slm – pks

Simple Learning Models (SLMs)

Description

Fits a simple learning model (SLM) for probabilistic knowledge structures by minimum discrepancy maximum likelihood estimation.

Usage

slm(K, N.R, method = c("MD", "ML", "MDML"), R = as.binmat(N.R),
    beta = rep(0.1, nitems), eta = rep(0.1, nitems),
    g = rep(0.1, nitems),
    betafix = rep(NA, nitems), etafix = rep(NA, nitems),
    betaequal = NULL, etaequal = NULL,
    randinit = FALSE, incradius = 0,
    tol = 1e-07, maxiter = 10000, zeropad = 16,
    checkK = TRUE)

getSlmPK(g, K, Ko)

## S3 method for class 'slm'
print(x, P.Kshow = FALSE, parshow = TRUE,
      digits=max(3, getOption("digits") - 2), ...)

Arguments

K a state-by-problem indicator matrix representing the knowledge space. An element is one if the problem is contained in the state, and else zero.

N.R a (named) vector of absolute frequencies of response patterns.

method MD for minimum discrepancy estimation, ML for maximum likelihood estimation, MDML for minimum discrepancy maximum likelihood estimation.

R a pattern-by-problem indicator matrix of unique response patterns. Per default inferred from the names of N.R.

beta, eta, g vectors of initial values for the error, guessing, and solvability parameters.

betafix, etafix vectors of fixed error and guessing parameter values; NA indicates a free parameter.

betaequal, etaequal lists of vectors of problem indices; each vector represents an equivalence class: it contains the indices of problems for which the error or guessing parameters are constrained to be equal. (See Examples.)

randinit logical, if TRUE then initial parameter values are sampled uniformly with constraints. (See Details.)

incradius include knowledge states of distance from the minimum discrepant states less than or equal to incradius.

tol tolerance, stopping criterion for iteration.

maxiter the maximum number of iterations.

zeropad the maximum number of items for which an incomplete N.R vector is completed and padded with zeros.

checkK logical, if TRUE K is checked for well-gradedness.

Ko a state-by-problem indicator matrix representing the outer fringe for each knowledge state in K; typically the result of a call to getKFringe.

x an object of class slm, typically the result of a call to slm.

P.Kshow logical, should the estimated distribution of knowledge states be printed?

parshow logical, should the estimates of error, guessing, and solvability parameters be printed?

digits a non-null value for digits specifies the minimum number of significant digits to be printed in values.

… additional arguments passed to other methods.

Details

See Doignon and Falmagne (1999) for details on the simple learning model (SLM) for probabilistic knowledge structures. The model requires a well-graded knowledge space K.

An slm object inherits from class blim. See blim for details on the function arguments. The helper function getSlmPK returns the distribution of knowledge states P.K.

Value

An object of class slm and blim. It contains all components of a blim object. In addition, it includes:

g the vector of estimates of the solvability parameters.

References

Doignon, J.-P., & Falmagne, J.-C. (1999). Knowledge spaces. Berlin: Springer.

Examples

library("pks")

data(DoignonFalmagne7)
K   <- DoignonFalmagne7$K     # well-graded knowledge space
N.R <- DoignonFalmagne7$N.R   # frequencies of response patterns

## Fit simple learning model (SLM) by different methods
slm(K, N.R, method = "MD")    # minimum discrepancy estimation


Simple learning models (SLMs)

Number of knowledge states: 9
Number of response patterns: 32
Number of respondents: 1000

Method: Minimum discrepancy
Number of iterations: 1
Goodness of fit (2 log likelihood ratio):
    G2(16) = 125.19, p = 0

Minimum discrepancy distribution (mean = 0.254)
  0   1   2 
760 226  14 

Mean number of errors (total = 0.25582)
careless error    lucky guess 
    0.16253301     0.09328252 

Error, guessing, and solvability parameters
      beta      eta       g
a 0.092089 0.000001 0.79633
b 0.088720 0.000001 0.78900
c 0.045058 0.040640 0.67817
d 0.000001 0.040858 0.51355
e 0.000001 0.054722 0.53343

slm(K, N.R, method = "ML")    # maximum likelihood estimation by EM


Simple learning models (SLMs)

Number of knowledge states: 9
Number of response patterns: 32
Number of respondents: 1000

Method: Maximum likelihood
Number of iterations: 751
Goodness of fit (2 log likelihood ratio):
    G2(16) = 27.525, p = 0.036008

Minimum discrepancy distribution (mean = 0.254)
  0   1   2 
760 226  14 

Mean number of errors (total = 0.43083)
careless error    lucky guess 
    0.41317121     0.01765381 

Error, guessing, and solvability parameters
       beta       eta       g
a 0.1775600 0.0000010 0.87909
b 0.1739680 0.0000010 0.87043
c 0.1832340 0.0000010 0.73031
d 0.0054960 0.0000104 0.48765
e 0.0045130 0.0240918 0.47873

slm(K, N.R, method = "MDML")  # MDML estimation


Simple learning models (SLMs)

Number of knowledge states: 9
Number of response patterns: 32
Number of respondents: 1000

Method: Minimum discrepancy maximum likelihood
Number of iterations: 138
Goodness of fit (2 log likelihood ratio):
    G2(16) = 116.74, p = 0

Minimum discrepancy distribution (mean = 0.254)
  0   1   2 
760 226  14 

Mean number of errors (total = 0.25522)
careless error    lucky guess 
    0.18088170     0.07433556 

Error, guessing, and solvability parameters
      beta      eta       g
a 0.105502 0.000001 0.80827
b 0.100550 0.000001 0.79938
c 0.035363 0.022415 0.66652
d 0.000001 0.023790 0.51486
e 0.000001 0.058872 0.51965

## Compare SLM and BLIM
m1 <-  slm(K, N.R, method = "ML")
m2 <- blim(K, N.R, method = "ML")
anova(m1, m2)

Analysis of Deviance Table

Model 1: m1
Model 2: m2
  Resid. Df Resid. Dev Df Deviance Pr(>Chi)   
1        16     27.525                        
2        13     12.623  3   14.902 0.001903 **
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

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