Weights for a continuous exposure are a ratio of densities. Both densities are evaluated on the standardized residual \(z_i = (A_i - \mu_i) / \sigma\), where \(A_i\) is the exposure, \(\mu_i\) the fitted conditional mean, and \(\sigma\) the residual spread, so a specification describes a density on a standardized scale rather than on the scale of the exposure itself.
These constructors name the family and record the parameters that identify
it. Pass one to the .density argument of wt_ate() or wt_cens(), which
also accept the strings "normal", "laplace", and "kernel" for the
families that need no parameters, and a bare function of one argument, which
is wrapped with dens_fn().
dens_normal()is the standard normal density, the default.dens_laplace()is the standard Laplace density, \(\exp(-|z|) / 2\), which puts more mass in the tails than the normal. Its scale is estimated under the Laplace itself, where it is the mean absolute residual, or by the root mean square of the residuals, which is the spread every family without an estimator of its own is read at.dens_t()is Student's t density withdfdegrees of freedom, heavier tailed still, and heavier the smallerdfis. Its scale is estimated under the t itself, by maximum likelihood, or by the root mean square of the residuals.dens_kernel()is a kernel density estimate of the standardized residuals, fit withstats::density()and interpolated to each observation. It assumes no family at all, at the cost of a density that is not a smooth function of the model's parameters: weights built from it have no closed-form standard error, andipw()reports none for them at all. Bootstrap the whole fit by hand to put an interval around such an estimate.dens_fn()is a density you write yourself.
Arguments
- sigma_method
How the spread of the conditional density is estimated from the residuals of the propensity score model, either
"rms", the root mean square, or"mle", the scale estimated under the family itself, which bothdens_laplace()anddens_t()take by default. See the section below.- df
Degrees of freedom for Student's t, a single positive, finite number.
- bw
The bandwidth passed to
stats::density(): a single positive number, or the name of one of its selection rules ("nrd0","nrd","ucv","bcv","SJ","SJ-ste", or"SJ-dpi").- adjust
A single positive number the bandwidth is multiplied by, as in
stats::density(). Values above 1 smooth the estimate further.- kernel
The smoothing kernel, one of the kernels
stats::density()accepts.- n
The number of grid points
stats::density()evaluates, at least 2. The estimate is interpolated between them, so a largernfollows the shape of the residuals more closely.- f
A function of one argument, the standardized residual, returning one non-negative, finite density value for each element it is given.
Value
An object of class propensity_density: a list with the elements
family, the name of the family; params, the parameters that identify
it, which is an empty list for a family that takes none; and fn, the
function that evaluates the density, which is NULL for dens_kernel().
dens_t() and dens_laplace() carry a fourth element, sigma_method,
the name of the estimator their scale is read with.
The spread of the conditional density
Both densities are evaluated on a residual standardized by a spread, and for
the conditional density that spread is estimated from the residuals of the
propensity score model. A family with no scale estimator of its own takes
the root mean square of those residuals, which is sigma_method = "rms" and
is the maximum likelihood estimator of the spread of a normal density.
It is not the maximum likelihood estimator of the scale of a heavier-tailed
family. The scale of such a family is smaller than its standard deviation,
and the root mean square is pulled outward by the large residuals a heavy
tail produces, which is what the family was chosen to accommodate.
sigma_method = "mle" estimates the scale under the family itself, and both
dens_laplace() and dens_t() take it by default, since the spread the
density is divided by is the scale parameter of the family reading it. For
dens_laplace() it is the mean absolute residual. For dens_t() it is the
root of
\(\sum_i \left[(\nu + 1) r_i^2 / (\nu \sigma^2 + r_i^2)\right] = n\),
where \(r_i\) is the residual and \(\nu\) is df. Each residual enters
that sum through a bounded term, so a residual far out in the tail moves the
estimate by less than it moves the root mean square. Prefer it when the
residuals are heavy tailed, which is the case these families are for; the two
estimators answer the same question, and answer it alike, as df grows and
the t approaches the normal.
dens_normal() offers no such choice, because the root mean square is
already its maximum likelihood estimator. Neither do dens_kernel() and
dens_fn(): a kernel is fit to the standardized residuals and divided by the
same spread, so the spread cancels up to the grid the estimate is
interpolated on, and a density you write yourself names no family an
estimator could be derived from.
The choice describes both densities of the ratio. The marginal density that
stabilizes the weights is the exposure's own, read at the exposure's mean and
at the spread the same estimator gives for it, so the two halves of the ratio
are densities of the same width. A scale estimated by maximum likelihood is
recorded by density_meta() as sigma = "mle", and ipw() estimates it
alongside the propensity score model's coefficients, solving the equation it
is the root of as part of its stacked system so that the standard errors
account for it. Supplying a .sigma says the spread is a number of your own
rather than one estimated from the residuals, so the two cannot be given
together.
Examples
dens_normal()
#> <density: normal>
dens_t(df = 4)
#> <density: t(df = 4)>
dens_t(df = 4, sigma_method = "mle")
#> <density: t(df = 4)>
dens_laplace(sigma_method = "rms")
#> <density: laplace>
dens_kernel(adjust = 1.5)
#> <density: kernel(bw = "nrd0", adjust = 1.5, kernel = "gaussian", n = 512)>
dens_fn(function(z) stats::dt(z, df = 4))
#> <density: function>