anchor {psychotools}

Anchor Methods for the Detection of Uniform DIF the Rasch Model

Description

The anchor function provides a variety of anchor methods for the detection of uniform differential item functioning (DIF) in the Rasch model between two pre-specified groups. These methods can be divided in an anchor class that determines characteristics of the anchor method and an anchor selection that determines the ranking order of candidate anchor items. The aim of the anchor function is to provide anchor items for DIF testing, e.g. with anchortest.

Usage

anchor(object, ...)
## Default S3 method:
anchor(object, object2,
  class = c("constant", "forward"), select = NULL,
  length = NULL, range = c(0.1, 0.8), ...)
## S3 method for class 'formula'
anchor(formula, data = NULL, subset = NULL,
  na.action = NULL, weights = NULL, model = raschmodel, ...)

Arguments

object, object2

Fitted model objects of class “raschmodel” estimated via conditional maximum likelihood using raschmodel.

...

further arguments passed over to an internal call of anchor.default in the formula method. In the default method, these additional arguments are currently not being used.

class

character. Available anchor classes are the constant anchor class implying a constant anchor length defined by length and the iterative forward anchor class, for an overview see Kopf et al. (2015a).

select

character. Several anchor selection strategies are available: "MTT", "MPT", "MT", "MP", "NST", "AO", "AOP" based on Kopf et al. (2015b) as well as "Gini", "CLF", "GiniT", "CLFT" based on Strobl et al. (2021). The latter four can only be combined with class = "constant" and length = 1. The default is select = "Gini" unless either length > 1 where "MPT" is used or class = "forward" where "MTT" is used. For more details see below.

length

integer. It pre-defines a maximum anchor length. Per default, the forward anchor grows up to the proportion of currently presumed DIF-free items specified in range and the constant anchor class selects one anchor item, unless an explicit limiting number is defined in length by the user.

range

numeric vector of length 2. The first element is the percentage of first anchor candidates to be excluded for consideration when the forward anchor class is used and the second element determines a percentage of currently presumed DIF-free items up to which the anchor from the forward anchor class is allowed to grow.

formula

formula of type y ~ x where y specifies a matrix of dichotomous item responses and x the grouping variable, e.g., gender, for which DIF should be tested for.

data

a data frame containing the variables of the specified formula.

subset

logical expression indicating elements or rows to keep: missing values are taken as false.

na.action

a function which indicates what should happen when the data contain missing values (NAs).

weights

an optional vector of weights (interpreted as case weights).

model

an IRT model fitting function with a suitable itempar method, by default raschmodel.

Details

The anchor methods provided consist of an anchor class that determines characteristics of the anchor and an anchor selection that determines the ranking order of candidate anchor items.

In the constant anchor class, the anchor length is pre-defined by the user within the argument length, defaulting to a length of one. In contrast, the iterative forward class starts with a single anchor item and includes items in the anchor as long as the anchor length is shorter than a certain percentage of the number of items that do not display statistically significant DIF (default: 0.8). Furthermore, a percentage of first anchor candidates is excluded from consideration (default: 0.1) and the user is allowed to set a maximum number of anchor items using the argument length. A detailed description of the anchor classes can be found in Kopf et al. (2015a).

In more recent work Strobl et al. (2021) suggest a simpler yet powerful anchor method based on inequality criteria like the Gini coefficient. A similar approach based on the component loss function (CLF) was suggested by Muthén & Asparouhov (2014). These criteria can be shown to attain their optimium for a single-anchor, thus correponding to a constant class of length 1. Due to the simple structure in combination with good empirical performance the Gini-based selection was made the default in version 0.7-0 of the package.

Both anchor classes require an explicit anchor selection strategy (as opposed to the all-other anchor class which is therefore not included in the function anchor). The anchor selection strategy determines the ranking order of candidate anchor items. In case of two groups, each item j, j = 1, \ldots, k (where k denotes the number of items in the test) obtains a criterion value c_j that is defined by the anchor selection strategy. The ranking order is determined by the rank of the criterion value rank(c_j).

The criterion values c_j for item j from the different anchor selection strategies are provided in the following equations: d_j denotes the difference of the item parameters, t_j the corresponding test statistic, and p_j the resulting p-values. In all cases, the anchor items are given in parentheses. Furthermore, \mathrm{Gini}(\cdot) denotes the Gini inequality index, \mathrm{CLF}(\cdot) the component loss function (sum of square root values), 1(\cdot) the indicator function, \lceil 0.5\cdot k \rceil the empirical 50% quantile, and A_\mathrm{purified} the anchor after purification steps. More detailed descriptions are available in Strobl et al. (2021) and Kopf et al. (2015b).

Gini selection (of item parameter differences) by Strobl et al. (2021):

c_j^\mathrm{Gini} = - \mathrm{Gini} (\{ |d_1(j)|, \ldots, |d_k(j)| \})

GiniT selection (of test statistics) similar to Strobl et al. (2021):

c_j^\mathrm{GiniT} = - \mathrm{Gini} (\{ |t_1(j)|, \ldots, |t_k(j)| \})

CLF selection (of item parameter differences) by Muthén & Asparouhov (2014):

c_j^\mathrm{CLF} = \mathrm{CLF} (\{ |d_1(j)|, \ldots, |d_k(j)| \})

CLFT selection (of test statistics) similar to Muthén & Asparouhov (2014):

c_j^\mathrm{CLFT} = \mathrm{CLF} (\{ |t_1(j)|, \ldots, |t_k(j)| \})

All-other selection by Woods (2009), here abbreviated AO:

c_j^\mathrm{AO} = | t_j (\{1,\ldots,k\}\setminus j) |

All-other purified selection by Wang et al. (2012), here abbreviated AOP:

c_j^\mathrm{AOP} = | t_j ( A_\mathrm{purified} ) |

Number of significant threshold selection based on Wang et al. (2004), here abbreviated NST:

c_j^\mathrm{NST} = \sum_{l \in \{1,\ldots,k\} \setminus j} 1 \left\{ p_j ( \{l\} ) \leq \alpha \right\}) |

Mean test statistic selection by Shih et al. (2009), here abbreviated MT:

c_j^\mathrm{MT} = \frac{1}{k-1} \sum_{l \in \{1,\ldots,k\} \setminus j} \left| t_j ( \{l\}) \right|

Mean p-value selection by Kopf et al. (2015b), here abbreviated MP:

c_j^\mathrm{MP} = - \frac{1}{k-1} \sum_{l \in \{1,\ldots,k\} \setminus j} p_j ( \{l\} )

Mean test statistic threshold selection by Kopf et al. (2015b), here abbreviated MTT:

c_j^\mathrm{MTT} = \sum_{l \in \{1,\ldots,k\} \setminus j} 1 \left\{ \left| t_j ( \{l\} ) \right| > \left( \left| \frac{1}{k-1} \sum_{l \in \{ 1, \ldots, k \} \setminus j} t_j ( \{l\} ) \right| \right)_{\left( \lceil 0.5\cdot k\rceil \right)} \right\}

Mean p-value threshold selection by Kopf et al. (2015b), here abbreviated MPT:

c_j^\mathrm{MPT} = - \sum_{l \in \{1,\ldots,k\} \setminus j} 1 \left\{ p_j ( \{l\} ) > \left( \frac{1}{k-1} \sum_{l \in \{ 1, \ldots, k \} \setminus j} p_j ( \{l\} ) \right)_{ \left( \lceil 0.5\cdot k\rceil \right)} \right\}

Kopf et al. (2015b) recommend to combine the class = "constant" with select = "MPT" and the class = "forward" with select = "MTT", respectively.

The all-other anchor class (that assumes that DIF is balanced i.e. no group has an advantage in the test) is here not considered as explicit anchor selection and, thus, not included in the anchor function (but in the anchortest function). Note that the all-other anchor class requires strong prior knowledge that DIF is balanced.

Value

An object of class anchor, i.e. a list including

anchor_items

the integer index for the selected anchor items

ranking_order

a ranking order (integer index) of the candidate anchor items by their criterion values

criteria

the criterion values obtained in the anchor selection for each item (unsorted)

References

Kopf J, Zeileis A, Strobl C (2015a). A Framework for Anchor Methods and an Iterative Forward Approach for DIF Detection. Applied Psychological Measurement, 39(2), 83–103. doi:10.1177/0146621614544195

Kopf J, Zeileis A, Strobl C (2015b). Anchor Selection Strategies for DIF Analysis: Review, Assessment, and New Approaches. Educational and Psychological Measurement, 75(1), 22–56. doi:10.1177/0013164414529792

Muthén B, Asparouhov T (2014). IRT Studies of Many Groups: The Alignment Method. Frontiers in Psychology, 5, 978. doi:10.3389/fpsyg.2014.00978

Shih CL, Wang WC (2009). Differential Item Functioning Detection Using the Multiple Indicators, Multiple Causes Method with a Pure Short Anchor. Applied Psychological Measurement, 33(3), 184–199.

Strobl C, Kopf J, Kohler L, von Oertzen T, Zeileis A (2021). Anchor Point Selection: Scale Alignment Based on an Inequality Criterion. Applied Psychological Measurement, 45(3), 214–230. doi:10.1177/0146621621990743

Wang WC (2004). Effects of Anchor Item Methods on the Detection of Differential Item Functioning within the Family of Rasch Models. Journal of Experimental Education, 72(3), 221–261.

Wang WC, Shih CL, Sun GW (2012). The DIF-Free-then-DIF Strategy for the Assessment of Differential Item Functioning. Educational and Psychological Measurement, 72(4), 687–708.

Woods C (2009). Empirical Selection of Anchors for Tests of Differential Item Functioning. Applied Psychological Measurement, 33(1), 42–57.

See Also

anchortest

Examples

## Verbal aggression data
data("VerbalAggression", package = "psychotools")

## Gini anchor (Strobl et al. 2021) for gender DIF in the self-to-blame situations
anchor(resp2[, 1:12] ~ gender , data = VerbalAggression)

## alternatively: based on fitted raschmodel objects
raschmodels <- with(VerbalAggression, lapply(levels(gender), function(i) 
  raschmodel(resp2[gender == i, 1:12])))
anchor(raschmodels[[1]], raschmodels[[2]])

if(requireNamespace("multcomp")) {

## four anchor items from constant anchor class using MPT-selection (Kopf et al. 2015b)
anchor(object = raschmodels[[1]], object2 = raschmodels[[2]], 
  class = "constant", select = "MPT", length = 4)

## iterative forward anchor class using MTT-selection (Kopf et al. 2015b)
set.seed(1)
fanchor <- anchor(object = raschmodels[[1]], object2 = raschmodels[[2]],
  class = "forward", select = "MTT", range = c(0.05, 1))
fanchor

## the same using the formula interface
set.seed(1)
fanchor2 <- anchor(resp2[, 1:12] ~ gender , data = VerbalAggression,
  class = "forward", select = "MTT", range = c(0.05, 1))

## criteria really the same?
all.equal(fanchor$criteria, fanchor2$criteria, check.attributes = FALSE)
}