Table of Contents

Overview - Expectancy Models

Overview - Program Layout

Navigating Between Windows

Entering Required Input

Calculating Results

Computational Accuracy


Overview 1 - Expectancy Models

Models

This program is used to estimate the amount of workforce improvement that can be realized by implementing a valid selection procedure in an organization. The program quickly and accurately computes institutional expectancies under three different models, based on information that you supply:

+ Under the Taylor-Russell model, you specify the percentage of employees considered successful prior to introducing a new selection tool, the selection ratio you plan to use, and the correlation between the selection tool and job performance. The program calculates the percentage of selected employees expected to be successful.

+ With the Lawshe et al. model, you decide what predictor score intervals you want information about, the percentage of employees considered successful prior to introducing a new selection tool, and the correlation between the selection tool and job performance. The program calculates the percentage of applicants within each score interval that would be expected to succeed on the job if hired.

+ Under the Naylor-Shine model, you provide the mean and standard deviation of job performance of employees that is obtained without the selection tool, the selection ratio you plan to use, and the correlation between the selection tool and job performance. The program calculates the expected mean performance level of the group that is hired using the specified selection ratio.

Assumptions
All of these models assume a bivariate normal relationship between applicant scores on the selection tool and job performance. In practice, this assumption is often satisfied reasonably well when professional standards are observed in the development of predictors and the measurement of job performance. Users should be familiar with the implications of this assumption when interpreting the results calculated by the program.

For More Information
For more discussion of these expectancy models and their underlying computations, see Theoretical expectancies: Replacing classic tables with flexible, accurate computing procedures by Richard A. McLellan, a paper presented at the 12th Annual Conference of the Society for Industrial and Organizational Psychology, April 1997.

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Taylor-Russell Model

The reference for the Taylor-Russell model is:

Taylor, H. C. & Russell, J. T. (1939). The relationship of validity coefficients to the practical effectiveness of tests in selection: Discussion and tables. Journal of Applied Psychology, 23, 565-578.

The question addressed by this model is, what proportion of the selected group can be expected to perform at or above the level that minimally defines success, given a specific selection ratio and correlation coefficient?

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Lawshe et al. Model

The reference for the Lawshe et al. model is:

Lawshe, C. H., Bolda, R. A., Brune, R. L., & Auclair, G. (1958). Expectancy charts II: Their theoretical development. Personnel Psychology, 11, 545-559.

The question addressed by this model is, what proportion of the group whose predictor scores fall between two points is expected to be successful, given a particular correlation coefficient and a specified prior rate of success?

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Naylor-Shine Model

The reference for the Naylor-Shine model is:

Naylor, J. C. & Shine, L. C. (1965). A table for determining the increase in mean criterion score obtained by using a selection device. Journal of Industrial Psychology, 3, 33-42.

The question addressed by this model is, what is the expected mean job performance level of a group of applicants with predictor scores at or above a particular minimum, given the mean and standard deviation of job performance without the predictor, and a particular correlation coefficient?

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Overview 2 - Program Layout

There are four main screens you will use in this program - a "home window" and one window for each of the three expectancy models. You start out in the home window, where you choose the model that you want to use, and enter descriptive information about the predictor (i.e., the selection procedure).

Then, clicking on the "Continue" button brings up the window for the model you have chosen.

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Navigating Between Windows

Starting out
When you first start the Theoretical Expectancy Calculator and clear the welcome screen, you will be in the home window. To get past this window, you must first select one of the three expectancy models and type in the mean and standard deviation of the predictor in the yellow entry fields (see Entering Required Input). After that, simply click on the "Continue" button and a window will open for the expectancy model you have selected.

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Entering Required Input

The basic concept in this program is that on each window, there is an area for user input, and an area for results. Each item is clearly labeled, and once all required input has been entered, clicking the "Calculate" button initiates the computation of results.

Any piece of information that you will need to type in is shown as a box like this one:

Raw Score Mean

On all windows, the specific characters that you can type into the yellow entry fields are only those that would lead to valid values. For example, you will not be allowed to enter a negative sign in a field that is asking for a percentage between zero and 100. Text can be pasted into these fields from the Windows clipboard too, although characters that are not appropriate for the field will not be flagged until calculations are attempted.

Entry of characters is limited to the width of the entry field itself. Once the maximum allowed number of characters has been reached, no more input will be accepted. This is to ensure that no characters will scroll out of view, so that all results and input data are correctly displayed at all times (especially important if you are retaining printouts). You can, of course, use the direction keys, the backspace key, and the delete key to edit the entry you are making.

Finally, you will notice that when you change values in many of the entry fields after computing results, any results that depend on the value you changed will be erased. This includes any changes to the predictor mean and standard deviation on the home window. If these values are changed, dependent results on other windows will be erased when you switch back to those windows. Again, this is to avoid any possibility of displaying results based on values that are no longer showing in the input fields of the window.


Calculating Results

When you have entered all of the input values required for your analysis, simply click on the "Calculate" button. The appropriate results will be displayed.

When you click on the calculate button, the first thing that happens is that the program checks all of your input values to see if they are legitimate for the particular analysis you are doing. If anything is not valid (or if a required piece of information is missing), you will be given a message about what is wrong.


Computational Accuracy

All calculations within this program are carried out in double precision (i.e., to sixteen significant digits), and the algorithms used (see Computational References) are all accurate to 15 digits. For display, the results are rounded to three decimal digits. Therfore, all results displayed in this program are precise to (more than) the number of digits shown.

Computational References

There are four "numerical approximation" algorithms at the heart of this program. Approximations are required since we are dealing with infinitely increasing and decreasing functions.

The approximation algorithms do the following:

+ convert z to a proportion
+ convert a proportion to z
+ calculate the bivariate normal probability given z1 (predictor critical score), z2 (criterion critical score), and r (correlation between predictor and criterion)
+ find the ordinate (height) of a normal distribution at z

Converting z to proportion
This algorithm is based on the following source:

Kennedy, W. J. & Gentle, J. E. (1980). Statistical computing. New York: Marcel Dekker. (see pp.90-92 in particular)

Converting proportion to z
This algorithm is based on the following source:

Algorithm AS241, in Wichura, M.J. (1988). The percentage points of the normal distribution. Applied Statistics, 37, 477-484.

Bivariate normal probability
This algorithm is based on the following source:

Donnely, T. G. (1973). Algorithm 462: Bivariate normal distribution. Communications of the ACM, 16, 638.

Normal distribution ordinate
This algorithm is based on the following source:

StatSoft, Inc. (1995). STATISTICA for Windows [Computer program manual]. Tulsa, OK: StatSoft, Inc.