An illustration with complete psychometric data

It is time to try out these ideas with (simulated) psychometric data. First, we examine the situation where both tests, the reference test and the new one, have been administered to the same respondents. This is not a typical situation because, in practice, we usually deal with data collected in an incomplete design where no candidates take both tests. In this situation, if we can estimate the parameters of the items, we can use the IRT model to fill in the missing data. Obviously, without observations we no longer know whether an individual has actually passed the reference test or not (see Bolsinova and Maris (2016)). We can, however, calculate sensitivity and specificity even in that situation.

We use the verbal aggression data with the first 14 items assumed to be the reference test, and the remaining 10 items to be the target test. The pass-fail score on the reference test is set to 10. All this is quite arbitrary, of course – the test is not intended to measure educational achievement, and we are certainly not handing out diplomas for verbal aggressiveness.

As a start, we produce a scatter plot of the scores:

The proportion of respondents who pass the reference test gradually increases with the score on the new test; below, it is plotted in red:

prob_pass = tp = rep(0,29)
for (i in seq_along(0:28)){
  prob_pass[i] = sum(ts$ref_test[ts$new_test==i]>=10) / sum(ts$new_test==i)
  tp[i] = sum(ts$ref_test[ts$new_test>=i]>=10) / sum(ts$new_test>=i)
}
plot(0:28, prob_pass, ylab="Proportion passing the reference test", xlab="New test score", ylim=c(0,1), type = "o", col="red",bty='l')
lines(0:28, tp, type="o", lty=2, col="blue")

The blue dashed line is the percentage of true positives. Among other things, it shows that if the new pass-fail score is zero and everybody passes, about 15 percent will be qualified. If this is high enough we can save ourselves the trouble of administering a new test.

Since the same respondents have taken both tests, it is straightforward to determine the sensitivity and specificity of the new test for each pass-fail score:

specificity = sensitivity = rep(0, 29)
for (i in seq_along(0:28)){
  sensitivity[i] = sum(ts$ref_test[ts$new_test>=i]>=10)/sum(ts$ref_test>=10)
  specificity[i] = sum(ts$ref_test[ts$new_test<i]<10)/(sum(ts$ref_test[ts$new_test<i]<10)+sum(ts$ref_test[ts$new_test>=i]<10)) 
}
plot(0:28, sensitivity, ylab="sensitivity/specificity", xlab="new test score", ylim=c(0,1), type = "o", col="red",bty='l')
lines(0:28, specificity, col="green", type="o")

If sensitivity and specificity are deemed equally important, the plot suggests 14 or 15 as appropriate pass-fail scores. Higher values would make the specificity go up (i.e., less false positives) at the expense of sensitivity; lower values would do the reverse. From the plots made earlier, we see that about half of the persons with a score equal to the passing score would actually pass the reference test.

As candidates are classified into two classes, we can readily show the trade-off between sensitivity and specificity with the ROC curve:

plot(1-specificity, sensitivity, col="green", xlim=c(0,1), ylim=c(0,1), type="l",bty='l')
textxy(1-specificity, sensitivity, as.character(0:28), cex=0.7, offset = 0)
abline(0,1,lty=2, col="grey")

Other packages and functions may produce prettier plots — but can they do it without complete data?

Incomplete Data

It is more involved, but practically more useful, to calculate sensitivity and specificity even though the two tests have not been both administered to the same persons. We first simulate a complete data set with responses to 60 items. Then, we select the first 500 persons and 40 items to represent the reference test, and the remaining 200 persons and items 21 through 60 to represent the target test. Note that the two respondent groups are sampled from two different ability populations and cannot be considered as equivalent.

We create a new dexter project, and we add the two booklets. We assume that the pass-fail limit for the reference test is 23 points.

The probability to pass

As a start, we use the function probability_to_pass to estimate, for each score on the target test, the proportion of persons who pass the reference test. We show this as a plot.

ou_e = probability_to_pass(db_sm, ref_items = ref_items, 
                                  design = tibble(booklet_id="bk2", item_id=target_items), 
                                  pass_fail = pass_fail)
plot(ou_e, what="equating")

Let the scores on the reference and new test be written, respectively, as \(R_+\) and \(N_+\), and let \(\mathbf{x}\) denote the available data. We calculate \(P(R_+ \geq c|N_+ = s, \mathbf{x})\), the probability that a person with a score of \(s\) on the target test will pass. Simply stated, we calculate this probability by averaging over what could have happened on the basis of the available data. A technical appendix explains a bit more.

A little extra in the plot are the horizontal red lines. These refer to the traditional ability-based pass-fail score obtained as follows: First, estimate an ability for each score on the reference and the new test. Then, determine the score on the new test corresponding to the smallest ability equal to or larger than the ability corresponding to the pass-fail score on the reference test. This will then be the pass-fail score in the new test. The horizontal lines are actually the probabilities of pass-fail scores thus obtained. While this is the usual procedure to equate a pass-fail score, we believe the procedure that we propose here is more informative.

Sensitivity and specificity

Having the probability to pass and the proportion of respondents, \(P(N_+ = s)\), for all possible scores \(s\), we can calculate sensitivity and specificity and plot them for each possible pass-fail score on the target test:

plot(ou_e, what='sens/spec')

If sensitivity and specificity are equally important, the plot suggests 23 as a pass-fail score on the new test.

Conclusion

An educational test equipped with a single pass-fail score is a binary test aimed at classifying candidates into those who pass, and those who fail. In fields like medical testing or machine learning, sensitivity and specificity are typically considered to evaluate the performance of such a test. Essentially, what we have tried to do is apply this logic to educational tests. Item response theory was used to deal with incomplete data as we typically see in educational measurement. Presumably, medical testing faces similar complications, but this is beyond our immediate scope.

In closing, we should mention that this is a first attempt to equate tests in a way that does not involve unobservable quantities; it uses item response theory to fill in missing data, no more. We would like to test the functions more thoroughly before attempting an extension to classification into more than two classes. Feedback from users is very welcome.

Technical Appendix

Let us consider how we determine the probability to pass the reference test given a score \(N_+=s\) on the target test. This probability can be defined as:
\[ P(R_+ \geq c|N_+ = s, \mathbf{x}) = \int_{\mathbf{b},\theta} P(R_+ \geq c|\theta,\mathbf{b}) f(\theta|N_+ = s, \mathbf{b}) f(\mathbf{b}|\mathbf{x}) d\mathbf{b} d\theta, \]

where \(Y_+\) denotes the score on the reference test, with pass-fail score \(c\), \(\theta\) denotes ability and \(\mathbf{b}\) is the vector of item parameters. The integrand has three parts:

• \(f(\mathbf{b}|\mathbf{x})\): The posterior of the item parameters, given all data \(\mathbf{x}\).
• \(f(\theta|N_+ = s, \mathbf{b})\): The posterior of ability given the score on the new test.
• \(P(R_+ \geq c|\theta,\mathbf{b})\): The probability to pass given ability equal to \[ P(R_+ \geq c|\theta,\mathbf{b}) = \sum_{s \geq c} P(R_+ = s |\theta, \mathbf{b}) = \sum_{s \geq c} \frac{\gamma_s(\mathbf{b}) e^{s \theta}} {\sum_h \gamma_h(\mathbf{b}) e^{h \theta}}, \] with \(\gamma_s(\mathbf{b})\) the elementary symmetric function of order s.

To calculate a Monte-Carlo estimate, repeat:

• sample item parameters \(\mathbf{b}^* \sim f(\mathbf{b}|\mathbf{x})\)
• sample a plausible value \(\theta^* \sim f(\theta|N_+ = s, \mathbf{b}^*)\)
• calculate \(P(R_+ \geq c|\theta^*,\mathbf{b}^*)\)

We do this as often as we can afford (currently, five hundred times), and we take the average as an estimate of \(P(R_+ \geq c|\theta^*,\mathbf{b}^*)\).

Sensitivity and specificity are calculated from the score distribution on the new-test and the probability to pass. The following code shows how:

  p_pass_given_s = out$probability_to_pass
  ps = ou_e$pnew
  tp = rev(cumsum(rev(p_pass_given_s*ps))) / rev(cumsum(rev(ps)))
  sensitivity = rev(cumsum(rev(p_pass_given_s*ps))) / tp[1]
  
  tn = rev(cumsum(rev((1-p_pass_given_s)*ps))) / rev(cumsum(rev(ps)))
  specificity =  1 - rev(cumsum(rev((1-p_pass_given_s)*ps))) / tn[1]