample from that matrix.

This means that, if your file contains raw data, and you have selected the default distributional options, the program will calculate the covariance matrix corresponding to your data, and will generate simulated data from a Multivariate Normal distribution having those characteristics.

If you have a raw data file, and you want a simulated population that exactly mirrors the characteristics of those raw data, you should use the Bootstrap option below.

Bootstrap. This option may only be selected if the data file contains raw data. In bootstrapping, a simulated sample of size N is created by treating the current data file as a discrete multivariate population, where each observation vector occurs equally often. The bootstrapped sample is obtained by sampling, randomly with replacement vectors of observations from the current data file. So, suppose the current data file contains 100 observations on 10 variables. If you request the Bootstrap option with Sample Size of 50, the program will, 50 times, sample randomly with replacement from the integers from 1 to 100. The resulting list of 50 integers is used to select which observations from the data file are included in the bootstrapped sample. Note, it is possible (likely, in fact) that some of the observations in the resulting data file will be identical. This occurs if, during the sampling of integers, the same integer is selected more than once.