Expected Ratings Matrix

Description

Expected ratings matrix given item measures, person measures and ordered rating category thresholds.

Usage

expdata(items, persons, thresholds, minRating)

Arguments

Details

It is assumed that the set of ordinal rating categories consists of all integers from the lowest rating category specified by minRating to the highest rating category, which is minRating + length(thresholds).

Value

A numeric matrix of expected ratings.

Note

Expected ratings are literally the expected value of the ordinal rating categories when treated as integers. Expected ratings that cannot be calculated return as NA (e.g., if either the person or item measure is NA). Intended use is for chi-squared tests or for calculating infit and outfit statistics.

Author(s)

Chris Bradley (cbradley05@gmail.com)

See Also

misfit

Examples

# Using randomly generated values with minimum rating set to zero
im <- runif(20, -2, 2)
pm <- runif(50, -2, 2)
th <- sort(runif(5, -2, 2))
m <- expdata(items = im, persons = pm, thresholds = th, minRating = 0)

Item Measures

Description

Estimates item measures assuming person measures are known and all persons use the same set of rating category thresholds.

Usage

ims(data, persons, thresholds, misfit = FALSE, minRating = NULL)

Arguments

Details

minRating must be specified if either the smallest or largest possible rating category is not in data (i.e., no person used one of the extreme rating categories). If minRating is specified, the ordinal rating scale is assumed to go from minRating to minRating + length(thresholds) in integer steps.

Value

A list whose elements are:

Note

Item measures estimated with ims differ from those estimated with msd because ims assumes all persons use the same rating category thresholds while msd does not. Intended use of ims is with an anchored set of persons and thresholds. Item measures that cannot be estimated will return as NA (e.g., if all responses to an item consist of only the highest rating category, or of only the lowest rating category, that item's item measure cannot be estimated).

Author(s)

Chris Bradley (cbradley05@gmail.com)

See Also

msd

Examples

# Simple example with randomly generated values and lowest rating category = 0.
d <- as.numeric(sample(0:4, 500, replace = TRUE))
dm <- matrix(d, nrow = 50, ncol = 10)
pm <- runif(50, -2, 2)
th <- sort(runif(4, -2, 2))
im <- ims(data = dm, persons = pm, thresholds = th, misfit = TRUE, minRating = 0)

Infit and Outfit Statistics

Description

Calculates infit and outfit statistics for items and persons.

Usage

misfit(data, items, persons, thresholds, minRating = NULL)

Arguments

Details

minRating must be specified if either the smallest or largest possible rating category is not in data (no person used one of the extreme rating categories). If minRating is specified, the ordinal rating scale is assumed to go from minRating to minRating + length(thresholds).

Value

A list whose elements are:

Author(s)

Chris Bradley (cbradley05@gmail.com)

Examples

# Using randomly generated values
d <- as.numeric(sample(0:5, 500, replace = TRUE))
dm <- matrix(d, nrow = 50, ncol = 10)
im <- runif(10, -2, 2)
pm <- runif(50, -2, 2)
th <- sort(runif(5, -2, 2))
m <- misfit(data = dm, items = im, persons = pm, thresholds = th)

# If the lowest or highest rating category is not in \code{data}, specify \code{minRating}
dm[dm == 0] <- NA
m2 <- misfit(data = dm, items = im, persons = pm, thresholds = th, minRating = 0)

Method of Successive Dichotomizations

Description

Estimates item measures, person measures, rating category thresholds and their standard errors using the method of successive dichotomizations. Option provided for anchoring certain items and persons while estimating the rest. Option also provided for estimating infit and outfit statistics.

Usage

msd(data, items = NULL, persons = NULL, misfit = FALSE)

Arguments

Details

items and persons are optional numeric vectors that specify item and person measures that are "anchored" and not estimated. The length of items must equal the number of columns in data and the length of persons must equal the number of rows in data. Only entries set to NA in items and persons are estimated. Default for both items and persons is NULL, which is equivalent to a vector of NA so that all items and persons are estimated.

Value

A list whose elements are:

Note

The axis origin is set by convention at the mean item measure. All item measures and person measures that cannot be estimated will return as NA (e.g., if a person responds with only the highest rating category, or with only the lowest rating category, to all items, that person's person measure cannot be estimated).

The accuracy of msd can be tested using the simdata function (see Examples).

Author(s)

Chris Bradley (cbradley05@gmail.com)

References

Bradley, C. and Massof, R. W. (2018) Method of successive dichotomizations: An improved method for estimating measures of latent variables from rating scale data. PLoS One, 13(10) doi:10.1371/journal.pone.0206106

See Also

simdata

Examples

# Simple example using a randomly generated ratings matrix
d <- as.numeric(sample(0:5, 200, replace = TRUE))
dm <- matrix(d, nrow = 20, ncol = 10)
m1 <- msd(dm, misfit = TRUE)

# Anchor first 5 item measures and first 10 person measures
im <- m1$item_measures
im[6:length(im)] <- NA
pm <- m1$person_measures
pm[11:length(pm)] <- NA
m2 <- msd(dm, items = im, persons = pm)

# To test the accuracy of msd using simdata, set the mean item measure to zero
# (axis origin in msd is the mean item measure) and the mean threshold to
# zero (any non-zero mean threshold is reflected in the person measures).
im <- runif(100, -2, 2)
im <- im - mean(im)
pm <- runif(100, -2, 2)
th <- sort(runif(5, -2, 2))
th <- th - mean(th)
d <- simdata(im, pm, th, missingProb = 0.15, minRating = 0)
m <- msd(d)

# Compare msd parameters to true values.  Linear regression should
# yield a slope very close to 1 and an intercept very close to 0.
lm(m$item_measures ~ im)
lm(m$person_measures ~ pm)
lm(m$thresholds ~ th)

Rating Category Probabilities

Description

Estimates the probability of observing each rating category given a set of ordered rating category thresholds.

Usage

msdprob(x, thresholds)

Arguments

Details

It is assumed that thresholds partitions the real line into length(thresholds)+1 ordered intervals that represent the rating categories.

Value

A matrix of probabilities where each of the length(thresholds)+1 rows represents a different rating category (lowest rating category is the top row) and each of the length(x) columns represents a different person minus item measure.

Note

msdprob can be used to create probability curves, which represent the probability of rating an item with each rating category as a function of the person measure minus item measure (see Examples).

Author(s)

Chris Bradley (cbradley05@gmail.com)

Examples

# Simple example
p <- msdprob(c(1.4, -2.2), thresholds = c(-1.1, -0.3, 0.5, 1.7, 2.2))

# Plot probability curves — each curve represents the probability of
# rating an item with a given rating category as a function of the
# person measure minus item measure.
x <- seq(-6, 6, 0.1)
p <- msdprob(x, thresholds = c(-3.2, -1.4, 0.5, 1.7, 3.5))
plot(0, 0, xlim = c(-6, 6), ylim = c(0, 1), type = "n",
    xlab = "Person minus item measure", ylab = "Probability")
for (i in seq(1, dim(p)[1])){
  lines(x, p[i,], type = "l", lwd = "2" , col = rainbow(6)[i])
}

Person Measures

Description

Estimates person measures assuming item measures are known and all persons use the same set of rating category thresholds.

Usage

pms(data, items, thresholds, misfit = FALSE, minRating = NULL)

Arguments

Details

minRating must be specified if either the smallest or largest possible rating category is not in data (i.e., no person used one of the extreme rating categories). If minRating is specified, the ordinal rating scale is assumed to go from minRating to minRating + length(thresholds) in integer steps.

Value

A list whose elements are:

Note

Person measures estimated with pms differ from those estimated with msd because pms assumes all persons use the same rating category thresholds while msd does not. Intended use of pms is with an anchored set of items and thresholds. Person measures that cannot be estimated will return as NA (e.g., if a person responds to all items with only the highest rating category, or with only the lowest rating category, that person's person measure cannot be estimated).

Author(s)

Chris Bradley (cbradley05@gmail.com)

See Also

msd

Examples

# Simple example with randomly generated values and lowest rating category = 0
d <- as.numeric(sample(0:4, 500, replace = TRUE))
dm <- matrix(d, nrow = 25, ncol = 20)
im <- runif(20, -2, 2)
th <- sort(runif(4, -2, 2))
pm <- pms(data = dm, items = im, thresholds = th, misfit = TRUE, minRating = 0)

Dichotomous Rasch Model

Description

Estimates item measures, person measures and their standard errors using the dichotomous Rasch model. A special case of the function msd when the rating scale consists of only two rating categories: 0 and 1. Option provided for anchoring certain items and persons while estimating the rest. Option also provided for estimating infit and outfit statistics.

Usage

rasch(data, items = NULL, persons = NULL, misfit = FALSE)

Arguments

Details

items and persons are optional numeric vectors that specify item and person measures that should be "anchored" and not estimated. The length of items must equal the number of columns in data and the length of persons must equal the number of rows in data. Only entries set to NA in items and persons are estimated. Default for both items and persons is NULL, which is equivalent to a vector of NA so that all items and persons are estimated.

Value

A list whose elements are:

Note

The axis origin is set by convention at the mean item measure. All item measures and person measures that cannot be estimated will return as NA (e.g., if a person responds with a single rating category to all items, that person's person measure cannot be estimated).

rasch is the basis for the "successive dichotomizations" in msd and is repeatedly called by msd when there are three or more rating categories.

The accuracy of rasch can be tested using the simdata function (see Examples).

Author(s)

Chris Bradley (cbradley05@gmail.com)

See Also

msd simdata

Examples

# Simple example using a randomly generated ratings matrix
d <- as.numeric(sample(0:1, 200, replace = TRUE))
dm <- matrix(d, nrow = 20, ncol = 10)
m1 <- rasch(dm, misfit = TRUE)

# Anchor first 5 item measures and first 10 person measures
im <- m1$item_measures
im[6:length(im)] <- NA
pm <- m1$person_measures
pm[11:length(pm)] <- NA
m2 <- rasch(dm, items = im, persons = pm)

# To test the accuracy of rasch using simdata, set the true mean item measure to
# zero (axis origin in rasch is the mean item measure).  Note that the threshold for
# dichotomous data is at 0.
im <- runif(100, -2, 2)
im <- im - mean(im)
pm <- runif(100, -2, 2)
th <- 0
d <- simdata(im, pm, th, missingProb = 0.15, minRating = 0)
m <- rasch(d)

# Compare rasch parameters to true values.  Linear regression should
# yield a slope very close to 1 and an intercept very close to 0.
lm(m$item_measures ~ im)
lm(m$person_measures ~ pm)

Simulated Rating Scale Data

Description

Generates simulated rating scale data given item measures, person measures and rating category thresholds.

Usage

simdata(items, persons, thresholds, missingProb = 0, minRating = 0)

Arguments

Details

It is assumed that the set of ordinal rating categories consists of all integers from the lowest rating category specified by minRating to the highest rating category, which is minRating + length(thresholds).

Value

A numeric matrix of simulated rating scale data.

Note

simdata can be used to test the accuracy of msd (see Examples).

Author(s)

Chris Bradley (cbradley05@gmail.com)

See Also

msd

Examples

# Use simdata to test the accuracy of msd. First, randomly generate item 
# measures, person measures and thresholds with 15 percent missing data and 
# ordinal rating categories from 0 to 5. Then, set mean item measure to zero 
# (axis origin in msd is the mean item measure) and mean threshold to zero 
# (any non-zero mean threshold is reflected in the person measures).
im <- runif(100, -2, 2)
pm <- runif(100, -2, 2)
th <- sort(runif(5, -2, 2))
im <- im - mean(im)
th <- th - mean(th)
d <- simdata(im, pm, th, missingProb = 0.15, minRating = 0)
m <- msd(d)

# Compare msd parameters to true values.  Linear regression should
# yield a slope very close to 1 and an intercept very close to 0.
lm(m$item_measures ~ im)
lm(m$person_measures ~ pm)
lm(m$thresholds ~ th)

Rating Category Thresholds

Description

Estimates rating category thresholds for msd given rating scale data, item measures and person measures.

Usage

thresh(data, items, persons)

Arguments

Details

The length of items must equal the number of columns in data and the length of persons must equal the number of rows in data. Neither items nor persons can consist of only NA.

Value

A list whose elements are:

Note

thresh is a special case of msd when item measures and person measures are known.

Author(s)

Chris Bradley (cbradley05@gmail.com)

See Also

msd

Examples

# Using randomly generated values
d <- as.numeric(sample(0:5, 1000, replace = TRUE))
m <- matrix(d, nrow = 50, ncol = 20)
im <- runif(20, -2, 2)
pm <- runif(50, -2, 2)
th1 <- thresh(m, items = im, persons = pm)

# Anchor first 10 item measures and first 10 person measures
im[11:length(im)] <- NA
pm[11:length(pm)] <- NA
th2 <- thresh(m, items = im, persons = pm)