# loglinb2: Loglinear Model for Two Binary Responses In VGAM: Vector Generalized Linear and Additive Models

## Description

Fits a loglinear model to two binary responses.

## Usage

 `1` ```loglinb2(exchangeable = FALSE, zero = "u12") ```

## Arguments

 `exchangeable` Logical. If `TRUE`, the two marginal probabilities are constrained to be equal. Should be set `TRUE` for ears, eyes, etc. data. `zero` Which linear/additive predictors are modelled as intercept-only? A `NULL` means none of them. See `CommonVGAMffArguments` for more information.

## Details

The model is

P(Y1=y1,Y2=y2) = exp(u0 + u1*y1 + u2*y2 + u12*y1*y2)

where y1 and y2 are 0 or 1, and the parameters are u1, u2, u12. The normalizing parameter u0 can be expressed as a function of the other parameters, viz.,

u0 = -log[1 + exp(u1) + exp(u2) + exp(u1 + u2 + u12)].

The linear/additive predictors are (eta1,eta2,eta3) = (u1,u2,u12).

## Value

An object of class `"vglmff"` (see `vglmff-class`). The object is used by modelling functions such as `vglm`, `rrvglm` and `vgam`.

When fitted, the `fitted.values` slot of the object contains the four joint probabilities, labelled as (Y1,Y2) = (0,0), (0,1), (1,0), (1,1), respectively.

## Note

The response must be a two-column matrix of ones and zeros only. This is more restrictive than `binom2.or`, which can handle more types of input formats. Note that each of the 4 combinations of the multivariate response need to appear in the data set. After estimation, the response attached to the object is also a two-column matrix; possibly in the future it might change into a four-column matrix.

Thomas W. Yee

## References

Yee, T. W. and Wild, C. J. (2001). Discussion to: “Smoothing spline ANOVA for multivariate Bernoulli observations, with application to ophthalmology data (with discussion)” by Gao, F., Wahba, G., Klein, R., Klein, B. Journal of the American Statistical Association, 96, 127–160.

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

`binom2.or`, `binom2.rho`, `loglinb3`.
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21``` ```coalminers <- transform(coalminers, Age = (age - 42) / 5) # Get the n x 4 matrix of counts fit0 <- vglm(cbind(nBnW,nBW,BnW,BW) ~ Age, binom2.or, data = coalminers) counts <- round(c(weights(fit0, type = "prior")) * depvar(fit0)) # Create a n x 2 matrix response for loglinb2() # bwmat <- matrix(c(0,0, 0,1, 1,0, 1,1), 4, 2, byrow = TRUE) bwmat <- cbind(bln = c(0,0,1,1), wheeze = c(0,1,0,1)) matof1 <- matrix(1, nrow(counts), 1) newminers <- data.frame(bln = kronecker(matof1, bwmat[, 1]), wheeze = kronecker(matof1, bwmat[, 2]), wt = c(t(counts)), Age = with(coalminers, rep(age, rep(4, length(age))))) newminers <- newminers[with(newminers, wt) > 0,] fit <- vglm(cbind(bln,wheeze) ~ Age, loglinb2(zero = NULL), weight = wt, data = newminers) coef(fit, matrix = TRUE) # Same! (at least for the log odds-ratio) summary(fit) # Try reconcile this with McCullagh and Nelder (1989), p.234 (0.166-0.131) / 0.027458 # 1.275 is approximately 1.25 ```