Fit the Bradley-Terry-Luce model to paired comparisons

Estimates object locations from paired-comparison data by conditional maximum likelihood. The Bradley-Terry-Luce model is the conditional form of the dichotomous Rasch model – within an item pair, given one correct response, the Rasch probability that it was the easier item is exactly of BTL form – so it belongs to the same measurement family and is estimated by the same conventions as the rest of the package: Newton-Raphson on the person-free likelihood, locations identified by the sum-zero constraint, and Godambe sandwich standard errors, clustered by judge when a judge column is given (so repeated comparisons by the same judge need not be independent). Objects that win or lose every comparison have no finite estimate and are removed with a note, exactly as extreme persons are set aside in a Rasch calibration; the comparison graph must remain connected. Beyond that, finite estimates exist only when the directed win graph is strongly connected (Ford 1957): if some subset of objects never concedes a point to the rest, the likelihood pushes the two clusters infinitely far apart, so the fit stops with an error naming the separated objects rather than presenting the optimiser's boundary values as measures – remove the separated cluster, or collect comparisons that cross the divide. Anchors relax this condition: an anchored object is pinned, so the fit only requires every free object to be tied to an anchor in both win directions (otherwise the constrained likelihood still recedes along the unanchored cluster, and the same error results).

Usage

btl(
  data,
  object_a,
  object_b,
  winner = NULL,
  response = NULL,
  margin = NULL,
  judge = NULL,
  count = NULL,
  order = NULL,
  position = FALSE,
  anchors = NULL,
  ties = c("drop", "half", "error"),
  thresholds = c("free", "pc"),
  maxit = 60,
  tol = 1e-08
)

Arguments

data: A data frame with one comparison per row.
object_a, object_b: Names of the columns holding the two objects compared.
winner: Name of the column holding the winner of each row: its value must equal the row's object_a or object_b entry; "tie" or "draw" marks a tie; anything else (including blanks) is treated as missing and dropped with a note. Ignored when response is given.
response: Optional name of a column holding a graded preference for object_a over object_b – an ORDERED factor (factor(..., ordered = TRUE), levels worst to best for object_a) or integer scores 0..m; a plain factor is refused, since its alphabetical level order would silently define (and can reverse) the response scale. Fits the adjacent-categories ordinal extension of BTL (Tutz 1986; Agresti 1992): a partial-credit structure on the difference of locations with thresholds constrained symmetric, tau_k = -tau_(m+1-k), so the model is invariant to presentation order. Two categories reproduce BTL exactly; three give the Davidson (1970) ties model.
margin: Optional name of a column holding the extent of the win ("a little", "much", ...), as an ordered factor or increasing values; combined with winner it assembles the graded response without any orientation bookkeeping ("B by much" means the same thing whichever column B sits in). Winner values matching neither object are ties and form the middle category.
judge: Optional name of a judge column; enables the judge fit table and clusters the sandwich standard errors by judge.
count: Optional name of a column of replication counts (a row standing for several identical comparisons).
order: Optional name of a column giving each judge's judgment sequence (timestamps or ranks; requires judge). Adds the within-judge dependence analysis: an exposure effect (the advantage, in logits, of an object the judge has seen before over one they have not) and a carry-over effect (the pull of the judge's own earlier verdicts on the same object – response dependence in the sense of Marais and Andrich 2008), estimated jointly with the locations and reported in dependence. Incompatible with ties = "half".
position: Logical: when TRUE, object_a is taken as the first-presented (left) object of every comparison and a first-position advantage is estimated – a single coefficient, in logits, added to every comparison's location difference, the pure positional form of the Davidson and Beaver (1977) within-pair order-effect device. It is reported in dependence with effect = "position" (every comparison is informative, so n_informative is the total weighted comparison count) and estimated jointly with the locations, alongside the exposure and carry-over effects when order is also given. Identification comes from triangle closure (K >= 3), so the constant oriented covariate is estimable even when each pair has a fixed orientation, though weakly. Note that ties = "half" duplicates rows in the same orientation, so the first position stays well defined.
anchors: Optional named numeric vector for equating: names are object names, values are fixed locations in logits. The named objects are held exactly at those locations and the remaining objects are estimated freely with no sum-zero constraint – the origin and scale come from the anchors, exactly as an anchored rasch calibration works. Anchored objects report a standard error of zero (their location is a constant, not an estimate). An anchored object that is undefeated or winless is an error, not silently removed as a free boundary object would be.
ties: How to treat ties in the dichotomous analysis: "drop" (default, removed with a note), "half" (half a win each way, a common pragmatic device – flagged in the notes because the halves are not independent Bernoulli trials), or "error". With graded responses, code ties as a middle category instead.
thresholds: "free" (default) estimates every symmetric threshold parameter; "pc" pools them to the spread (linear) principal component – the symmetric case of the principal-component threshold structure, whose even skewness component is structurally zero here – so thinly used categories borrow strength from every response. Both modes report the component decomposition.
maxit, tol: Newton-Raphson iteration cap and convergence tolerance.

Value

A list of class "rasch_btl": objects (location, se, comparisons, wins – or the graded score – infit and outfit mean squares, fit residual and its df), pairs (per pair: n, observed and expected win proportions – or mean graded responses – standardised residual, chi-square component – the pair chi-squares treat comparisons as independent and are descriptive under judge clustering; the object and judge fit residuals and the clustered standard errors carry the robust inference), judges (when given: per judge n, infit, outfit, fit residual, df), total_chisq, total_df, total_p, the object separation index osi, loglik, cl (the composite-likelihood information ingredients used by compare_fits: the Godambe effective parameter count and the independent-unit count), convergence details, and notes. Graded fits add thresholds (the symmetric threshold estimates with standard errors), m, and categories. With an order column the within-judge dependence effects table carries an n_informative count, and dependence_data holds every comparison with its per-comparison exposure and carry-over covariates (see plot_btl_dependence).

Details

Fit is reported at three levels, mirroring the Rasch diagnostics. Per object and (when given) per judge: the log-of-mean-square fit residual of Andrich and Marais (2019, ch. 23) over their comparisons, with apportioned degrees of freedom – an erratic judge or an object of inconsistent quality shows exactly as an erratic person or misfitting item does. Per pair: the classical goodness-of-fit table comparing observed and expected win proportions, with the total chi-square on (pairs used) minus (free location parameters) degrees of freedom. The object separation index is the analogue of the PSI: the proportion of observed location variance not due to error. Anchored objects enter it with their fixed locations and zero error – they are separated with certainty by construction.

References

Bradley, R. A. and Terry, M. E. (1952). Rank analysis of incomplete block designs: I. The method of paired comparisons. Biometrika, 39, 324-345. Luce, R. D. (1959). Individual Choice Behavior. Wiley. Andrich, D. (1978). Relationships between the Thurstone and Rasch approaches to item scaling. Applied Psychological Measurement, 2, 451-462.

Tutz, G. (1986). Bradley-Terry-Luce models with an ordered response. Journal of Mathematical Psychology, 30(3), 306-316. Agresti, A. (1992). Analysis of ordinal paired comparison data. Journal of the Royal Statistical Society C, 41(2), 287-297. Davidson, R. R. (1970). On extending the Bradley-Terry model to accommodate ties in paired comparison experiments. Journal of the American Statistical Association, 65(329), 317-328. Ford, L. R. (1957). Solution of a ranking problem from binary comparisons. American Mathematical Monthly, 64(8), 28-33.

Davidson, R. R., & Beaver, R. J. (1977). On extending the Bradley-Terry model to incorporate within-pair order effects. Biometrics, 33(4), 693-702.

Examples

set.seed(1)
beta <- c(A = -1, B = -0.3, C = 0.4, D = 0.9)
pairs <- t(combn(names(beta), 2))
d <- data.frame(a = rep(pairs[, 1], each = 30),
                b = rep(pairs[, 2], each = 30))
p <- plogis(beta[d$a] - beta[d$b])
d$win <- ifelse(runif(nrow(d)) < p, d$a, d$b)
btl(d, object_a = "a", object_b = "b", winner = "win")
#> Bradley-Terry-Luce analysis: 4 objects, 180 comparisons
#> Conditional ML: converged in 6 iterations; sandwich SEs
#> Object separation index 0.963; pairwise chi-square 1.07 on 3 df, p = 0.783
#>  object location    se comparisons wins fit_resid
#>       A   -1.238 0.214          90   16    -0.111
#>       B   -0.354 0.186          90   36     0.526
#>       C    0.448 0.180          90   56    -0.056
#>       D    1.144 0.209          90   72     0.037