fitted.btfit returns the fitted values from a fitted btfit model object.
# S3 method for btfit fitted(object, subset = NULL, as_df = FALSE, ...)
| object | An object of class "btfit", typically the result |
|---|---|
| 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 |
| as_df | Logical scalar, determining class of output. If |
| ... | Other arguments |
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.
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 $$p_{ij} = \frac{\pi_i}{\pi_i + \pi_j},$$ where \(\pi_i\) and \(\pi_j\) are positive-valued parameters representing the skills of items \(i\) and \(j\), for \(1 \le i, j, \le K\).
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.
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.
citations_btdata <- btdata(BradleyTerryScalable::citations) fit1 <- btfit(citations_btdata, 1) fitted(fit1)#> 4 x 4 sparse Matrix of class "dgCMatrix" #> citing #> cited JRSS-B Biometrika JASA Comm Statist #> JRSS-B 188.00000 286.4629 317.26219 281.7444 #> Biometrika 218.53715 714.0000 505.35443 725.0322 #> JASA 149.73781 312.6456 1072.00000 812.2472 #> Comm Statist 11.25564 37.9678 68.75276 425.0000fitted(fit1, as_df = TRUE)#> # A tibble: 6 x 5 #> component cited citing fit1 fit2 #> <chr> <chr> <chr> <dbl> <dbl> #> 1 full_dataset JRSS-B Biometrika 286.4629 218.53715 #> 2 full_dataset JRSS-B JASA 317.2622 149.73781 #> 3 full_dataset Biometrika JASA 505.3544 312.64557 #> 4 full_dataset JRSS-B Comm Statist 281.7444 11.25564 #> 5 full_dataset Biometrika Comm Statist 725.0322 37.96780 #> 6 full_dataset JASA Comm Statist 812.2472 68.75276toy_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)#> $`2` #> 3 x 3 sparse Matrix of class "dgCMatrix" #> player2 #> player1 Han Gal Fin #> Han . 1.1406148 0.8586132 #> Gal 0.8593852 . 1.6412871 #> Fin 0.1413868 0.3587129 . #> #> $`3` #> 4 x 4 sparse Matrix of class "dgCMatrix" #> player2 #> player1 Cyd Amy Ben Dan #> Cyd . 1.2728582 . 0.7259617 #> Amy 0.7271418 . 0.5684605 1.2042516 #> Ben . 0.4315395 . 1.0692677 #> Dan 0.2740383 0.7957484 0.9307323 . #>fitted(fit2a, as_df = TRUE)#> # A tibble: 8 x 5 #> component player1 player2 fit1 fit2 #> <chr> <chr> <chr> <dbl> <dbl> #> 1 2 Han Gal 1.1406148 0.8593852 #> 2 2 Han Fin 0.8586132 0.1413868 #> 3 2 Gal Fin 1.6412871 0.3587129 #> 4 3 Cyd Amy 1.2728582 0.7271418 #> 5 3 Amy Ben 0.5684605 0.4315395 #> 6 3 Cyd Dan 0.7259617 0.2740383 #> 7 3 Amy Dan 1.2042516 0.7957484 #> 8 3 Ben Dan 1.0692677 0.9307323fitted(fit2a, subset = function(x) "Amy" %in% names(x))#> $`3` #> 4 x 4 sparse Matrix of class "dgCMatrix" #> player2 #> player1 Cyd Amy Ben Dan #> Cyd . 1.2728582 . 0.7259617 #> Amy 0.7271418 . 0.5684605 1.2042516 #> Ben . 0.4315395 . 1.0692677 #> Dan 0.2740383 0.7957484 0.9307323 . #>#> # A tibble: 12 x 5 #> component player1 player2 fit1 fit2 #> <chr> <chr> <chr> <dbl> <dbl> #> 1 full_dataset Cyd Amy 1.2677729 0.73222715 #> 2 full_dataset Eve Gal 0.8811003 0.11889971 #> 3 full_dataset Han Gal 1.1718336 0.82816645 #> 4 full_dataset Eve Ben 0.9108535 0.08914648 #> 5 full_dataset Amy Ben 0.5857476 0.41425236 #> 6 full_dataset Eve Dan 0.9196439 0.08035613 #> 7 full_dataset Cyd Dan 0.7327768 0.26722321 #> 8 full_dataset Amy Dan 1.2259480 0.77405202 #> 9 full_dataset Ben Dan 1.0566477 0.94335225 #> 10 full_dataset Eve Fin 0.9671479 0.03285215 #> 11 full_dataset Han Fin 0.8489711 0.15102890 #> 12 full_dataset Gal Fin 1.5978031 0.40219689