Table 24.1, 24.11, ... Principal components/contrast plots of person loadings

Please do not interpret this as a usual factor analysis. These plots show contrasts between opposing factors, identified as "A,B,.." and "a,b,...", not loadings on one factor. For more discussion, see dimensionality and contrasts.

 

Quick summary:

(a) the X-axis is the measurement axis. So we are not concerned about quadrants, we are concerned about vertical differences. The Table 24 plots show contrasts between types of persons: those at the top vs. those at the bottom.

 

(b) "How much" is important. See the Variance Table explained in Table 24.0. Important differences have eigenvalues greater than 2.0.

 

(c) If the difference is important, it suggests that we divide the dataset into two pieces: the persons in the top half of the plot and the persons in the bottom half. Perform two separate analyses and cross-plot and correlate the item calibrations. We will then see for which items the differences are important. Usually, for a carefully designed instrument, it is such a small segment that we decide it is not worth thinking of the test as measuring two dimensions. Table 24.4 also helps us think about this.

 

1. Put a code into the person label to indicate the subset to which the item belongs.

2a. Use PSELECT= for each subset code, and produce a person measure (IFILE=). Cross-plot the item calibrations.

or

2b. Do a Differential Item Functioning (DIF=) analysis based on the subset code. Table 30.1 will give you an inter-person-subset t-test for each item.

 

Alternatively, transpose the dataset, and investigate persons as though they are items.

 

Table 24.0 Variance components scree plot for persons

Table 24.1, 24.11 Principal components plots of person loadings

Table 24.2, 24.12 Person Principal components analysis/contrast of residuals

Table 24.3, 24.13 Person contrast by items

Table 24.4, 24.14 Person contrast loadings sorted by measure

Table 24.5, 24.15 Person contrast loadings sorted by entry number

Table 24.99 Largest residual correlations for persons

 


 

These plots show the contrasts by plotting the unstandardized "raw" loading on each component against the person measure. The contrast shows persons with different residual patterns. A random pattern with few high loadings is expected.

 

The horizontal axis is the Rasch dimension. This has been extracted from the data prior to the analysis of residuals.

 

Letters "A,B,C,..." and "a,b,c,..." identify persons with the most opposed loadings on the first contrast in the residuals. On subsequent contrasts, the items retain their first contrast identifying letters. When there are 9 persons or less, the person number is displayed.

 

In the residuals, each person is modeled to contribute one unit of randomness. Thus, there are as many residual variance units as there are persons. For comparison, the amount of variance explained by the measures is approximated as units of that same size.

 

In the Figure below , the first contrast in the standardized residuals separates the persons into 3 clusters. To identify the persons, see Table 24.3. In this example, the dimension is noticeable, with strength of around 18 out of 74 persons. This is in the residual variance, i.e., in the part of the observations unexplained by the measurement model. But, hopefully, most of the variance in the observations has been explained by the model. The part of that explained variance attributable to the persons is shown in variance units locally-rescaled to accord with the residual variances. In this example, the variance explained by the person measures is equivalent to 30 persons. Consequently, the secondary dimension (or whatever) in the persons is noticeable.

 

For persons:

 

     Table of STANDARDIZED RESIDUAL variance in Eigenvalue units = KID information units

                                           Eigenvalue   Observed   Expected

Total raw variance in observations     =     150.8183 100.0%         100.0%

  Raw variance explained by measures   =      76.8183  50.9%          50.7%

    Raw variance explained by persons  =      30.5375  20.2%          20.2%

    Raw Variance explained by items    =      46.2809  30.7%          30.6%

  Raw unexplained variance (total)     =      74.0000  49.1% 100.0%   49.3%

    Unexplned variance in 1st contrast =      18.2333  12.1%  24.6%

 

       STANDARDIZED RESIDUAL CONTRAST 1 PLOT

 

      -4     -3     -2     -1      0      1      2      3      4      5

      -+------+------+------+------+------+------+------+------+------+- COUNT  CLUSTER

   .9 +                   A        |                                   + 1      1

      |                           B|                                   | 1      1

   .8 +                       C D  |                                   + 2      1

      |                            | E                                 | 1      1

   .7 +                   I  G    H|    F                              + 4      1

      |                            KMJL                                | 4      1

   .6 +                            |                                   +

      |                            | N                                 | 1      1

C  .5 +                 P          |      O                            + 2      1

O     |                            |                                   |

N  .4 +                           Q|                                   + 1      1

T     |                            S        R                          | 2      2

R  .3 +                     T      |                                   + 1      2

A     |                            |    U                              | 1      2

S  .2 +                           V|                                   + 1      2

T     |                            | W                                 | 1      2

   .1 +                            |Y X                                + 2      2

1     |                           1|  1    Z                           | 3      2

   .0 +----------------------------|---1--------------1----------------+ 2      2

L     |                            |   1      1  1                     | 3      2

O -.1 +                       1    |  1      1                         + 3      2

A     |                            |1                                  | 1      2

D -.2 +                            |  1    1                           + 2      2

I     |                            |1 1                                | 2      2

N -.3 +                            |  1  2                             + 3      3

G     |                            |  2      1                         | 3      3

  -.4 +                            |x    1   y     z                   + 4      3

      |                            |      v  w                         | 2      3

  -.5 +                            | u q  o   s r  p                 t + 7      3

      |                            |       ln             m            | 3      3

  -.6 +                            |        k    ji                    + 3      3

      |                            |                        h          | 1      3

  -.7 +                            |             g      f      e       + 3      3

      |                            |            c b d                  | 3      3

  -.8 +                            |                a                  + 1      3

      -+------+------+------+------+------+------+------+------+------+-

      -4     -3     -2     -1      0      1      2      3      4      5

                                 KID MEASURE

 COUNT:                 1 2 112 1 5346832333342 23222 1 1 1 1  1     1

 

ACT          1    1  1   11 111 21  21221  1   21 1 1

               T         S         M         S         T

%TILE        0      10   20  30 40 50 70  80   90  99

 

The plot shows a contrast in the residuals for PERSONS. Each letter is a person up to a maximum of 52 persons, A-Z and a-z. For persons 53-74, "1" means that there is one person at that location on the plot. "2" means that there are two persons, etc.


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