Simulate person-by-item Rasch data — simulate

Generates dichotomous, partial credit, or rating scale data. Optional arguments introduce item misfit, guessing, multidimensionality, local dependence, DIF, response styles, or missingness. Generating values are stored in attr(x, "truth"). A positive planted proportion selects at least one person or response cell when the requested departures can coexist.

Usage

simulate_rasch(
  n_persons = 500,
  n_items = 20,
  model = c("dichotomous", "PCM", "RSM"),
  n_categories = 3,
  theta_mean = 0,
  theta_sd = 1,
  theta_dist = "normal",
  difficulty = c(-2.5, 2.5),
  threshold_spread = 1.2,
  discrimination = 1,
  guessing = 0,
  second_dim = NULL,
  dependence = NULL,
  dif = NULL,
  careless = 0,
  response_style = NULL,
  speeded = 0,
  disordered = NULL,
  n_groups = 1,
  missing = 0,
  seed = NULL
)

Arguments

n_persons, n_items: Sample size and test length.
model: "dichotomous", "PCM", or "RSM". Under "RSM" every item shares one category-threshold pattern (items differ by location only); under "PCM" each item's threshold spacings and span are drawn afresh, as the partial credit model allows.
n_categories: Response categories for polytomous models (>= 3).
theta_mean, theta_sd: Realised sample mean and standard deviation of the person distribution.
theta_dist: Shape of the person distribution: "normal", "uniform", "skew", or "bimodal".
difficulty: Either the two endpoints of an evenly spaced location range, or one location per item.
threshold_spread: Half-range of the category thresholds about each item location (polytomous).
discrimination: The item slope, supplied as one value or one per item. One common value changes the logit unit without departing from a Rasch model. Differences between items are a deliberate slope departure: values above the common pattern produce steeper responses and values below it produce flatter responses, and require theta_sd > 0.
guessing: Scalar or length-n_items lower asymptote (dichotomous): low-location persons answer correctly by chance. A positive asymptote requires theta_sd > 0.
second_dim: NULL, or list(items=, rho=): the named items load on a second trait whose realised sample correlation with the first is rho. It lies in [-1, 1); a correlation of 1 would reproduce the primary trait rather than plant another dimension. At least three persons are needed unless rho is -1. Each item is named once, and at least one item must remain on the primary trait; moving every item to the second trait is still a one-dimensional scale.
dependence: NULL, or list(pairs=, strength=): each pair's second item responds partly to the first. This departure feeds the residual-dependence diagnostics. Each directed pair is listed once.
dif: NULL, or list(items=, uniform=, nonuniform=): the named items function differently for the last person group: a location shift (uniform) and/or a slope change (nonuniform). Needs n_groups >= 2; each item is named once and at least one item must remain invariant. A common shift or slope change on every item changes the group origin or unit rather than defining item DIF. A non-uniform effect also requires theta_sd > 0.
careless: Proportion of persons who answer at random. At least one person in each generated group must retain model-based responses. Careless and response-style assignments are disjoint; their requested counts must fit.
response_style: NULL, or list(type=, prop=, strength=) with type "extreme" or "middle": a proportion prop of persons favour the end (or middle) categories regardless of the trait, with distortion strength (default 1.6) on the log-probability scale (polytomous). At least one person must remain without the style; applying one category weighting to everyone merely changes the fitted thresholds.
speeded: Proportion not reached at the last item: a growing tail of missing responses over the final items. These cells are kept distinct from any completely-at-random missing cells. Persons are selected independently of their ability and responses; this does not simulate non-ignorable missingness or change the response model.
disordered: NULL or item names/indices given disordered thresholds (polytomous; feeds the threshold diagnostics).
n_groups: Number of equal person groups (a group factor column is added when > 1, for DIF).
missing: Proportion of responses set missing completely at random, drawn from cells not already missing through speededness. The requested count must fit among those cells and leave at least one observed response.
seed: Optional non-negative whole-number RNG seed. See rasch_rng for generator support.

Value

A data frame of class "rasch_sim" (item columns I01..., an id column, and a group column when grouped), with attr(x, "truth") holding the generating parameters and the planted departures.

Examples

# a clean scale with one over-discriminating item and one DIF item
d <- simulate_rasch(400, 12, discrimination = c(3, rep(1, 11)),
                    dif = list(items = "I06", uniform = 1), n_groups = 2,
                    seed = 1)
fit <- rasch(d, id = "id", factors = "group")
fit$items[c("item", "infit_ms", "outfit_ms")]   # item 1 misfits
#>  item infit_ms outfit_ms
#>   I01    1.013     0.428
#>   I02    1.043     0.955
#>   I03    0.982     0.822
#>   I04    1.054     1.124
#>   I05    1.040     0.946
#>   I06    1.091     1.134
#>   I07    1.027     0.993
#>   I08    1.113     1.126
#>   I09    1.045     1.052
#>   I10    1.046     0.949
#>   I11    1.036     0.984
#>   I12    0.985     0.781
dif_anova(fit)$summary                           # item 6 flags
#>  item  term F_uniform p_uniform p_uniform_adj eta2_uniform uniform_DIF
#>   I01 group     4.320     0.038         0.805        0.012            
#>   I02 group     1.858     0.174         1.000        0.005            
#>   I03 group     0.033     0.856         1.000        0.000            
#>   I04 group     2.029     0.155         1.000        0.005            
#>   I05 group     0.856     0.356         1.000        0.002            
#>   I06 group    25.647   < 0.001       < 0.001        0.062           *
#>   I07 group     3.186     0.075         1.000        0.008            
#>   I08 group     0.083     0.773         1.000        0.000            
#>   I09 group     0.754     0.386         1.000        0.002            
#>   I10 group     0.520     0.471         1.000        0.001            
#>   I11 group     4.891     0.028         0.620        0.012            
#>   I12 group     0.018     0.895         1.000        0.000            
#>  F_nonuniform p_nonuniform p_nonuniform_adj eta2_nonuniform nonuniform_DIF
#>         1.550        0.173            1.000           0.020               
#>         0.020        1.000            1.000           0.000               
#>         1.761        0.120            1.000           0.022               
#>         1.711        0.131            1.000           0.022               
#>         1.192        0.313            1.000           0.015               
#>         0.473        0.797            1.000           0.006               
#>         2.561        0.027            0.620           0.032               
#>         0.954        0.446            1.000           0.012               
#>         0.822        0.534            1.000           0.011               
#>         0.512        0.767            1.000           0.007               
#>         0.455        0.810            1.000           0.006               
#>         1.208        0.305            1.000           0.015               
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