3 Parameters
Once a data set has been identified, a number of relevant parameters need to be considered before using EpiNow2. As these will affect any results, it is worth spending some time investigating what their values should be.
3.1 Delay distributions
EpiNow2 works with different delays that apply to different
parts of the infection and observation process. They are defined using a
common interface that involves functions that are named after the
probability distributions, i.e. LogNormal(),
Gamma(), etc. For help with this function, see its manual
page
In all cases, the distributions given can be fixed (i.e. have no uncertainty) or variable (i.e. have associated uncertainty). For example, to define a fixed gamma distribution with mean 3, standard deviation (sd) 1 and maximum value 10, you would write
fixed_gamma <- Gamma(mean = 3, sd = 1, max = 10)
fixed_gamma
#> - gamma distribution (max: 10):
#> shape:
#> 9
#> rate:
#> 3which looks like this when plotted
If distributions are variable, the values with uncertainty are treated as prior probability densities in the Bayesian inference framework used by EpiNow2, i.e. they are estimated as part of the inference. For example, to define a variable gamma distribution where uncertainty in the mean is given by a normal distribution with mean 3 and sd 2, and uncertainty in the standard deviation is given by a normal distribution with mean 1 and sd 0.1, with a maximum value 10, you would write
uncertain_gamma <- Gamma(mean = Normal(3, 2), sd = Normal(1, 0.1), max = 10)
#> Warning: ! Uncertain gamma distribution specified in terms of parameters that are not
#> the "natural" parameters of the distribution shape and rate.
#> ℹ Converting using a crude and very approximate method that is likely to
#> produce biased results.
#> ℹ If possible it is preferable to specify the distribution directly in terms of
#> the natural parameters.
uncertain_gamma
#> - gamma distribution (max: 10):
#> shape:
#> - normal distribution:
#> mean:
#> 9
#> sd:
#> 2.5
#> rate:
#> - normal distribution:
#> mean:
#> 3
#> sd:
#> 1.4which looks like this when plotted
Note the warning about parameters. We used the mean and standard deviation to define this distribution with uncertain parameters, but it would be better to use the “natural” parameters of the gamma distribution, shape and rate, for example using the values estimate and reported back after calling the previous command.
There are various ways the specific delay distributions mentioned
below might be obtained. Often, they will come directly from the
existing literature reviewed by the user and studies conducted
elsewhere. Sometimes it might be possible to obtain them from existing
databases, e.g. using the epiparameter R
package. Alternatively they might be obtainable from raw data,
e.g. line-listed individual-level records. The EpiNow2 package
contains functionality for estimating delay distributions from observed
delays in the estimate_delays() function. For a more
comprehensive treatment of delays and their estimation avoiding common
biases one can consider, for example, the dynamicaltruncation
R package and associated paper.
3.1.1 Generation intervals
The generation interval is a delay distribution that describes the
amount of time that passes between an individual becoming infected and
infecting someone else. In EpiNow2, the generation time
distribution is defined by a call to gt_opts(), a function
that takes a single argument defined as a dist_spec object
(returned by the function corresponding to the probability distribution,
i.e. LogNormal(), Gamma(), etc.). For example,
to define the generation time as gamma distributed with uncertain mean
centered on 3 and sd centered on 1 with some uncertainty, a maximum
value of 10 and weighted by the number of case data points we could use
the shape and rate parameters suggested above (though notes that this
will only very approximately produce the uncertainty in mean and
standard deviation stated there):
3.1.2 Reporting delays
EpiNow2 calculates reproduction numbers based on the
trajectory of infection incidence. Usually this is not observed
directly. Instead, we calculate case counts based on, for example, onset
of symptoms, lab confirmations, hospitalisations, etc. In order to
estimate the trajectory of infection incidence from this we need to
either know or estimate the distribution of delays from infection to
count. Often, such counts are composed of multiple delays for which we
only have separate information, for example the incubation period (time
from infection to symptom onset) and reporting delay (time from symptom
onset to being a case in the data, e.g. via lab confirmation, if counts
are not by the date of symptom onset). In this case, we can combine
multiple delays with the plus (+) operator, e.g.
incubation_period <- LogNormal(
meanlog = Normal(1.6, 0.05),
sdlog = Normal(0.5, 0.05),
max = 14
)
reporting_delay <- LogNormal(meanlog = 0.5, sdlog = 0.5, max = 10)
combined_delays <- incubation_period + reporting_delay
combined_delays
#> Composite distribution:
#> - lognormal distribution (max: 14):
#> meanlog:
#> - normal distribution:
#> mean:
#> 1.6
#> sd:
#> 0.05
#> sdlog:
#> - normal distribution:
#> mean:
#> 0.5
#> sd:
#> 0.05
#> - lognormal distribution (max: 10):
#> meanlog:
#> 0.5
#> sdlog:
#> 0.5We can visualise this combined delay
In EpiNow2, the reporting delay distribution is defined by a
call to delay_opts(), a function that takes a single
argument defined as a dist_spec object (returned by
LogNormal(), Gamma() etc.). For example, if
our observations were by symptom onset we would use
If they were by date of lab confirmation that happens with a delay
given by reporting_delay, we would use
3.1.3 Truncation
Besides the delay from infection to the event that is recorded in the data, there can also be a delay from that event to being recorded in the data. For example, data reported by symptom onset may only become part of the dataset once lab confirmation has occurred, or even a day or two after that confirmation. Statistically, this means our data is right-truncated. In practice, it means that recent data will be unlikely to be complete.
The amount of such truncation that exists in the data can be
estimated from multiple snapshots of the data, i.e. what the data looked
like at multiple past dates. One can then use methods that use the
amount of backfilling that occurred 1, 2, … days after data for a date
are first reported. In EpiNow2, this can be done using the
estimate_truncation() method which returns, amongst others,
posterior estimates of the truncation distribution. For more details on
the model used for this, see the estimate_truncation vignette.
In the estimate_infections() function, the truncation
distribution is defined by a call to trunc_opts(), a
function that takes a single argument defined as a
dist_spec (either defined by the user or obtained from a
call to estimate_truncation() or any other method for
estimating right truncation). This will then be used to correct for
right truncation in the data.
The separation of estimation of right truncation on the one hand and estimation of the reproduction number on the other may be attractive for practical purposes but is questionable statistically as it separates two processes that are not strictly separable, potentially introducing a bias. An alternative approach where these are estimated jointly is being implemented in the epinowcast package, which is being developed by the EpiNow2 developers with collaborators.
3.2 Completeness of reporting
Another issue affecting the progression from infections to reported
outcomes is underreporting, i.e. the fact that not all infections are
reported as cases. This varies both by pathogen and population (and
e.g. the proportion of infections that are asymptomatic) as well as the
specific outcome used as data and where it is located on the severity
pyramid (e.g. hospitalisations vs. community cases). In EpiNow2
we can specify the proportion of infections that we expect to be
observed (with uncertainty assumed represented by a truncated normal
distribution with bounds at 0 and 1) using the scale
argument to the obs_opts() function. For example, if we
think that 40% (with standard deviation 1%) of infections end up in the
data as observations we could specify.
3.3 Initial reproduction number
The default model that estimate_infections() uses to
estimate reproduction numbers requires specification of a prior
probability distribution for the initial reproduction number. This
represents the user’s initial belief of the value of the reproduction
number, where there is no data yet to inform its value. By default this
is assumed to be represented by a lognormal distribution with mean and
standard deviation of 1. It can be changed using the
rt_opts() function. For example, if the user believes that
at the very start of the data the reproduction number was 2, with
uncertainty in this belief represented by a standard deviation of 1,
they would use
3.4 Weighing delay priors
When providing uncertain delay distributions one can end up in a
situation where the estimated means are shifted a long way from the
given distribution means, and possibly further than is deemed realistic
by the user. In that case, one could specify narrower prior
distributions (e.g., smaller mean_sd) in order to keep the
estimated means closer to the given mean, but this can be difficult to
do in a principled manner in practice. As a more straightforward
alternative, one can choose to weigh the generation time priors by the
number of data points in the case data set by setting
weigh_delay_priors = TRUE (the default).
