# anchor

##### Anchor Methods for the Detection of Uniform DIF the Rasch Model

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`

.

- Keywords
- regression

##### Usage

```
anchor(object, …)
# S3 method for default
anchor(object, object2,
class = c("constant", "forward"), select = NULL,
length = NULL, range = c(0.1, 0.8), …)
# S3 method for 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. (2014a).- select
character. Several anchor selection strategies are available:

`"MTT"`

,`"MPT"`

,`"MT"`

,`"MP"`

,`"NST"`

,`"AO"`

,`"AOP"`

. For details see below and Kopf et al. (2014b). Following their recommendations, the default`selection = "MPT"`

is used for`class = "constant"`

whereas`selection = "MTT"`

is used for`class = "forward"`

.- 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 four anchor items, 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 (

`NA`

s).- 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`

. The default is four anchor
items. 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. (2014a).

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 are defined by the respective anchor selection strategy as in the following equations, where \(1\) denotes the indicator function, \(t_j\) denotes the test statistic for item \(j\), followed by the underlying anchor items in parenthesis, \(p_j\) the corresponding p-values (also with anchor items in parenthesis), \(\lceil 0.5\cdot k\rceil\) denotes the empirical 50% quantile and \(A_{purified}\) the anchor after purification steps (a detailed description of the anchor selection strategies implemented in this function is provided in Kopf et al., 2014b).

All-other selection by Woods(2009), here abbreviated AO: $$ c_j^{AO} = | t_j (\{1,\ldots,k\}\textbackslash j) |$$

All-other purified selection by Wang et al. (2012), here abbreviated AOP: $$ c_j^{AOP} = | t_j ( A_{purified} ) |$$

Number of significant threshold selection based on Wang et al. (2004), here abbreviated NST: $$ c_j^{NST} = \sum_{l \in \{1,\ldots,k\} \textbackslash j} 1 \left\{ p_j ( \{l\} ) \leq \alpha \right\}) |$$

Mean test statistic selection by Shih et al. (2009), here abbreviated MT: $$ c_j^{MT} = \frac{1}{k-1} \sum_{l \in \{1,\ldots,k\} \textbackslash j} \left| t_j ( \{l\}) \right| $$

Mean p-value selection by Kopf et al. (2014b), here abbreviated MP: $$ c_j^{MP} = - \frac{1}{k-1} \sum_{l \in \{1,\ldots,k\} \textbackslash j} p_j ( \{l\} ) $$

Mean test statistic threshold selection by Kopf et al. (2014b), here abbreviated MTT: $$ c_j^{MTT} = \sum_{l \in \{1,\ldots,k\} \textbackslash j} 1 \left\{ \left| t_j ( \{l\} ) \right| > \left( \left| \frac{1}{k-1} \sum_{l \in \{ 1, \ldots, k \} \textbackslash j} t_j ( \{l\} ) \right| \right)_{\left( \lceil 0.5\cdot k\rceil \right)} \right\} $$

Mean p-value threshold selection by Kopf et al. (2014b), here abbreviated MPT: $$ c_j^{MPT} = - \sum_{l \in \{1,\ldots,k\} \textbackslash j} 1 \left\{ p_j ( \{l\} ) > \left( \frac{1}{k-1} \sum_{l \in \{ 1, \ldots, k \} \textbackslash j} p_j ( \{l\} ) \right)_{ \left( \lceil 0.5\cdot k\rceil \right)} \right\} $$

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

the anchor items for DIF analysis

a ranking order of candidate anchor items

the criterion values obtained by the respective anchor selection

##### 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.
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.
10.1177/0013164414529792

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.

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

##### Examples

```
# NOT RUN {
if(requireNamespace("multcomp")) {
## Verbal aggression data
data("VerbalAggression", package = "psychotools")
## Rasch model for the self-to-blame situations; gender DIF test
raschmodels <- with(VerbalAggression, lapply(levels(gender), function(i)
raschmodel(resp2[gender == i, 1:12])))
## four anchor items from constant anchor class using MPT-selection
canchor <- anchor(object = raschmodels[[1]], object2 = raschmodels[[2]],
class = "constant", select = "MPT", length = 4)
canchor
summary(canchor)
## iterative forward anchor class using MTT-selection
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))
## really the same?
all.equal(fanchor, fanchor2, check.attributes = FALSE)
}
# }
```

*Documentation reproduced from package psychotools, version 0.5-1, License: GPL-2 | GPL-3*