categories is large. Suppose that an achievement growth model includes five control variables—prior achievement, parental education, parental income, race, and special education status. If each variable is split into five different values, the total number of reporting categories would equal 55 = 3,125.9 Clearly, it would be advantageous to find some creative ways of presenting these data without producing information overload.
How should the total performance indicator be interpreted? One interpretation is that it captures the effect at some past date of enrolling one additional student in a school, holding all school-level factors constant, including the composition of the student group that attends the school. If these characteristics are relatively stable from year to year, the total school performance indicator could provide reliable information on the future performance of schools and thus be very helpful to students and parents who are in the process of choosing a neighborhood to live in and/or a school to attend. In short, the total performance indicator is appropriate for purposes of informing school choice, but it is not the most appropriate indicator for holding schools accountable for their performance because it fails to exclude components of school performance that are external to the school.
For an indicator that serves the accountability function, let us turn to the second level of the value-added model. This equation captures the school-level factors that contribute to growth in student achievement. A simple level-two equation is:
ηs = δ1 Externals + δ2 Internals + us (3)
where ηs is the school effect (and total school performance indicator) for school s from the level-one equation; Externals and Internals represent all observed school-level characteristics assumed to determine growth in student achievement plus a constant term; us is the unobserved determinant of total school performance; and δ1 and δ2 are parameters that must be estimated. To be consistent with the level-one equation, it is assumed that the internal school characteristics all have mean zero in the benchmark year.
The distinction between the external and internal variables is crucial for the purpose of measuring school performance. Externals includes all observed school-level characteristics that could be considered external to the school (plus a constant term), including neighborhood and community characteristics and aggregate student characteristics such as the average socioeconomic status of all students in a school. Internals includes all observed school-level characteristics that could be considered internal to the school, principally school policies and inputs.