# A Function to Perform Nonparametric Rasch Model Tests A variety of nonparametric tests as proposed by Ponocny (2001), Koller and Hatzinger (2012), and an exact version of the Martin-Löf test are implemented. The function operates on random binary matrices that have been generated using an MCMC algorithm (Verhelst, 2008) from the `RaschSampler` package (Hatzinger, Mair, and Verhelst, 2009). UTF-8 ```r NPtest(obj, n = NULL, method = "T1", ...) ``` ## Arguments - `obj`: A binary data matrix (or data frame) or an object containing the output from the `RaschSampler` package. - `n`: If `obj` is a matrix or a data frame, `n` is the number of sampled matrices (default is 500) - `method`: One of the test statistics. See Details below. - ``...``: Further arguments according to `method`. See Details below. Additionally, the sampling routine can be controlled by specifying `burn_in`, `step`, and `seed` (for details see below and `rsctrl`). A summary of the sampling object may be obtained using the option `RSinfo = TRUE`. ## Details The function uses the `RaschSampler` package, which is now packaged with `eRm` for convenience. It can, of course, still be accessed and downloaded separately via CRAN. As an input the user has to supply either a binary data matrix or a `RaschSampler` output object. If the input is a data matrix, the `RaschSampler` is called with default values (i.e., `rsctrl(burn_in = 256, n_eff = n, step = 32)`, see `rsctrl`), where `n` corresponds to `n_eff` (the default number of sampled matrices is 500). By default, the starting values for the random number generators (`seed`) are chosen randomly using system time. Methods other than those listed below can easily be implemented using the `RaschSampler` package directly. The currently implemented methods (following Ponocny's notation of $T$-statistics) and their options are: - **$T_1$:**: `method = "T1"` Checks for local dependence via increased inter-item correlations. For all item pairs, cases are counted with equal responses on both items. - **$T_1m$:**: `method = "T1m"` Checks for multidimensionality via decreased inter-item correlations. For all item pairs, cases are counted with equal responses on both items. - **$T_1l$:**: `method = "T1l"` Checks for learning. For all item pairs, cases are counted with response pattern (1,1). - **$T_md$:**: `method = "Tmd", idx1, idx2` `idx1` and `idx2` are vectors of indices specifying items which define two subscales, e.g., `idx1 = c(1, 5, 7)` and `idx2 = c(3, 4, 6)` Checks for multidimensionality based on correlations of person raw scores for the subscales. - **$T_2$:**: `method = "T2", idx = NULL, stat = "var"` `idx` is a vector of indices specifying items which define a subscale, e.g., `idx = c(1, 5, 7)` `stat` defines the used statistic as a character object which can be: `"var"` (variance), `"mad1"` (mean absolute deviation), `"mad2"` (median absolute deviation), or `"range"` (range) Checks for local dependence within model deviating subscales via increased dispersion of subscale person rawscores. - **$T_2m$:**: `method = "T2m", idx = NULL, stat = "var"` `idx` is a vector of indices specifying items which define a subscale, e.g., `idx = c(1, 5, 7)` `stat` defines the used statistic as a character object which can be: `"var"` (variance), `"mad1"` (mean absolute deviation), `"mad2"` (median absolute deviation), `"range"` (range) Checks for multidimensionality within model deviating subscales via decreased dispersion of subscale person rawscores. - **$T_4$:**: `method = "T4", idx = NULL, group = NULL, alternative = "high"` `idx` is a vector of indices specifying items which define a subscale, e.g., `idx = c(1, 5, 7)` `group` is a logical vector defining a subject group, e.g., `group = ((age >= 20) & (age < 30))` `alternative` specifies the alternative hypothesis and can be: `"high"` or `"low"`. Checks for group anomalies (DIF ) via too high (low) raw scores on item(s) for specified group. - **$T_10$:**: `method = "T10", splitcr = "median"` `splitcr` defines the split criterion for subject raw score splitting. `"median"` uses the median as split criterion, `"mean"` performs a mean-split. Optionally, `splitcr` can also be a vector which assigns each person to one of two subgroups (e.g., following an external criterion). This vector can be numeric, character, logical, or a factor. Global test for subgroup-invariance. Checks for different item difficulties in two subgroups (for details see Ponocny, 2001). - **$T_11$:**: `method = "T11"` Global test for local dependence. The statistic calculates the sum of absolute deviations between the observed inter-item correlations and the expected correlations. - **$T_pbis$:**: `method = "Tpbis", idxt, idxs` Test for discrimination. The statistic calculates a point-biserial correlation for a test item (specified via `idxt`) with the person row scores for a subscale of the test sum (specified via `idxs`). If the correlation is too low, the test item shows different discrimination compared to the items of the subscale. - ****Martin-Löf****: The exact version of the **Martin-Löf** statistic is specified via `method = "MLoef"` and optionally `splitcr` (see `MLoef`). - **$Q_3h$:**: `method = "Q3h"` Checks for local dependence by detecting an increased correlation of inter-item residuals. Low p-values correspond to a high ("h") correlation between two items. - **$Q_3l$:**: `method = "Q3l"` Checks for local dependence by detecting a decreased correlation of inter-item residuals. Low p-values correspond to a low ("l") correlation between two items. ## Returns Depending on the `method` argument, a list is returned which has one of the following classes: `'T1obj'`, `'T1mobj'`, `'T1lobj'`, `'Tmdobj'`, `'T2obj'`, `'T2mobj'`, `'T4obj'`, `'T10obj'`, `'T11obj'`, `'Tpbisobj'`, `'Q3hobj'` or `'Q3lobj'`. The main output element is `prop` giving the one-sided $p$-value, i.e., the number of statistics from the sampled matrices which are equal or exceed the statistic based on the observed data. For $T_1$, $T_1m$, and $T_1l$, `prop` is a vector. For the Martin-Löf test, the returned object is of class `'MLobj'`. Besides other elements, it contains a `prop` vector and `MLres`, the output object from the asymptotic Martin-Löf test on the input data. ## Note The `RaschSampler` package is no longer required to use `NPtest` since `eRm` version 0.15-0. ## References Ponocny, I. (2001). Nonparametric goodness-of-fit tests for the Rasch model. **Psychometrika, 66**(3), 437--459. tools:::Rd_expr_doi("10.1007/BF02294444") Verhelst, N. D. (2008). An efficient MCMC algorithm to sample binary matrices with fixed marginals. **Psychometrika, 73**(4), 705--728. tools:::Rd_expr_doi("10.1007/s11336-008-9062-3") Verhelst, N., Hatzinger, R., & Mair, P. (2007). The Rasch sampler. **Journal of Statistical Software, 20**(4), 1--14. [https://www.jstatsoft.org/v20/i04](https://www.jstatsoft.org/v20/i04) Koller, I., & Hatzinger, R. (2013). Nonparametric tests for the Rasch model: Explanation, development, and application of quasi-exact tests for small samples. **Interstat, 11**, 1--16. [http://interstat.statjournals.net/YEAR/2013/abstracts/1311002.php](http://interstat.statjournals.net/YEAR/2013/abstracts/1311002.php) Koller, I., Maier, M. J., & Hatzinger, R. (2015). An Empirical Power Analysis of Quasi-Exact Tests for the Rasch Model: Measurement Invariance in Small Samples. **Methodology, 11**(2), 45--54. tools:::Rd_expr_doi("10.1027/1614-2241/a000090") Debelak, R., & Koller, I. (2019). Testing the Local Independence Assumption of the Rasch Model With Q3-Based Nonparametric Model Tests. **Applied Psychological Measurement** tools:::Rd_expr_doi("10.1177/0146621619835501") ## Author(s) Reinhold Hatzinger ## Examples ```r ### Preparation: # data for examples below X <- as.matrix(raschdat1) # generate 100 random matrices based on original data matrix rmat <- rsampler(X, rsctrl(burn_in = 100, n_eff = 100, seed = 123)) ## the following examples can also directly be used by setting ## rmat <- as.matrix(raschdat1) ## without calling rsampler() first t1 <- NPtest(rmat, n = 100, method = "T1") ### Examples ################################################################### ###--- T1 ---------------------------------------------------------------------- t1 <- NPtest(rmat, method = "T1") # choose a different alpha for selecting displayed values print(t1, alpha = 0.01) ###--- T2 ---------------------------------------------------------------------- t21 <- NPtest(rmat, method = "T2", idx = 1:5, burn_in = 100, step = 20, seed = 7654321, RSinfo = TRUE) # default stat is variance t21 t22 <- NPtest(rmat, method = "T2", stat = "mad1", idx = c(1, 22, 5, 27, 6, 9, 11)) t22 ###--- T4 ---------------------------------------------------------------------- age <- sample(20:90, 100, replace = TRUE) # group MUST be a logical vector # (value of TRUE is used for group selection) age <- age < 30 t41 <- NPtest(rmat, method = "T4", idx = 1:3, group = age) t41 sex <- gl(2, 50) # group can also be a logical expression (generating a vector) t42 <- NPtest(rmat, method = "T4", idx = c(1, 4, 5, 6), group = sex == 1) t42 ###--- T10 --------------------------------------------------------------------- t101 <- NPtest(rmat, method = "T10") # default split criterion is "median" t101 ## Not run: split <- runif(100) t102 <- NPtest(rmat, method = "T10", splitcr = split > 0.5) t102 t103 <- NPtest(rmat, method = "T10", splitcr = sex) t103 ## End(Not run) ###--- T11 --------------------------------------------------------------------- t11 <- NPtest(rmat, method = "T11") t11 ###--- Tpbis ------------------------------------------------------------------- tpb <- NPtest(X[, 1:5], method = "Tpbis", idxt = 1, idxs = 2:5) tpb ###--- Martin-Löf -------------------------------------------------------------- ## Not run: # takes a while ... split <- rep(1:3, each = 10) NPtest(raschdat1, n = 100, method = "MLoef", splitcr = split) ## End(Not run) ```

NPtest function