﻿ Item scores and correlations

# Item scores and correlations

A person gets a score for each item s/he answers.  For example, for a cognitive item, the custom is to give one point for a right answer, and zero points otherwise.  For affective items, each possible response usually has a certain number of points associated with it.  "Strongly agree", for example, might equate to a score of 5 points, while "strongly disagree" might be made equal to just a single point.

In Lertap, a person gets a score on each item even when no answer is given.  For cognitive items, a non-response usually equates to zero points, while for affective items Lertap will apply MDO, the "missing data option", to non-responses.  MDO usually equals the mean of the possible scores for an item -- for example, if the possible range is 1.00, 2.00, 3.00, 4.00, 5.00, then MDO=3.00 (note).

When asked to do so, Lertap will get all the item scores together, and write them to a new worksheet named IStats.   An IStats worksheet may have two sections -- the first section has rows whose cells contain the score each person earned on each item.

The second section is optional, depending on a setting in the System worksheet.

If row 22 of the System sheet has Yes as its "Present setting", then you'll find IStats will include rows of descriptive statistics for each item, such as the median, mean, and standard deviation.  These rows will be followed by an interitem correlation matrix.  Under "normal" conditions, a single matrix of Pearson product-moment correlation coefficients is made, with a row of average correlations included at the bottom of the matrix.  Lertap uses standard Excel functions for all of these calculations.  Average correlations are computed by using the n-1 non-diagonal entries in each column, where "n" is the number of items.

There are two settings in the System worksheet which affect the information found under this section of the IStats worksheet.  One of them replaces the diagonal element of the correlation matrix with the "SMC", the squared multiple correlation.  The SMC for an item is often used in factor analysis as an initial estimate of the item's common variance.

Prior to October 2004, Lertap used Excel's MINVERSE matrix function to invert the correlation matrix, part of the process of calculating SMC values.  It was found, however, that MINVERSE would regularly fail when asked to invert matrices with 50 or more subtest items, and a switch was made to the M_INV function found in the Foxes Group's package of matrix routines.  (For more about the Foxes Group, see the eigenvalue topic.)

Both MINVERSE and M_INV require a scratch area to work with, and Lertap uses the IStats worksheet itself for this (on slow computers, or cases where many items are involved, you might see Lertap scratching about).

The standard setting for SMC calculations is "off"; to activate SMC output, change the setting seen in Row 21, Column 2 of the System worksheet.  If this setting is changed, it takes effect immediately.  (However, this doesn't mean that any correlations matrices you may have already made will immediately change; in fact, they won't -- you'll have to delete or rename the IStats worksheet, and then get Lertap to make a new IStats sheet.)

For additional information about Lertap's calculation of SMC values, just page ahead to the next topic.

You'll also find that Lertap will endeavor to find the eigenvalues of the correlation matrix.  Eigenvalues are also known as "latent roots".  Read more about them by paying a visit to this topic.

Other settings in the System worksheet determine whether or not Lertap might add a matrix of tetrachoric correlations to the IStats output, and possibly make worksheets designed for export to other data analysis systems.  There's more about these matters in the following topics.

A special macro, the "IStatsPruner" is available for dissecting an IStats summary. It creates an IScores worksheet with only the item scores, and an ICorrs worksheet with only the correlations.

For more about IStats sheets, please refer to the manual.  You'll find an example in Chapter 7, and a fairly thorough discussion under the "Item scores matrix" section in Chapter 10.  Note, however, that the SMC, tetrachoric, export worksheets, and eigens options were added after the manual was printed.