# S-X2 Fit Statistic Computes the $S\text{-}X^2$ item fit statistic proposed by Orlando and Thissen (2000, 2003). This statistic evaluates the fit of IRT models by comparing observed and expected item response frequencies across summed score groups. ```r sx2_fit(x, ...) ## Default S3 method: sx2_fit( x, data, D = 1, alpha = 0.05, min.collapse = 1, norm.prior = c(0, 1), nquad = 30, weights, pcm.loc = NULL, ... ) ## S3 method for class 'est_item' sx2_fit( x, alpha = 0.05, min.collapse = 1, norm.prior = c(0, 1), nquad = 30, weights, pcm.loc = NULL, ... ) ## S3 method for class 'est_irt' sx2_fit( x, alpha = 0.05, min.collapse = 1, norm.prior = c(0, 1), nquad = 30, weights, pcm.loc = NULL, ... ) ``` ## Arguments - `x`: A data frame containing item metadata (e.g., item parameters, number of categories, IRT model types, etc.); or an object of class `est_irt` obtained from `est_irt()`, or `est_item` from `est_item()`. See `est_irt()` or `simdat()` for more details about the item metadata. This data frame can be easily created using the `shape_df()` function. - `...`: Additional arguments passed to or from other methods. - `data`: A matrix of examinees' item responses corresponding to the items specified in the `x` argument. Rows represent examinees and columns represent items. - `D`: A scaling constant used in IRT models to make the logistic function closely approximate the normal ogive function. A value of 1.7 is commonly used for this purpose. Default is 1. - `alpha`: A numeric value specifying the significance level ($\alpha$) for the hypothesis test associated with the $S\text{-}X^2$ statistic. Default is 0.05. - `min.collapse`: An integer specifying the minimum expected frequency required per cell before adjacent cells are collapsed. Default is 1. See Details . - `norm.prior`: A numeric vector of length two specifying the mean and standard deviation of the normal prior distribution. These values are used to generate the Gaussian quadrature points and weights. Ignored if `method` is `"ML"`, `"MLF"`, `"WL"`, or `"INV.TCC"`. Default is `c(0, 1)`. - `nquad`: An integer specifying the number of Gaussian quadrature points used to approximate the normal prior distribution. Default is 30. - `weights`: A two-column matrix or data frame containing the quadrature points (first column) and their corresponding weights (second column) for the latent ability distribution. If omitted, default values are generated using `gen.weight()` according to the `norm.prior` and `nquad` arguments. - `pcm.loc`: An optional integer vector indicating the row indices of items that follow the partial credit model (PCM), where slope parameters are fixed. Default is `NULL`. ## Returns A list containing the following components: - **fit_stat**: A data frame summarizing the $S\text{-}X^2$ fit statistics for all items, including the chi-square value, degrees of freedom, critical value, and p-value. - **item_df**: A data frame containing the item metadata as specified in the input argument `x`. - **exp_freq**: A list of collapsed expected frequency tables for all items. - **obs_freq**: A list of collapsed observed frequency tables for all items. - **exp_prob**: A list of collapsed expected probability tables for all items. - **obs_prop**: A list of collapsed observed proportion tables for all items. ## Details The accuracy of the $\chi^{2}$ approximation in item fit statistics can be compromised when expected cell frequencies in contingency tables are too small (Orlando & Thissen, 2000). To address this issue, Orlando and Thissen (2000) proposed collapsing adjacent summed score groups to ensure a minimum expected frequency of at least 1. However, applying this collapsing approach directly to polytomous item data can result in excessive information loss (Kang & Chen, 2008). To mitigate this, Kang and Chen (2008) instead collapsed adjacent response categories **within** each summed score group, maintaining a minimum expected frequency of 1 per category. The same collapsing strategies are implemented in `sx2_fit()`. If a different minimum expected frequency is desired, it can be specified via the `min.collapse` argument. When an item is labeled as "DRM" in the item metadata, it is treated as a 3PLM item when computing the degrees of freedom for the $S\text{-}X^2$ statistic. Additionally, any missing responses in the `data` are automatically replaced with incorrect responses (i.e., 0s). ## Methods (by class) * `sx2_fit(default)`: Default method for computing $S\text{-}X^{2}$ fit statistics from a data frame `x` containing item metadata. * `sx2_fit(est_item)`: An object created by the function `est_item()`. * `sx2_fit(est_irt)`: An object created by the function `est_irt()`. ## Examples ```r ## Example 1: All five polytomous IRT items follow the GRM ## Import the "-prm.txt" output file from flexMIRT flex_sam <- system.file("extdata", "flexmirt_sample-prm.txt", package = "irtQ") # Select the item metadata x <- bring.flexmirt(file = flex_sam, "par")$Group1$full_df # Generate examinees' abilities from N(0, 1) set.seed(23) score <- rnorm(500, mean = 0, sd = 1) # Simulate response data data <- simdat(x = x, theta = score, D = 1) # Compute fit statistics fit1 <- sx2_fit(x = x, data = data, nquad = 30) # Display fit statistics fit1$fit_stat ## Example 2: Items 39 and 40 follow the GRM, and items 53, 54, and 55 ## follow the PCM (with slope parameters fixed to 1) # Replace the model names with "GPCM" and # set the slope parameters of items 53–55 to 1 x[53:55, 3] <- "GPCM" x[53:55, 4] <- 1 # Generate examinees' abilities from N(0, 1) set.seed(25) score <- rnorm(1000, mean = 0, sd = 1) # Simulate response data data <- simdat(x = x, theta = score, D = 1) # Compute fit statistics fit2 <- sx2_fit(x = x, data = data, nquad = 30, pcm.loc = 53:55) # Display fit statistics fit2$fit_stat ``` ## References Kang, T., & Chen, T. T. (2008). Performance of the generalized S-X2 item fit index for polytomous IRT models. **Journal of Educational Measurement, 45**(4), 391-406. Orlando, M., & Thissen, D. (2000). Likelihood-based item-fit indices for dichotomous item response theory models. **Applied Psychological Measurement, 24**(1), 50-64. Orlando, M., & Thissen, D. (2003). Further investigation of the performance of S-X2: An item fit index for use with dichotomous item response theory models. **Applied Psychological Measurement, 27**(4), 289-298. ## See Also `irtfit()`, `simdat()`, `shape_df()`, `est_irt()`, `est_item()` ## Author(s) Hwanggyu Lim hglim83@gmail.com

sx2_fit function