R: Bivariate Binary Regression with an Odds Ratio (Family...

binom2.or {VGAM} | R Documentation

## Bivariate Binary Regression with an Odds Ratio (Family Function)

### Description

Fits a Palmgren (bivariate odds-ratio model, or bivariate logistic regression) model to two binary responses. Actually, a bivariate logistic/probit/cloglog/cauchit model can be fitted. The odds ratio is used as a measure of dependency.

### Usage

    binom2.or(lmu = "logitlink", lmu1 = lmu, lmu2 = lmu, loratio = "loglink",
              imu1 = NULL, imu2 = NULL, ioratio = NULL, zero = "oratio",
              exchangeable = FALSE, tol = 0.001, more.robust = FALSE)

### Arguments

`lmu` | Link function applied to the two marginal probabilities. See `Links` for more choices. See the note below.
`lmu1, lmu2` | Link function applied to the first and second of the two marginal probabilities.
`loratio` | Link function applied to the odds ratio. See `Links` for more choices.
`imu1, imu2, ioratio` | Optional initial values for the marginal probabilities and odds ratio. See `CommonVGAMffArguments` for more details. In general good initial values are often required so use these arguments if convergence failure occurs.
`zero` | Which linear/additive predictor is modelled as an intercept only? The default is for the odds ratio. A `NULL` means none. See `CommonVGAMffArguments` for more details.
`exchangeable` | Logical. If `TRUE`, the two marginal probabilities are constrained to be equal.
`tol` | Tolerance for testing independence. Should be some small positive numerical value.
`more.robust` | Logical. If `TRUE` then some measures are taken to compute the derivatives and working weights more robustly, i.e., in an attempt to avoid numerical problems. Currently this feature is not debugged if set `TRUE`.

### Details

Also known informally as the Palmgren model, the bivariate logistic model is a full-likelihood based model defined as two logistic regressions plus `log(oratio) = eta3` where `eta3` is the third linear/additive predictor relating the odds ratio to explanatory variables. Explicitly, the default model is

logit[P(Y_j=1)] = eta_j,\ \ \ j=1,2

for the marginals, and

log[P(Y_{00}=1) P(Y_{11}=1) / (P(Y_{01}=1) P(Y_{10}=1))] = eta_3,

specifies the dependency between the two responses. Here, the responses equal 1 for a success and a 0 for a failure, and the odds ratio is often written psi=p00 p11 / (p10 p01). The model is fitted by maximum likelihood estimation since the full likelihood is specified. The two binary responses are independent if and only if the odds ratio is unity, or equivalently, the log odds ratio is 0. Fisher scoring is implemented.

The default models eta3 as a single parameter only, i.e., an intercept-only model, but this can be circumvented by setting `zero = NULL` in order to model the odds ratio as a function of all the explanatory variables. The function `binom2.or()` can handle other probability link functions such as `probitlink`, `clogloglink` and `cauchitlink` links as well, so is quite general. In fact, the two marginal probabilities can each have a different link function. A similar model is the bivariate probit model (`binom2.rho`), which is based on a standard bivariate normal distribution, but the bivariate probit model is less interpretable and flexible.

The `exchangeable` argument should be used when the error structure is exchangeable, e.g., with eyes or ears data.

### Value

An object of class `"vglmff"` (see `vglmff-class`). The object is used by modelling functions such as `vglm` 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. These estimated probabilities should be extracted with the `fitted` generic function.

### Note

At present we call `binom2.or` families a bivariate odds-ratio model. The response should be either a 4-column matrix of counts (whose columns correspond to (Y1,Y2) = (0,0), (0,1), (1,0), (1,1) respectively), or a two-column matrix where each column has two distinct values, or a factor with four levels. The function `rbinom2.or` may be used to generate such data. Successful convergence requires at least one case of each of the four possible outcomes.

By default, a constant odds ratio is fitted because `zero = 3`. Set `zero = NULL` if you want the odds ratio to be modelled as a function of the explanatory variables; however, numerical problems are more likely to occur.

The argument `lmu`, which is actually redundant, is used for convenience and for upward compatibility: specifying `lmu` only means the link function will be applied to `lmu1` and `lmu2`. Users who want a different link function for each of the two marginal probabilities should use the `lmu1` and `lmu2` arguments, and the argument `lmu` is then ignored. It doesn't make sense to specify `exchangeable = TRUE` and have different link functions for the two marginal probabilities.

Regarding Yee and Dirnbock (2009), the `xij` (see `vglm.control`) argument enables environmental variables with different values at the two time points to be entered into an exchangeable `binom2.or` model. See the author's webpage for sample code.

### Author(s)

Thomas W. Yee

### References

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

le Cessie, S. and van Houwelingen, J. C. (1994). Logistic regression for correlated binary data. Applied Statistics, 43, 95–108.

Palmgren, J. (1989). Regression Models for Bivariate Binary Responses. Technical Report no. 101, Department of Biostatistics, University of Washington, Seattle.

Yee, T. W. and Dirnbock, T. (2009). Models for analysing species' presence/absence data at two time points. Journal of Theoretical Biology, 259(4), 684–694.

### See Also

`rbinom2.or`, `binom2.rho`, `loglinb2`, `zipebcom`, `coalminers`, `binomialff`, `logitlink`, `probitlink`, `clogloglink`, `cauchitlink`.

### Examples

    # Fit the model in Table 6.7 in McCullagh and Nelder (1989)
    coalminers <- transform(coalminers, Age = (age - 42) / 5)
    fit <- vglm(cbind(nBnW, nBW, BnW, BW) ~ Age,
                binom2.or(zero = NULL), data = coalminers)
    fitted(fit)
    summary(fit)
    coef(fit, matrix = TRUE)
    c(weights(fit, type = "prior")) * fitted(fit)  # Table 6.8

    ## Not run:  with(coalminers, matplot(Age, fitted(fit), type = "l", las = 1,
                             xlab = "(age - 42) / 5", lwd = 2))
    with(coalminers, matpoints(Age, depvar(fit), col=1:4))
    legend(x = -4, y = 0.5, lty = 1:4, col = 1:4, lwd = 2,
           legend = c("1 = (Breathlessness=0, Wheeze=0)",
                      "2 = (Breathlessness=0, Wheeze=1)",
                      "3 = (Breathlessness=1, Wheeze=0)",
                      "4 = (Breathlessness=1, Wheeze=1)"))
    ## End(Not run)

    # Another model: pet ownership
    ## Not run:  data(xs.nz, package = "VGAMdata")
    # More homogeneous:
    petdata <- subset(xs.nz, ethnicity == "European" & age < 70 & sex == "M")
    petdata <- na.omit(petdata[, c("cat", "dog", "age")])
    summary(petdata)
    with(petdata, table(cat, dog))  # Can compute the odds ratio

    fit <- vgam(cbind((1-cat) * (1-dog), (1-cat) * dog,
                         cat  * (1-dog),    cat  * dog) ~ s(age, df = 5),
                binom2.or(zero =    3), data = petdata, trace = TRUE)
    colSums(depvar(fit))
    coef(fit, matrix = TRUE)

    ## End(Not run)

    ## Not run:  # Plot the estimated probabilities
    ooo <- order(with(petdata, age))
    matplot(with(petdata, age)[ooo], fitted(fit)[ooo, ], type = "l",
            xlab = "Age", ylab = "Probability", main = "Pet ownership",
            ylim = c(0, max(fitted(fit))), las = 1, lwd = 1.5)
    legend("topleft", col=1:4, lty = 1:4, leg = c("no cat or dog ",
           "dog only", "cat only", "cat and dog"), lwd = 1.5)
    ## End(Not run)

[Package VGAM version 1.1-3 Index]