Newton-Raphson steps towards the maximum likelihood estimate of ability

Computes the log-likelihood of a response pattern under the Rasch model on a fine grid of abilities, fits a smoothing spline to it, and takes Newton-Raphson steps \(\theta_1 = \theta_0 - LL'(\theta_0) / LL''(\theta_0)\) with the derivatives of the spline, as the MLE Rasch tab of the app does. For a raw score of 0 or of all items the log-likelihood has no maximum, and the steps move away.

Usage

newton_raphson_rasch(
  responses,
  b = c(-1.9, -0.6, -0.25, 0.3, 0.45),
  start = 0,
  iterations = 1,
  theta = seq(-6, 6, by = 0.001)
)

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, p. 22).

start

Starting value of the estimate.

iterations

Number of steps.

theta

Grid of abilities on which the log-likelihood is computed.

Value

A list with theta and loglik, the grid and its log-likelihoods, and steps, a data frame with one row per step: start and end, the estimates the step starts from and ends at; loglik_start and loglik_end, the log-likelihood there; d1 and d2, the first and second derivatives at the start; intercept and crossing, where the tangent at the start crosses the vertical and horizontal axes; and angle, the acute angle in degrees between the tangent and the horizontal axis.

References

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

Examples

nr <- newton_raphson_rasch(response_pattern(2), iterations = 5)
nr$steps[, c("start", "end")]
#>        start        end
#> 1  0.0000000 -0.8340437
#> 2 -0.8340437 -0.8378275
#> 3 -0.8378275 -0.8378288
#> 4 -0.8378288 -0.8378288
#> 5 -0.8378288 -0.8378288