Generate pairwise comparison data for one or more items given absolute latent scores

To add a single item, theta should be a vector of latent scores. To add multiple items at a time, theta should be a matrix with one item in each column. Item names can be given as the colnames of theta.

The interpretation of theta depends on the context where the data were generated. For example, in chess, theta represents unobserved chess skill that is partially revealed by match outcomes.

The graph can be regarded as undirected, but data are generated relative to the order of vertices in the row. Vertices do not commute. For example, a -1 for vertices ‘a’ and ‘b’ is the same as 1 for vertices ‘b’ and ‘a’.

generateItem(df, theta, th = 0.5, scale = 1, name)

a data frame with pairs of vertices given in columns pa1 and pa2, and item response data in other columns


a vector or matrix of absolute latent scores. See details below.


a vector of thresholds


the scaling constant


a vector of item names


The pairwise comparison item response model has thresholds and a scale parameter similar to the partial credit model (Masters, 1982). The model is cumbersome to describe in traditional mathematical notation, but the R code is fairly straightforward,

softmax <- function(y) exp(y) / sum(exp(y))

cmp_probs <- function(scale, pa1, pa2, thRaw) { th <- cumsum(thRaw) diff <- scale * (pa2 - pa1) unsummed <- c(0, c(diff + rev(th)), c(diff - th), use.names = FALSE) softmax(cumsum(unsummed)) }

The function cmp_probs takes a scale constant, the latent scores for two objects pa1 and pa2, and a vector of thresholds thRaw. The thresholds are parameterized as the difference from the previous threshold. For example, thresholds c(0.5, 0.5) are not at the same location but are at locations c(0.5, 1.0). Thresholds are symmetric. If there is one thresholds then the model admits three possible response outcomes (e.g. win, tie, and lose). Responses are always stored centered with zero representing a tie. Therefore, it is necessary to add one plus the number of thresholds to response data to index into the vector returned by cmp_probs. For example, if our response data (-1, 0, 1) has one threshold then we would add 2 (1 + 1 threshold) to obtain the indices (1, 2, 3).

Use itemModelExplorer to explore the item model. In this shiny app, the discrimination parameter does what is customary in item response models. However, it is not difficult to show that discrimination is a function of thresholds and scale. That is, discrimination is not an independent parameter and cannot be estimated. In pairwise comparison models, discrimination and measurement error are confounded.


Masters, G. N. (1982). A Rasch model for partial credit scoring. Psychometrika, 47, 149<U+2013>174. doi: 10.1007/BF02296272

See Also

Other item generators: generateCovItems, generateFactorItems

  • generateItem
df <- roundRobinGraph(letters[1:5], 40)
df <- generateItem(df)
# }
Documentation reproduced from package pcFactorStan, version 0.11, License: GPL (>= 3)

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