fitted.btfit: Fitted Method for "btfit"
In BradleyTerryScalable: Fits the Bradley-Terry Model to Potentially Large and Sparse Networks of Comparison Data

Description Usage Arguments Details Value Author(s) References See Also Examples

fitted.btfit returns the fitted values from a fitted btfit model object.

1 2	## S3 method for class 'btfit' fitted(object, subset = NULL, as_df = FALSE, ...)

`object`	An object of class "btfit", typically the result `ob` of `ob <- btfit(..)`. See `btfit`.
`subset`	A condition for selecting one or more subsets of the components. This can either be a character vector of names of the components (i.e. a subset of `names(object$pi)`), a single predicate function (that takes a vector of `object$pi` as its argument), or a logical vector of the same length as the number of components, (i.e. `length(object$pi)`).
`as_df`	Logical scalar, determining class of output. If `TRUE`, the function returns a data frame. If `FALSE` (the default), the function returns a matrix (or list of matrices). Note that setting `as_df = TRUE` can have a significant computational cost when any of the components have a large number of items.
`...`	Other arguments

Consider a set of K items. Let the items be nodes in a graph and let there be a directed edge (i, j) when i has won against j at least once. We call this the comparison graph of the data, and denote it by G_W. Assuming that G_W is fully connected, the Bradley-Terry model states that the probability that item i beats item j is

p_{ij} = \frac{π_i}{π_i + π_j},

where π_i and π_j are positive-valued parameters representing the skills of items i and j, for 1 ≤ i, j, ≤ K.

The expected, or fitted, values under the Bradley-Terry model are therefore:

m_{ij} = n_{ij}p_{ij},

where n_{ij} is the number of comparisons between item i and item j.

If there are values on the diagonal in the original btdata$wins matrix, then these appear as the values on the diagonal of the fitted matrix. These values do not appear in the data frame if the as_df argument is set to TRUE.

The function btfit is used to fit the Bradley-Terry model. It produces a "btfit" object that can then be passed to fitted.btfit to obtain the fitted values m_{ij}. Note that the Bradley-Terry probabilities p_{ij} can be calculated using btprob.

If G_W is not fully connected, then a penalised strength parameter can be obtained using the method of Caron and Doucet (2012) (see btfit, with a > 1), which allows for a Bradley-Terry probability of any of the K items beating any of the others. Alternatively, the MLE can be found for each fully-connected component of G_W (see btfit, with a = 1), and the probability of each item in each component beating any other item in that component can be found.

If as_df = FALSE and the model has been fit on the full dataset, returns a matrix where the i,j-th element is the Bradley-Terry expected value m_{ij} (See Details). Otherwise, a list of such matrices is returned, one for each fully-connected component. If as_df = TRUE, returns a five-column data frame, where the first column is the component that the two items are in, the second column is item1, the third column is item2, the fourth column, fit1, is the expected number of times that item 1 beats item 2 and the fifth column, fit2, is the expected number of times that item 2 beats item 1. If btdata$wins has named dimnames, these will be the colnames for columns one and two. Otherwise these colnames will be item1 and item2. See Details.

Ella Kaye

Bradley, R. A. and Terry, M. E. (1952). Rank analysis of incomplete block designs: 1. The method of paired comparisons. Biometrika, 39(3/4), 324-345.

Caron, F. and Doucet, A. (2012). Efficient Bayesian Inference for Generalized Bradley-Terry Models. Journal of Computational and Graphical Statistics, 21(1), 174-196.

btfit, btprob, btdata

citations_btdata <- btdata(BradleyTerryScalable::citations)
fit1 <- btfit(citations_btdata, 1)
fitted(fit1)
fitted(fit1, as_df = TRUE)
toy_df_4col <- codes_to_counts(BradleyTerryScalable::toy_data, c("W1", "W2", "D"))
toy_btdata <- btdata(toy_df_4col)
fit2a <- btfit(toy_btdata, 1)
fitted(fit2a)
fitted(fit2a, as_df = TRUE)
fitted(fit2a, subset = function(x) "Amy" %in% names(x))
fit2b <- btfit(toy_btdata, 1.1)
fitted(fit2b, as_df = TRUE)