CDM (version 1.0-0)

din: Main Function for Parameter Estimation in Cognitive Diagnosis Models

Description

din provides parameter estimation for cognitive diagnosis models of the types ``DINA'', ``DINO'' and ``mixed DINA and DINO''. It is the main function of this package.

Usage

din(data, q.matrix, conv.crit = 0.001, maxit = 100, 
	  constraint.guess = NULL, constraint.slip = NULL,
	  guess.init = rep(0.2, ncol(data)), slip.init = guess.init,
	  weights = rep(1, nrow(data)), rule = "DINA", progress = TRUE)

Arguments

data

a required $N$ times $J$ data matrix containing the binary responses, 0 or 1, of $N$ respondants to $J$ test items, where 1 denotes a correct anwer and 0 an incorrect one. The nth row of the matrix represents the binary response pattern o

q.matrix

a required binary $J$ times $K$ containing the attributes not required or required, 0 or 1, to master the items. The jth row of the matrix is a binary indicator vector indicating which attributes are not required (coded by 0) and which a

conv.crit

termination criterion of iterations in the parameter estimation process. Iteration ends if the maximal change in parameter estimates is below this value.

maxit

maximal number of iterations in the estimation process.

constraint.guess

an optional matrix of fixed guessing parameters. The first column of this matrix indicates the numbers of the items whose guessing parameters are fixed and the second column the values the guessing parameters are fixed to.

constraint.slip

an optional matrix of fixed slipping parameters. The first column of this matrix indicates the numbers of the items whose guessing parameters are fixed and the second column the values the guessing parameters are fixed to.

guess.init

an optional initial vector of guessing parameters. Guessing parameters are bounded between 0 and 1.

slip.init

an optional initial vector of guessing parameters. Slipping parameters are bounded between 0 and 1.

weights

an optional vector of weights for the response pattern. Non-integer weights allow for different sampling schemes.

rule

an optional character string or vector of character strings specifying the model rule that is used. The character strings must be of "DINA" or "DINO". If a vector of character strings is specified, implying an ite

progress

an optional logical indicating whether the function should print the progress of iteration in the estimation process.

Value

• coef
  a dataframe giving for each item condensation rule, the estimated guessing and slipping parameters and their standard errors. All entries are rounded to 3 digits.
• guess
  a dataframe giving the estimated guessing parameters and their standard errors for each item.
• slip
  a dataframe giving the estimated slipping parameters and their standard errors for each item.
• loglike
  a numeric giving the value of the maximized log likelihood.
• AIC
  a numeric giving the AIC value of the model.
• BIC
  a numeric giving the BIC value of the model.
• posterior
  a matrix given the posterior skill distribution for all respondents. The nth row of the matrix gives the probabilities for respondent n to possess skills 1 to K.
• like
  a matrix giving the values of the maximized likelihood for all respondents.
• data
  the input matrix of binary response data.
• q.matrix
  the input matrix of the required attributes.
• pattern
  a matrix giving the frequency of observed response patterns stored in item. patt.split, the attribute classes leading to highest endorsement probability for the respective response pattern (mle.est) with the corresponding posterior class probability (mle.post), the attribute classes having the highest occurrence probability given the response pattern (map.est) with the corresponding posterior class probability (mle.post), and the estimated posterior for each response pattern.
• attribute.patt
  a data frame giving the estimated occurrence probabilities of the skill classes and the expected frequency of the attribute classes given the model.
• skill.patt
  a matrix given the population prevalences of the skills.
• subj.pattern
  a vector of strings indicating the item response pattern for each subject.
• attribute.patt.splitted
  a dataframe giving the response pattern of the respondents.
• display
  a character giving the model specified under rule.
• item.patt.split
  a matrix giving the splitted response pattern.
• item.patt.freq
  a numeric vector given the frequencies of the response pattern in item.patt.split.

concept

• diagnosis models
• binary response data

Details

In the CDM DINA (deterministic-input, noisy-and-gate; de la Torre and Douglas, 2004) and DINO (deterministic-input, noisy-or-gate; Templin and Henson, 2006) models endorsement probabilities are modeled based on guessing and slipping parameters, given the different skill patterns. The probability of respondent $n$ for solving item $j$ is calculated as a function of the respondent's latent response $\eta_{nj}$ and the guessing and slipping rates $g_j$ and $s_j$ for item $j$ conditional on the respondent's skill pattern $\alpha_n$: $$P_j(\alpha_n) = P(X_{nj} = 1 | \alpha_n) = g_j^{(1- \eta_{nj})} (1 - s_j) ^{\eta_{nj}}.$$ The respondent's latent response $\eta_{nj}$ is a binary number, 0 or 1, indicating absence or presence of all (rule = "DINO") or at least one (rule = "DINO") required skill(s) for item $j$, respectively. DINA and DINO parameter estimation is performed by maximization of the marginal likelihood of the data. The a priori distribution of the skill vectors is a uniform distribution. The implementation follows the EM algorithm by de la Torre (2009). An additional condition in parameter estimation in DINA and DINO models is $$g_j < 1 - s_j$$ for each item, that is, the chance of guessing an item is supposed to be smaller than the chance of mastering the required skills for that item and not slipping. However, the EM algorithm needs not satisfy that constraint. In that cases there will be a warning during the estimation algorithm. Possible problem solving strategies are to adjust the convergence criteria conv.crit, maxit, guess.init and slip.init or to put constraints on the guessing and slipping parameters (constraint.guess and constraint.slip) of the items that violate the additional condition. The function din returns an object of the class din (see Value), for which plot, print, and summary methods are provided; plot.din, print.din, and summary.din, respectively.

References

de la Torre, J. (2009) Dina model parameter estimation: A didactic. Journal of Educational and Behavioral Statistics, 34, 115--130. de la Torre, J. and Douglas, J. (2004) Higher-order latent trait models for cognitive diagnosis. Psychometrika, 69, 333--353. Rupp, A. A., Templin, J. and Henson, R. A. (2010) Diagnostic Measurement: Theory, Methods, and Applications. New York: The Guilford Press. Templin, J. and Henson, R. (2006) Measurement of psychological disorders using cognitive diagnosis models. Psychological Methods, 11, 287--305.

See Also

plot.din, the S3 method for plotting objects of the class din; print.din, the S3 method for printing objects of the class din; summary.din, the S3 method for summarizing objects of the class din, which creates objects of the class summary.din; din, the main function for DINA and DINO parameter estimation, which creates objects of the class din. See also CDM-package for general information about this package.

Examples

##
## (1) examples based on dataset fractions.subtraction.data
##

## dataset fractions.subtraction.data and corresponding Q-Matrix
fraction.subtraction.data
fraction.subtraction.qmatrix

## Misspecification in parameter specification for method din()
## leads to warnings and terminates estimation procedure. E.g.,

# See Q-Matrix specification
fractions.dina.warning1 <- din(data = fraction.subtraction.data,
  q.matrix = t(fraction.subtraction.qmatrix)) 
  
# See guess.init specification
fractions.dina.warning2 <- din(data = fraction.subtraction.data,
  q.matrix = fraction.subtraction.qmatrix, guess.init = rep(1.2,
  ncol(fraction.subtraction.data)))
  
# See rule specification   
fractions.dina.warning3 <- din(data = fraction.subtraction.data,
  q.matrix = fraction.subtraction.qmatrix, rule = c(rep("DINA",
  10), rep("DINO", 9)))

## Parameter estimation of DINA model
# rule = "DINA" is default
fractions.dina <- din(data = fraction.subtraction.data,
  q.matrix = fraction.subtraction.qmatrix, rule = "DINA")
fractions.dina # see print.din

attributes(fractions.dina)
str(fractions.dina)	  

## For instance accessing the guessing parameters through
## assignment
fractions.dina$guess

## corresponding summaries, including diagnostic accuracies,
## summary of skill pattern distribution and information 
## criteria AIC and BIC
summary(fractions.dina)

## In particular, accessing detailed summary through assignment
detailed.summary.fs <- summary(fractions.dina)
str(detailed.summary.fs)

## Diagnostic accuracy of item 8 seems to low. This is also
## visualized in the first plot 
plot(fractions.dina)

## The reason therefore is a high guessing parameter
round(fractions.dina$guess[,1], 2)

## Set an upper boundary for the guessing parameter of 
## item 5, 8 and 9
fractions.dina.bound <- din(data = fraction.subtraction.data, 
  q.matrix = fraction.subtraction.qmatrix, constraint.guess =
  matrix(c(5,8,9, rep(0.2, 3)), ncol = 2))
fractions.dina.bound
detailed.summary.fs.bound <- summary(fractions.dina.bound)

## This improves the diagnostic accuracies
summary(detailed.summary.fs$IDI[1,])
summary(detailed.summary.fs.bound$IDI[1,])

## The second plot shows the expected (MAP) and observed skill 
## probabilities. The third plot visualizes the skill pattern
## occurrence probabilities; Only the 'highest' are labeled; it
## is obvious that the skill class '11111111' (all skills are
## mastered) is the most probable in this population. The fourth
## plot shows the skill probabilities conditional on response
## patterns; in this population the skills 3 and 6 seem to be
## mastered easier than the others. The fifth plot shows the
## skill probabilities conditional on a specified response
## pattern; it is shown whether a skill is mastered (above 
## .5+'uncertainty') unclassifiable (within the boundaries) or
## not mastered (below .5-'uncertainty'). In this case, the
## fifteenth respondent was chosen; if no response pattern is 
## specified, the plot will not be shown (of course)
pattern <- paste(fraction.subtraction.data[15,], collapse = "")

#uncertainty = 0.1, highest = 0.05 are default
plot(fractions.dina.bound, uncertainty = 0.1, highest = 0.05, 
  pattern = pattern)

##
## (2) examples based on dataset sim.dina
##

# DINA Model
d1 <- din(sim.dina, q.matr = sim.qmatrix, rule = "DINA",
  conv.crit = 0.01, maxit = 500, progress = TRUE)
summary(d1)

# Mixed DINA and DINO Model
d1b <- din(sim.dina, q.matr = sim.qmatrix, rule = 
  c(rep("DINA", 7), rep("DINO", 2)), conv.crit = 0.01,
  maxit = 500, progress = FALSE)
summary(d1b)

# DINO Model
d2 <- din(sim.dina, q.matr = sim.qmatrix, rule = "DINO",
  conv.crit = 0.01, maxit = 500, progress = FALSE)
summary(d2)

# Comparison of DINA and DINO estimates
lapply(list("guessing" = rbind("DINA" = d1$guess[,1],
  "DINO" = d2$guess[,1]), "slipping" = rbind("DINA" = 
  d1$slip[,1], "DINO" = d2$slip[,1])), round, 2)

# Comparison of the information criteria
c("DINA"=d1$AIC, "MIXED"=d1b$AIC, "DINO"=d2$AIC)

# following estimates:
d1$coef            # guessing and slipping parameter
d1$guess           # guessing parameter
d1$slip            # slipping parameter
d1$skill.patt      # probabilities for skills
d1$attribute.patt  # attribute pattern with probabilities
d1$subj.pattern    # pattern per subject

# posterior probabilities for every response pattern
d1$posterior       

##
## (3) examples based on dataset sim.dino
##

# DINO Estimation
d3 <- din(sim.dino, q.matr = sim.qmatrix, rule = "DINO",
  conv.crit = 0.005, progress = FALSE)

# Mixed DINA and DINO Model
d3b <- din(sim.dino, q.matr = sim.qmatrix, rule = 
  c(rep("DINA", 4), rep("DINO", 5)), conv.crit = 0.001, 
  progress = FALSE)
                        
# DINA Estimation
d4 <- din(sim.dino, q.matr = sim.qmatrix, rule = "DINA",
  conv.crit = 0.005, progress = FALSE)
            
# Comparison of DINA and DINO estimates
lapply(list("guessing" = rbind("DINO" = d3$guess[,1],
  "DINA" = d4$guess[,1]), "slipping" = rbind("DINO" = 
  d3$slip[,1], "DINA" = d4$slip[,1])), round, 2)

# Comparison of the information criteria
c("DINO"=d3$AIC, "MIXED"=d3b$AIC, "DINA"=d4$AIC)

##
## (4) example estimation with weights based on dataset sim.dina
##

# Here, a weighted maximum likelihood estimation is used 
# This could be useful for survey data.

# i.e. first 200 persons have weight 2, the other have weight 1
(weights <- c(rep(2, 200), rep(1, 200)))

d5 <- din(sim.dina, sim.qmatrix, rule = "DINA", conv.crit = 
  0.005, weights = weights, progress = FALSE)
        
# Comparison of the information criteria
c("DINA"=d1$AIC, "WEIGHTS"=d5$AIC)


##
## (5) example estimation within a Balanced Incomplete 
##     Block (BIB) Design generated on dataset sim.dina
##

# generate BIB data

# The next example shows that the din and nida functions
# work for (relatively arbitrary) missing value pattern

# Here, a missing by design is generated in the dataset dinadat.bib
sim.dina.bib <- sim.dina
sim.dina.bib[1:100, 1:3] <- NA
sim.dina.bib[101:300, 4:8] <- NA
sim.dina.bib[301:400, c(1,2,9)] <- NA


d6 <- din(sim.dina.bib, sim.qmatrix, rule = "DINA", 
  conv.crit = 0.0005, weights = weights, maxit=200)

d7 <- din(sim.dina.bib, sim.qmatrix, rule = "DINO",
  conv.crit = 0.005, weights = weights)

# Comparison of DINA and DINO estimates
lapply(list("guessing" = rbind("DINA" = d6$guess[,1],
  "DINO" = d7$guess[,1]), "slipping" = rbind("DINA" =
  d6$slip[,1], "DINO" = d7$slip[,1])), round, 2)