University of Illinois IRT Laboratory

Investigating Unidimensionality for Dichotomous Data

Once again, for the models we are presenting on this website, unidimensionality is necessary. This page will walk you through an example of a PAF analysis with data from the Agreeableness scale (dichotomously scored; e.g., right-wrong or 0,1). It is important to note that with dichotomous data, one must conduct a factor analysis on tetrachoric correlations. This is due to the fact that while 0's and 1's may have been obtained from the measure, a continuum underlies the data.

Note, that if one dichotomizes raw data, by selecting a threshold where values less than the threshold are coded as 0 and those greater than the threshold are coded as 1, tetrachoric correlations will not exhibit artifacts of this process. That is, they will be unaffected by item difficulty, they will satisfy the assumptions for factor analysis, and the resulting factor analysis will not possess difficulty factors (Hulin, Drasgow, & Parsons, 1983).

Of the three most popular statistical analyses packages, it is our understanding that only SYSTAT can provide tetrachoric correlations. Hence, we will be using the SYSTAT 8.0 software package in this example. The procedure is as follows:

Opening the data in SYSTAT

Open the dichotomous data file in SYSTAT

From the menu...
Go to 'File'
'Open' the appropriate file, selecting the appropriate file type

View the data

Go to 'View'
Go to 'Data'

Obtain tetrachoric correlations

Go to 'Statistics'
Select 'Correlations'
Then 'Simple'
In the dialogue box, select 'Binary data' and choose 'Tetra' under the pull-down menu
Check the box for 'Save file' (as we wish to save the correlation matrix that SYSTAT will create)
Add scale items in the 'Variable(s)' list (e.g., the 10 items from the Agreeableness scale)
Click 'OK' and indicate a filename for your correlation matrix

View of the Correlations dialogue box in SYSTAT

Systat 8.0

This procedure should save a correlation matrix based on the tetrachorics that will later be analyzed with the PAF procedure described next. Note that in some versions of SYSTAT, one may need to redeclare this file as a correlation matrix instead of the default "similarity matrix."

Initiating a factor analysis

First, open the saved correlation matrix in SYSTAT. This file should be a triangular matrix and displayed in the Data Viewer.

Perform a factor analysis to determine number of factors underlying the data

From the menu....
Go to 'Statistics'
Go to 'Data Reduction'
Go to 'Factor Analysis'

Factor analysis in the SYSTAT menu

Systat 8.0

Select all the items that are within the subtest or scale (i.e., the 10 items from the Agreeableness scale) and add them to the list of 'Model variables'
Set 'Method' to 'Iterative principle axis'
Indicate the sample size for matrix input (i.e., 1500 in our example)
Keep 'Rotation' set to the default which is 'no rotation' (other rotations can be explored later)
If desired, go to 'Save' and select any additional information in the output to be saved if desired
Click 'OK'

Specifying the settings for a factor analysis in SYSTAT

Systat 8.0

Sample view of the syntax (partial)

USE "C:\Program Files\SYSTAT 10\tetracor.syd"
REM -- Following commands were produced by the FACTOR dialog:
FACTOR
MODEL A1 A2 A3 A4 A5 A6 A7 A8 A9 A10
 PRINT SHORT
ESTIMATE  / METHOD=IPA LISTWISE N=1500 EIGEN=0.0
REM -- End of commands from the FACTOR dialog

Interpreting the SYSTAT output file

View the Factor Pattern

View of the factor patterns for the Agreeableness scale (SYSTAT output)

Factor pattern
 
                         1           2           3           4           5
 
   A1                    0.767       0.156       0.132       0.011      -0.122
   A2                    0.618       0.156      -0.206       0.126       0.015
   A3                    0.511      -0.077       0.005      -0.188      -0.041
   A4                    0.794      -0.228      -0.147      -0.070       0.028
   A5                    0.794       0.036       0.040       0.004       0.079
   A6                    0.552      -0.249       0.162       0.110      -0.012
   A7                    0.745       0.020       0.035       0.131       0.032
   A8                    0.656       0.042      -0.073      -0.031      -0.165
   A9                    0.588      -0.113      -0.015      -0.004       0.060
   A10                   0.617       0.223       0.081      -0.118       0.122
 
                         6           7           8
 
   A1                   -0.010      -0.092      -0.020
   A2                   -0.006      -0.048       0.036
   A3                   -0.109       0.035       0.062
   A4                    0.028      -0.082       0.014
   A5                    0.112       0.114       0.036
   A6                    0.049      -0.041       0.036
   A7                   -0.144       0.066       0.013
   A8                    0.065       0.080      -0.033
   A9                   -0.033       0.016      -0.139
   A10                   0.022      -0.051      -0.007

Examine the Eigenvalues

Note how large each factor is and the differences between them
In our example, the first factor is large and subsequent factors small, which supports our assumption of unidimensionality

View of the Eigenvalues: Agreeableness scale

Latent Roots (Eigenvalues)
 
                         1           2           3           4           5
 
                         4.508       0.235       0.123       0.100       0.071
 
                         6           7           8           9          10
 
                         0.055       0.047       0.029      -0.002      -0.005

Examine how much of variance is explained by each factor

Variance explained by factor

Variance Explained by Factors
 
                         1           2           3           4           5
 
                         4.508       0.235       0.123       0.100       0.071
 
                         6           7           8
 
                         0.055       0.047       0.029

Examine the percentage of variance explained by each factor

Percent of variance explained

Percent of Total Variance Explained
 
                         1           2           3           4           5
 
                        45.080       2.347       1.227       1.003       0.709
 
                         6           7           8
 
                         0.546       0.472       0.291

Examine the scree plot

Each factor from the second factor onward should be a minor contributor to the data
In this example one primary factor acts as the underlying trait
This is adequate for our assumption of unidimensionality

Scree plot

About the scree plot: Note how the first factor dominates the other factors, that is, there is a large difference between the first and second factors. A significant drop in the contribution of the factors between the first and second factors can be seen as evidence for unidimensionality. Thus, the absence of scree, or "debris" at the bottom of the slope in the plot is desirable because it indicates that the second factor is small. Because our sample data is simulated, the actual scree may vary according to the data.

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