Multi-logit Link Function

Computes the multilogit transformation, including its inverse and the first two derivatives.

Usage

multilogitlink(theta, refLevel = "(Last)", M = NULL, whitespace = FALSE,
       bvalue = NULL, sumcon = FALSE, inverse = FALSE, deriv = 0,
       all.derivs = FALSE, short = TRUE, tag = FALSE)

Arguments

theta

Numeric or character. See below for further details.

refLevel, M, whitespace

See multinomial.

bvalue

See Links.

sumcon

Logical. Use summation constraints? Details are at multinomial.

all.derivs

Logical. This is currently experimental only.

inverse, deriv, short, tag

Details at Links.

Details

The multilogitlink() link function is a generalization of the logitlink link to \(M\) levels/classes. It forms the basis of the multinomial logit model. It is sometimes called the multi-logit link or the multinomial logit link; some people use softmax too. When its inverse function is computed it returns values which are positive and add to unity.

Value

For multilogitlink with deriv = 0, the multilogit of theta, i.e., log(theta[, j]/theta[, M+1]) when inverse = FALSE, and if inverse = TRUE then exp(theta[, j])/(1+rowSums(exp(theta))).

For deriv = 1, then the function returns d eta / d theta as a function of theta if inverse = FALSE, else if inverse = TRUE then it returns the reciprocal.

Here, all logarithms are natural logarithms, i.e., to base e.

References

McCullagh, P. and Nelder, J. A. (1989). Generalized Linear Models, 2nd ed. London: Chapman & Hall.

Author

Thomas W. Yee

Note

Numerical instability may occur when theta is close to 1 or 0 (for multilogitlink). One way of overcoming this is to use, e.g., bvalue. Currently care.exp() is used to avoid NAs being returned if the probability is too close to 1.

Warning

Argument sumcon is a recent addition and may have limited functionality in other parts of VGAM. It has not been tested extensively.

Examples

pneumo <- transform(pneumo, let = log(exposure.time))
fit <- vglm(cbind(normal, mild, severe) ~ let,  # For illustration only!
            multinomial, trace = TRUE, data = pneumo)
#> Iteration 1: deviance = 5.407271
#> Iteration 2: deviance = 5.34745
#> Iteration 3: deviance = 5.347382
#> Iteration 4: deviance = 5.347382
fitted(fit)
#>      normal        mild      severe
#> 1 0.9927503 0.005875947 0.001373768
#> 2 0.9329702 0.043219077 0.023810688
#> 3 0.8488899 0.085745054 0.065365011
#> 4 0.7485338 0.128835331 0.122630879
#> 5 0.6393787 0.168725388 0.191895881
#> 6 0.5334715 0.201127232 0.265401245
#> 7 0.4313692 0.226188995 0.342441766
#> 8 0.3581471 0.239824757 0.402028109
predict(fit)
#>   log(mu[,1]/mu[,3]) log(mu[,2]/mu[,3])
#> 1          6.5829217         1.45330985
#> 2          3.6682387         0.59614746
#> 3          2.5639424         0.27139129
#> 4          1.8089375         0.04935622
#> 5          1.2035440        -0.12868047
#> 6          0.6981629        -0.27730511
#> 7          0.2308628        -0.41473071
#> 8         -0.1155781        -0.51661353

multilogitlink(fitted(fit))
#>         [,1]        [,2]
#> 1  6.5829217  1.45330985
#> 2  3.6682387  0.59614746
#> 3  2.5639424  0.27139129
#> 4  1.8089375  0.04935622
#> 5  1.2035440 -0.12868047
#> 6  0.6981629 -0.27730511
#> 7  0.2308628 -0.41473071
#> 8 -0.1155781 -0.51661353
multilogitlink(fitted(fit)) - predict(fit)  # Should be all 0s
#>            [,1]          [,2]
#> 1  0.000000e+00  4.440892e-16
#> 2  0.000000e+00  2.220446e-16
#> 3  0.000000e+00 -1.665335e-16
#> 4  0.000000e+00 -7.632783e-17
#> 5  0.000000e+00 -1.387779e-16
#> 6  2.220446e-16  5.551115e-17
#> 7  1.665335e-16  0.000000e+00
#> 8 -5.551115e-17  1.110223e-16

multilogitlink(predict(fit), inverse = TRUE)  # rowSums() add to unity
#>        [,1]        [,2]        [,3]
#> 1 0.9927503 0.005875947 0.001373768
#> 2 0.9329702 0.043219077 0.023810688
#> 3 0.8488899 0.085745054 0.065365011
#> 4 0.7485338 0.128835331 0.122630879
#> 5 0.6393787 0.168725388 0.191895881
#> 6 0.5334715 0.201127232 0.265401245
#> 7 0.4313692 0.226188995 0.342441766
#> 8 0.3581471 0.239824757 0.402028109
multilogitlink(predict(fit), inverse = TRUE, refLevel = 1)
#>          [,1]      [,2]        [,3]
#> 1 0.001373768 0.9927503 0.005875947
#> 2 0.023810688 0.9329702 0.043219077
#> 3 0.065365011 0.8488899 0.085745054
#> 4 0.122630879 0.7485338 0.128835331
#> 5 0.191895881 0.6393787 0.168725388
#> 6 0.265401245 0.5334715 0.201127232
#> 7 0.342441766 0.4313692 0.226188995
#> 8 0.402028109 0.3581471 0.239824757
multilogitlink(predict(fit), inverse = TRUE) -
fitted(fit)  # Should be all 0s
#>   [,1] [,2] [,3]
#> 1    0    0    0
#> 2    0    0    0
#> 3    0    0    0
#> 4    0    0    0
#> 5    0    0    0
#> 6    0    0    0
#> 7    0    0    0
#> 8    0    0    0

multilogitlink(fitted(fit), deriv = 1)
#>       normal       mild     severe
#> 1 138.943754 171.191239 728.926256
#> 2  15.990591  24.183102  43.022338
#> 3   7.795702  12.756267  16.368641
#> 4   5.312622   8.909735   9.294324
#> 5   4.337011   7.129762   6.448624
#> 6   4.018006   6.223741   5.129167
#> 7   4.076810   5.713387   4.440982
#> 8   4.350138   5.485197   4.159708
multilogitlink(fitted(fit), deriv = 2)
#>         normal         mild        severe
#> 1 19025.449830 -28962.03409 -5.298736e+05
#> 2   221.420110   -534.27143 -1.762778e+03
#> 3    42.406154   -134.81708 -2.329056e+02
#> 4    14.029214    -58.92861 -6.519766e+01
#> 5     5.243333    -33.67970 -2.562486e+01
#> 6     1.080754    -23.15364 -1.234382e+01
#> 7    -2.281339    -17.87591 -6.214829e+00
#> 8    -5.368762    -15.65599 -3.390448e+00