class: left, middle, inverse, title-slide # The Rasch Model - Extensions via TAM and the MRCML ## Justifications and Considerations </span> ### --- <style> pre { white-space: pre !important; overflow-y: scroll !important; max-height: 50vh !important; } </style> # What we'll be doing... + Why? -- + Review of the Linear Model (GLM) with "normal" assumptions to introduce design matrices in the GLM -- + An introduction to MRCML - the Multidimensional Random Coefficients Multinomial Logit Model + An introduction to arrays in R + Working with design matrices in TAM -- + Next time - the depths of MRCML and design matrices for different model types + Linear Logistic Test Model (item side) + Alternative to TAM (Random Item IRT, Continuous predictors, controlling priors, etc...) --- class: middle # The Rasch model + We use the Rasch model for checking various properties of our items as evidence for having measured (or other hypotheses) -- + We also may have designs that include raters (essays, observation protocols, etc) - so you have participants, items, & raters -- + We may have multidimensionality -- + We may want to include the effect of item type or format, person properties, or their interactions -- + We may want to include item dependencies, item covariates, or test form covariates (is including a specific property in item or item category reducing or increasing the probability of category or item endorsement) --- class: middle # The Generalized Linear Model Classic example, borrowing from Andy... .pull-left[ + Let's say I tell you that the average time it takes to get to the store from UCSB is 20 minutes. That's all I tell you. + When you go to the store (any store), how long would you guess your journey is about to take? -- + We can write, `\(\mathbb{E}[Time] = 20\)` minutes ] -- .pull-right[ + Now, let's say you're on some sort of schedule... + You might also be worried about the variance of these trips (you might want to know about the probability of the trip taking more than 30 minutes) ] --- class: middle # The GLM, cont'd + To account for this, you request I describe your trip times with a distribution, namely, a normal distribution + These trip times now have a mean and variance to describe expectations and uncertainty + `\(Time \sim \mathcal N(\mu = 20, \sigma^2 = 25)\)` + We can plot this --- <img src="MRCML_xaringan_files/figure-html/unnamed-chunk-2-1.png" width="648" style="display: block; margin: auto;" /> --- class: middle # Let's write this as a linear model now, `$$y = \beta_0*X_0 + \epsilon$$` -- `\(X_0\)` is just defined to equal 1 + `\(\epsilon\)` is normally, independently, and identically distributed with mean 0 and variance - let's say the mean is also 20 minutes --- class: middle # Running this model In R this model looks like ```r lm(data = df, y ~ 1) ``` + The model now has a "linear" or deterministic part, and a "random" part - that the outcome will never exactly equal the mean ("the error distribution") --- class: middle # The GLM cont'd Rewritten in matrix form - this model looks like: `$$\hat{y} = X\beta$$` or... `$$\mathbb{E}[Time] = X\beta$$` --- class: middle # Representing this model .pull-left[ + Using the notation of De Boeck and Wilson (2004)  ] -- .pull-right[ + The squiggly line I tried to draw (sorry...) represents the random component of the model - the distribution + Observed time is not the same as the linear model predicted time - there's some uncertainty ] --- # In Matrix Form If you have data from 5 trips your model looks like this $$ `\begin{bmatrix} 20 \\ 20 \\ 20 \\ 20 \\ 20 \\ \end{bmatrix} = \begin{bmatrix} 1 \\ 1 \\ 1 \\ 1 \\ 1 \\ \end{bmatrix} * \begin{bmatrix}20\\ \end{bmatrix}` \quad $$ --- class: middle # The GLM continued .pull-left[ + What I didn't tell you is that the average trip takes 20 minutes by bike ... + But it actually takes 5 minutes less time by car (on average) + Now we have additional information - but still need a distribution for modeling each mode of transportation! `$$\mathbb{E}[Time|Mode] = X\beta$$` ] -- .pull-right[ + The distribution is now relative to each mode of transport - the conditional distribution + Normally distributed error, mean 0. + Two normally distributed populations - mean 20, and mean 15. + Could fit the same variance or different variances ] --- class: middle # The GLM continued + Given the description above, in non matrix notation, with bikes as reference group `\(E[Time|Mode] = 20 * X_0 + (-5*X_1)\)` + Finally in matrix form, if two trips were by bike, and three by car, bike coded as 0, is the second column of the matrix. + We also have a new vector, the beta vector, which holds the values of our coefficients. + The matrix with 1s and 0s is called the design matrix $$ `\begin{bmatrix} 20 \\ 15 \\ 20 \\ 15 \\ 15 \\ \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 1 & 1 \\ 1 & 0 \\ 1 & 1\\ 1 & 1 \\ \end{bmatrix} * \begin{bmatrix}20\\ -5 \end{bmatrix}` \quad $$ --- <img src="MRCML_xaringan_files/figure-html/unnamed-chunk-4-1.png" width="648" style="display: block; margin: auto;" /> --- class: middle # Binary or categorical outcomes? + Our conditional distribtions aren't really Normal anymore - but... + The outcome data data might have a bernoulli, binomial, poisson, exponential distribution which are all part of the same family of distribution + This common family of distributions is known as the exponential family of distributions - the lifeblood of the GLM. --- class: middle # Aside: The common family for the GLM - The exponential family + When you write these conditional distributions in this exponential family form: + Given a linear function, `\(\eta = B_0 + B_1*X_1..\)` + There is a function `\(g(\mu) = \eta\)` + A function can be inverted so `\(g^{-1}(\eta) = \mu\)` -- When this parameter/function comes directly from the distribution written in the common exponential family form, this is called the "cannonical parameter" such that -- `\(\eta \equiv\theta\)` & `\(g^{-1}(\theta)\)` will get you `\(\mu\)` --- # For logistic regression The outcome is defined as being drawn from a Bernoulli (or Binomial): [Click Here for my Derivation](https://dbkatz.com/assets/Presentations/Reminar_MRCML/linkfun.pdf) .pull-left[ `\(X_i \sim Bern(p)\)`. The mean of the Bernoulli distribution is the probability of one of the two outcomes (coded as 1 or 0) which we'll call `p`. Conditional expectation is: `$$P(y=1|x)$$` ] -- .pull-right[ `$$g(\mu)=\eta =\theta = log(\frac{p}{1-p})$$` The inverse ... `$$p = \frac{exp(\eta)}{1+exp(\eta)}$$` ] --- class: middle # Ok, enough of this... + The point isn't to memorize this - it's to see the similarities in item response models + What else can we add to the Rasch model? + Item response data is comprised of categorical data + Pet peeve - calling categorical data "normally distributed" because the middle categories have the highest frequencies --- class: middle # Symbolizing our model We add a straight line between the linear model and eta (they're equivalent) and now we add a link function which is symbolized by the straight arrow between `\(\eta\)` and the probability. + We still have the random component (of course) .center[  ] --- ## The Rasch model From De Boeck & Wilson (2004): .pull-left[  + `\(Y_{pi}\)`: response of person p on item i (categorical) ] .pull-right[ + `\(\beta_i\)`: item or item category difficulty - it'll be a vector + `\(X_{ik}\)`: design matrix containing 1s and 0s pertaining to if the particular `\(\beta\)` corresponds to item i in category k + `\(\theta_p\)`: the person ability - it is a random effect (next slide) + `\(Z_{p0}\)` is a constant (here) but later a set of predictors - basically showing that `\(\theta_p\)` is sampled from a distribution - a random effect ] --- class: middle # Formally: 1. `$$\pi_{pi} = P(Y_{ik}=y|\theta_p, \beta_{ik})$$` -- 1. `$$\pi_{pi} = \frac{exp(\theta_p-\beta_{ik})}{1 + exp(\theta_p-\beta_{ik})}$$` -- 1. `$$\theta_p \sim N(0, \sigma^2_\theta)$$` -- This is a version of the Random Coefficient Multinomial Logit Model (RCML) --- class: middle ### Writing the Rasch model as a linear model + `\(\eta_{pi} = \sum_{p=1}^n\theta_pZ_{pij} + \sum_{k=1}^K\beta_{ik}X_{pik}\)` + Not pretty, but we just wrote this out as a linear model like you're used to. But `\(\beta\)` is now reversed sign - this is the item "easiness" instead of difficulty. + `\(X\)` is a design matrix of item predictors - this is the A matrix in `TAM` + `\(Z\)` is a design matrix of person predictors. --- class: middle # Predictors in the Rasch model? Why? + You want to know about the influence of item property Y (word problem vs non-word problem, specific word use on a self-report survey) -- + Differential item functioning - is an item (category) relatively harder (to endorse) for certain student groups even after matching on the level of the property of interest? -- + There are person properties that you want to account for or know about (being in a specific classroom or not) - what is the effect of this property on the same logit scale as the items? --- # It would be great if... We could have some form of the model so it looks the same regardless of ... -- + Item type (dichotomous, polytomous) + Item covariates -- + Dimensionality (single or multidimensional, within or between item dimensionality) + Person covariates included (get person abilities or person property effects on abilities) -- + Rater effects included -- + That is the Multidimensional Random Coefficients Multinomial Logit Model (MRCML) --- class: middle # Rasch Model revisited using the design matrices + We can now use design matrices with dummy variables, effectively, like we did in our earlier regression `$$P(X_{ik} = 1; A, B | \delta, \theta) = \frac{exp(b_{ik}\theta + a'_{ik}\delta)}{\sum_{k=1}^{K_i}exp(b_{ik}\theta + a'_{ik}\delta)}$$` + a and b are elements of the design matrices `A` and `B` + this form of the model allows for all sorts of item types... --- class: middle ## Partial Credit Model - Three Response Options 1. `$$P(x_i =0) = \frac{1}{1 + exp(\theta-\beta_{i1}) + exp(2\theta-(\beta_{i1}+\beta_{i2})}$$` 1. `$$P(x_i=1) = \frac{ exp(\theta-\beta_{i1}) }{1 + exp(\theta-\beta_{i1}) + exp(2\theta-(\beta_{i1}+\beta_{i2})}$$` 1. `$$P(x_i=2) = \frac{ exp(2\theta-(\beta_{i1}+\beta_{i2}))}{1 + exp(\theta-\beta_{i1}) + exp(2\theta-(\beta_{i1}+\beta_{i2}))}$$` --- ## Design Matrices `$$P(X_{ik} = 1; A, B | \delta, \theta) = \frac{exp(x'(b_{ik}\theta + a'_{ik}\delta)}{\sum_{k=1}^{K_i}exp(z'(b_{ik}\theta + a'_{ik}\delta))}$$` .pull-left[ B = $$ `\begin{bmatrix} 0 \\ 1 \\ 2 \\ \end{bmatrix}` $$ A = $$ `\begin{bmatrix} 0 & 0 \\ 1 & 0 \\ 1 & 1 \\ \end{bmatrix}` $$ ] .pull-right[ + Person responds in category 2 (the highest category - they get a dummy variable of 1 for the third category/row of the item in the x matrix) + Numerator - invokes the 2 in the B design matrix - so we start with `\(2\theta\)` + Numerator - invokes the third row of the A matrix - the 1 and 1, so we get `\(\beta_{i1} + \beta_{i2}\)` ] --- # Yielding...  --- ### Multidimensional Rasch example + Using `\(\delta\)` for item difficulty instead of `\(\beta\)`; Items 1 and 2 indicator for first dim, items 1 & 3 correct. (from Kennedy (2005) , p. 17) . .center[ <img src="dich_des.jpg" width="1733" height="10%" /> ] --- ### First covariate, or facet in the Rasch model - DIF + Checking to see if items or item categories are relatively harder (or easier) for students of different subgroups at the same "level" of the property We can express the model: `$$\eta_{ip} = \theta_p + \delta_ik + \beta_1*group_p + \gamma_i(item*group_p)$$` And we introduce pseudo items in the design matrix, A, for two dichotmous items and one group with two categories I1_G1 = Pseudo Item ``` ## I1 I2 G1 I1xG1 ## I1_G1 1 0 0 0 ## I1_G2 1 0 1 1 ## I2_G1 0 1 0 0 ## I2_G2 0 1 1 0 ``` --- class: middle ## Thinking about attempts to measure (could make this more generic) .center[  ] --- # Arrays + Unfortunately, the package `TAM` uses arrays instead of matrices -- + Well, they use matrices - just multidimensional matrices -- + Matrices in R have two dimensions + Arrays have more than two dimensions - they're like multiple dimensions --- # Arrays, cont'd + An array with dimensions [4, 4, 2] has 2 matrices, with 4 rows and 4 columns. ```r # Create two vectors of different lengths. v <- c(1:16) # Make it an array ary <- array(c(v),dim = c(4,4,2)) ary ``` ``` ## , , 1 ## ## [,1] [,2] [,3] [,4] ## [1,] 1 5 9 13 ## [2,] 2 6 10 14 ## [3,] 3 7 11 15 ## [4,] 4 8 12 16 ## ## , , 2 ## ## [,1] [,2] [,3] [,4] ## [1,] 1 5 9 13 ## [2,] 2 6 10 14 ## [3,] 3 7 11 15 ## [4,] 4 8 12 16 ``` ```r #If you wanted to select only the second matrix #ary[ , , 2] ``` --- Get only the second columns - turn it into one matrix ```r ary[,2,] ``` ``` ## [,1] [,2] ## [1,] 5 5 ## [2,] 6 6 ## [3,] 7 7 ## [4,] 8 8 ``` -- Get only the first rows ```r ary[1, , ] ``` ``` ## [,1] [,2] ## [1,] 1 1 ## [2,] 5 5 ## [3,] 9 9 ## [4,] 13 13 ``` -- get the first rows and second columns ```r ary[1, 2, ] ``` ``` ## [1] 5 5 ``` --- ### Thank Dog, some TAM code + We'll use some simulated data to understand the different models + `TAM` is a package for running versions of the MRCML model + For dichotomous and polytomous data ```r #install.packages(TAM) #install.packages library(TAM) # ?tam.mml ``` --- # Data for running the model ```r # Load data items <- read.csv("https://raw.githubusercontent.com/danielbkatz/danielbkatz.github.io/master/assets/Presentations/Reminar_MRCML/sim_data.csv") # remove id column and resp <- items[-c(1,42)] head(resp) ``` ``` ## Math.1 Math.2 Math.3 Math.4 Math.5 Math.6 Math.7 Math.8 Math.9 Math.10 ## 1 1 1 1 0 0 0 1 0 1 1 ## 2 1 1 1 0 0 1 0 1 1 0 ## 3 1 0 0 0 0 0 1 0 0 0 ## 4 1 1 0 0 0 0 1 0 1 0 ## 5 0 1 0 0 0 0 1 0 0 0 ## 6 0 1 1 0 0 1 1 0 1 1 ## Science.1 Science.2 Science.3 Science.4 Science.5 Science.6 Science.7 ## 1 1 1 0 0 0 1 0 ## 2 1 1 0 0 0 1 1 ## 3 0 1 0 0 0 1 1 ## 4 0 1 0 1 0 1 1 ## 5 1 1 0 0 1 1 0 ## 6 0 1 0 0 0 1 0 ## Science.8 Science.9 Science.10 ELA.1 ELA.2 ELA.3 ELA.4 ELA.5 ELA.6 ELA.7 ## 1 1 0 1 1 1 1 0 1 1 1 ## 2 1 1 1 1 1 1 1 1 1 0 ## 3 1 0 1 1 0 0 1 1 1 0 ## 4 1 0 1 0 1 1 0 0 1 1 ## 5 1 0 0 1 1 1 1 1 1 1 ## 6 0 0 0 1 1 1 1 0 1 1 ## ELA.8 ELA.9 ELA.10 WordProb.1 WordProb.2 WordProb.3 WordProb.4 WordProb.5 ## 1 1 1 1 0 1 0 1 0 ## 2 1 1 0 1 1 1 0 0 ## 3 0 0 1 0 1 0 0 1 ## 4 1 1 1 0 1 0 0 0 ## 5 1 0 1 0 1 0 1 0 ## 6 1 1 1 0 1 0 0 1 ## WordProb.6 WordProb.7 WordProb.8 WordProb.9 WordProb.10 ## 1 0 0 1 0 1 ## 2 1 0 1 0 0 ## 3 0 1 1 0 1 ## 4 1 0 1 0 1 ## 5 1 0 0 0 0 ## 6 1 0 0 0 1 ``` --- # Running the Rasch Model (first ten items) ```r # Get first ten items df <- resp[,1:10] # Run the model mod1 <- tam.mml(resp = df) ``` ```r # item difficulties head(mod1$xsi) ``` ``` ## xsi se.xsi ## Math.1 -1.3301167 0.06494537 ## Math.2 -0.6036252 0.05876371 ## Math.3 0.2329543 0.05746165 ## Math.4 0.8544774 0.06033535 ## Math.5 0.9465537 0.06105641 ## Math.6 -0.2136443 0.05742465 ``` --- # Extract Design Matrices + TAM uses contrast coding - so items get a -1 instead of a 1. ```r A_mat <- mod1$A B_mat <- mod1$B # array with 10 rows, 2 columns, and 10 matrices dim(A_mat) ``` ``` ## [1] 10 2 10 ``` ```r # array with 10 rows, 2 columns, 1 matrix dim(B_mat) ``` ``` ## [1] 10 2 1 ``` --- # A mat ```r # First 5 rows, First two matrices A_mat[, , 1:2] ``` ``` ## , , Math.1 ## ## Category0 Category1 ## Item01 0 -1 ## Item02 0 0 ## Item03 0 0 ## Item04 0 0 ## Item05 0 0 ## Item06 0 0 ## Item07 0 0 ## Item08 0 0 ## Item09 0 0 ## Item10 0 0 ## ## , , Math.2 ## ## Category0 Category1 ## Item01 0 0 ## Item02 0 -1 ## Item03 0 0 ## Item04 0 0 ## Item05 0 0 ## Item06 0 0 ## Item07 0 0 ## Item08 0 0 ## Item09 0 0 ## Item10 0 0 ``` --- # A matrix continued - ```r A_mat[,2,] ``` ``` ## Math.1 Math.2 Math.3 Math.4 Math.5 Math.6 Math.7 Math.8 Math.9 Math.10 ## Item01 -1 0 0 0 0 0 0 0 0 0 ## Item02 0 -1 0 0 0 0 0 0 0 0 ## Item03 0 0 -1 0 0 0 0 0 0 0 ## Item04 0 0 0 -1 0 0 0 0 0 0 ## Item05 0 0 0 0 -1 0 0 0 0 0 ## Item06 0 0 0 0 0 -1 0 0 0 0 ## Item07 0 0 0 0 0 0 -1 0 0 0 ## Item08 0 0 0 0 0 0 0 -1 0 0 ## Item09 0 0 0 0 0 0 0 0 -1 0 ## Item10 0 0 0 0 0 0 0 0 0 -1 ``` --- Transform to get item "easiness" params - multiply by -1 ```r A_mat1 <- A_mat A_mat1[,2,] <- A_mat[,2,] *-1 A_mat1[,, 1:2] ``` ``` ## , , Math.1 ## ## Category0 Category1 ## Item01 0 1 ## Item02 0 0 ## Item03 0 0 ## Item04 0 0 ## Item05 0 0 ## Item06 0 0 ## Item07 0 0 ## Item08 0 0 ## Item09 0 0 ## Item10 0 0 ## ## , , Math.2 ## ## Category0 Category1 ## Item01 0 0 ## Item02 0 1 ## Item03 0 0 ## Item04 0 0 ## Item05 0 0 ## Item06 0 0 ## Item07 0 0 ## Item08 0 0 ## Item09 0 0 ## Item10 0 0 ``` ```r # model with new A matrix mod2 <- tam.mml(df, A=A_mat1) ``` --- # Compare the item parameters ```r # model with default design matrices head(mod1$xsi) ``` ``` ## xsi se.xsi ## Math.1 -1.3301167 0.06494537 ## Math.2 -0.6036252 0.05876371 ## Math.3 0.2329543 0.05746165 ## Math.4 0.8544774 0.06033535 ## Math.5 0.9465537 0.06105641 ## Math.6 -0.2136443 0.05742465 ``` ```r #model with transformed A matrix head(mod2$xsi) ``` ``` ## xsi se.xsi ## Math.1 1.3300881 0.06494508 ## Math.2 0.6035996 0.05876352 ## Math.3 -0.2329764 0.05746155 ## Math.4 -0.8544967 0.06033529 ## Math.5 -0.9465727 0.06105636 ## Math.6 0.2136203 0.05742450 ``` --- # B Matrix Note the name of the matrix here is Dim 1 - hold that thought ```r B_mat ``` ``` ## , , Dim01 ## ## Cat0 Cat1 ## Math.1 0 1 ## Math.2 0 1 ## Math.3 0 1 ## Math.4 0 1 ## Math.5 0 1 ## Math.6 0 1 ## Math.7 0 1 ## Math.8 0 1 ## Math.9 0 1 ## Math.10 0 1 ``` --- class: middle # The linear logistic test model (LLTM) - base of many models + Note the data we have here has four "dimensions" + What if we wanted to know the overall difficulty of item type? -- + We'll use the design matrices -- + Note - this is not really a "multidimensional model" in the sense of multiple factors - this models each type of item as a fixed effect --- # The item side base for + For dichotomous or polytomous items `$$P(Y=y|\theta_s, \delta_i) = \frac{exp(\theta_s-\delta_i)}{1+exp(\theta_s-\delta_i)}$$` `$$log\biggl[P/(1-P)\biggr] = \eta = \theta_p-\delta_i$$` `$$\delta_i = \sum(q_{ij}\nu_j)$$` + q = "weights" for item i and item effect j -- + `\(\nu_j\)` = item effect/variable/indicator if item i is related to some effect -- + In our data, whether an item is a math item or not. --- # How do we indicate this? A matrix For 4 items, treat the first two items as the same and second two items as the same ```r A2 <- matrix(data = 0, nrow = 4, ncol = 2) A2[1:2,1] <- 1 A2[3:4,2] <- 1 A2 ``` ``` ## [,1] [,2] ## [1,] 1 0 ## [2,] 1 0 ## [3,] 0 1 ## [4,] 0 1 ``` --- ## Doing this in TAM + work with the full `resp` data frame -- + Code all item types together so that all math items, all ela items...used for generating area specific (fixed) difficulties ```r #use TAM to generate the design matrices for us des_lltm <- designMatrices(resp = resp) # we want the A matrix A <- des_lltm$A ``` --- ### Looking at the design matrix + Let's look at the matrices for the first four items + We'll keep only the first four matrices since we're going to have only four parameters (math difficulty, ela difficulty, science difficuly, word problem) ```r A[ , , 1:4] ``` ``` ## , , Math.1 ## ## Category0 Category1 ## Item01 0 -1 ## Item02 0 0 ## Item03 0 0 ## Item04 0 0 ## Item05 0 0 ## Item06 0 0 ## Item07 0 0 ## Item08 0 0 ## Item09 0 0 ## Item10 0 0 ## Item11 0 0 ## Item12 0 0 ## Item13 0 0 ## Item14 0 0 ## Item15 0 0 ## Item16 0 0 ## Item17 0 0 ## Item18 0 0 ## Item19 0 0 ## Item20 0 0 ## Item21 0 0 ## Item22 0 0 ## Item23 0 0 ## Item24 0 0 ## Item25 0 0 ## Item26 0 0 ## Item27 0 0 ## Item28 0 0 ## Item29 0 0 ## Item30 0 0 ## Item31 0 0 ## Item32 0 0 ## Item33 0 0 ## Item34 0 0 ## Item35 0 0 ## Item36 0 0 ## Item37 0 0 ## Item38 0 0 ## Item39 0 0 ## Item40 0 0 ## ## , , Math.2 ## ## Category0 Category1 ## Item01 0 0 ## Item02 0 -1 ## Item03 0 0 ## Item04 0 0 ## Item05 0 0 ## Item06 0 0 ## Item07 0 0 ## Item08 0 0 ## Item09 0 0 ## Item10 0 0 ## Item11 0 0 ## Item12 0 0 ## Item13 0 0 ## Item14 0 0 ## Item15 0 0 ## Item16 0 0 ## Item17 0 0 ## Item18 0 0 ## Item19 0 0 ## Item20 0 0 ## Item21 0 0 ## Item22 0 0 ## Item23 0 0 ## Item24 0 0 ## Item25 0 0 ## Item26 0 0 ## Item27 0 0 ## Item28 0 0 ## Item29 0 0 ## Item30 0 0 ## Item31 0 0 ## Item32 0 0 ## Item33 0 0 ## Item34 0 0 ## Item35 0 0 ## Item36 0 0 ## Item37 0 0 ## Item38 0 0 ## Item39 0 0 ## Item40 0 0 ## ## , , Math.3 ## ## Category0 Category1 ## Item01 0 0 ## Item02 0 0 ## Item03 0 -1 ## Item04 0 0 ## Item05 0 0 ## Item06 0 0 ## Item07 0 0 ## Item08 0 0 ## Item09 0 0 ## Item10 0 0 ## Item11 0 0 ## Item12 0 0 ## Item13 0 0 ## Item14 0 0 ## Item15 0 0 ## Item16 0 0 ## Item17 0 0 ## Item18 0 0 ## Item19 0 0 ## Item20 0 0 ## Item21 0 0 ## Item22 0 0 ## Item23 0 0 ## Item24 0 0 ## Item25 0 0 ## Item26 0 0 ## Item27 0 0 ## Item28 0 0 ## Item29 0 0 ## Item30 0 0 ## Item31 0 0 ## Item32 0 0 ## Item33 0 0 ## Item34 0 0 ## Item35 0 0 ## Item36 0 0 ## Item37 0 0 ## Item38 0 0 ## Item39 0 0 ## Item40 0 0 ## ## , , Math.4 ## ## Category0 Category1 ## Item01 0 0 ## Item02 0 0 ## Item03 0 0 ## Item04 0 -1 ## Item05 0 0 ## Item06 0 0 ## Item07 0 0 ## Item08 0 0 ## Item09 0 0 ## Item10 0 0 ## Item11 0 0 ## Item12 0 0 ## Item13 0 0 ## Item14 0 0 ## Item15 0 0 ## Item16 0 0 ## Item17 0 0 ## Item18 0 0 ## Item19 0 0 ## Item20 0 0 ## Item21 0 0 ## Item22 0 0 ## Item23 0 0 ## Item24 0 0 ## Item25 0 0 ## Item26 0 0 ## Item27 0 0 ## Item28 0 0 ## Item29 0 0 ## Item30 0 0 ## Item31 0 0 ## Item32 0 0 ## Item33 0 0 ## Item34 0 0 ## Item35 0 0 ## Item36 0 0 ## Item37 0 0 ## Item38 0 0 ## Item39 0 0 ## Item40 0 0 ``` -- # We'll keep only the first 4 matrices ```r A4 <- A[ , , 1:4] # rename the dimensions dimnames(A4)[[3]] <- c("Math", "Science", "ELA", "Word_Prob") A4 ``` ``` ## , , Math ## ## Category0 Category1 ## Item01 0 -1 ## Item02 0 0 ## Item03 0 0 ## Item04 0 0 ## Item05 0 0 ## Item06 0 0 ## Item07 0 0 ## Item08 0 0 ## Item09 0 0 ## Item10 0 0 ## Item11 0 0 ## Item12 0 0 ## Item13 0 0 ## Item14 0 0 ## Item15 0 0 ## Item16 0 0 ## Item17 0 0 ## Item18 0 0 ## Item19 0 0 ## Item20 0 0 ## Item21 0 0 ## Item22 0 0 ## Item23 0 0 ## Item24 0 0 ## Item25 0 0 ## Item26 0 0 ## Item27 0 0 ## Item28 0 0 ## Item29 0 0 ## Item30 0 0 ## Item31 0 0 ## Item32 0 0 ## Item33 0 0 ## Item34 0 0 ## Item35 0 0 ## Item36 0 0 ## Item37 0 0 ## Item38 0 0 ## Item39 0 0 ## Item40 0 0 ## ## , , Science ## ## Category0 Category1 ## Item01 0 0 ## Item02 0 -1 ## Item03 0 0 ## Item04 0 0 ## Item05 0 0 ## Item06 0 0 ## Item07 0 0 ## Item08 0 0 ## Item09 0 0 ## Item10 0 0 ## Item11 0 0 ## Item12 0 0 ## Item13 0 0 ## Item14 0 0 ## Item15 0 0 ## Item16 0 0 ## Item17 0 0 ## Item18 0 0 ## Item19 0 0 ## Item20 0 0 ## Item21 0 0 ## Item22 0 0 ## Item23 0 0 ## Item24 0 0 ## Item25 0 0 ## Item26 0 0 ## Item27 0 0 ## Item28 0 0 ## Item29 0 0 ## Item30 0 0 ## Item31 0 0 ## Item32 0 0 ## Item33 0 0 ## Item34 0 0 ## Item35 0 0 ## Item36 0 0 ## Item37 0 0 ## Item38 0 0 ## Item39 0 0 ## Item40 0 0 ## ## , , ELA ## ## Category0 Category1 ## Item01 0 0 ## Item02 0 0 ## Item03 0 -1 ## Item04 0 0 ## Item05 0 0 ## Item06 0 0 ## Item07 0 0 ## Item08 0 0 ## Item09 0 0 ## Item10 0 0 ## Item11 0 0 ## Item12 0 0 ## Item13 0 0 ## Item14 0 0 ## Item15 0 0 ## Item16 0 0 ## Item17 0 0 ## Item18 0 0 ## Item19 0 0 ## Item20 0 0 ## Item21 0 0 ## Item22 0 0 ## Item23 0 0 ## Item24 0 0 ## Item25 0 0 ## Item26 0 0 ## Item27 0 0 ## Item28 0 0 ## Item29 0 0 ## Item30 0 0 ## Item31 0 0 ## Item32 0 0 ## Item33 0 0 ## Item34 0 0 ## Item35 0 0 ## Item36 0 0 ## Item37 0 0 ## Item38 0 0 ## Item39 0 0 ## Item40 0 0 ## ## , , Word_Prob ## ## Category0 Category1 ## Item01 0 0 ## Item02 0 0 ## Item03 0 0 ## Item04 0 -1 ## Item05 0 0 ## Item06 0 0 ## Item07 0 0 ## Item08 0 0 ## Item09 0 0 ## Item10 0 0 ## Item11 0 0 ## Item12 0 0 ## Item13 0 0 ## Item14 0 0 ## Item15 0 0 ## Item16 0 0 ## Item17 0 0 ## Item18 0 0 ## Item19 0 0 ## Item20 0 0 ## Item21 0 0 ## Item22 0 0 ## Item23 0 0 ## Item24 0 0 ## Item25 0 0 ## Item26 0 0 ## Item27 0 0 ## Item28 0 0 ## Item29 0 0 ## Item30 0 0 ## Item31 0 0 ## Item32 0 0 ## Item33 0 0 ## Item34 0 0 ## Item35 0 0 ## Item36 0 0 ## Item37 0 0 ## Item38 0 0 ## Item39 0 0 ## Item40 0 0 ``` --- # Recode + All items of the same type get dummy coded the same, zero otherwise + We do this by assigning -1 to the pertinent dimensions + For instance, two items, two facets A = $$ `\begin{bmatrix} 0 & -1 \\ 0 & -1 \\ -1 & 0 \\ -1 & 0 \\ \end{bmatrix}` $$ --- ### Let's do this in TAM ```r # zero it all out A4[,,] <- 0 A4[1:10,2, 1] <- -1 A4[11:20,2,2] <- -1 A4[21:30,2,3] <- -1 A4[31:40,2,4] <- -1 A4 ``` ``` ## , , Math ## ## Category0 Category1 ## Item01 0 -1 ## Item02 0 -1 ## Item03 0 -1 ## Item04 0 -1 ## Item05 0 -1 ## Item06 0 -1 ## Item07 0 -1 ## Item08 0 -1 ## Item09 0 -1 ## Item10 0 -1 ## Item11 0 0 ## Item12 0 0 ## Item13 0 0 ## Item14 0 0 ## Item15 0 0 ## Item16 0 0 ## Item17 0 0 ## Item18 0 0 ## Item19 0 0 ## Item20 0 0 ## Item21 0 0 ## Item22 0 0 ## Item23 0 0 ## Item24 0 0 ## Item25 0 0 ## Item26 0 0 ## Item27 0 0 ## Item28 0 0 ## Item29 0 0 ## Item30 0 0 ## Item31 0 0 ## Item32 0 0 ## Item33 0 0 ## Item34 0 0 ## Item35 0 0 ## Item36 0 0 ## Item37 0 0 ## Item38 0 0 ## Item39 0 0 ## Item40 0 0 ## ## , , Science ## ## Category0 Category1 ## Item01 0 0 ## Item02 0 0 ## Item03 0 0 ## Item04 0 0 ## Item05 0 0 ## Item06 0 0 ## Item07 0 0 ## Item08 0 0 ## Item09 0 0 ## Item10 0 0 ## Item11 0 -1 ## Item12 0 -1 ## Item13 0 -1 ## Item14 0 -1 ## Item15 0 -1 ## Item16 0 -1 ## Item17 0 -1 ## Item18 0 -1 ## Item19 0 -1 ## Item20 0 -1 ## Item21 0 0 ## Item22 0 0 ## Item23 0 0 ## Item24 0 0 ## Item25 0 0 ## Item26 0 0 ## Item27 0 0 ## Item28 0 0 ## Item29 0 0 ## Item30 0 0 ## Item31 0 0 ## Item32 0 0 ## Item33 0 0 ## Item34 0 0 ## Item35 0 0 ## Item36 0 0 ## Item37 0 0 ## Item38 0 0 ## Item39 0 0 ## Item40 0 0 ## ## , , ELA ## ## Category0 Category1 ## Item01 0 0 ## Item02 0 0 ## Item03 0 0 ## Item04 0 0 ## Item05 0 0 ## Item06 0 0 ## Item07 0 0 ## Item08 0 0 ## Item09 0 0 ## Item10 0 0 ## Item11 0 0 ## Item12 0 0 ## Item13 0 0 ## Item14 0 0 ## Item15 0 0 ## Item16 0 0 ## Item17 0 0 ## Item18 0 0 ## Item19 0 0 ## Item20 0 0 ## Item21 0 -1 ## Item22 0 -1 ## Item23 0 -1 ## Item24 0 -1 ## Item25 0 -1 ## Item26 0 -1 ## Item27 0 -1 ## Item28 0 -1 ## Item29 0 -1 ## Item30 0 -1 ## Item31 0 0 ## Item32 0 0 ## Item33 0 0 ## Item34 0 0 ## Item35 0 0 ## Item36 0 0 ## Item37 0 0 ## Item38 0 0 ## Item39 0 0 ## Item40 0 0 ## ## , , Word_Prob ## ## Category0 Category1 ## Item01 0 0 ## Item02 0 0 ## Item03 0 0 ## Item04 0 0 ## Item05 0 0 ## Item06 0 0 ## Item07 0 0 ## Item08 0 0 ## Item09 0 0 ## Item10 0 0 ## Item11 0 0 ## Item12 0 0 ## Item13 0 0 ## Item14 0 0 ## Item15 0 0 ## Item16 0 0 ## Item17 0 0 ## Item18 0 0 ## Item19 0 0 ## Item20 0 0 ## Item21 0 0 ## Item22 0 0 ## Item23 0 0 ## Item24 0 0 ## Item25 0 0 ## Item26 0 0 ## Item27 0 0 ## Item28 0 0 ## Item29 0 0 ## Item30 0 0 ## Item31 0 -1 ## Item32 0 -1 ## Item33 0 -1 ## Item34 0 -1 ## Item35 0 -1 ## Item36 0 -1 ## Item37 0 -1 ## Item38 0 -1 ## Item39 0 -1 ## Item40 0 -1 ``` --- ## Run the model ```r mod_l <- tam.mml(resp=resp, A = A4) # get "model"item" fit itfit <- tam.fit(mod_l) ``` + Get item difficulties (you may get slightly different values) ```r mod_l$xsi ``` ``` ## xsi se.xsi ## Math -0.109231488 0.01678528 ## Science 0.006363148 0.01676390 ## ELA -1.287357593 0.01988335 ## Word_Prob 0.024911944 0.01676495 ``` -- ```r itfit$itemfit ``` ``` ## parameter Outfit Outfit_t Outfit_p Outfit_pholm Infit Infit_t ## 1 Math 1.4770249 112.09548 0.000000e+00 0.000000e+00 1.4830318 113.34048 ## 2 Science 0.8510156 -43.26278 0.000000e+00 0.000000e+00 0.8482414 -44.11350 ## 3 ELA 1.1884167 16.70814 1.143463e-62 1.143463e-62 1.2131581 18.77354 ## 4 Word_Prob 0.8736461 -36.35628 2.090765e-289 4.181530e-289 0.8731073 -36.51659 ## Infit_p Infit_pholm ## 1 0.000000e+00 0.000000e+00 ## 2 0.000000e+00 0.000000e+00 ## 3 1.243123e-78 1.243123e-78 ## 4 6.049716e-292 1.209943e-291 ``` + This is a strange model but describes the "difficulty effect" of item type --- class: middle #### What about individual item difficulties plus overall effect of item type? + I'm still trying to figure out if I'm doing this correctly (need to simulate) -- + I want the item difficulty of the item "adjusting for" the item type effect (this is relevant to differential bundle functioning or testlets as well Wang and Wilson (2005)) -- + Start with design matrix A and add item types to the matrix - four columns to each matrix - these are pseudo items - not sure if I need to leave one as all zeros for identification or code as `-1` and `1` instead --- # Setting up design matrices ```r # create a new array with 40 rows, 2 columns, and 4 matrices # 40 items to get coded, still two categories, 4 matrices A_type <- array(data=0, dim = c(40, 2, 4)) A_type[1:10, 2, 1] <- -1 A_type[11:20, 2, 2] <- -1 A_type[21:30, 2, 3] <- -1 A_type[31:40, 2, 4] <- -1 dimnames(A_type)[[3]] <- c("Math", "Science", "ELA", "Word_Prob") A_ex <- abind::abind(A, A_type) ``` --- # Check out the array ```r A_ex[ , 2, ] ``` ``` ## Math.1 Math.2 Math.3 Math.4 Math.5 Math.6 Math.7 Math.8 Math.9 Math.10 ## Item01 -1 0 0 0 0 0 0 0 0 0 ## Item02 0 -1 0 0 0 0 0 0 0 0 ## Item03 0 0 -1 0 0 0 0 0 0 0 ## Item04 0 0 0 -1 0 0 0 0 0 0 ## Item05 0 0 0 0 -1 0 0 0 0 0 ## Item06 0 0 0 0 0 -1 0 0 0 0 ## Item07 0 0 0 0 0 0 -1 0 0 0 ## Item08 0 0 0 0 0 0 0 -1 0 0 ## Item09 0 0 0 0 0 0 0 0 -1 0 ## Item10 0 0 0 0 0 0 0 0 0 -1 ## Item11 0 0 0 0 0 0 0 0 0 0 ## Item12 0 0 0 0 0 0 0 0 0 0 ## Item13 0 0 0 0 0 0 0 0 0 0 ## Item14 0 0 0 0 0 0 0 0 0 0 ## Item15 0 0 0 0 0 0 0 0 0 0 ## Item16 0 0 0 0 0 0 0 0 0 0 ## Item17 0 0 0 0 0 0 0 0 0 0 ## Item18 0 0 0 0 0 0 0 0 0 0 ## Item19 0 0 0 0 0 0 0 0 0 0 ## Item20 0 0 0 0 0 0 0 0 0 0 ## Item21 0 0 0 0 0 0 0 0 0 0 ## Item22 0 0 0 0 0 0 0 0 0 0 ## Item23 0 0 0 0 0 0 0 0 0 0 ## Item24 0 0 0 0 0 0 0 0 0 0 ## Item25 0 0 0 0 0 0 0 0 0 0 ## Item26 0 0 0 0 0 0 0 0 0 0 ## Item27 0 0 0 0 0 0 0 0 0 0 ## Item28 0 0 0 0 0 0 0 0 0 0 ## Item29 0 0 0 0 0 0 0 0 0 0 ## Item30 0 0 0 0 0 0 0 0 0 0 ## Item31 0 0 0 0 0 0 0 0 0 0 ## Item32 0 0 0 0 0 0 0 0 0 0 ## Item33 0 0 0 0 0 0 0 0 0 0 ## Item34 0 0 0 0 0 0 0 0 0 0 ## Item35 0 0 0 0 0 0 0 0 0 0 ## Item36 0 0 0 0 0 0 0 0 0 0 ## Item37 0 0 0 0 0 0 0 0 0 0 ## Item38 0 0 0 0 0 0 0 0 0 0 ## Item39 0 0 0 0 0 0 0 0 0 0 ## Item40 0 0 0 0 0 0 0 0 0 0 ## Science.1 Science.2 Science.3 Science.4 Science.5 Science.6 Science.7 ## Item01 0 0 0 0 0 0 0 ## Item02 0 0 0 0 0 0 0 ## Item03 0 0 0 0 0 0 0 ## Item04 0 0 0 0 0 0 0 ## Item05 0 0 0 0 0 0 0 ## Item06 0 0 0 0 0 0 0 ## Item07 0 0 0 0 0 0 0 ## Item08 0 0 0 0 0 0 0 ## Item09 0 0 0 0 0 0 0 ## Item10 0 0 0 0 0 0 0 ## Item11 -1 0 0 0 0 0 0 ## Item12 0 -1 0 0 0 0 0 ## Item13 0 0 -1 0 0 0 0 ## Item14 0 0 0 -1 0 0 0 ## Item15 0 0 0 0 -1 0 0 ## Item16 0 0 0 0 0 -1 0 ## Item17 0 0 0 0 0 0 -1 ## Item18 0 0 0 0 0 0 0 ## Item19 0 0 0 0 0 0 0 ## Item20 0 0 0 0 0 0 0 ## Item21 0 0 0 0 0 0 0 ## Item22 0 0 0 0 0 0 0 ## Item23 0 0 0 0 0 0 0 ## Item24 0 0 0 0 0 0 0 ## Item25 0 0 0 0 0 0 0 ## Item26 0 0 0 0 0 0 0 ## Item27 0 0 0 0 0 0 0 ## Item28 0 0 0 0 0 0 0 ## Item29 0 0 0 0 0 0 0 ## Item30 0 0 0 0 0 0 0 ## Item31 0 0 0 0 0 0 0 ## Item32 0 0 0 0 0 0 0 ## Item33 0 0 0 0 0 0 0 ## Item34 0 0 0 0 0 0 0 ## Item35 0 0 0 0 0 0 0 ## Item36 0 0 0 0 0 0 0 ## Item37 0 0 0 0 0 0 0 ## Item38 0 0 0 0 0 0 0 ## Item39 0 0 0 0 0 0 0 ## Item40 0 0 0 0 0 0 0 ## Science.8 Science.9 Science.10 ELA.1 ELA.2 ELA.3 ELA.4 ELA.5 ELA.6 ELA.7 ## Item01 0 0 0 0 0 0 0 0 0 0 ## Item02 0 0 0 0 0 0 0 0 0 0 ## Item03 0 0 0 0 0 0 0 0 0 0 ## Item04 0 0 0 0 0 0 0 0 0 0 ## Item05 0 0 0 0 0 0 0 0 0 0 ## Item06 0 0 0 0 0 0 0 0 0 0 ## Item07 0 0 0 0 0 0 0 0 0 0 ## Item08 0 0 0 0 0 0 0 0 0 0 ## Item09 0 0 0 0 0 0 0 0 0 0 ## Item10 0 0 0 0 0 0 0 0 0 0 ## Item11 0 0 0 0 0 0 0 0 0 0 ## Item12 0 0 0 0 0 0 0 0 0 0 ## Item13 0 0 0 0 0 0 0 0 0 0 ## Item14 0 0 0 0 0 0 0 0 0 0 ## Item15 0 0 0 0 0 0 0 0 0 0 ## Item16 0 0 0 0 0 0 0 0 0 0 ## Item17 0 0 0 0 0 0 0 0 0 0 ## Item18 -1 0 0 0 0 0 0 0 0 0 ## Item19 0 -1 0 0 0 0 0 0 0 0 ## Item20 0 0 -1 0 0 0 0 0 0 0 ## Item21 0 0 0 -1 0 0 0 0 0 0 ## Item22 0 0 0 0 -1 0 0 0 0 0 ## Item23 0 0 0 0 0 -1 0 0 0 0 ## Item24 0 0 0 0 0 0 -1 0 0 0 ## Item25 0 0 0 0 0 0 0 -1 0 0 ## Item26 0 0 0 0 0 0 0 0 -1 0 ## Item27 0 0 0 0 0 0 0 0 0 -1 ## Item28 0 0 0 0 0 0 0 0 0 0 ## Item29 0 0 0 0 0 0 0 0 0 0 ## Item30 0 0 0 0 0 0 0 0 0 0 ## Item31 0 0 0 0 0 0 0 0 0 0 ## Item32 0 0 0 0 0 0 0 0 0 0 ## Item33 0 0 0 0 0 0 0 0 0 0 ## Item34 0 0 0 0 0 0 0 0 0 0 ## Item35 0 0 0 0 0 0 0 0 0 0 ## Item36 0 0 0 0 0 0 0 0 0 0 ## Item37 0 0 0 0 0 0 0 0 0 0 ## Item38 0 0 0 0 0 0 0 0 0 0 ## Item39 0 0 0 0 0 0 0 0 0 0 ## Item40 0 0 0 0 0 0 0 0 0 0 ## ELA.8 ELA.9 ELA.10 WordProb.1 WordProb.2 WordProb.3 WordProb.4 ## Item01 0 0 0 0 0 0 0 ## Item02 0 0 0 0 0 0 0 ## Item03 0 0 0 0 0 0 0 ## Item04 0 0 0 0 0 0 0 ## Item05 0 0 0 0 0 0 0 ## Item06 0 0 0 0 0 0 0 ## Item07 0 0 0 0 0 0 0 ## Item08 0 0 0 0 0 0 0 ## Item09 0 0 0 0 0 0 0 ## Item10 0 0 0 0 0 0 0 ## Item11 0 0 0 0 0 0 0 ## Item12 0 0 0 0 0 0 0 ## Item13 0 0 0 0 0 0 0 ## Item14 0 0 0 0 0 0 0 ## Item15 0 0 0 0 0 0 0 ## Item16 0 0 0 0 0 0 0 ## Item17 0 0 0 0 0 0 0 ## Item18 0 0 0 0 0 0 0 ## Item19 0 0 0 0 0 0 0 ## Item20 0 0 0 0 0 0 0 ## Item21 0 0 0 0 0 0 0 ## Item22 0 0 0 0 0 0 0 ## Item23 0 0 0 0 0 0 0 ## Item24 0 0 0 0 0 0 0 ## Item25 0 0 0 0 0 0 0 ## Item26 0 0 0 0 0 0 0 ## Item27 0 0 0 0 0 0 0 ## Item28 -1 0 0 0 0 0 0 ## Item29 0 -1 0 0 0 0 0 ## Item30 0 0 -1 0 0 0 0 ## Item31 0 0 0 -1 0 0 0 ## Item32 0 0 0 0 -1 0 0 ## Item33 0 0 0 0 0 -1 0 ## Item34 0 0 0 0 0 0 -1 ## Item35 0 0 0 0 0 0 0 ## Item36 0 0 0 0 0 0 0 ## Item37 0 0 0 0 0 0 0 ## Item38 0 0 0 0 0 0 0 ## Item39 0 0 0 0 0 0 0 ## Item40 0 0 0 0 0 0 0 ## WordProb.5 WordProb.6 WordProb.7 WordProb.8 WordProb.9 WordProb.10 Math ## Item01 0 0 0 0 0 0 -1 ## Item02 0 0 0 0 0 0 -1 ## Item03 0 0 0 0 0 0 -1 ## Item04 0 0 0 0 0 0 -1 ## Item05 0 0 0 0 0 0 -1 ## Item06 0 0 0 0 0 0 -1 ## Item07 0 0 0 0 0 0 -1 ## Item08 0 0 0 0 0 0 -1 ## Item09 0 0 0 0 0 0 -1 ## Item10 0 0 0 0 0 0 -1 ## Item11 0 0 0 0 0 0 0 ## Item12 0 0 0 0 0 0 0 ## Item13 0 0 0 0 0 0 0 ## Item14 0 0 0 0 0 0 0 ## Item15 0 0 0 0 0 0 0 ## Item16 0 0 0 0 0 0 0 ## Item17 0 0 0 0 0 0 0 ## Item18 0 0 0 0 0 0 0 ## Item19 0 0 0 0 0 0 0 ## Item20 0 0 0 0 0 0 0 ## Item21 0 0 0 0 0 0 0 ## Item22 0 0 0 0 0 0 0 ## Item23 0 0 0 0 0 0 0 ## Item24 0 0 0 0 0 0 0 ## Item25 0 0 0 0 0 0 0 ## Item26 0 0 0 0 0 0 0 ## Item27 0 0 0 0 0 0 0 ## Item28 0 0 0 0 0 0 0 ## Item29 0 0 0 0 0 0 0 ## Item30 0 0 0 0 0 0 0 ## Item31 0 0 0 0 0 0 0 ## Item32 0 0 0 0 0 0 0 ## Item33 0 0 0 0 0 0 0 ## Item34 0 0 0 0 0 0 0 ## Item35 -1 0 0 0 0 0 0 ## Item36 0 -1 0 0 0 0 0 ## Item37 0 0 -1 0 0 0 0 ## Item38 0 0 0 -1 0 0 0 ## Item39 0 0 0 0 -1 0 0 ## Item40 0 0 0 0 0 -1 0 ## Science ELA Word_Prob ## Item01 0 0 0 ## Item02 0 0 0 ## Item03 0 0 0 ## Item04 0 0 0 ## Item05 0 0 0 ## Item06 0 0 0 ## Item07 0 0 0 ## Item08 0 0 0 ## Item09 0 0 0 ## Item10 0 0 0 ## Item11 -1 0 0 ## Item12 -1 0 0 ## Item13 -1 0 0 ## Item14 -1 0 0 ## Item15 -1 0 0 ## Item16 -1 0 0 ## Item17 -1 0 0 ## Item18 -1 0 0 ## Item19 -1 0 0 ## Item20 -1 0 0 ## Item21 0 -1 0 ## Item22 0 -1 0 ## Item23 0 -1 0 ## Item24 0 -1 0 ## Item25 0 -1 0 ## Item26 0 -1 0 ## Item27 0 -1 0 ## Item28 0 -1 0 ## Item29 0 -1 0 ## Item30 0 -1 0 ## Item31 0 0 -1 ## Item32 0 0 -1 ## Item33 0 0 -1 ## Item34 0 0 -1 ## Item35 0 0 -1 ## Item36 0 0 -1 ## Item37 0 0 -1 ## Item38 0 0 -1 ## Item39 0 0 -1 ## Item40 0 0 -1 ``` --- # Final Array ```r A_ex[, , 41:44] ``` ``` ## , , Math ## ## Category0 Category1 ## Item01 0 -1 ## Item02 0 -1 ## Item03 0 -1 ## Item04 0 -1 ## Item05 0 -1 ## Item06 0 -1 ## Item07 0 -1 ## Item08 0 -1 ## Item09 0 -1 ## Item10 0 -1 ## Item11 0 0 ## Item12 0 0 ## Item13 0 0 ## Item14 0 0 ## Item15 0 0 ## Item16 0 0 ## Item17 0 0 ## Item18 0 0 ## Item19 0 0 ## Item20 0 0 ## Item21 0 0 ## Item22 0 0 ## Item23 0 0 ## Item24 0 0 ## Item25 0 0 ## Item26 0 0 ## Item27 0 0 ## Item28 0 0 ## Item29 0 0 ## Item30 0 0 ## Item31 0 0 ## Item32 0 0 ## Item33 0 0 ## Item34 0 0 ## Item35 0 0 ## Item36 0 0 ## Item37 0 0 ## Item38 0 0 ## Item39 0 0 ## Item40 0 0 ## ## , , Science ## ## Category0 Category1 ## Item01 0 0 ## Item02 0 0 ## Item03 0 0 ## Item04 0 0 ## Item05 0 0 ## Item06 0 0 ## Item07 0 0 ## Item08 0 0 ## Item09 0 0 ## Item10 0 0 ## Item11 0 -1 ## Item12 0 -1 ## Item13 0 -1 ## Item14 0 -1 ## Item15 0 -1 ## Item16 0 -1 ## Item17 0 -1 ## Item18 0 -1 ## Item19 0 -1 ## Item20 0 -1 ## Item21 0 0 ## Item22 0 0 ## Item23 0 0 ## Item24 0 0 ## Item25 0 0 ## Item26 0 0 ## Item27 0 0 ## Item28 0 0 ## Item29 0 0 ## Item30 0 0 ## Item31 0 0 ## Item32 0 0 ## Item33 0 0 ## Item34 0 0 ## Item35 0 0 ## Item36 0 0 ## Item37 0 0 ## Item38 0 0 ## Item39 0 0 ## Item40 0 0 ## ## , , ELA ## ## Category0 Category1 ## Item01 0 0 ## Item02 0 0 ## Item03 0 0 ## Item04 0 0 ## Item05 0 0 ## Item06 0 0 ## Item07 0 0 ## Item08 0 0 ## Item09 0 0 ## Item10 0 0 ## Item11 0 0 ## Item12 0 0 ## Item13 0 0 ## Item14 0 0 ## Item15 0 0 ## Item16 0 0 ## Item17 0 0 ## Item18 0 0 ## Item19 0 0 ## Item20 0 0 ## Item21 0 -1 ## Item22 0 -1 ## Item23 0 -1 ## Item24 0 -1 ## Item25 0 -1 ## Item26 0 -1 ## Item27 0 -1 ## Item28 0 -1 ## Item29 0 -1 ## Item30 0 -1 ## Item31 0 0 ## Item32 0 0 ## Item33 0 0 ## Item34 0 0 ## Item35 0 0 ## Item36 0 0 ## Item37 0 0 ## Item38 0 0 ## Item39 0 0 ## Item40 0 0 ## ## , , Word_Prob ## ## Category0 Category1 ## Item01 0 0 ## Item02 0 0 ## Item03 0 0 ## Item04 0 0 ## Item05 0 0 ## Item06 0 0 ## Item07 0 0 ## Item08 0 0 ## Item09 0 0 ## Item10 0 0 ## Item11 0 0 ## Item12 0 0 ## Item13 0 0 ## Item14 0 0 ## Item15 0 0 ## Item16 0 0 ## Item17 0 0 ## Item18 0 0 ## Item19 0 0 ## Item20 0 0 ## Item21 0 0 ## Item22 0 0 ## Item23 0 0 ## Item24 0 0 ## Item25 0 0 ## Item26 0 0 ## Item27 0 0 ## Item28 0 0 ## Item29 0 0 ## Item30 0 0 ## Item31 0 -1 ## Item32 0 -1 ## Item33 0 -1 ## Item34 0 -1 ## Item35 0 -1 ## Item36 0 -1 ## Item37 0 -1 ## Item38 0 -1 ## Item39 0 -1 ## Item40 0 -1 ``` --- ### Testing this LLTM (may be easier with the eRm package) + Might want to use do in long data form (easier to include facets this way) + Can include more categories that `-1` and `1` in `TAM` - could take on ordinal values + Can add now covariates for things like individual categories for polytomous items + Item or category position effects + "Severity" or wording --- # Run the new model ```r test_run <- tam.mml(resp=resp, A = A_ex) ``` ```r # Item difficulties test_run$xsi ``` ``` ## xsi se.xsi ## Math.1 -1.15759447 0.06169386 ## Math.2 -0.50611041 0.05551036 ## Math.3 0.23894943 0.05423183 ## Math.4 0.79407082 0.05712653 ## Math.5 0.87667246 0.05785140 ## Math.6 -0.15860188 0.05418143 ## Math.7 -0.52154221 0.05559979 ## Math.8 1.02067324 0.05930785 ## Math.9 -1.68471685 0.07074748 ## Math.10 0.01382643 0.05399926 ## Science.1 -0.05131973 0.05399823 ## Science.2 -2.17697719 0.08096956 ## Science.3 2.98424286 0.11495551 ## Science.4 -0.52781745 0.05521237 ## Science.5 0.48307885 0.05549410 ## Science.6 -1.77160114 0.07095983 ## Science.7 0.67829992 0.05681226 ## Science.8 -0.97811139 0.05865354 ## Science.9 2.28750653 0.08703684 ## Science.10 -0.41609880 0.05471005 ## ELA.1 -1.25898560 0.07493500 ## ELA.2 -0.92513028 0.06778220 ## ELA.3 -0.35647695 0.05935776 ## ELA.4 -0.32843485 0.05905203 ## ELA.5 -0.17561230 0.05755321 ## ELA.6 -1.87207302 0.09327876 ## ELA.7 -0.36353326 0.05943621 ## ELA.8 -1.51555798 0.08172172 ## ELA.9 -0.27660831 0.05851230 ## ELA.10 -0.58892256 0.06227373 ## WordProb.1 -0.03976762 0.05400046 ## WordProb.2 -2.18875250 0.08096955 ## WordProb.3 2.90818047 0.11187031 ## WordProb.4 -0.46087876 0.05484437 ## WordProb.5 0.54267409 0.05592667 ## WordProb.6 -1.68555128 0.06895528 ## WordProb.7 0.67298598 0.05686322 ## WordProb.8 -1.02798526 0.05905355 ## WordProb.9 2.33773872 0.08906916 ## WordProb.10 -0.44285619 0.05476956 ## Math -0.03279081 0.01818586 ## Science 0.04404676 0.02004367 ## ELA -0.64374851 0.02055893 ## Word_Prob 0.05582184 0.02002690 ``` --- #Compare the model fits ```r fit_comp <- CDM::IRT.compareModels(test_run, mod_l) fit_comp$IC ``` ``` ## Model loglike Deviance Npars Nobs AIC BIC AIC3 AICc ## 1 test_run -32087.06 64174.11 45 1500 64264.11 64503.21 64309.11 64266.96 ## 2 mod_l -38564.85 77129.71 5 1500 77139.71 77166.27 77144.71 77139.75 ## CAIC ## 1 64548.21 ## 2 77171.27 ``` ```r fit_comp$LRtest ``` ``` ## Model1 Model2 Chi2 df p ## 1 mod_l test_run 12955.59 40 0 ``` --- ## Learning from differential item functioning + Another way to do this is to use differential item functioning A matrices + We can use the design matrices again `$$\eta_{ip} = \theta_p + \delta_ik + \beta_1*group_p + \gamma_i(item*group_p)$$` + We use `formulaA` because it creates the `A matrix` for us in TAM ```r treat <- items[42] treat$treat <- ifelse(treat == "treat", 1, 0) formA <- ~ item*treat # Restrict items to 5 items to make output shorter dif_des_A <- designMatrices.mfr(resp = resp[1:5], facets = treat, formulaA = formA) ``` --- # DIF Design Matrix ```r dif_des_A$A$A.3d ``` ``` ## , , Math.1 ## ## _step0 _step1 ## Math.1-treat0 0 -1 ## Math.1-treat1 0 -1 ## Math.2-treat0 0 0 ## Math.2-treat1 0 0 ## Math.3-treat0 0 0 ## Math.3-treat1 0 0 ## Math.4-treat0 0 0 ## Math.4-treat1 0 0 ## Math.5-treat0 0 0 ## Math.5-treat1 0 0 ## ## , , Math.2 ## ## _step0 _step1 ## Math.1-treat0 0 0 ## Math.1-treat1 0 0 ## Math.2-treat0 0 -1 ## Math.2-treat1 0 -1 ## Math.3-treat0 0 0 ## Math.3-treat1 0 0 ## Math.4-treat0 0 0 ## Math.4-treat1 0 0 ## Math.5-treat0 0 0 ## Math.5-treat1 0 0 ## ## , , Math.3 ## ## _step0 _step1 ## Math.1-treat0 0 0 ## Math.1-treat1 0 0 ## Math.2-treat0 0 0 ## Math.2-treat1 0 0 ## Math.3-treat0 0 -1 ## Math.3-treat1 0 -1 ## Math.4-treat0 0 0 ## Math.4-treat1 0 0 ## Math.5-treat0 0 0 ## Math.5-treat1 0 0 ## ## , , Math.4 ## ## _step0 _step1 ## Math.1-treat0 0 0 ## Math.1-treat1 0 0 ## Math.2-treat0 0 0 ## Math.2-treat1 0 0 ## Math.3-treat0 0 0 ## Math.3-treat1 0 0 ## Math.4-treat0 0 -1 ## Math.4-treat1 0 -1 ## Math.5-treat0 0 0 ## Math.5-treat1 0 0 ## ## , , Math.5 ## ## _step0 _step1 ## Math.1-treat0 0 0 ## Math.1-treat1 0 0 ## Math.2-treat0 0 0 ## Math.2-treat1 0 0 ## Math.3-treat0 0 0 ## Math.3-treat1 0 0 ## Math.4-treat0 0 0 ## Math.4-treat1 0 0 ## Math.5-treat0 0 -1 ## Math.5-treat1 0 -1 ## ## , , treat0 ## ## _step0 _step1 ## Math.1-treat0 0 -1 ## Math.1-treat1 0 1 ## Math.2-treat0 0 -1 ## Math.2-treat1 0 1 ## Math.3-treat0 0 -1 ## Math.3-treat1 0 1 ## Math.4-treat0 0 -1 ## Math.4-treat1 0 1 ## Math.5-treat0 0 -1 ## Math.5-treat1 0 1 ## ## , , Math.1:treat0 ## ## _step0 _step1 ## Math.1-treat0 0 -1 ## Math.1-treat1 0 1 ## Math.2-treat0 0 0 ## Math.2-treat1 0 0 ## Math.3-treat0 0 0 ## Math.3-treat1 0 0 ## Math.4-treat0 0 0 ## Math.4-treat1 0 0 ## Math.5-treat0 0 1 ## Math.5-treat1 0 -1 ## ## , , Math.2:treat0 ## ## _step0 _step1 ## Math.1-treat0 0 0 ## Math.1-treat1 0 0 ## Math.2-treat0 0 -1 ## Math.2-treat1 0 1 ## Math.3-treat0 0 0 ## Math.3-treat1 0 0 ## Math.4-treat0 0 0 ## Math.4-treat1 0 0 ## Math.5-treat0 0 1 ## Math.5-treat1 0 -1 ## ## , , Math.3:treat0 ## ## _step0 _step1 ## Math.1-treat0 0 0 ## Math.1-treat1 0 0 ## Math.2-treat0 0 0 ## Math.2-treat1 0 0 ## Math.3-treat0 0 -1 ## Math.3-treat1 0 1 ## Math.4-treat0 0 0 ## Math.4-treat1 0 0 ## Math.5-treat0 0 1 ## Math.5-treat1 0 -1 ## ## , , Math.4:treat0 ## ## _step0 _step1 ## Math.1-treat0 0 0 ## Math.1-treat1 0 0 ## Math.2-treat0 0 0 ## Math.2-treat1 0 0 ## Math.3-treat0 0 0 ## Math.3-treat1 0 0 ## Math.4-treat0 0 -1 ## Math.4-treat1 0 1 ## Math.5-treat0 0 1 ## Math.5-treat1 0 -1 ``` --- # Polytomous Data + Just so you can see the A design matrix for polytomous data + We'll use the partial credit model (by convention, the Rating Scale Model uses a different parameterization) + The item category difficulties - item step difficulties - are allowed to differ across items ```r data(data.sim.mfr) # 100 people 5 items dim(data.sim.mfr) ``` ``` ## [1] 100 5 ``` ```r head(data.sim.mfr) ``` ``` ## [,1] [,2] [,3] [,4] [,5] ## V1 3 3 3 3 1 ## V1.1 2 3 2 3 2 ## V1.2 1 3 0 3 0 ## V1.3 1 3 3 3 0 ## V1.4 0 1 3 2 1 ## V1.5 1 3 1 3 0 ``` ```r apply(as.data.frame(data.sim.mfr), 2, table) ``` ``` ## V1 V2 V3 V4 V5 ## 0 52 18 22 6 64 ## 1 28 17 10 5 21 ## 2 14 13 15 19 11 ## 3 6 52 53 70 4 ``` --- ## Get design matrices for polytomous data + Reversed design matrices because of contrast coding... ```r polyt_des <- designMatrices(resp = data.sim.mfr, modeltype = "PCM") # Creat or have the model create A_poly <- polyt_des$A # poly_mod <- tam.mml(data.sim.mfr) # Different parameterization (item + item*step) poly_mod2 <- tam.mml(data.sim.mfr, irtmodel = "PCM2") ``` --- # In case of curiosity For more see Kim and Wilson (2020) ```r poly_mod$A[1:4, , ] ``` ``` ## , , I1_Cat1 ## ## Category0 Category1 Category2 Category3 ## Item01 0 -1 -1 -1 ## Item02 0 0 0 0 ## Item03 0 0 0 0 ## Item04 0 0 0 0 ## ## , , I1_Cat2 ## ## Category0 Category1 Category2 Category3 ## Item01 0 0 -1 -1 ## Item02 0 0 0 0 ## Item03 0 0 0 0 ## Item04 0 0 0 0 ## ## , , I1_Cat3 ## ## Category0 Category1 Category2 Category3 ## Item01 0 0 0 -1 ## Item02 0 0 0 0 ## Item03 0 0 0 0 ## Item04 0 0 0 0 ## ## , , I2_Cat1 ## ## Category0 Category1 Category2 Category3 ## Item01 0 0 0 0 ## Item02 0 -1 -1 -1 ## Item03 0 0 0 0 ## Item04 0 0 0 0 ## ## , , I2_Cat2 ## ## Category0 Category1 Category2 Category3 ## Item01 0 0 0 0 ## Item02 0 0 -1 -1 ## Item03 0 0 0 0 ## Item04 0 0 0 0 ## ## , , I2_Cat3 ## ## Category0 Category1 Category2 Category3 ## Item01 0 0 0 0 ## Item02 0 0 0 -1 ## Item03 0 0 0 0 ## Item04 0 0 0 0 ## ## , , I3_Cat1 ## ## Category0 Category1 Category2 Category3 ## Item01 0 0 0 0 ## Item02 0 0 0 0 ## Item03 0 -1 -1 -1 ## Item04 0 0 0 0 ## ## , , I3_Cat2 ## ## Category0 Category1 Category2 Category3 ## Item01 0 0 0 0 ## Item02 0 0 0 0 ## Item03 0 0 -1 -1 ## Item04 0 0 0 0 ## ## , , I3_Cat3 ## ## Category0 Category1 Category2 Category3 ## Item01 0 0 0 0 ## Item02 0 0 0 0 ## Item03 0 0 0 -1 ## Item04 0 0 0 0 ## ## , , I4_Cat1 ## ## Category0 Category1 Category2 Category3 ## Item01 0 0 0 0 ## Item02 0 0 0 0 ## Item03 0 0 0 0 ## Item04 0 -1 -1 -1 ## ## , , I4_Cat2 ## ## Category0 Category1 Category2 Category3 ## Item01 0 0 0 0 ## Item02 0 0 0 0 ## Item03 0 0 0 0 ## Item04 0 0 -1 -1 ## ## , , I4_Cat3 ## ## Category0 Category1 Category2 Category3 ## Item01 0 0 0 0 ## Item02 0 0 0 0 ## Item03 0 0 0 0 ## Item04 0 0 0 -1 ## ## , , I5_Cat1 ## ## Category0 Category1 Category2 Category3 ## Item01 0 0 0 0 ## Item02 0 0 0 0 ## Item03 0 0 0 0 ## Item04 0 0 0 0 ## ## , , I5_Cat2 ## ## Category0 Category1 Category2 Category3 ## Item01 0 0 0 0 ## Item02 0 0 0 0 ## Item03 0 0 0 0 ## Item04 0 0 0 0 ## ## , , I5_Cat3 ## ## Category0 Category1 Category2 Category3 ## Item01 0 0 0 0 ## Item02 0 0 0 0 ## Item03 0 0 0 0 ## Item04 0 0 0 0 ``` --- # Multidimensional Model + "MIRT" + Closest thing to factor analysis in the Rasch world + Items load onto dimensions - we use a Q matrix + In the case of our `resp` data ... + Correlated factors + Compensatory --- class: center # MIRT Model  --- ## Setting up the multidimensional model ```r # set up the Q matrix - dimensionality assignment Q <- matrix(0, nrow = 40, ncol = 4) Q[1:10 , 1] <- Q[11:20, 2] <- Q[21:30, 3]<- Q[31:40, 4] <- 1 # If you want to run the model # tam.mml(resp, Q = Q, control=list(snodes=500, maxiter=10)) multi_des <- designMatrices(resp = resp, Q = Q, ndim = 4) ``` --- # Design Matrix B ```r # The person side matrix is changed multi_des$B ``` ``` ## , , Dim01 ## ## Cat0 Cat1 ## Math.1 0 1 ## Math.2 0 1 ## Math.3 0 1 ## Math.4 0 1 ## Math.5 0 1 ## Math.6 0 1 ## Math.7 0 1 ## Math.8 0 1 ## Math.9 0 1 ## Math.10 0 1 ## Science.1 0 0 ## Science.2 0 0 ## Science.3 0 0 ## Science.4 0 0 ## Science.5 0 0 ## Science.6 0 0 ## Science.7 0 0 ## Science.8 0 0 ## Science.9 0 0 ## Science.10 0 0 ## ELA.1 0 0 ## ELA.2 0 0 ## ELA.3 0 0 ## ELA.4 0 0 ## ELA.5 0 0 ## ELA.6 0 0 ## ELA.7 0 0 ## ELA.8 0 0 ## ELA.9 0 0 ## ELA.10 0 0 ## WordProb.1 0 0 ## WordProb.2 0 0 ## WordProb.3 0 0 ## WordProb.4 0 0 ## WordProb.5 0 0 ## WordProb.6 0 0 ## WordProb.7 0 0 ## WordProb.8 0 0 ## WordProb.9 0 0 ## WordProb.10 0 0 ## ## , , Dim02 ## ## Cat0 Cat1 ## Math.1 0 0 ## Math.2 0 0 ## Math.3 0 0 ## Math.4 0 0 ## Math.5 0 0 ## Math.6 0 0 ## Math.7 0 0 ## Math.8 0 0 ## Math.9 0 0 ## Math.10 0 0 ## Science.1 0 1 ## Science.2 0 1 ## Science.3 0 1 ## Science.4 0 1 ## Science.5 0 1 ## Science.6 0 1 ## Science.7 0 1 ## Science.8 0 1 ## Science.9 0 1 ## Science.10 0 1 ## ELA.1 0 0 ## ELA.2 0 0 ## ELA.3 0 0 ## ELA.4 0 0 ## ELA.5 0 0 ## ELA.6 0 0 ## ELA.7 0 0 ## ELA.8 0 0 ## ELA.9 0 0 ## ELA.10 0 0 ## WordProb.1 0 0 ## WordProb.2 0 0 ## WordProb.3 0 0 ## WordProb.4 0 0 ## WordProb.5 0 0 ## WordProb.6 0 0 ## WordProb.7 0 0 ## WordProb.8 0 0 ## WordProb.9 0 0 ## WordProb.10 0 0 ## ## , , Dim03 ## ## Cat0 Cat1 ## Math.1 0 0 ## Math.2 0 0 ## Math.3 0 0 ## Math.4 0 0 ## Math.5 0 0 ## Math.6 0 0 ## Math.7 0 0 ## Math.8 0 0 ## Math.9 0 0 ## Math.10 0 0 ## Science.1 0 0 ## Science.2 0 0 ## Science.3 0 0 ## Science.4 0 0 ## Science.5 0 0 ## Science.6 0 0 ## Science.7 0 0 ## Science.8 0 0 ## Science.9 0 0 ## Science.10 0 0 ## ELA.1 0 1 ## ELA.2 0 1 ## ELA.3 0 1 ## ELA.4 0 1 ## ELA.5 0 1 ## ELA.6 0 1 ## ELA.7 0 1 ## ELA.8 0 1 ## ELA.9 0 1 ## ELA.10 0 1 ## WordProb.1 0 0 ## WordProb.2 0 0 ## WordProb.3 0 0 ## WordProb.4 0 0 ## WordProb.5 0 0 ## WordProb.6 0 0 ## WordProb.7 0 0 ## WordProb.8 0 0 ## WordProb.9 0 0 ## WordProb.10 0 0 ## ## , , Dim04 ## ## Cat0 Cat1 ## Math.1 0 0 ## Math.2 0 0 ## Math.3 0 0 ## Math.4 0 0 ## Math.5 0 0 ## Math.6 0 0 ## Math.7 0 0 ## Math.8 0 0 ## Math.9 0 0 ## Math.10 0 0 ## Science.1 0 0 ## Science.2 0 0 ## Science.3 0 0 ## Science.4 0 0 ## Science.5 0 0 ## Science.6 0 0 ## Science.7 0 0 ## Science.8 0 0 ## Science.9 0 0 ## Science.10 0 0 ## ELA.1 0 0 ## ELA.2 0 0 ## ELA.3 0 0 ## ELA.4 0 0 ## ELA.5 0 0 ## ELA.6 0 0 ## ELA.7 0 0 ## ELA.8 0 0 ## ELA.9 0 0 ## ELA.10 0 0 ## WordProb.1 0 1 ## WordProb.2 0 1 ## WordProb.3 0 1 ## WordProb.4 0 1 ## WordProb.5 0 1 ## WordProb.6 0 1 ## WordProb.7 0 1 ## WordProb.8 0 1 ## WordProb.9 0 1 ## WordProb.10 0 1 ``` --- ## Alternative to TAM + `eRm` is a conditional maximum likelihood based package that has special functions devoted to the LLTM + Random Item or Random Item covariate models can be run in `lme4` De Boeck (2008); De Boeck, Bakker, Zwitser, Nivard, Hofman, Tuerlinckx, Partchev, and others (2011) + I think multilevel models can be run in TAM but definitely `lme4` + We can use Bayesian approaches (`MCMCpack` or `mirt`) - `brms` uses `lme4` modeling language to translate code into `Stan` files and uses Stan to model and estimate + Check out papers around Adams, Wilson, and Wang (1997) --- ## `Stan` + Stan allows ultimate flexibility but is complicated to use -- + Fully Bayesian - takes a while but you can really express models as you'd like -- + There is a large and growing R eco system being built around Stan -- + A package called `edStan` built by Daniel Furr (a student of Sophia Rabe-Hasketh) -- + There are case studies in the Stan manual for IRT models in R -- + Can express any model so long as its probabilistically sound - `Stan is a probabilistic programming language` --- ## Example - express the Rasch model in a Bayesian way "hyperprior for persons" `$$\theta \sim N(0, \sigma^2)$$` "hyperprior for items" `$$\delta \sim N(0, 1)$$` "prior for `\(\sigma\)`" `$$sigma \sim exp(.1)$$` "Likelihood" `$${Pr}[y_n = 1] = {logit}^{-1}(\theta - \delta)$$` $$ y \sim bernoulli(Pr[y_n=1])$$ --- # References Adams, R. J., M. Wilson, and W. Wang (1997). "The multidimensional random coefficients multinomial logit model". In: _Applied psychological measurement_ 21.1, pp. 1-23. De Boeck, P. (2008). "Random item IRT models". In: _Psychometrika_ 73.4, pp. 533-559. De Boeck, P., M. Bakker, R. Zwitser, et al. (2011). "The estimation of item response models with the lmer function from the lme4 package in R". In: _Journal of Statistical Software_ 39.12, pp. 1-28. De Boeck, P. and M. Wilson (2004). _Explanatory item response models_. Wiley Online Library. Kennedy, C. A. (2005). "Constructing measurement models for MRCML estimation: A primer for using the BEAR scoring engine". In: _Berkeley Eval Assess Res Cent_. Kim, J. and M. Wilson (2020). "Polytomous item explanatory item response theory models". In: _Educational and Psychological Measurement_ 80.4, pp. 726-755. --- Wang, W. and M. Wilson (2005). "The Rasch testlet model". In: _Applied Psychological Measurement_ 29.2, pp. 126-149.