Angoff's Delta plot (Angoff and Ford, 1973) is a straightforward test-score method to detect DIF among dichotomously scored items. Proportions of correct responses are computed first per item and per group of respondents, and are successively transformed onto z-scores and then onto \(\Delta\) scores. The pairs of \(\Delta\) scores can
be displayed onto a scatter plot, called the Delta plot, and the majr axis of the ellipsoid of Delta points is
derived. Eventually, items whose perpendicular distance (from the major axis) is too large are flagged as DIF. See Angoff and Ford (1973) for further details.
The data must be passed through the argument data and can be of three types. Each type is defined by the
type argument and can take three values: "response", "prop" and "delta".
If type is "response", the input data consist in a matrix with one row per respondent and
\(J+1\) columns, where \(J\) is the number of items. In the colmuns coding for the items, only possible entries are 0
(for incorrect responses), 1 (for corect responses) and NA (for missing values). The extra column is used to
define group membership: all respondents of the reference group take the same value (either numeric or character), and all respondents in the focal group take the same (numeric or character) value but different from the reference group. Note
that the group membership column can be located anywhere in the data set (not especially in first or last position).
If type is "prop", the input data consist in a two-column matrix with one row per item. Each
row contains the proportions of correct responses, respectively in the reference group (first column) and in the focal group (second column).
If type is "delta", the input data consist in a two-column matrix that is similar to that
provided with the "prop" type of input, but with the Delta scores provided instead of the proportions of correct
responses.
If the type of input is either "prop" or "delta", not anymore input information is required and the
arguments group and focal.bname are ignored. Otherwise, the group membership column in the data
matrix is specified by giving to argument group either the column number (1 for first column, etc.) or the column
name (provided the data matrix has argument names). Moreover, the focal group is specified by giving to the
argument focal.name the value that was used in the group membership column to code for the focal group.
If the input type is not "delta", then extreme proportions of correct responses (either provided when
type is "prop" or computed from the data if type is "response") are adjusted by specifying
the arguments extreme, const.range and nrAdd with appropriate values. See the adjustExtreme function for further details (note that the cuyrrent extreme argument corresponds to the method argument in this function).
The threshold for flaging items as DIF can be of two types and is specified by the thr argument.
It can be fixed to some arbitrary positive value by the user, for instance 1.5 (Angoff and Ford, 1973). In this
case, thr takes the required numeric threshold value.
Alternatively, it can be derived from the bivariate normal approximation of the Delta points (Magis and Facon, 2012). In this case, thr must be given the character value "norm" (which is the default value).
This threshold equals
$$\Phi^{-1}(1-\alpha/2) \; \sqrt{\frac{b^2\,{s_0}^2-2\,b\,s_{01}+{s_1}^2}{b^2+1}}$$
where \(\Phi\) is the density of the standard normal distribution, \(\alpha\) is the significance level (set by the argument alpha with default value 0.05), \(b\) is the slope parameter of the major axis, \(s_0\) and \(s_1\)
are the sample standard deviations of the Delta scores in the reference group and the focal group, respecively, and
\(s_{01}\) is the sample covariance of the Delta scores (see Magis and Facon, 2012, for further details).
Item purification can be performed by setting the argument purify to TRUE (by default it is FALSE so
no purification is performed). The item purification process (IPP) starts when at least one item was flagged as DIF after
the first run of the Delta plot, and proceeds as follows.
The intercept and slope parameters of the major axis are re-calculated by removing all DIF that are currently
flagged as DIF. This yields updated values \(a^*\), \(b^*\), \(s_0^*\), \(s_1^*\) and \(s_{01}^*\) of the
intercept and slope parameters, sample stanbdard deviations and sample covariance of the Delta scores.
Perpendicular distances (for all items) are updated with respect to the updated major axis.
Detection threshold is also updated. Three possible updates are possible: see below.
All items are now tested for the presence of DIF, given the updated perpendicular distances and major axis.
If the set of items flagged as DIF is the same as the one from the previous loop, stop the process. Otherwise go
back to step 1.
Unlike traditional DIF methods, the detection threshold may also be updated since it depends on the sample estimates (when
the normal approximation is considered). Three approaches are currently implemented and are specified by the purType argument.
Method 1 (purType=="IPP1"): the same threshold is used throughout the purification process, it is not
iteratively updated. The threshold is the one obtained after the first run of the Delta plot.
Method 2 (purType=="IPP2"): only the slope parameter is updated in the threshold formula. By this way, one keeps the full data structure (i.e. neither the sample variances nor the sample covariance of the Delta scores are
modified) but only the slope parameter is adjusted to lessen the impact of DIF items.
Method 3 (purType=="IPP3"): all adjusted parameters are plugged in the threshold formula. This approach
completely discards the effect of items flagged as DIF from the computation of the threshold.
See Magis and Facon (2013) for further details. Note that purification can also be performed with fixed threshold (i.e.
specified by the user), but then only IPP1 process is performed.
In order to avoid possible infinite loops in the purification process, a maximal number of iterations must be specified
through the argument maxIter. The default maximal number of iterations is 10.
The output contains all input information, the Delta scores and perpendicular distances, the parameter of the major axis
and the items flagged as DIF (if none, a character sentence is returned). In addition, the detection threshold and the
type of threshold (fixed or normal approximation) is provided.
If item purification was run, several additional elements are returned: the number of iterations, a logical indicator
whether the convergence was reached (or not, meaning that the process stopped because of reaching the maximal number of
allowed iterations), a matrix with indicators of which items were flagged as DIF at each iteration, and the type of item
purification process. Moreover, perpendicular distances are returned in a matrix format (one column per iteration), as
well as successive major axis parameters (one row per iteration) and successive thresholds (as a vector).
The output is managed and printed in a more user-friendly way. When item purification is performed, only the first and
last steps are displayed. Specifying the argument only.final to FALSE prints in addition all intermediate steps of the process (successive perpendicular distances, parameters of the major axis, and detection thresholds).
The output can be saved into na text file by specifying the argument save.output to TRUE (by default the
output is not captured). If so, the argument output can be specified as a vector of two character values. The first
one gives the desired name of the text file, and the second one specifies the directory where the file will be saved (full
path is required but without the final "/" symbol, see Examples below). By default, the output will be saved in the current working directory as "out.txt" file.