Table 23.2, 24.2 Principal components plots of item or 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 23 plots show contrasts between types of items: those at the top vs. those at the bottom. 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 23.0. Important differences have eigenvalues greater than 2.0.

(c) If the difference is important, it suggests that we divide the test into two pieces: the items in the top half of the plot and the items in the bottom half. Perform two separate analyses and cross-plot and correlate the person measures. We will then see for whom 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. Tables 23.4 and 24.4 also help us think about this.

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

2a. Use ISELECT= for each subset code, and produce a person measure (PFILE=). Cross-plot the person measures.

2b. Do a Differential Person Functioning (DPF=) analysis based on the subset code. Table 31.1 will give you an inter-item-subset t-test for each person.

These plots show the contrasts by plotting the loading on each component against the item calibration (or person measure). The contrast shows items (or 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 items (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 items (persons) or less, the item number is displayed.

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

In this example, based on the FIMÔ data, Example 10 using exam10hi.txt data, the first contrast in the standardized residuals separates the items into 3 clusters. To identify the items, see Tables 23.3, 24.3. You will see that items A and B have a psycho-social aspect.

In this example, the dimension is noticeable, with strength of around 3 out of 13 items. 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 measures is equivalent to 16 items. Consequently, though the secondary dimension in the items is noticeable, we do not expect it to have much practical impact on person measurement.

For items:

STANDARDIZED RESIDUAL CONTRAST 1 PLOT

-2 -1 0 1 2

-+---------------+---------------+---------------+---------------+- COUNT

.8 + | +

| A | | 1

.7 + B | + 1

| | |

.6 + | +

| | |

C .5 + | +

O | | |

N .4 + C | + 1

T | | D | 1

R .3 + | +

A | | |

S .2 + | +

T | | E | 1

.1 + | +

1 | | |

.0 +--------------------------------|------------F-------------------+ 1

L | | |

O -.1 + | +

A | |G | 1

D -.2 + e f| + 2

I | | |

N -.3 + | +

G | | |

-.4 + | +

| | |

-.5 + | +

| | |

-.6 + d | c + 2

| b | a | 2

-.7 + | +

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

-2 -1 0 1 2

ITEM MEASURE

COUNT: 1 1 1 12 1 11 2 1 1

For persons:

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

-+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+- COUNT

.9 + | A B + 2

| | C | 1

.8 + F E | D + 3

| H | G | 2

.7 + K I | J + 3

| | L | 1

.6 + | +

| N | M | 2

.5 + O P | + 2

C | | |

O .4 + Q R | + 2

N | | |

T .3 + | T S + 2

R | | U | 1

A .2 + V | + 1

S | W | X | 2

T .1 + 1 | Y Z + 3

| 1 | | 1

1 .0 +------------------------1-1---|----1------1------------------+ 4

| 1 1| 1 | 3

L -.1 + 1 | 1 1 + 3

O | 1 1 | | 2

A -.2 + | +

D | 1 | 1 | 2

I -.3 + | +

N | 1 | z 1 | 3

G -.4 + xy w | + 3

| | s v t u | 4

-.5 + n q | r po + 5

| | m l | 2

-.6 + j |k + 2

| e i h | gf | 5

-.7 + d c | + 2

| | b | 1

-.8 + a | + 1

| | |

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

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

PERSON MEASURE

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 53-181, "1" means that there is one person at that location on the plot. "2" means that there are two persons, etc.

Example: A teacher survey had hundreds of items contributed by numerous stakeholders (with special agendas), but almost no hypothesized constructs. In other words, it was a mess! PCA of residuals was used to discover subsets of items that cooperated. A useful strategy is to use IWEIGHT=. Weight a core subset of items on a construct of interest "1", and weight all the other items "0". Then do a Winsteps analysis. All the items will be calibrated, but only the weighted items will contribute to the person measures. Then do a PCA of residuals. All the items will participate. The core items will cluster. Unweighted items that cluster with the core can be inspected, and added to the core if suitable. Then the process is repeated. When all the items for one construct have been identified, those items can be deleted from the dataset. The process begins again with the next core subset of items.