Paired-Comparison Design Matrix — pcX • eba

Computes a paired-comparison design matrix.

pcX(nstimuli, omitRef = TRUE)

Arguments

nstimuli	number of stimuli in the paired-comparison design
omitRef	logical, if `TRUE` (default), the first column corresponding to the reference category is omitted

Details

The design matrix can be used when fitting a Bradley-Terry-Luce (BTL) model or a Thurstone-Mosteller (TM) model by means of glm or lm. See Critchlow and Fligner (1991) for more details.

Value

A matrix having (nstimuli - 1)*nstimuli/2 rows and nstimuli - 1 columns (if the reference category is omitted).

References

Critchlow, D.E., & Fligner, M.A. (1991). Paired comparison, triple comparison, and ranking experiments as generalized linear models, and their implementation in GLIM. Psychometrika, 56, 517--533. doi: 10.1007/bf02294488

See also

eba, thurstone, glm, balanced.pcdesign, linear2btl.

Examples

data(drugrisk)               # absolute choice frequencies
btl <- eba(drugrisk[, , 1])  # fit Bradley-Terry-Luce model using eba
summary(btl)
#> 
#> Parameter estimates:
#>   Estimate Std. Error z value Pr(>|z|)    
#> 1 0.013430   0.003359   3.998 6.39e-05 ***
#> 2 0.010447   0.002689   3.886 0.000102 ***
#> 3 0.006606   0.001794   3.683 0.000231 ***
#> 4 0.118607   0.021072   5.629 1.82e-08 ***
#> 5 0.427059   0.037799  11.298  < 2e-16 ***
#> 6 0.130363   0.022718   5.738 9.56e-09 ***
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> Model tests:
#>           Df1 Df2 logLik1 logLik2 Deviance Pr(>Chi)    
#> Overall     1  30 -300.17  -66.87   466.61   <2e-16 ***
#> EBA         5  15  -29.89  -24.02    11.73    0.304    
#> Effect      0   5 -257.33  -29.89   454.88   <2e-16 ***
#> Imbalance   1  15  -42.84  -42.84     0.00    1.000    
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> AIC:  69.775 
#> Pearson X2: 10.19

y1 <- t(drugrisk[, , 1])[lower.tri(drugrisk[, , 1])]
y0 <-   drugrisk[, , 1][ lower.tri(drugrisk[, , 1])]

## Fit Bradley-Terry-Luce model using glm
btl.glm <- glm(cbind(y1, y0) ~ 0 + pcX(6), binomial)
summary(btl.glm)
#> 
#> Call:
#> glm(formula = cbind(y1, y0) ~ 0 + pcX(6), family = binomial)
#> 
#> Deviance Residuals: 
#>     Min       1Q   Median       3Q      Max  
#> -2.2811  -0.4484  -0.1199   0.4798   1.2335  
#> 
#> Coefficients:
#>         Estimate Std. Error z value Pr(>|z|)    
#> pcX(6)1  -0.2512     0.2145  -1.171  0.24161    
#> pcX(6)2  -0.7095     0.2239  -3.169  0.00153 ** 
#> pcX(6)3   2.1784     0.2589   8.413  < 2e-16 ***
#> pcX(6)4   3.4595     0.3044  11.365  < 2e-16 ***
#> pcX(6)5   2.2729     0.2618   8.682  < 2e-16 ***
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> (Dispersion parameter for binomial family taken to be 1)
#> 
#>     Null deviance: 466.61  on 15  degrees of freedom
#> Residual deviance:  11.73  on 10  degrees of freedom
#> AIC: 69.775
#> 
#> Number of Fisher Scoring iterations: 4
#> 

## Fit Thurstone Case V model using glm
tm.glm <- glm(cbind(y1, y0) ~ 0 + pcX(6), binomial(probit))
summary(tm.glm)
#> 
#> Call:
#> glm(formula = cbind(y1, y0) ~ 0 + pcX(6), family = binomial(probit))
#> 
#> Deviance Residuals: 
#>      Min        1Q    Median        3Q       Max  
#> -2.24897  -0.54660   0.00718   0.64243   1.63449  
#> 
#> Coefficients:
#>         Estimate Std. Error z value Pr(>|z|)    
#> pcX(6)1  -0.1574     0.1250  -1.259  0.20800    
#> pcX(6)2  -0.4202     0.1297  -3.241  0.00119 ** 
#> pcX(6)3   1.2120     0.1347   8.999  < 2e-16 ***
#> pcX(6)4   1.9005     0.1567  12.131  < 2e-16 ***
#> pcX(6)5   1.2519     0.1357   9.229  < 2e-16 ***
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> (Dispersion parameter for binomial family taken to be 1)
#> 
#>     Null deviance: 466.614  on 15  degrees of freedom
#> Residual deviance:  15.181  on 10  degrees of freedom
#> AIC: 73.226
#> 
#> Number of Fisher Scoring iterations: 5
#>