Nonparametric model

library(serosv)

Local estimation by polynomial

Refer to Chapter 7.1

Proposed model

Within the local polynomial framework, the linear predictor η(a) is approximated locally at one particular value a0 for age by a line (local linear) or a parabola (local quadratic).

The estimator for the k-th derivative of η(a0), for k = 0, 1, …, p (degree of local polynomial) is as followed:

η̂(k)(a0) = k!β̂k(a0)

The estimator for the prevalence at age a0 is then given by

π̂(a0) = g−1{β̂0(a0)}

  • Where g is the link function

The estimator for the force of infection at age a0 by assuming p ≥ 1 is as followed

λ̂(a0) = β̂1(a0)δ{β̂0(a0)}

  • Where $\delta \{ \hat{\beta}_0(a_0) \} = \frac{dg^{-1} \{ \hat{\beta}_0(a_0) \} } {d\hat{\beta}_0(a_0)}$

Fitting data

mump <- mumps_uk_1986_1987
age <- mump$age
pos <- mump$pos
tot <- mump$tot
y <- pos/tot

Use plot_gcv() to show GCV curves for the nearest neighbor method (left) and constant bandwidth (right).

plot_gcv(
   age, pos, tot,
   nn_seq = seq(0.2, 0.8, by=0.1),
   h_seq = seq(5, 25, by=1)
 )

Use lp_model() to fit a local estimation by polynomials.

lp <- lp_model(age, pos = pos, tot = tot, kern="tcub", nn=0.7, deg=2)
plot(lp)