qvcalc.BTabilities: Quasi Variances for Estimated Abilities

In BradleyTerry2: Bradley-Terry Models

Description

A method for qvcalc::qvcalc() to compute a set of quasi variances (and corresponding quasi standard errors) for estimated abilities from a Bradley-Terry model as returned by BTabilities().

Usage

## S3 method for class 'BTabilities'
qvcalc(object, ...)

Arguments

object	a "BTabilities" object as returned by BTabilities().
...	additional arguments, currently ignored.

Details

For details of the method see Firth (2000), Firth (2003) or Firth and de Menezes (2004). Quasi variances generalize and improve the accuracy of “floating absolute risk” (Easton et al., 1991). This device for economical model summary was first suggested by Ridout (1989).

Ordinarily the quasi variances are positive and so their square roots (the quasi standard errors) exist and can be used in plots, etc.

Value

A list of class "qv", with components

covmat	The full variance-covariance matrix for the estimated abilities.
qvframe	A data frame with variables estimate, SE, quasiSE and quasiVar, the last two being a quasi standard error and quasi-variance for each ability.
dispersion	NULL (dispersion is fixed to 1).
relerrs	Relative errors for approximating the standard errors of all simple contrasts.
factorname	The name of the ID factor identifying players in the BTm formula.
coef.indices	NULL (no required for this method).
modelcall	The call to BTm to fit the Bradley-Terry model from which the abilities were estimated.

Author(s)

David Firth

References

Easton, D. F, Peto, J. and Babiker, A. G. A. G. (1991) Floating absolute risk: an alternative to relative risk in survival and case-control analysis avoiding an arbitrary reference group. Statistics in Medicine 10, 1025–1035.

Firth, D. (2000) Quasi-variances in Xlisp-Stat and on the web. Journal of Statistical Software 5.4, 1–13. https://www.jstatsoft.org/article/view/v005i04.

Firth, D. (2003) Overcoming the reference category problem in the presentation of statistical models. Sociological Methodology 33, 1–18.

Firth, D. and de Menezes, R. X. (2004) Quasi-variances. Biometrika 91, 65–80.

Menezes, R. X. de (1999) More useful standard errors for group and factor effects in generalized linear models. D.Phil. Thesis, Department of Statistics, University of Oxford.

Ridout, M.S. (1989). Summarizing the results of fitting generalized linear models to data from designed experiments. In: Statistical Modelling: Proceedings of GLIM89 and the 4th International Workshop on Statistical Modelling held in Trento, Italy, July 17–21, 1989 (A. Decarli et al., eds.), pp 262–269. New York: Springer.

See Also

qvcalc::worstErrors(), qvcalc::plot.qv().

Examples

    
example(baseball)
baseball.qv <- qvcalc(BTabilities(baseballModel2))
print(baseball.qv)
plot(baseball.qv, xlab = "team",
     levelNames = c("Bal", "Bos", "Cle", "Det", "Mil", "NY", "Tor"))

Example output

basbll> ##  This reproduces the analysis in Sec 10.6 of Agresti (2002).
basbll> data(baseball) # start with baseball data as provided by package

basbll> ##  Simple Bradley-Terry model, ignoring home advantage:
basbll> baseballModel1 <- BTm(cbind(home.wins, away.wins), home.team, away.team,
basbll+                       data = baseball, id = "team")

basbll> ##  Now incorporate the "home advantage" effect
basbll> baseball$home.team <- data.frame(team = baseball$home.team, at.home = 1)

basbll> baseball$away.team <- data.frame(team = baseball$away.team, at.home = 0)

basbll> baseballModel2 <- update(baseballModel1, formula = ~ team + at.home)

basbll> ##  Compare the fit of these two models:
basbll> anova(baseballModel1, baseballModel2)
Analysis of Deviance Table

Response: cbind(home.wins, away.wins)

Model 1: ~team
Model 2: ~team + at.home
  Resid. Df Resid. Dev Df Deviance
1        36     44.053            
2        35     38.643  1   5.4106
           estimate        SE   quasiSE   quasiVar
Baltimore 0.0000000 0.0000000 0.2593386 0.06725653
Boston    1.1438027 0.3378422 0.2179751 0.04751313
Cleveland 0.7046945 0.3350014 0.2234723 0.04993988
Detroit   1.4753572 0.3445518 0.2224204 0.04947085
Milwaukee 1.6195550 0.3473653 0.2258201 0.05099472
New York  1.2813404 0.3404034 0.2189600 0.04794348
Toronto   1.3271104 0.3403222 0.2188008 0.04787379

BradleyTerry2 documentation built on Feb. 3, 2020, 5:08 p.m.