normalLink: Link function using a normal regression.
In ralmond/CPTtools: Tools for Creating Conditional Probability Tables
normalLink R Documentation
Link function using a normal regression.
Description
This link function assumes that the effective theta for this
distribution defines the mean of a normal distribution in a
generalized regression model. The link scale parameter describes the
residual variance.
Usage
normalLink(et, linkScale = NULL, obsLevels = NULL)
Arguments
et
A matrix of effective theta values. There should be one row in this
table for each configuration of the parent variables of the
conditional probability table and one column for each state of the
child variables except for the last.
linkScale
The residual standard deviation parameter. This
value must be supplied and should be a positive number; the default
value of NULL generates an error.
obsLevels
An optional character vector giving the names of the
child variable states. If supplied, it should have length
ncol(et)+1.
Details
This function takes care of the third step in the algorithm of
calcDPCTable. Its input is a matrix of effective theta
values (comparable to the last column of the output of
eThetaFrame), one column for each of the child variable
states (obsLevels) except for the last one. Each row
represents a different configuration of the parent variables. The
output is the conditional probability table. The use of this function
makes calcDPCTable behave like calcDNTable.
The idea behind this link function was first proposed in Almond
(2010), and it is more completely described in Almond et al. (2015).
The motivation comes from assuming that the child variable is created
by taking cuts on an underlying continuous variable. The marginal
distribution of this variable is a standard normal. The conditional
distribution is like a regression prediction with the effective theta
from the parent variables (the et argument) as the expected
value and the linkScale parameter as the residual standard
deviation.
The calculation works as follows: First cut points are set related to
the categories of the child variable. Let m be the number of
categories (this should be one more than the number of columns of
et) and the length of obsLevels if that is supplied).
Then the cut points are set at cuts <- qnorm(((m -
1):1)/m).
Then for each row of the conditional probability table i, the
probability of being in state k is calculated by
pnorm(cuts[k]-et[i, 1])/linkScale) -
pnorm(cuts[k-1]-et[i, 1])/linkScale) with the pnorm
expression set to 0 or 1 at the endpoints. Note that only the first
column of et is used in the calculation.
Value
A matrix with one more column than et giving the conditional
probabilities for each configuration of the parent variables (which
correspond to the rows).
Note
The motivation for the normal link function originally came from the
observation of odd behavior when variables given a DiBello-Samejima
distribution (that is, using the gradedResponse link
function) were used as parent variables for other variables with a
DiBello-Samejima distribution.
One potential reason for the odd behavior was that the graded response
link function was not an inverse of the procedure used to assign the
effectiveThetas to the parent variables. Thus, using a
probit link function (normalLink) was thought to be better for
parent variables than using a logistic link function
(gradedResponse), at the same time the convention of
assigning parent values based on quantiles of the normal distribution
started. This made the normalLink and effectiveThetas
approximate inverses (information is still lost through
discritization). Note that in the current implementation the scale
factor of 1.7 has been added to both the partialCredit
and gradedResponse functions to make the logistic
function closer to the normal distribution and a better inverse for
the effective theta procedure.
Author(s)
Russell Almond
References
Almond, R. G. (2010). ‘I can name that Bayesian network in two
matrixes.’ International Journal of Approximate Reasoning.
51, 167-178.
Almond, R.G., Mislevy, R.J., Steinberg, L.S., Yan, D. and Williamson, D.M.
(2015) Bayesian Networks in Educational Assessment.
Springer. Chapter 8.
See Also
Other link functions: partialCredit,
gradedResponse.
Functions which directly use the link function:
eThetaFrame, calcDPCTable,
mapDPC
Earlier version of the graded response link:
calcDNTable
Examples
skill1l <- c("High","Medium","Low")
correctL <- c("Correct","Incorrect")
pcreditL <- c("Full","Partial","None")
gradeL <- c("A","B","C","D","E")
## Get some effective theta values.
et <- effectiveThetas(3)
normalLink(matrix(et,ncol=1),.5,correctL)
normalLink(matrix(et,ncol=1),.3,correctL)
normalLink(matrix(et,nrow=3,ncol=2),.5,pcreditL)
normalLink(matrix(et,nrow=3,ncol=2),.8,pcreditL)
normalLink(matrix(et,nrow=3,ncol=4),.5,gradeL)
normalLink(matrix(et,nrow=3,ncol=4),.25,gradeL)
ralmond/CPTtools documentation built on Dec. 27, 2024, 7:15 a.m.
R Package Documentation
Browse R Packages
We want your feedback!
Note that we can't provide technical support on individual packages. You should contact the package authors for that.
| normalLink | R Documentation |
Link function using a normal regression.
Description
This link function assumes that the effective theta for this distribution defines the mean of a normal distribution in a generalized regression model. The link scale parameter describes the residual variance.
Usage
normalLink(et, linkScale = NULL, obsLevels = NULL)
Arguments
et |
A matrix of effective theta values. There should be one row in this table for each configuration of the parent variables of the conditional probability table and one column for each state of the child variables except for the last. |
linkScale |
The residual standard deviation parameter. This
value must be supplied and should be a positive number; the default
value of |
obsLevels |
An optional character vector giving the names of the
child variable states. If supplied, it should have length
|
Details
This function takes care of the third step in the algorithm of
calcDPCTable. Its input is a matrix of effective theta
values (comparable to the last column of the output of
eThetaFrame), one column for each of the child variable
states (obsLevels) except for the last one. Each row
represents a different configuration of the parent variables. The
output is the conditional probability table. The use of this function
makes calcDPCTable behave like calcDNTable.
The idea behind this link function was first proposed in Almond
(2010), and it is more completely described in Almond et al. (2015).
The motivation comes from assuming that the child variable is created
by taking cuts on an underlying continuous variable. The marginal
distribution of this variable is a standard normal. The conditional
distribution is like a regression prediction with the effective theta
from the parent variables (the et argument) as the expected
value and the linkScale parameter as the residual standard
deviation.
The calculation works as follows: First cut points are set related to
the categories of the child variable. Let m be the number of
categories (this should be one more than the number of columns of
et) and the length of obsLevels if that is supplied).
Then the cut points are set at cuts <- qnorm(((m -
1):1)/m).
Then for each row of the conditional probability table i, the
probability of being in state k is calculated by
pnorm(cuts[k]-et[i, 1])/linkScale) -
pnorm(cuts[k-1]-et[i, 1])/linkScale) with the pnorm
expression set to 0 or 1 at the endpoints. Note that only the first
column of et is used in the calculation.
Value
A matrix with one more column than et giving the conditional
probabilities for each configuration of the parent variables (which
correspond to the rows).
Note
The motivation for the normal link function originally came from the
observation of odd behavior when variables given a DiBello-Samejima
distribution (that is, using the gradedResponse link
function) were used as parent variables for other variables with a
DiBello-Samejima distribution.
One potential reason for the odd behavior was that the graded response
link function was not an inverse of the procedure used to assign the
effectiveThetas to the parent variables. Thus, using a
probit link function (normalLink) was thought to be better for
parent variables than using a logistic link function
(gradedResponse), at the same time the convention of
assigning parent values based on quantiles of the normal distribution
started. This made the normalLink and effectiveThetas
approximate inverses (information is still lost through
discritization). Note that in the current implementation the scale
factor of 1.7 has been added to both the partialCredit
and gradedResponse functions to make the logistic
function closer to the normal distribution and a better inverse for
the effective theta procedure.
Author(s)
Russell Almond
References
Almond, R. G. (2010). ‘I can name that Bayesian network in two matrixes.’ International Journal of Approximate Reasoning. 51, 167-178.
Almond, R.G., Mislevy, R.J., Steinberg, L.S., Yan, D. and Williamson, D.M. (2015) Bayesian Networks in Educational Assessment. Springer. Chapter 8.
See Also
Other link functions: partialCredit,
gradedResponse.
Functions which directly use the link function:
eThetaFrame, calcDPCTable,
mapDPC
Earlier version of the graded response link:
calcDNTable
Examples
skill1l <- c("High","Medium","Low")
correctL <- c("Correct","Incorrect")
pcreditL <- c("Full","Partial","None")
gradeL <- c("A","B","C","D","E")
## Get some effective theta values.
et <- effectiveThetas(3)
normalLink(matrix(et,ncol=1),.5,correctL)
normalLink(matrix(et,ncol=1),.3,correctL)
normalLink(matrix(et,nrow=3,ncol=2),.5,pcreditL)
normalLink(matrix(et,nrow=3,ncol=2),.8,pcreditL)
normalLink(matrix(et,nrow=3,ncol=4),.5,gradeL)
normalLink(matrix(et,nrow=3,ncol=4),.25,gradeL)
R Package Documentation
Browse R Packages
We want your feedback!
Note that we can't provide technical support on individual packages. You should contact the package authors for that.
Embedding an R snippet on your website
Add the following code to your website.
For more information on customizing the embed code, read Embedding Snippets.