# Using Predictive Value Confidence Intervals to "Predict" Outcomes

Here's the quick version:

Say I have a confusion matrix with the following data based on a proficiency cut score on a pretest and outcomes on (passing/failing) a class. The cut score was determined using an ROC curve. The confusion matrix below represents the TP, FP, FN, and TN at the cut score identified using the ROC curve. (The data below is adjusted for easy conversation.)

TP: 550 FP: 200
FN: 280 TN: 825

Here are the stats based on the data above:

Sensitivity: 66.27%
1 - Specificity: 19.51%
Accuracy: 74.12%
Positive Predictive Value (PPV; Bayes' Thm.): 73.33% with 95% CI (70.64%, 75.86%)
1 - Negative Predictive Value (1 - NPV): 25.34% with 95% CI (23.49%, 27.28%)

Is it acceptable to use the minimum and maximum values in the PPV confidence interval with minimum and maximum values in the 1 - NPV confidence interval to "forecast/expect" a range of values for the probability that a group of students in a class will pass given the results of the pretest? The goal is to have an idea of how many students will pass/fail the class based on students' pretest scores.

For example, 75% of students on the first day of class scored "proficient" on the pretest, based on our proficiency cut score. If the past data (as shown in the confusion matrix) shows that the probability of passing the class given that a student is proficient on the pretest (PPV) is 73.33% with 95% CI (70.64%, 75.86%) and that the probability of passing the class given that a student is not proficient on the pretest (1 - NPV) is 25.34% with 95% CI (23.49%, 27.28%), could we use the min and max of both confidence intervals to say:

0.7064 * 75 + 0.2349 * 25 [the min on both CIs] = 58.9%
0.7586 * 75 + 0.2728 * 25 [the max on both CIs] = 63.7%

We expect to see 58.9% to 63.7% of the current group of students pass the class?

This is effectively using Bayes' Theorem/conditional probability using pretest and class pass rate data gathered in the past to answer our question. Is there an issue with doing this using the confidence intervals instead of the actual conditional probabilities 73.33% and 25.34%? Of course, "predicting"/expecting here is used loosely. We know the real world always has something interesting in store. (Side note: We are adding a prerequisite course in the future to hopefully aid in increasing pass rates over time.) It'll be interesting to see how class outcomes actually change over time once data becomes available for students who take the prerequisite course before this course, comparing prerequisite vs. no prerequisite group data, scores from both groups on the pretest and pass/fail rates of the class, etc., but for now...

Is it acceptable to do the calculations above and say that, according to past data, we'd expect to see 58.9% to 63.7% of the current group of students pass the class given this class's performance on the pretest?