Manifest Monotonicity (Rest-Score Regression) — ManifestMonotonicity • exametrika

Checks, item by item, whether the probability of reaching each category rises with the rest score, the observable counterpart of latent monotonicity in Mokken scale analysis (Molenaar & Sijtsma, 2000; van der Ark, 2007). A single-peaked or U-shaped item, which the item-rest correlation and Scalability can show as merely weak, appears here as a run of decreases.

Usage

ManifestMonotonicity(
  U,
  na = NULL,
  Z = NULL,
  w = NULL,
  minvi = 0.03,
  minsize = NULL
)

Arguments

U: Either an object of class "exametrika" or raw data. When raw data is given, it is converted to the exametrika class with the dataFormat function.
na: Values to be treated as missing values.
Z: Missing indicator matrix of type matrix or data.frame. Values of 1 indicate observed responses, while 0 indicates missing data.
w: Item weight vector. Not used.
minvi: Smallest decrease counted as a violation. Default 0.03.
minsize: Smallest rest-score group. NULL (default) uses the rule described in Details.

Value

An object of class c("exametrika", "ManifestMonotonicity") with

ItemLabel: Item labels.
summary: Data frame, one row per item: the item scalability coefficient Hj and the totals over item steps of active pairs, violations, violations per active pair, largest violation, sum of violations, sum per active pair, largest $z$ and number of significant violations.
groups: List, one data frame per item: rest-score range, size, category frequencies, mean item score and $P(X \ge x)$ for each group.
violations: List, one matrix per item: the same statistics per item step, with the two groups involved in the largest $z$.
minsize, minvi: The settings used.
n_dropped: Number of respondents dropped for missing responses.

Details

For item $j$ the rest score $R_{-j}$ is the sum of the other items. Respondents are sorted by rest score and cut into groups of at least minsize people, never splitting a rest score; a remainder smaller than minsize is merged into the last group. Within each group the item-step proportions $P(X_j \ge x)$ are computed for every category $x$ above the lowest. Manifest monotonicity requires $$P(X_j \ge x \mid R_{-j} \in \text{higher group}) \ge P(X_j \ge x \mid R_{-j} \in \text{lower group})$$ for every pair of groups. A decrease larger than minvi counts as a violation, and each violation gets the normal approximation $$z = \frac{2\left|\sqrt{(a+1)(d+1)} - \sqrt{bc}\right|}{\sqrt{n-1}},$$ where $a$ and $b$ are the numbers at or above and below $x$ in the higher group, $c$ and $d$ the same in the lower group, and $n = a + b + c + d$. A $z$ above 1.6449 (one-sided 5\ multiplicity correction) is counted as significant.

The default minsize follows the mokken package: $N/10$ for $N \ge 500$, $N/5$ for $250 < N < 500$, $N/3$ for $N \le 250$ and 50 for $N < 150$ (all rounded down). It is advisable to repeat the check with a few values of minsize. Below 100 respondents the default exceeds half the sample and the function stops; such a sample is too small for the regression at its default resolution, and minsize must then be chosen deliberately.

Respondents with any missing response are dropped (listwise), because a rest score over different item sets is not comparable; a message reports how many. With complete data the results agree with check.monotonicity() of the mokken package.

For polytomous items latent monotonicity does not strictly imply manifest monotonicity (Junker & Sijtsma, 2000); the check is used in practice as if it did. The function only reports; it never recodes or drops items.

References

Junker, B. W., & Sijtsma, K. (2000). Latent and manifest monotonicity in item response models. Applied Psychological Measurement, 24, 65–81.

Molenaar, I. W., & Sijtsma, K. (2000). User's manual MSP5 for Windows. IEC ProGAMMA.

van der Ark, L. A. (2007). Mokken scale analysis in R. Journal of Statistical Software, 20(11), 1–19.

Examples

# \donttest{
mm <- ManifestMonotonicity(J5S1000)
#> ManifestMonotonicity: 26 respondents with missing responses were dropped.
mm$summary
#>   Item        Hj n_active n_violation vi_per_active max_violation sum_violation
#> 1   V1 0.2344842       63           1    0.01587302    0.03103070    0.03103070
#> 2   V2 0.2591719       42           0    0.00000000    0.00000000    0.00000000
#> 3   V3 0.2244265       63           1    0.01587302    0.07020516    0.07020516
#> 4   V4 0.2887626       42           0    0.00000000    0.00000000    0.00000000
#> 5   V5 0.2345297       63           1    0.01587302    0.04356061    0.04356061
#>   sum_per_active     z_max n_significant
#> 1   0.0004925508 0.5967635             0
#> 2   0.0000000000 0.0000000             0
#> 3   0.0011143676 1.0828149             0
#> 4   0.0000000000 0.0000000             0
#> 5   0.0006914382 0.7464788             0
# }