Lightweight family objects used by panglm(), analogous in spirit to
stats::family objects but keyed to the integer codes the C++ backend
expects (see src/families.h).
Details
gammafamily() is the gamma GLM with a log link, the usual
choice for skewed positive outcomes such as health care costs: the
variance is proportional to the squared mean and the coefficients act
multiplicatively on the mean, with no retransformation needed.
orderedfamily() is the ordered logit or ordered probit model
for ordinal outcomes such as self-assessed health:
P(y <= j) = F(c_j - x'beta - u_i). The response may be a factor, an
ordered factor or integer-valued; categories follow the factor levels
or the sorted values. The intercept is absorbed in the cutpoints c_j,
which are reported after the slopes. model = "pooling" and
model = "random" (Gaussian random intercept by adaptive
Gauss-Hermite quadrature) are available for both links, model = "within" for the
logit link only (the "blow-up and cluster" fixed-effects ordered logit
of Baetschmann, Staub and Winkelmann 2015). The fit has class
c("panordered", "panglm"); see predict.panordered().
betafamily() is beta regression for continuous outcomes in
the open interval (0, 1), such as utility index scores rescaled to
(0, 1), proportions of days in hospital or adherence fractions:
y ~ Beta(mu * phi, (1 - mu) * phi) with g(mu) = x'beta, so E(y) = mu
and Var(y) = mu (1 - mu) / (1 + phi). The precision phi is
reported as fit$precision. model = "pooling" and model = "random"
(Gaussian random intercept on the link scale, adaptive Gauss-Hermite
quadrature) are available; there is no fixed-effects estimator.
Outcomes equal to 0 or 1 must be transformed first, e.g. with
(y * (n - 1) + 0.5) / n (Smithson and Verkuilen 2006).
tobitfamily() is the censored normal (Tobit) model for
outcomes piled up at a floor or a ceiling, such as health scores at
the ceiling of a scale (motor function scores, EQ-5D utility at 1) or
spending at zero: the latent outcome is y* = x'beta + u_i + e_it
with e_it ~ N(0, sigma_e^2), and y = left when y* <= left,
y = right when y* >= right, y = y* otherwise. Use left = -Inf
or right = Inf for one-sided censoring. model = "pooling" and
model = "random" (Gaussian random intercept u_i ~ N(0, sigma_u^2),
adaptive Gauss-Hermite quadrature) are available. There is no
fixed-effects Tobit (incidental parameters); use model = "random"
with cre = TRUE. predict(type = "link") returns the latent mean
x'beta and predict(type = "response") the mean of the observed,
censored outcome (averaged over the random intercept), which is the
scale of partialeffects(). fit$sigma_e and fit$sigma (random
intercept SD) are stored on the fit.
References
Baetschmann, G., Staub, K. E. and Winkelmann, R. (2015). Consistent estimation of the fixed effects ordered logit model. Journal of the Royal Statistical Society A, 178, 685-703. doi:10.1111/rssa.12090
Ferrari, S. and Cribari-Neto, F. (2004). Beta regression for modelling rates and proportions. Journal of Applied Statistics, 31, 799-815. doi:10.1080/0266476042000214501
Smithson, M. and Verkuilen, J. (2006). A better lemon squeezer? Maximum-likelihood regression with beta-distributed dependent variables. Psychological Methods, 11, 54-71. doi:10.1037/1082-989X.11.1.54
Tobin, J. (1958). Estimation of relationships for limited dependent variables. Econometrica, 26, 24-36. doi:10.2307/1907382
See also
panglm(), whose family argument takes these objects, and
panglm-deprecated for the former names.
Examples
gaussianfamily()
#> $family
#> [1] "gaussian"
#>
#> $link
#> [1] "identity"
#>
#> $family_id
#> [1] 0
#>
#> $link_id
#> [1] 0
#>
poissonfamily()
#> $family
#> [1] "poisson"
#>
#> $link
#> [1] "log"
#>
#> $family_id
#> [1] 1
#>
#> $link_id
#> [1] 1
#>
binomialfamily("probit")
#> $family
#> [1] "binomial"
#>
#> $link
#> [1] "probit"
#>
#> $family_id
#> [1] 2
#>
#> $link_id
#> [1] 3
#>
negbinfamily()
#> $family
#> [1] "negbin"
#>
#> $link
#> [1] "log"
#>
#> $family_id
#> [1] 1
#>
#> $link_id
#> [1] 1
#>
gammafamily()
#> $family
#> [1] "gamma"
#>
#> $link
#> [1] "log"
#>
#> $family_id
#> [1] 3
#>
#> $link_id
#> [1] 1
#>
orderedfamily("probit")
#> $family
#> [1] "ordered"
#>
#> $link
#> [1] "probit"
#>
#> $family_id
#> [1] NA
#>
#> $link_id
#> [1] 3
#>
betafamily()
#> $family
#> [1] "beta"
#>
#> $link
#> [1] "logit"
#>
#> $family_id
#> [1] NA
#>
#> $link_id
#> [1] 2
#>
tobitfamily(left = 0)
#> $family
#> [1] "tobit"
#>
#> $link
#> [1] "identity"
#>
#> $family_id
#> [1] NA
#>
#> $link_id
#> [1] 0
#>
#> $left
#> [1] 0
#>
#> $right
#> [1] Inf
#>
data(copd)
fit <- panglm(hospitalized ~ crp, data = copd, index = c("id", "visit"),
model = "within", family = binomialfamily("logit"))
#> 17 group(s) with no within-group outcome variation dropped (all-0 or all-1) -- they carry no information for the conditional likelihood.
coef(fit)
#> crp
#> 0.09667869