# Adding Hypotheses

# Walkthrough

This vignette introduces the addition of new hypothesis to allow for a power analysis for the Wald, LR, score and gradient test. It is intended for advanced R users. The necessary steps are:

- Function to set up the restricted model
- Function to set up the unrestricted model
- Function to maximize the likelihood of the restricted parameters
- Putting it together

As an example, we will set up a simple hypothesis, testing whether the difficulty of the first item is equal to 0.

Additional templates for hypothesis objects are included in the “hypothesis_templates” vignette.

## Function to set up the restricted model

In our restricted model, we define the A Matrix and the c vector. They formally represent the hypothesis. Then we setup the instructions for mirt to fit the model, which imply to keep the first item difficulty fixed at 0.

    res <- function(altpars, nullpars = NULL) {

        n.items <- length(altpars[[1]]) # we can read off the number of items from the altpars object

        # the A matrix represents the calculations that need to be performed on the item parameters according to the hypothesis. in This case, only the difficulty of the first item needs to be extracted.
        Amat <- c(0, 1, rep(0, (n.items - 1) * 2)) |>
                (function(x) matrix(x, ncol = n.items *
                    2, byrow = TRUE))()
        # the c vector is the value that the item parameters are compared to after transformation by the A matrix. In this case, the difficulty parameter is only compared against 0.
        cvec = 0
        # By specifying a mirt.model, we instruct mirt on how to fit the restricted model. In this case, it is a model where the first difficulty parameter is kept at 0.
        model = mirt::mirt.model(paste("F = 1-",
                n.items, "
                               FIXED = (1, d)
                               START = (1,d,0)"))

        re <- list(n.items = n.items, itemtype = "2PL",
            Amat = Amat, cvec = cvec, model = model)
        return(re)
    }

## Function to set up the unrestricted model

The unrestricted model is a basic 2PL model. Note that we generate a longpars object that represents beta in the multiplication “A * beta = c”.

    unres <- function(altpars) {

       # We first transform the parameters altpars from a list to a vector using the longpars argument. It results from a concatenation of the discrimination and difficulty parameters.
        longpars = pars.long(pars = altpars, itemtype = "2PL")
        # For the unrestricted model, we fit a simple 2PL model. This is specified in mirt using a 1 and the "2PL" itemtype.
        model = 1
        itemtype = "2PL"
        re <- list(parsets = altpars, model = model, itemtype = itemtype, longpars = longpars)

        return(re)
    }

## Function to maximize the likelihood of the restricted parameters

We define a function that calculates the maximum likelihood parameters under the restricted model. Note that this function can be left blank if you don’t want to setup the analytical approach. The sampling-based approach can still be used without this function.

In this specific case, we know that the restricted model parameters can be expected to result in the true parameters for all but the first item. The only parameter in question is the discrimination of the first item parameter. We therefore ask the question: If we set the first item difficulty to 0, what is the most likely item discrimination given the data follows the true parameters.

    maximizeL <- function(hyp) {
        # Hypothesis-specific algorithm to find the
        # maximum likelihood restricted parameter set

      # in this case, the a parameter of the first item is searched for.

        maxlpreload <- function(pars) {
            # returns the density for each response
            # pattern under the model parameters pars

          # setting up all response patterns
            patterns <- as.matrix(expand.grid(lapply(seq_len(length(pars$a)),
                function(x) c(0, 1))))

            pre <- c()
            #calculating the density
            for (i in seq_len(nrow(patterns))) {
                pre[i] <- funs$g(patterns[i, ], pars)
            }

            return(pre)
        }

        maxl <- function(x, pars, pre) {
            # calculates the likelihood of parameters
            # x given model 'pars'

          # setting up all response patterns
            patterns <- as.matrix(expand.grid(lapply(seq_len(length(pars$a)),
                function(x) c(0, 1))))

            # collecting all parameters in a list, inserting x as the first a parameter
            x <- list(a = c(x, pars$a[2:length(pars$a)]),
                d = c(0, pars$d[2:length(pars$d)]))

            res <- c()
            # calculating the likelihoods for each pattern under this parameter set
            for (i in seq_len(nrow(patterns))) {
                px <- pre[i]
                qx <- funs$g(patterns[i, ], x)
                res[i] <- {
                    px * log(qx)
                }
            }
            re <- -sum(res)
        }
        resmod <- hyp$resmod
        unresmod <- hyp$unresmod

        pars <- unresmod$parsets
        # loading the model specific density functions
        funs <- load.functions(unresmod$itemtype)

        # setting some starting value for the optimization
        startval <- pars$a[1]

        # calculating the densities as definied above
        maxlpre <- maxlpreload(pars)

        # finding the a parameter with the highest likelihood
        optpar <- stats::optim(startval, function(x) {
            maxl(x, pars, maxlpre)
        }, method = "BFGS")
        re <- pars
        # saving the resulting item parameters
        re$a <- c(optpar$par[1], pars$a[2:length(pars$a)])
        re$d <- c(0, pars$d[2:length(pars$d)])

        return(re)
    }

## Putting it together

We put the functions defined above in a list.

    h_2PL_basic <- list(res = res, unres = unres, maximizeL = maximizeL)

# Testing the new hypothesis

The new hypothesis can be set up using the h_2PL_basic list object as the type argument of the setup.hypothesis function. We may then calculate the power at arbitrary sample sizes.

    altpars <- list(a = rlnorm(5, sdlog = 0.4), d = rnorm(5))

    altpars$d[1] <- 0.2

    hyp <- setup.hypothesis(type = h_2PL_basic, altpars = altpars)

    res <- irtpwr(hyp = hyp, alpha = 0.05, power = 0.8)
    summary(res)
    #>
    #>  Sample sizes for power = 0.8 (alpha = 0.05):
    #>
    #>  Statistic    N
    #>       Wald 1330
    #>         LR 1179
    #>      Score 1181
    #>   Gradient 1176
    #>
    #> Method: Analytical