The mode of the posterior under a normal prior. The prior makes
the posterior log-concave in the tails whatever the pattern, so a
MAP estimate EXISTS for every pattern including perfect ones,
where the ML estimate does not – the reason MAP is the usual
production choice. The price is SHRINKAGE toward the prior mean,
largest where information is smallest, and shrinkage_vs_ml
reports the gap rather than leaving it assumed away. The standard
error uses the posterior curvature, so the prior's information
narrows it.
Usage
morie_psy_map_theta(
y,
a = NULL,
b = NULL,
c = NULL,
prior = c(0, 1),
bounds = c(-6, 6)
)
Arguments
- y
binary responses.
- a, b, c
item parameters; b required.
- prior
c(mean, sd).
- bounds
search interval.
Value
list: theta, se, prior_mean, prior_sd, information,
posterior_information, shrinkage_vs_ml,
exists_for_perfect_patterns, n_items, method.
References
Samejima (1969); Bock and Aitkin (1981),
Psychometrika 46:443-459; Mislevy (1986).
Examples
morie_psy_map_theta(rep(1, 4), b = c(-1, -0.5, 0.5, 1))$theta
#> [1] 1.103758
