Description Details References See Also Examples
A support function called by drm
that parses from the argument
dep
the covariates and functional forms for the association
model. This function should not be used directly by the user.
In the argument dep
, the user needs to specify the covariates and
functional forms on the association parameters. The following list
describes the notation and concise interpretation of the association
parameters. For more details, see the reference below.
nu
For "N"-structure: the proportion of susceptibles in the
population, i.e. proportion of units that can have a value greater
than the smallest category. Covariates can be specified. See examples
below.
nu1, nu2, ..., nuk
For "L"-structure: the proportion in
the population in the latent class 1, 2, ..., k. The proportion in the
latent class 0 can be obtained by calculating
1-\code{nu1}-\code{nu2}-... Note that
for binary responses, the number of latent classes can be specified
with an argument Lclass
(Default 2). Covariates can be
specified. See examples below.
kappa
(binary response)For "L"-structure with
Lclass=2
(default), the success probability in the latent class
0 divided by the success probability in the latent class 1,
i.e. kappa = pr(Y=1|L=0)/pr(Y=1|L=1). Covariates can be specified;
examples below.
kappa0, kappa1,...,kappak-1
(binary response)For
"L"-structure with Lclass=k+1
:
the success probabilities in the latent class 0, 1, ..., k-1
divided by the success probability in the latent class k. For example,
kappa0 = pr(Y=1|L=0)/pr(Y=1|L=k). Covariates can be specified. See
examples below.
kappa1, kappa2,...,kappak-1
(multicategorical response with k
levels)For "L"-structure with two latent classes
(i.e. Lclass=2
): the category probabilities in categories 1,
2,..., k-1 in the latent class 0 divided by the corresponding
probabilities in the latent class 1. For example, kappa2 =
pr(Y=2|L=0)/pr(Y=2|L=1). The smallest response value is regarded as
the baseline, denoted by 0
xi1, xi0
(binary response)For "B"-structure: the shape
parameters of the Beta-distribution (sometimes also noted as xi1=p
and
xi0=q
). Covariates can be specified. See examples below.
xi0, xi1, ..., xik
(multicategorical response)For "D"-structure:
the shape parameters of the Dirichlet distribution
tau
(binary responses)For "M"-structure: adjacent second order
dependence ratio. If the number of repeated measurements is greater
than two, the adjacent tau
's are assumed to be equal
(i.e. stationarity of the dependence ratios). In order to specify
equalities or functional forms (i.e. non-stationary overlapping
dependence ratios), see examples below.
tau12, tau13, tau123
(binary responses)For "M2"-structure:
adjacent second order dependence ratio (tau12
), adjacent third
order dependence ratio (tau123
) and the second order dependence
ratio between first and third response (tau13
). If the number
of repeated measurements is greater than three, the tau
's are
assumed to be equal (i.e. stationarity of the dependence
ratios). Equalities and functional forms can be specified. See
examples below.
tau11, tau12, ..., tau21, tau22, ...,taukk
(multicategorical
responses)For "M"-structure: adjacent second order
dependence ratios for categories 1, 2, ..., k, where the smallest
response value is regarded as the baseline, denoted by 0. Equalities
and functional forms can be specified as for the binary "M"-structures
above. See examples below.
Jokinen J. Fast estimation algorithm for likelihood-based analysis of repeated categorical responses. Computational Statistics and Data Analysis 2006; 51:1509-1522.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 | ### Example of functional forms:
## non-stationary second order Markov structure
## initial values of the dependence ratios are set to 1.
## Not run:
data(wheeze)
assoc <- list("M2",
tau12 ~ function(a78=1, a89=1, a910=1)c(a78, a89, a910),
tau123 ~ function(a789=1, a8910=1)c(a789, a8910),
tau13 ~ function(a79=1, a810=1)c(a79, a810))
fit1 <- drm(wheeze~I(age>9)+smoking+cluster(id)+Time(age),
data=wheeze, dep=assoc, print=0)
### Example of other parameter restrictions:
## fixing parameters to a known value: ~tau12==1, ~tau21==1
## setting parameters to equal: ~tau11==tau22
data(marijuana)
assoc <- list("M", ~tau12==1, ~tau21==1, ~tau11==tau22)
fit2 <- drm(y~age+cluster(id)+Time(age), data=marijuana,
subset=sex=="female", dep=assoc, print=0)
## setting all parameters to equal:
assoc <- list("M", ~tau11==tau12, ~tau11==tau21, ~tau11==tau22)
fit3 <- drm(y~age+cluster(id)+Time(age), data=marijuana,
subset=sex=="female", dep=assoc, print=0)
## End(Not run)
### Example of covariates for the association parameters:
## allow the probabilities within the latent class
## vary by sex. Note: covariate needs to be a factor.
data(obese)
assoc <- list("L", kappa ~ kappa:factor(sex))
fit4 <- drm(obese~age+cluster(id)+Time(age), data=obese,
dep=assoc, print=0)
### Example how to derive conditional probabilities from marginals
## Fit a model with three latent classes:
data(wheeze)
latent3 <- drm(wheeze~I(age>9)+smoking+cluster(id),data=wheeze,
dep="L",Lclass=3, print=0)
## calculate conditional probabilities:
## pr(Y=1|L=2) = pr(Y=1)/(nu2+kappa1*nu1+kappa0*(1-nu1-nu2))
## pr(Y=1|L=1) = kappa1*pr(Y=1|L=2)
## pr(Y=1|L=0) = kappa0*pr(Y=1|L=2)
est <- coef(latent3)
psi2 <- latent3$fitted.marginals/
(est["nu2"]+est["kappa1"]*est["nu1"]+
est["kappa0"]*(1-est["nu1"]-est["nu2"]))
psi1 <- psi2*est["kappa1"]
psi0 <- psi2*est["kappa0"]
## check the model validity, i.e. require that 0 < psi_i <1:
range(cbind(psi0,psi1,psi2))
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