Elimination-by-Aspects (EBA) Models

Fits a (multi-attribute) probabilistic choice model by maximum likelihood.

eba(M, A = 1:I, s = rep(1/J, J), constrained = TRUE)

OptiPt(M, A = 1:I, s = rep(1/J, J), constrained = TRUE)

# S3 method for class 'eba'
summary(object, ...)

# S3 method for class 'eba'
anova(object, ..., test = c("Chisq", "none"))

Arguments

M: a square matrix or a data frame consisting of absolute choice frequencies; row stimuli are chosen over column stimuli
A: a list of vectors consisting of the stimulus aspects; the default is 1:I, where I is the number of stimuli
s: the starting vector with default 1/J for all parameters, where J is the number of parameters
constrained: logical, if TRUE (default), parameters are constrained to be positive
object: an object of class eba, typically the result of a call to eba
test: should the p-values of the chi-square distributions be reported?
...: additional arguments; none are used in the summary method; in the anova method they refer to additional objects of class eba.

Details

eba is a wrapper function for OptiPt. Both functions can be used interchangeably. See Wickelmaier and Schmid (2004) for further details.

The probabilistic choice models that can be fitted to paired-comparison data are the Bradley-Terry-Luce (BTL) model (Bradley, 1984; Luce, 1959), preference tree (Pretree) models (Tversky and Sattath, 1979), and elimination-by-aspects (EBA) models (Tversky, 1972), the former being special cases of the latter.

A represents the family of aspect sets. It is usually a list of vectors, the first element of each being a number from 1 to I; additional elements specify the aspects shared by several stimuli. A must have as many elements as there are stimuli. When fitting a BTL model, A reduces to 1:I (the default), i.e. there is only one aspect per stimulus.

The maximum likelihood estimation of the parameters is carried out by nlm. The Hessian matrix, however, is approximated by nlme::fdHess. The likelihood functions L.constrained and L are called automatically.

See group.test for details on the likelihood ratio tests reported by summary.eba.

Value

coefficients: a vector of parameter estimates
estimate: same as coefficients
logL.eba: the log-likelihood of the fitted model
logL.sat: the log-likelihood of the saturated (binomial) model
goodness.of.fit: the goodness of fit statistic including the likelihood ratio fitted vs. saturated model (-2logL), the degrees of freedom, and the p-value of the corresponding chi-square distribution
u.scale: the unnormalized utility scale of the stimuli; each utility scale value is defined as the sum of aspect values (parameters) that characterize a given stimulus
hessian: the Hessian matrix of the likelihood function
cov.p: the covariance matrix of the model parameters
chi.alt: the Pearson chi-square goodness of fit statistic
fitted: the fitted paired-comparison matrix
y1: the data vector of the upper triangle matrix
y0: the data vector of the lower triangle matrix
n: the number of observations per pair (y1 + y0)
mu: the predicted choice probabilities for the upper triangle
nobs: the number of pairs

Author

Florian Wickelmaier

References

Bradley, R.A. (1984). Paired comparisons: Some basic procedures and examples. In P.R. Krishnaiah & P.K. Sen (eds.), Handbook of Statistics, Volume 4. Amsterdam: Elsevier. doi:10.1016/S0169-7161(84)04016-5

Luce, R.D. (1959). Individual choice behavior: A theoretical analysis. New York: Wiley.

Tversky, A. (1972). Elimination by aspects: A theory of choice. Psychological Review, 79, 281–299. doi:10.1037/h0032955

Tversky, A., & Sattath, S. (1979). Preference trees. Psychological Review, 86, 542–573. doi:10.1037/0033-295X.86.6.542

Wickelmaier, F., & Schmid, C. (2004). A Matlab function to estimate choice model parameters from paired-comparison data. Behavior Research Methods, Instruments, and Computers, 36, 29–40. doi:10.3758/BF03195547

Examples

data(celebrities)                     # absolute choice frequencies
btl1 <- eba(celebrities)              # fit Bradley-Terry-Luce model
A <- list(c(1,10), c(2,10), c(3,10),
          c(4,11), c(5,11), c(6,11),
          c(7,12), c(8,12), c(9,12))  # the structure of aspects
eba1 <- eba(celebrities, A)           # fit elimination-by-aspects model

summary(eba1)                         # goodness of fit
#> 
#> Parameter estimates:
#>    Estimate Std. Error z value Pr(>|z|)    
#> 1  0.223609   0.024875   8.989  < 2e-16 ***
#> 2  0.121112   0.019596   6.181 6.39e-10 ***
#> 3  0.087820   0.016368   5.365 8.08e-08 ***
#> 4  0.040326   0.009573   4.212 2.53e-05 ***
#> 5  0.016307   0.004648   3.509 0.000451 ***
#> 6  0.040139   0.010089   3.979 6.93e-05 ***
#> 7  0.036679   0.006549   5.601 2.13e-08 ***
#> 8  0.093101   0.012080   7.707 1.29e-14 ***
#> 9  0.143109   0.015352   9.322  < 2e-16 ***
#> 10 0.071635   0.028935   2.476 0.013295 *  
#> 11 0.054775   0.009606   5.702 1.19e-08 ***
#> 12 0.056997   0.011805   4.828 1.38e-06 ***
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> Model tests:
#>           Df1 Df2 logLik1 logLik2 Deviance Pr(>Chi)    
#> Overall     1  72 -763.65 -235.22  1056.85   <2e-16 ***
#> EBA        11  36 -119.01 -103.93    30.17    0.218    
#> Effect      0  11 -632.36 -119.01  1026.68   <2e-16 ***
#> Imbalance   1  36 -131.29 -131.29     0.00    1.000    
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> AIC:  260.03 
#> Pearson X2: 30.05
plot(eba1)                            # residuals versus predicted values

anova(btl1, eba1)                     # model comparison based on likelihoods
#> Analysis of Deviance Table
#> 
#> Model 1: btl1
#> Model 2: eba1
#>   Resid. Df Resid. Dev Df Deviance  Pr(>Chi)    
#> 1        28     78.217                          
#> 2        25     30.166  3   48.051 2.077e-10 ***
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
confint(eba1)                         # confidence intervals for parameters
#>          2.5 %     97.5 %
#> 1  0.174854300 0.27236430
#> 2  0.082705521 0.15951920
#> 3  0.055739127 0.11990028
#> 4  0.021563110 0.05908841
#> 5  0.007197544 0.02541685
#> 6  0.020366095 0.05991280
#> 7  0.023843499 0.04951394
#> 8  0.069424905 0.11677799
#> 9  0.113018831 0.17319870
#> 10 0.014924518 0.12834594
#> 11 0.035946383 0.07360297
#> 12 0.033860426 0.08013390
uscale(eba1)                          # utility scale
#>        LBJ         HW        CDG         JU         CY        AJF         BB 
#> 0.21830768 0.14252010 0.11790307 0.07031851 0.05255887 0.07018075 0.06926518 
#>         ET         SL 
#> 0.11098488 0.14796095 

ci <- 1.96 * sqrt(diag(cov.u(eba1)))      # 95% CI for utility scale values
dotchart(uscale(eba1), xlim=c(0, .3), main="Choice among celebrities",
         xlab="Utility scale value (EBA model)", pch=16)    # plot the scale
arrows(uscale(eba1)-ci, 1:9, uscale(eba1)+ci, 1:9, .05, 90, 3)  # error bars
abline(v=1/9, lty=2)                      # indifference line
mtext("(Rumelhart and Greeno, 1971)", line=.5)


## See data(package = "eba") for application examples.