# Model Fit based on Person Fit Index Q function for performing a model fit test based on calculating the person fit index `Q` for sub samples. Persons are ranked according their fit and in turn a Kruskal-Wallis rank sum test (see `kruskal.test`) is performed to test if the parameters of the distribution of Q are the same in each group (sample). ```r Qtest( obj = NULL, data = NULL, threshold = NULL, split = "median", splitseed = "no", ... ) ``` ## Arguments - `obj`: an object of class `"pers"` or class `"pair"`as a result from function `pers` or `pair` respectively. - `data`: optional response data when object of class `"pers"` or class `"pair"` is not provided. - `threshold`: optional in case that object of class `"pers"` or class `"pair"` is not provided. Threshold values as matrix with row and columnnames !! -- items as rows and thresholds as columns. Thresholds should be ordered from left to right, some items may have less thresholds than the others, in this case the respective row/column is filled with an NA value - see examples. - `split`: Specifies the splitting criterion. Basically there are three different options available - each with several modes - which are controlled by passing the corresponding character expression to the argument. 1) Using the rawscore for splitting into subsamples with the following modes: `split = "median"` median raw score split - high score group and low score group; `split = "mean"` mean raw score split - high score group and low score group. Finaly `split = "score"` that is splitting `daten` into as many subsamples as there are raw score groups - discarding min and max (theoretical) score group - which matches the concept proposed by Andersen (1973). 2) Dividing the persons in `daten` into subsamples with equal size by random allocation with the following modes: `split = "random"` (which is equivalent to `split = "random.2"`) divides persons into two subsamples with equal size. In general the number of desired subsamples must be expressed after the dot in the character expression - e.g. `split = "random.6"` divides persons into 6 subsamples (with equal size) by random allocation etc. 3) The third option is using a manifest variable as a splitting criterion. In this case a vector with the same length as number of cases in `daten` must be passed to the argument grouping the data into subsamples. This vector should be coded as `"factor"` or a `"numeric"` integer vector with min = 1. - `splitseed`: numeric, used for `set.seed(splitseed)` for random splitting - see argument `split`. - `...`: not used so far. ## Returns A list with class "htest" containing the following components as returned by `kruskal.test`. ## Details The person Q-index proposed by Tarnai and Rost, (1990) is solely based on the empirical responses and the item parameters. Thus the computation of person parameters using the function `pers` is not required - see examples. But for convenience return objects of both functions are accepted in function `Q`. This offers the opportunity to get some inference on model fit only based on item parameters. ## Examples ```r ####################### data(bfiN) # get some data ip <- pair(daten = bfiN,m = 6) # item parameters according the partial credit model Q(ip) Qtest(ip) ### with data an thresholds as external objects ##### threshold <- matrix(seq(-3,3,length.out = 9),ncol = 3) dimnames(threshold) <- list(c("I1","I2","I3"),c("1","2","2")) threshold resp_vec <- c(3,0,2,1,2,2,2,2,1,3,0,NA,NA,0,2,3,NA,2,NA,2,1,2,NA,1,2,2,NA) resp_emp <- matrix(resp_vec,ncol = 3,byrow = TRUE) colnames(resp_emp) <- c("I1","I2","I3") resp_emp Qindex <- Q(data = resp_emp,threshold = threshold) cbind(resp_emp,Qindex) Qtest_res <- Qtest(data = resp_emp,threshold = threshold) Qtest_res #### unequal number of thresholds ################### threshold <- matrix(seq(-3,3,length.out = 9),ncol = 3) dimnames(threshold) <- list(c("I1","I2","I3"),c("1","2","2")) threshold[2,3] <- NA resp_vec <- c(3,0,2,1,2,2,2,2,1,3,0,NA,NA,0,2,3,NA,2,NA,2,1,2,NA,1,2,2,NA) resp_emp <- matrix(resp_vec,ncol = 3,byrow = TRUE) colnames(resp_emp) <- c("I1","I2","I3") resp_emp Qindex <- Q(data = resp_emp,threshold = threshold) cbind(resp_emp,Qindex) Qtest_res <- Qtest(data = resp_emp,threshold = threshold) Qtest_res ``` ## References Tarnai, C., & Rost, J. (1990). **Identifying aberrant response patterns in the Rasch model: the Q index**. Münster: ISF. Andersen, E. B. (1973). A goodness of fit test for the rasch model. **Psychometrika, 38**(1), 123–140.

Qtest function