and have ignored how the distributions and the relationships among them change over time.2
The top box in the figure, labeled “ET: All Students—True Eligibility (Unobserved),” represents the distribution of all enrolled students by their true eligibility status, including those who are eligible for free meals using program rules described below; those who are eligible for reduced-price meals using those rules; and all other students, who are eligible only for full-price meals. As noted, this distribution is not observed. The process by which students are identified and approved as being eligible for free or reduced-price meals is known as certification.3 Students who are found to be eligible through the certification process become approved students.
The second box in the figure, labeled “CT: Approved Students—Truth (Unobserved),” represents the distribution of all enrolled students according to a certification process with no errors. Some students who are eligible for free or reduced-price meals decline to participate in the certification process. All students who either do not apply or are not directly certified for free meals are eligible only for full-price meals, as are those students who apply but are found to be ineligible for free or reduced-price meals. The distribution, CT, is not observed. The number of students in the free category of CT will be less than or equal to the number in the free category of ET by the number of students who were not directly certified and who did not apply for benefits. Enrolled students who were not directly certified and did not apply for benefits will be in the full-price category of CT. Likewise, students in the reduced-price category of ET who did not apply will be in the full-price category of CT.
The first oval (and third item) in the figure, labeled “CO: Approved Students—Observed,” represents the distribution of enrolled students into categories of approved for free meals, approved for reduced-price meals, and eligible for full-price meals, in which the categories of approved for free or reduced-price meals are as determined by the actual operation of the certification process and maintained in school records. The certification process is described below. The difference between CT and CO is due to misclassification of students (errors) during the certification process. For example, some students who are eligible for free meals may have been approved for reduced-price meals.
2 As discussed below, a student paying full price at the beginning of the school year can be approved for free meals later in the year if, for example, the family’s income falls. Once approved, the student can continue to receive free meals for the remainder of the year (and up to 30 days into the next year until a new eligibility determination is made), even if the family’s income rises above the eligibility threshold for free meals.
3 The certification process encompasses both direct certification by comparison of student enrollment lists with state and local lists of participants in several means-tested programs and the solicitation, submission, and review of applications.