powLawOrd: Power law with ordinal observations

In eehh-stanford/yada: Yet Another Demographic Analysis package

Description

powLawOrd calculates the mean (h). powLawOrdSigma calculates the noise (sigma, or sig for short). powLawOrdNegLogLikVect calculates a vector of the negative log-likelihoods. powLawOrdNegLogLik calculates the negative log-likelihood (sum of powLawOrdNegLogLikVect). fitPowLawOrd returns the maximum likelihood fit. simPowLawOrd creates simulated data.

Usage

powLawOrd(x, th_v, transformVar = F)

Arguments

x	Vector of independent variable observations
th_v	Vector of parameters with ordering [rho,tau_1,...,tau_M,s,kappa]
transformVar	Whether a transformation of the parameterization is needed [Default FALSE]
vstar	Vector of latent dependent variable observations (for v^*)
v	Vector of dependent variable observations (ordinal)
rho	Scaling exponent
s	Baseline noise
kappa	Slope of noise [Optional]
hetSpec	Specification for the heteroskedasticity [Default 'none']

Details

We assume that the latent response variable vstar (for v^*) is distributed as

vstar ~ N(g(x),sig(x)^2)

where x is the independent variable, g(x) the mean, sig(x) the noise, and N denotes a normal distribution. If sig is independent of x, the model is homoskedastic. Otherwise, it is heteroskedastic. What is observed is not the latent response vstar, but rather an ordinal response variable v that is derived from vstar per

-Inf < vstar <= tau_1 --> m = 0

tau_1 < vstar <= tau_2 --> m = 1

...

tau_M < vstar <= Inf --> m = M

That is, there are M+1 ordinal categories m = 0,1,...M, and the ordered vector tau = [tau_1,...,tau_M] provides the cut-offs to derive the ordinal response, v, from the latent continuous response, vstar. Given this model, the likelihood of the observation (x,v) is

l_v = Phi((tau_{m+1} - g)/sig) - Phi((tau_m - g)/sig)

where Phi is the cumulative distribution function of the standard univariate normal with a mean of 0 and standard deviation of 1. We adopt the convention tau_0 = -Inf and tau_M+1 = Inf and adopt the notation that tau_m is tau_lo and tau_m+1 is tau_hi (and similarly for Phi and other variables). The negative log likelihood of the observation (x,v) is

eta_v = -log(Phi((tau_hi - g)/sig) - Phi((tau_lo - g)/sig))

= -log(Phi_hi-Phi_lo)

For the mean and noise, we adopt the parametric forms

h = x^rho

and

sig = s

[hetSpec = 'none']

sig = s*(1 + kappa*x)

[hetSpec = 'sd_x']

sig = s*(1 + kappa*x^rho)

[hetSpec = 'sd_resp']

The choice of x^rho comes from using an offset power law, alpha*x^rho + beta, and requiring alpha=1 and beta=0 for identifiability. For optimization, it is often preferable to work with an unconstrained variable. This is supported via the optional input transformVar. rho, s, kappa, and the differences between successive values of tau must be positive. This is accomplished by using, for example, rho_bar = log(rho) and rho = exp(rho_bar).

Author(s)

Michael Holton Price <[e-mail]>

eehh-stanford/yada documentation built on June 18, 2020, 8:05 p.m.