pcIRT (version 0.1)

CRSM: Estimation of Continuous Rating Scale Model (Mueller, 1987)

Description

Estimation of the Rating Scale Model for continuous data by Mueller (1987).

$$P_{vi}(a \leq X \leq b) = \frac{\int_a^b exp[x \mu + x(2c-x) \theta] dx}{\int_{c-\frac{d}{2}}^{c+\frac{d}{2}} exp[t \mu + t(2c-t) \theta] dt}$$

Parameters are estimated by a pairwise algorithm.

Usage

CRSM(data, start, min, max)
## S3 method for class 'CRSM':
summary(object, \dots)
## S3 method for class 'CRSM':
print(x, \dots)

Arguments

data

Data matrix or data frame; rows represent observations (persons), columns represent the items.

start

Starting values for parameter estimation. If missing, a vector of 0 is used as starting values.

min

The minimum value of the response scale (on which the data are based).

max

The maximum value of the response scale (on which the data are based).

object

object of class CRSM

x

{object of class CRSM}

...

...

Value

• data
  data matrix according to the input
• data_p
  data matrix with data transformed to a response interval between 0 and 1
• itempar
  estimated item parameters
• itempar_se_low
  estimated lower boundary for standard errors of estimated item parameters
• itempar_se_up
  estimated upper boundary for standard errors of estimated item parameters
• itempar_se
  estimated mean standard errors of estimated item parameters
• distrpar
  estimated distribution parameter
• distrpar_se_low
  estimated lower boundary for standard errors of estimated distribution parameter
• distrpar_se_up
  estimated upper boundary for standard errors of estimated distribution parameter
• itempar_se
  estimated mean standard errors of estimated distribution parameter
• iterations
  Number of Newton-Raphson iterations for each item pair
• call
  call of the CRSM function

Details

The parameters of the Continuous Rating Scale Model are estimated by a pairwise algorithm (the algorithm is described in detail in Mueller, 1999) using Newton-Raphson iterations for optimizing. For the pairwise algorithm no assumption on the person parameter distribution is necessary.

References

Mueller, H. (1987). A Rasch model for continuous ratings. Psychometrika, 52, 165-181.

Mueller, H. (1999). Probabilistische Testmodelle fuer diskrete und kontinuierliche Ratingskalen. [Probabilistic models for discrete and continuous rating scales]. Bern: Huber.

Examples

#estimate CRSM item parameters
data(example1)
res_crsm <- CRSM(example1, min=0, max=1)

summary(res_crsm)