[R] Create a vector from another vector

[Doran, Harold]

Hi Duncan

Here is a bit more detail, this is a bit tough to explain, sorry for not
being clear. Ordering is not important because the vector I am creating
is used as a sufficient statistic in an optimization routine to get some
MLEs. So, any combination of the vector that sums to X is OK. But, the
condition that x2[i] <= x[i] must be maintained. So, the example below
would not work because x2[1] > x[1] as you note below.

> I don't think it's really clear what you mean by "ordering is 
> not important".  Would
> 
> x2 <- c(6,5,2,4,2)
> be acceptable (a re-ordering of your first two examples), 
> even though x2[1] > x1[1]?

To be concrete, the following is the optimization function. This is a
psychometric problem where the goal is to get the MLE for a test taker
conditional on their response pattern (i.e., number of points on the
test) and the item parameters.

pcm.max3 <- function(score, d){
    pcm <- function(theta, d, score)
exp(sum(theta-d[1:score]))/sum(exp(cumsum(theta-d)))
    opt <- function(theta) -sum(log(mapply(pcm, d, theta = theta, score=
score )))
    start_val <- log(sum(score-1)/(length(score-1)/sum(score-1)))
    out <- optim(start_val, opt, method = "BFGS", hessian = TRUE)
    cat('theta is about', round(out$par, 2), ', se',
1/sqrt(out$hes),'\n')
  } 

Suppose we have a three item test. I store the item parameters in a list
as

items <- list(c(0,.5,1), c(0,1), c(0, -1, .5, 1))

We can get the total possible number correct as

(x <- sapply(items, length))
[1] 3 2 4

But, you cannot actually get the MLE for this because the likelihood is
unbounded in this case. 

So, let's say the student scored in the following categories for each
item:

x2 <- c(3,1,4)

By x2[i] <= x[i], I mean that there are 3 possible categories for item 1
above. So, a student can only score in categories 1,2 or 3. He cannot
score in category 4. This is why the condition that x2[i] <= x[i] is
critical. 

But, because total score is a sufficient statistic, (i.e., "ordering is
not important") we could either vector in the function pcm. 

x3 <- c(3,2,3)

Using the function 

pcm.max3(x2, items)
pcm.max3(x3, items)

Gives the same MLE.

But, the vector 

X_bad <- c(4,1,3)

Would not work. You can see that the elements of this vector actually
serve as indices denoting which category a test taker scored in for each
item in the list "items"

I hope this is helpful and appreciate your time.

Harold


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