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Nonparametric model

Source: vignettes/nonparametric_model.Rmd
nonparametric_model.Rmd

Local estimation by polynomial

Refer to Chapter 7.1

Proposed model

Within the local polynomial framework, the linear predictor η(a)\eta(a) is approximated locally at one particular value a0a_0 for age by a line (local linear) or a parabola (local quadratic).

The estimator for the kk-th derivative of η(a0)\eta(a_0), for k=0,1,…,pk = 0,1,…,p (degree of local polynomial) is as followed:

η̂(k)(a0)=k!β̂k(a0) \hat{\eta}^{(k)}(a_0) = k!\hat{\beta}_k(a_0)

The estimator for the prevalence at age a0a_0 is then given by

π̂(a0)=g−1{β̂0(a0)} \hat{\pi}(a_0) = g^{-1}\{ \hat{\beta}_0(a_0) \}

  • Where gg is the link function

The estimator for the force of infection at age a0a_0 by assuming p≥1p \ge 1 is as followed

λ̂(a0)=β̂1(a0)δ{β̂0(a0)} \hat{\lambda}(a_0) = \hat{\beta}_1(a_0) \delta \{ \hat{\beta}_0 (a_0) \}

  • Where δ{β̂0(a0)}=dg−1{β̂0(a0)}dβ̂0(a0)\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)