Conceptualizing rating scales and partial-credit response structures for communication can be challenging. Rasch measurement provides several approaches. Choose the one that is most meaningful for you.

 

Look at this edited excerpt of Table 3.2:

 

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|      DATA            |  QUALITY CONTROL  |   Andrich   |  EXPECTATION  |50% Cumulat|

| Category Counts  Cum.| Avge  Exp.  OUTFIT| Thresholds  |  Measure at   |Probability|

|Score   Used   %    % | Meas  Meas   MnSq |Measure  S.E.|Category  Zone |      at   |

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|  0      891   6%   6%|  -.04   -.07  1.3 |             |( -2.27)       |    low    |

|  1      383   3%   9%|   .47    .51  1.3 |  -.91    .04|   -.68   -1.53|   -1.23   |

|  2     1017   7%  15%|  1.07   1.17   .8 | -.15    .04|    .63   -.03|  -.07  |

|  3    12683  85% 100%|  2.15   2.14  1.0 |  1.06    .03|(  2.34)   1.55|    1.29   |

------------------------------------------------------------(Mean)---------(Median)---

 

Here are three ways of conceptualizing and communicating the transition, threshold, boundary between category 1 and category 2:

 

1.1) Rasch-half-point thresholds (Zone). Someone at the boundary between "1" and "2" would have an expected rating of 1.5, or 1000 persons at the boundary between "1" and "2" would have an average rating of 1.5. This boundary is the "zone" = "Expected Measure at 1+0.5 or 2 -0.5" which is -.03 logits, the Rasch-half-point threshold. To illustrate this, use the model item characteristic curve. The expected score ogive / model ICC (Table 21.2 - second on list in Graphs menu). The CAT+.25, CAT-0.5, AT CAT, and CAT-.25 columns in the ISFILE= plot points on this ogive. The expected score ogive relates most directly to the estimation of the Rasch parameters. Since it is only one line, it is also convenient for summarizing performance at any point on the latent variable by one number. Crucial points are the points on the variable corresponding to the lower category value + 0.5, i..e, more than the higher adjacent category value - 0.5. These Rasch-half-point thresholds are "average score thresholds" or "Rasch-ICC thresholds".

 

1.2) Rasch-Thurstonian thresholds (50% Cumulative Probability). Someone at the boundary between "1" and "2" would have a 50% chance of being rated 1 or below, and a 50% chance of being rated 2 or above. This is the Rasch-Thurstonian threshold of -.07 logits. To illustrate this, use the cumulative probability curves. The cumulative probability curves (Table 21.3 - and third on list in Graphs menu). The 50%PRB columns in the ISFILE= are the crucial points on these curves. and are the Rasch-Thurstonian thresholds, useful for identifying whether a person is most likely to respond below, at or above a certain category.

 

1.3) Rasch-Andrich thresholds. Someone at the boundary between "1" and "2" would have an equal chance of being rated 1 or 2. This is the Rasch--Andrich Threshold of -.15 logits. To illustrate this, use the category probability curves. The probability curves (Table 21.1 - and top of list in Graphs menu). The Andrich Threshold in the ISFILE= gives the point of equal probability between adjacent categories. The points of highest probability of intermediate categories are given by the AT CAT values. These probability curves relate most directly to the Rasch parameter values, also called Rasch-Andrich thresholds. They are at the intersection of adjacent probability curves, and indicate when the probability of being observed in the higher category starts to exceed that of being observed in the adjacent lower one. This considers the categories two at a time, but can lead to misinference if there is Rasch-Andrich threshold disordering.

 

Here are two ways of conceptualizing the measure corresponding to a category:

2.1) The peak of the category probability curve and

2.2) The point on the latent variable at which the expected score on the item has the value of the category number.

2.1) and 2.2) are the same location on the latent variable. For extreme categories, these locations are infinite, so they are usually replaced by the location for 0.25 score units away from the extreme. For example, the locations on the latent variable for a 1,2,3 rating scale are the measures corresponding to 1.25, 2, 2.75.

 

3.1) Empirical average measures. For any particular sample, there is the average ability of the people who scored in any particular category of any particular item. This is the "Average Measure" reported in Table 3.2. This is entirely sample-dependent. It is not reported in ISFILE=. In the Table, the empirical average measure of the persons responding in category 1 is .47 logits, and in category 2 is 1.07 logits. An empirical boundary between these two categories could be half-way between those values, which is .77 logits.

 

In summary,

 

What some authors call the "Category Gap" is the interval on the latent variable in which the category is the most likely category to be observed. Ben Wright used to call this the "modal category interval", because the mode is the most likely value in a distribution to be observed. The interval is between Andrich thresholds, and requires advancing Andrich thresholds, because the threshold is the point on the latent variable at which adjacent categories are equally probable to be observed. This interval is particularly useful if we want to make the interpretation "ability level <=> most likely category on the item".

 

There are also "mean category intervals", which is the interval on the latent variable in which the expected score (= average score for an ability level) is in the range (category number - 0.5) to (category number + 0.5). The expected scores on the items are used in the Rasch-estimation process. They are also useful when the interpretation is something like "this ability level averages 3 on this item", "ability level <=> average score on the item" . The ends of this category interval can be called the half-point thresholds.

 

The "median category interval" is the interval on the latent variable from the location at which there is a 50% or more chance of being observed in a category below the target category up to a 50% or more chance of being observed above the target category. The ends of this interval are called the Thurstonian Thresholds. The Thurstonian Thresholds are particularly useful when there are many rarely observed categories in a rating scale. This causes the Andrich thresholds to be jumbled and uninterpretable. The Thurstonian Thresholds are always ordered. "ability level <=> more than the categories below, less than the categories above."

 

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Category Intervals and Rasch-model Parameters

 

Let's distinguish between "parameters of the Rasch model" and "category intervals on the latent variable (Rasch dimension)".

 

Parameters of the Rasch model: the Andrich thresholds (ordered or disordered) are parameters of the usual formulations of Rasch polytomous models. There is no mathematical requirement that they are ordered, and there is a mathematical requirement that they are allowed to take any value, including -infinity and +infinity, that accords with the matching sufficient statistic, which is the count of the observations in the category, whether that produces ordered Andrich thresholds or not.

 

Category intervals on the latent variable: these can be defined in several ways and their values obtained from the parameter values of the Rasch model and vice-versa. For instance, suppose we start by looking at the latent variable. We choose a point for the end of a category. We say "everyone to the left of this point is probably in a lower category" and "everyone to the right of this point is probably in this category or above". We give this point a numerical value. We do the same thing for every other category boundary on the latent variable. Then we decide to simulate data that matches these point definitions. Now we need to change the definition of those points on the latent variable into a mathematical formula from which the generating parameters of the Rasch model can be obtained. Yes, we can do it. This definition of the category boundaries matches the Rasch-Thurstone thresholds, and the matching Rasch-Andrich thresholds (parameter values) are given by the formula in www.rasch.org/rmt/rmt164e.htm