items i; j by the Loevinger coefficient Hij = 1 Observed Nij (1; 0)

Expected Nij (1; 0): The ‘expected’ value is calculated under the null model that the items are independent. If no errors are observed, Hij = 1;

if as many errors are observed as expected under independence, then

Hij = 0. For example with two items with means _Xi: = 0:2; _Xj: = 0:6,

for a sample size of n = 100 the expected table is

Xjh = 0 Xjh = 1

Xih = 0 32 48 80

Xih = 1 8 12 20

40 60

There are 8 errors in the above table. Now suppose the errors were reduced to just 2 (i.e., 2 in the cell which contains 8). Then Hij = 1 - (2/8) = 0:75. Thus, a good scale should have Loevinger H coefficients that are large enough for all pairs i; j with i < j. Rules of thumb that have been found useful are as follows: Hij < 0:3 indicates poor/no scalability;

0:3 < Hij < 0:4 indicates useful but weak scalability;

0:4 < Hij < 0:5 indicates medium scalability;

0:5 < Hij indicates good scalability.

Similarly, Loevinger’s coefficients can be defined for all pairwise errors for a given item (Hi ) and for all pairwise errors for the entire scale (H).

Although you can run the procedure without specifying a value for Loevinger’s H, you can set levels as in the following command (“c” is the value set). msp a3f a3h a3i a3k, c(.4)

Below are some additional commands that can be used:

msp a3a-a3o, c(.4)

msp a3a-a3o, pairwise c(.4)

loevh a3a-a3o, pairwise

Mokken scaling can be used in a confirmatory way, with a given set of

items (where the order can be determined empirically) as well as in an exploratory way. In the exploratory method, a set of items is given,

and it is tried to find a well-scalable subset. This is done by first finding the pair with the highest Hij as the starting point for the scale; and by then consecutively adding items that have the highest values of the Loevinger coefficients with the items already included in the scale. This procedure can then be repeated with the remaining items to find a further scale among those. The reliability can be estimated also from the inter-item correlations. The Mokken scaling module is not part of the normal Stata program and must be downloaded. In the command line type: findit msp

Multidimensional Scaling: Assume we have information about the American

electorate’s perceptions of thirteen prominent political figures from the period of the 2004 presidential election. Specifically, we have the perceived dissimilarities between all pairs of political figures. With 13 figures, there will be 78 distinct pairs of figures. Rank-order pairs of political figures, according to their dissimilarity (from least to most dissimilar). Multidimensional Scaling (MDS) tries to find a set of k points in m-dimensional space such that the distances between pairs of points