EAP and MAP estimates of ability by quadrature

Bayesian estimation of a person's ability from a response pattern, at quadrature points \(X_r\) spread evenly from -4 to 4. With prior weights \(A(X_r)\) and likelihood \(L(X_r)\), the expected a posteriori estimate $$EAP = \frac{\sum_r X_r L(X_r) A(X_r)}{\sum_r L(X_r) A(X_r)}$$ is the mean of the posterior distribution, and the maximum a posteriori estimate is the quadrature point where the posterior \(L(X_r) A(X_r)\) is highest. The defaults replicate de Ayala (2009, p. 79).

Usage

eap_map(
  responses,
  b = c(-2.155, -0.245, 0.206, 0.984, 1.211),
  a = rep(1, length(b)),
  quadrature = 10,
  mean = 0,
  sd = 1,
  skew = 1
)

Arguments

responses

Ones and zeros, the responses to the items.

b

Difficulties of the items; the default is the five items of de Ayala (2009, Table 4.4).

a

Discriminations of the items: 1 for the Rasch model.

quadrature

Number of quadrature points.

mean, sd

Mean and standard deviation of the prior.

skew

Skewness parameter of the prior, xi in fGarch::dsnorm().

Value

A list with x, the quadrature points; prior, the prior weights; trace, the probability of each observed response at each point (one column per item); likelihood; posterior, the product of prior and likelihood; eap, psd (the posterior standard deviation) and map; and responses and prior_mean, as given.

References

de Ayala, R. J. (2009). The Theory and Practice of Item Response Theory. New York: Guilford Press.

Examples

estimate <- eap_map(response_pattern(2))
c(EAP = estimate$eap, MAP = estimate$map)
#>        EAP        MAP 
#> -0.2271434 -0.4444444