Classical Test Theory Item Analysis

Computes traditional classical test theory (CTT) item statistics - difficulty, an item-total correlation (discrimination), and Cronbach's alpha with the item removed - for dichotomous or polytomous item response data, along with optional flagging of items that fall outside commonly used quality thresholds.

Usage

ctt_item(
  data,
  item.id = NULL,
  cats = NULL,
  correct = FALSE,
  missing = NA,
  flag = TRUE,
  crit.p = c(0.1, 0.95),
  crit.dis = 0.2
)

Arguments

data: A data frame or matrix of already-scored item responses, with examinees in rows and items in columns. Item scores must range from 0 to cats[j] - 1 for each item j (0/1 for a dichotomous item; 0, 1, 2, ... for a polytomous/partial-credit item).
item.id: A character vector of item identifiers, in the same order as the columns of data. If NULL (default), item IDs are generated automatically as paste0("V", 1:ncol(data)), following the convention used elsewhere in irtQ (e.g., shape_df(), est_irt()). Note that this generated ID does not fall back to colnames(data); pass item.id explicitly to label items using the column names of data or any other identifier scheme.
cats: A numeric vector giving the number of score categories for each item (e.g., 2 for a dichotomous item), following the cats convention used elsewhere in irtQ (see, e.g., shape_df()). If NULL (default), the number of categories for each item is inferred from the observed maximum score in data (i.e., max(data[, j], na.rm = TRUE) + 1); supply cats explicitly whenever the maximum possible score may not have been observed in the sample.
correct: Logical. Both the raw (uncorrected) item-total correlation - where an item (or, in ctt_distr(), a response option/category) is correlated with the total score that includes its own contribution - and the corrected item-total correlation - excluding its own contribution - are always computed and reported as separate columns (see Details/Value). This argument only controls which of the two is used for flagging: if TRUE, flagging (in ctt_item()'s flag column, ctt_alpha()'s summary discrimination figure, and, in ctt_distr(), the crit.distractor check) is based on the corrected value; if FALSE (default), on the raw value.
missing: A value indicating missing responses in data, analogous to the missing argument in est_irt() and score_resp(). Any cell equal to missing is recoded to NA before analysis. Default is NA. Examinees with any remaining missing item response are excluded listwise from all statistics computed by this function.
flag: Logical. If TRUE (default), items are flagged when their difficulty or discrimination falls outside the thresholds given in crit.p and crit.dis.
crit.p: A numeric vector of length two giving the lower and upper difficulty bounds used for flagging: difficulty below the first value is flagged as too difficult, and difficulty above the second value is flagged as too easy. Default is c(0.10, 0.95).
crit.dis: A single numeric value giving the minimum acceptable discrimination (item-total correlation); items strictly below this value are flagged as poorly discriminating. Default is 0.20.

Value

A list with two elements:

item: A data frame with one row per item, containing the item label, number of score categories, difficulty, the raw (uncorrected) item-total correlation (discrimination_raw), the corrected (item-excluded) item-total correlation (discrimination_corrected), and alpha-with-item-removed, plus a flag column (a character description of which criteria were triggered, or "" when none were; present only when flag = TRUE). Flagging is based on whichever of discrimination_raw/discrimination_corrected is selected by correct.
crit: A list echoing the crit.p and crit.dis thresholds used, for reference in downstream reporting.

Details

Difficulty for item j is defined generally as the mean observed score divided by the item's maximum possible score, mean(data[, j], na.rm = TRUE) / (cats[j] - 1). For a dichotomous item (cats[j] = 2), this reduces to the familiar proportion-correct difficulty index. For a polytomous item, this expresses the average score as a proportion of the maximum attainable score, so that difficulty remains interpretable on the same 0-1 scale regardless of the number of score categories.

Discrimination for item j is the Pearson correlation between the item score and the total score, which for a dichotomous item is mathematically equivalent to the point-biserial correlation. Both the uncorrected (raw) item-total correlation and the corrected (item-excluded) item-total correlation are always computed and returned as separate columns; see, e.g., Crocker and Algina (1986) for discussion of both conventions. This mirrors how some software reports both side by side (e.g., psych::alpha() reports the raw item-total correlation as raw.r and the corrected version as r.drop) rather than defaulting to one or the other. The correct argument only selects which of the two feeds the discrimination flagging criterion (see crit.dis).

Alpha-with-item-removed for item j is Cronbach's alpha recomputed on the remaining ncol(data) - 1 items, using the same variance-based formula used for an overall, test-level alpha (k / (k - 1) * (1 - sum(item variances) / total variance)). A value of NA is returned wherever a needed variance is zero (e.g., a constant item, or fewer than two items remaining), since the relevant ratio is then undefined.

References

Crocker, L., & Algina, J. (1986). Introduction to classical and modern test theory. Holt, Rinehart and Winston.

Author

Hwanggyu Lim hglim83@gmail.com