OptiPt: Elimination-by-Aspects (EBA) Models

In eba: Elimination-by-Aspects Models

Description

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

Usage

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(s)

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

See Also

strans, uscale, cov.u, group.test, wald.test, plot.eba, residuals.eba, logLik.eba, simulate.eba, kendall.u, circular, trineq, thurstone, nlm.

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
plot(eba1)                            # residuals versus predicted values
anova(btl1, eba1)                     # model comparison based on likelihoods
confint(eba1)                         # confidence intervals for parameters
uscale(eba1)                          # utility scale

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.

Example output

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   3.508 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 0.013296 *  
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
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
         2.5 %     97.5 %
1  0.174854396 0.27236420
2  0.082705367 0.15951936
3  0.055739133 0.11990028
4  0.021563011 0.05908851
5  0.007197446 0.02541694
6  0.020365925 0.05991297
7  0.023843491 0.04951395
8  0.069424871 0.11677803
9  0.113018925 0.17319861
10 0.014924158 0.12834630
11 0.035946311 0.07360305
12 0.033860384 0.08013394
       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 

eba documentation built on Jan. 13, 2021, 10:12 a.m.