twinSIR.Rd
twinSIR
is used to fit additive-multiplicative intensity models for
epidemics as described in Höhle (2009). Estimation is driven
by (penalized) maximum likelihood in the point process frame work. Optimization
(maximization) of the (penalized) likelihood function is performed by means of
optim
.
The implementation is illustrated in Meyer et al. (2017, Section 4),
see vignette("twinSIR")
.
twinSIR(formula, data, weights, subset,
knots = NULL, nIntervals = 1, lambda.smooth = 0, penalty = 1,
optim.args = list(), model = TRUE, keep.data = FALSE)
an object of class "formula"
(or one that can be coerced to
that class): a symbolic description of the intensity model to be estimated.
The details of the model specification are given below.
an object inheriting from class "epidata"
.
an optional vector of weights to be used in the fitting process. Should be
NULL
(the default, i.e. all observations have unit weight) or a
numeric vector.
an optional vector specifying a subset of observations to be used in the
fitting process. The subset atRiskY == 1
is automatically chosen,
because the likelihood only depends on those observations.
numeric vector or NULL
(the default). Specification of the knots,
where we suppose a step of the log-baseline. With the current
implementation, these must be existing "stop"
time points in the
selected subset
of the data
, which is always
restricted to atRiskY == 1
rows.
The intervals of constant log-baseline
hazard rate then are \((minTime;knots_1]\), \((knots_1;knots_2]\),
..., \((knots_K;maxTime]\).
By default, the knots
are automatically chosen at the quantiles of
the infection time points such that nIntervals
intervals result.
Non-NULL knots
take precedence over nIntervals
.
the number of intervals of constant log-baseline hazard. Defaults to 1, which means an overall constant log-baseline hazard will be fitted.
numeric, the smoothing parameter \(\lambda\). By default it is 0 which
leads to unpenalized likelihood inference.
In case lambda.smooth=-1
, the automatic smoothing parameter
selection based on a mixed model approach is used (cf.
Höhle, 2009).
either a single number denoting the order of the difference used to penalize the log-baseline coefficients (defaults to 1), or a more specific penalty matrix \(K\) for the parameter sub-vector \(\beta\). In case of non-equidistant knots -- usually the case when using quantile based knot locations -- only a 1st order differences penalty matrix as in Fahrmeir and Lang (2001) is implemented.
a list with arguments passed to the optim
function.
Especially useful are the following ones:
par
:to specify initial parameter values. Those must be in the order
c(alpha, h0, beta)
, i.e. first the coefficients of the epidemic
covariates in the same order as they appear in the formula
, then
the log-baseline levels in chronological order and finally the
coefficients of the endemic covariates in the same order
as they appear in the cox
terms of the formula
. The default
is to start with 1's for alpha
and 0's for h0
and
beta
.
control
:for more detailed trace
-ing (default: 1), another REPORT
-ing
frequency if trace
is positive (default: 10), higher maxit
(maximum number of iterations, default: 300) or another factr
value
(default: 1e7, a lower value means higher precision).
method
:the optimization algorithm defaults to "L-BFGS-B"
(for
box-constrained optimization), if there are any epidemic (non-cox
)
variables in the model, and to "BFGS"
otherwise.
lower
:if method = "L-BFGS-B"
this defines the lower bounds for the
model coefficients. By default, all effects \(\alpha\) of epidemic
variables are restricted to be non-negative. Normally, this is exactly
what one would like to have, but there might be reasons for other lower
bounds, see the Note below.
hessian
:An estimation of the Expected Fisher Information matrix is always
part of the return value of the function. It might be interesting to see
the Observed Fisher Information (= negative Hessian at the maximum), too.
This will be additionally returned if hessian = TRUE
.
logical indicating if the model frame, the weights
,
lambda.smooth
, the penalty matrix \(K\) and the list of used
distance functions f
(from attributes(data)
) should be
returned for further computation. This defaults to TRUE
as this
information is necessary e.g. in the profile
and plot
methods.
logical indicating if the "epidata"
object (data
)
should be part of the return value. This is only necessary for use of the
simulate
-method for "twinSIR"
objects. The reason is that the twinSIR
function only uses and
stores the rows with atRiskY == 1
in the model
component, but
for the simulation of new epidemic data one needs the whole data set with
all individuals in every time block. The default value is FALSE
, so
if you intent to use simulate.twinSIR
, you have to set this to
TRUE
.
A model is specified through the formula
, which has the form
~ epidemicTerm1 + epidemicTerm2 + cox(endemicVar1) *
cox(endemicVar2)
,
i.e. the right hand side has the usual form as in lm
with
some variables marked as being endemic by the special function
cox
. The left hand side of the formula is empty and will be
set internally to cbind(start, stop, event)
, which is similar to
Surv(start, stop, event, type="counting")
in package survival.
Basically, the additive-multiplicative model for the infection intensity \(\lambda_i(t)\) for individual \(i\) is $$\lambda_i(t) = Y_i(t) * (e_i(t) + h_i(t))$$ where
is the at-risk indicator, indicating if individual \(i\) is
“at risk” of becoming infected at time point \(t\).
This variable is part of the event history data
.
is the epidemic component of the infection intensity, defined as
$$e_i(t) = \sum_{j \in I(t)} f(||s_i - s_j||)$$
where \(I(t)\) is the set of infectious individuals just before time
point \(t\), \(s_i\) is the coordinate vector of individual \(i\)
and the function \(f\) is defined as
$$f(u) = \sum_{m=1}^p \alpha_m B_m(u)$$
with unknown transmission parameters \(\alpha\) and known distance
functions \(B_m\). This set of distance functions results in the set of
epidemic variables normally calculated by the converter function
as.epidata
, considering the equality
$$e_i(t) = \sum_{m=1}^p \alpha_m x_{im}(t)$$
with \(x_{im}(t) = \sum_{j \in I(t)} B_m(||s_i - s_j||)\) being the
\(m\)'th epidemic variable for individual \(i\).
is the endemic (cox
) component of the infection intensity, defined
as
$$h_i(t) = \exp(h_0(t) + z_i(t)' \beta)$$
where \(h_0(t)\) is the log-baseline hazard function, \(z_i(t)\)
is the vector of endemic covariates of individual \(i\) and \(\beta\)
is the vector of unknown coefficients.
To fit the model, the log-baseline hazard function is approximated by a
piecewise constant function with known knots, but unknown levels,
which will be estimated. The approximation is specified by the arguments
knots
or nIntervals
.
If a big number of knots
(or nIntervals
) is chosen, the
corresponding log-baseline parameters can be rendered identifiable by
the use of penalized likelihood inference. At present, it is the job
of the user to choose an adequate value of the smoothing parameter
lambda.smooth
. Alternatively, a data driven
lambda.smooth
smoothing parameter selection based on a mixed
model representation of an equivalent truncated power spline is offered (see
reference for further details). The following two steps are iterated
until convergence:
Given fixed smoothing parameter, the penalized likelihood is optimized for the regression components using a L-BFGS-B approach
Given fixed regression parameters, a Laplace approximation of the marginal likelihood for the smoothing parameter is numerically optimized.
Depending on the data, convergence might take a couple of iterations.
Note also that it is unwise to include endemic covariates with huge values,
as they affect the intensities on the exponential scale (after
multiplication by the parameter vector \(\beta\)).
With large covariate values, the
optim
method "L-BFGS-B" will likely terminate due to an infinite
log-likelihood or score function in some iteration.
twinSIR
returns an object of class
"twinSIR"
, which is a list containing the following components:
a named vector of coefficients.
the maximum of the (penalized) log-likelihood function.
the number of log-likelihood and score function evaluations.
logical indicating convergence of the optimization algorithm.
if requested, the negative Hessian from
optim
.
an estimation of the Expected Fisher Information matrix.
the optimization algorithm used.
a numeric vector (c(minTime, knots, maxTime)
)
representing the consecutive intervals of constant log-baseline.
a numeric vector containing the number of infections in each of
the above intervals
.
if requested, the model information used. This is a list with
components "survs"
(data.frame with the id, start, stop and event
columns), "X"
(matrix of the epidemic variables), "Z"
(matrix
of the endemic variables), "weights"
(the specified weights
),
"lambda.smooth"
(the specified lambda.smooth
), "K"
(the penalty matrix used), and "f"
and "w"
(the functions to generate the used epidemic covariates).
Be aware that the model only contains those rows with atRiskY == 1
!
if requested, the supplied "epidata"
data
.
the matched call.
the specified formula
.
the terms
object used.
Höhle, M. (2009), Additive-multiplicative regression models for spatio-temporal epidemics, Biometrical Journal, 51 (6), 961-978.
Meyer, S., Held, L. and Höhle, M. (2017): Spatio-temporal analysis of epidemic phenomena using the R package surveillance. Journal of Statistical Software, 77 (11), 1-55. doi: 10.18637/jss.v077.i11
Michael Höhle and Sebastian Meyer
There are some restrictions to modelling the infection intensity
without a baseline hazard rate, i.e. without an intercept in the
formula
.
Reason: At some point, the optimization algorithm L-BFGS-B tries to set all
transmission parameters \(\alpha\) to the boundary value 0 and to calculate
the (penalized) score function with this set of parameters (all 0). The problem
then is that the values of the infection intensities \(lambda_i(t)\) are 0
for all \(i\) and \(t\) and especially at observed event times, which is
impossible. Without a baseline, it is not allowed to have all alpha's set to 0,
because then we would not observe any infections. Unfortunately, L-BFGS-B can
not consider this restriction. Thus, if one wants to fit a model without
baseline hazard, the control parameter lower
must be specified in
optim.args
so that some alpha is strictly positive, e.g.
optim.args = list(lower = c(0,0.001,0.001,0))
and the initial parameter
vector par
must not be the zero vector.
as.epidata
for the necessary data input structure,
plot.twinSIR
for plotting the path of the infection intensity,
profile.twinSIR
for profile likelihood estimation.
and simulate.twinSIR
for the simulation of epidemics following
the fitted model.
Furthermore, the standard extraction methods
vcov
, logLik
,
AIC
and
extractAIC
are implemented for
objects of class "twinSIR"
.
data("hagelloch")
summary(hagelloch)
# simple model with an overall constant baseline hazard rate
fit1 <- twinSIR(~ household + cox(AGE), data = hagelloch)
fit1
summary(fit1) # see also help("summary.twinSIR")
plot(fit1) # see also help("plot.twinSIR")
checkResidualProcess(fit1) # could be better
# fit a piecewise constant baseline hazard rate with 3 intervals using
# _un_penalized ML and estimated coefs from fit1 as starting values
fit2 <- twinSIR(~ household, data = hagelloch, nIntervals = 3,
optim.args = list(par = coef(fit1)[c(1,2,2,2)]))
summary(fit2)
# fit a piecewise constant baseline hazard rate with 7 intervals
# using _penalized_ ML
fit3 <- twinSIR(~ household, data = hagelloch, nIntervals = 7,
lambda.smooth = 0.1, penalty = 1)
summary(fit3)
checkResidualProcess(fit3)
# plot the estimated log-baseline levels
plot(x=fit2$intervals, y=coef(fit2)[c(2,2:4)], type="S", ylim=c(-6, -1))
lines(x=fit3$intervals, y=coef(fit3)[c(2,2:8)], type="S", col=2)
legend("right", legend=c("unpenalized 3", "penalized 7"), lty=1, col=1:2, bty="n")
## special use case: fit the model to a subset of the events only,
## while preserving epidemic contributions from the remainder
## (maybe some buffer area nodes)
fit_subset <- twinSIR(~ household, data = hagelloch, subset = CL=="preschool")
summary(fit_subset)
# \dontshow{
## the eventTimes attribute was wrong in surveillance <= 1.15.0
stopifnot(
length(residuals(fit_subset)) == sum(fit_subset$model$survs$event)
)
# }