cptChi2: Compare and observed to an expected conditional probability...

In ralmond/CPTtools: Tools for Creating Conditional Probability Tables

Compare and observed to an expected conditional probability table.

Description

Each row of a conditional probability frame (CPF) is a probability distribution. Observed data comes in the form of a contingency table (or expected contingency table) where the rows represent configurations of the parent variables and the columns represent the state of the child variable. The cptChi2 calculates the chi-squared data fit statistic for each row of the table, and sums them together. This provides a test of the plausibility that the data came from the proposed model.

Usage

cptChi2(obs, exp)
cptKL(est, exp)
cptG2(obs, exp)

Arguments

obs	A CPF which contains the observed data.
est	A CPF which contains the estimated distribution.
exp	A CPF which contains the candidate distribution.

Details

The cannonical use for this plot is to test the conditional probability model used in particular Bayesian network. The exp argument is the proposed table. If the parents are fully observed, then the obs argument would be a contingency table with the rows representing parent configurations and the columns the data. If the parents are not fully observed, then obs can be replaced by an expected contingincy table (see Almond, et al, 2015).

To avoid potential problems with zero cells, the data table is modified by adding the prior. That is the observed value is taken as obs+exp. This should not have a big effect if the sample size is large.

The function 'cptG2' is an alternate version which uses the deviance statistic to produce a chi-squared value.

The function 'cptKL' gives the Kulbach-Liebler distance between two probability distributions, so requires the first argument to be normalized.

Value

The for 'cptChi2' and 'cptG2', the sum of the chi-squres for all of the tables with the degrees of freedom. The degrees of freedom is given as an attribute df.

For 'cptKL', it is the Kulbach-Leibler distance.

Note

This is called a chi-square statistic, and the percentage points of the chi-square distribution can be used for reference. However, it is unclear whether the statistic is actually distributed according to the chi-square distribution under the hypothesis that the data come from the distribution given in exp. In particular, if an expected table is used instead of an actual table, the distribution is likely to not be exactly chi-squared.

Author(s)

Russell Almond

References

Almond, R.G., Mislevy, R.J., Steinberg, L.S., Williamson, D.M. and Yan, D. (2015) Bayesian Networks in Educational Assessment. Springer. Chapter 10.

Sinharay, S. and Almond, R.G. (2006). Assessing Fit of Cognitively Diagnostic Models: A case study. Educational and Psychological Measurement. 67(2), 239–257.

See Also

betaci, OCP, barchart.CPF, OCP.CPF

Examples

  skill1l <- c("High","Medium","Low") 
  skill2l <- c("High","Medium","Low","LowerYet") 
  correctL <- c("Correct","Incorrect") 
  pcreditL <- c("Full","Partial","None")
  gradeL <- c("A","B","C","D","E") 

  cpfTheta <- calcDPCFrame(list(),skill1l,numeric(),0,rule="Compensatory",
                           link="normalLink",linkScale=.5)

  ## Compatable data  
  datTheta <- cpfTheta ## Copy to get the shape
  datTheta[1,] <- rmultinom(1,25,cpfTheta[1,])

  cptChi2(datTheta,cpfTheta)
  cptG2(datTheta,cpfTheta)
  cptKL(normalize(datTheta),cpfTheta)

  ## Incompatable data
  datThetaX <- cpfTheta ## Copy to get the shape
  datThetaX[1,] <- rmultinom(1,100,c(.05,.45,.5))

  cptChi2(datThetaX,cpfTheta)
  cptG2(datThetaX,cpfTheta)
  cptKL(normalize(datThetaX),cpfTheta)

  cptComp <- calcDPCFrame(list(S2=skill2l,S1=skill1l),correctL,
                          lnAlphas=log(c(1.2,.8)), betas=0,
                          rule="Compensatory")

  cptConj <- calcDPCFrame(list(S2=skill2l,S1=skill1l),correctL,
                          lnAlphas=log(c(1.2,.8)), betas=0,
                          rule="Conjunctive")

  datComp <- cptComp
  for (i in 1:nrow(datComp))
      ## Each row has a random sample size.
      datComp[i,3:4] <- rmultinom(1,rnbinom(1,mu=15,size=1)+1,cptComp[i,3:4])
  datComp <- as.CPF(datComp)

  datConj <- cptConj
  for (i in 1:nrow(datConj))
      ## Each row has a random sample size.
      datConj[i,3:4] <- rmultinom(1,rnbinom(1,mu=15,size=1)+1,cptConj[i,3:4])
  datConj <- as.CPF(datConj)
  
  ## Compatable
  cptChi2(datConj,cptConj)
  cptG2(datConj,cptConj)
  cptKL(normalize(datConj),cptConj)

  ## Incompatable
  cptChi2(datComp,cptConj)
  cptG2(datComp,cptConj)
  cptKL(normalize(datComp),cptConj)


  cptPC1 <- calcDPCFrame(list(S1=skill1l,S2=skill2l),pcreditL,
                         lnAlphas=log(1),
                         betas=list(full=c(S1=0,S2=999),partial=c(S2=999,S2=0)),
                         rule="OffsetDisjunctive")
  cptPC2 <- calcDPCFrame(list(S1=skill1l,S2=skill2l),pcreditL,
                         lnAlphas=log(1),
                         betas=list(full=c(S1=0,S2=999),partial=c(S2=1,S2=0)),
                         rule="OffsetDisjunctive")


  datPC1 <- cptPC1
  for (i in 1:nrow(datPC1))
      ## Each row has a random sample size.
      datPC1[i,3:5] <- rmultinom(1,rnbinom(1,mu=25,size=1),cptPC1[i,3:5])
  datPC1 <- as.CPF(datPC1)
  
  datPC2 <- cptPC2
  for (i in 1:nrow(datPC2))
      ## Each row has a random sample size.
      datPC2[i,3:5] <- rmultinom(1,rnbinom(1,mu=25,size=1),cptPC2[i,3:5])
  datPC2 <- as.CPF(datPC2)
  
  ## Compatable
  cptChi2(datPC1,cptPC1)
  cptG2(datPC1,cptPC1)
  cptKL(normalize(datPC1),cptPC1)
   ## Inompatable
  cptChi2(datPC2,cptPC1)
  cptG2(datPC2,cptPC1)
  cptKL(normalize(datPC2),cptPC1)

ralmond/CPTtools documentation built on Dec. 27, 2024, 7:15 a.m.