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Ordered Logistic random walk model

Source: R/model_doc.R
OrderedRW.Rd

This is a state-space model defined by a Ordered logistic measurement error distribution and a latent random walk. For more details see the BinRW vignette.

Arguments

max_score

Maximum value that the score can take

prior

Named list of the model's priors. If NULL, uses the default prior for the model (see default_prior()).

Details

Details of the model are available in the paper.

Parameters

Population parameters:

  • sigma_lat: Standard deviation of the random walk

  • sigma_meas: Standard deviation (not scale) of the logistic distribution (in [0, max_score] space)

  • sigma_tot: Total standard deviation for prediction one step ahead

  • rho2: Proportion of measurement variance to the total variance. It can be interpreted similarly to an R-squared, the proportion of the explained variance (the variance of the measurements) in the total variance.

  • mu_y0: Population mean of y0 (initial condition).

  • sigma_y0: Population standard deviation of y0 (initial condition).

  • delta: Relative difference between cutpoints (simplex of length max_score - 1)

  • ct: Cutpoints (vector of length max_score, in [0, max_score] space)

Patient-dependent parameters:

  • y0: initial latent score (y_lat at t0).

Observation-dependent (patient- and time-dependent) parameters:

  • y_lat: Latent score (in [0, max_score] space)

See list_parameters(model = "OrderedRW") for more details.

Priors

The priors are passed as a named list with elements delta, sigma_lat, sigma_meas, mu_y0 and sigma_y0 specifying priors for the corresponding parameters.

The element delta should be a vector X1 of length max_score - 1, such as all all elements of X1 are positive and delta ~ dirichlet(X1).

The latent score can be interpreted in the original [0, max_score] space, the priors for the other parameters are specified normalised max_score. Their priors are defined by a vector of length 2, containing values for x1 and x2, x2 > 0, such as:

  • sigma_meas / max_score ~ lognormal(x1, x2)

  • sigma_lat / max_score ~ lognormal(x1, x2)

  • mu_y0 ~ normal(x1, x2)

  • sigma_y0 ~ normal+(x1, x2)

NB: for the lognormal distribution, x1 corresponds to the mean of the log and x2 to the sd of the log. NB: sigma_y0 is constrained to be positive so x1 are usually set to 0 to define a half-normal distribution.

Default priors

  • The default prior for delta is a uniform symmetric Dirichlet distribution with concentration 2.

  • The default priors for sigma_meas and sigma_lat are lognormal distribution which translate to a 95% CI that is approximately [.02, 0.40] * M. The prior for sigma_lat thus allows fast or slow transitions from a state where y = 0 is the most likely outcome to a state where y = M is the most likely outcome. The prior for sigma_meas allows very precise or imprecise measurements.

  • The default priors for mu_y0 and sigma_y0 have reasonable ranges and translate to an approximately uniform prior over the range of the score for y0.

Examples

EczemaModel("OrderedRW", max_score = 10)
#> OrderedRW model (discrete)
#> max_score = 10 
#> Prior: 
#> - delta ~ dirichlet(2,2,2,2,2,2,2,2,2)
#> - sigma_meas / max_score ~ lognormal(-2.3,0.69)
#> - sigma_lat / max_score ~ lognormal(-2.3,0.69)
#> - mu_y0 / max_score ~ normal(0.5,0.25)
#> - sigma_y0 / max_score ~ normal+(0,0.12)