Charter Schools: Hope or Hype?

By Jack Buckley; Mark Schneider | Go to book overview

APPENDIX 9.1

The CHOPIT Model

MATHEMATICALLY, THE CHOPIT MODEL is straightforward (although in practice estimation can be time-consuming). To begin with, let

denote the latent self-assessment measure (the parent's true grade for their child's school's parent-teacher relationship, for example). As in the case of a standard ordinal probit model, , where μi = Xiβ. The actual reported self-assessment, yi is a choice of K ordered categories. The latent variable is assumed to be converted to the observed measure via a vector of thresholds, τi such that yi = k if where and . Note that the cutpoints are indexed over the individual observations—the crucial difference from the standard model. These individual-level cutpoints are themselves defined as a function of another vector of covariates Vi and parameters γ. The first cutpoint is defined: , and the subsequent cutpoints are constrained to increase monotonically by assuming the functional form:

As discussed above, the vignettes are used to identify (anchor) the cutpoints. Let θj denote the mean response or evaluation by subjects (parents in the data discussed here) to vignette j. We assume normal variability in i's perception of θj,

, where . is measured using a survey instrument with the same K response options again partitioned by cutpoints so that the categorical response zij + k if These cutpoints are determined using the same coefficient vector γ as in the mean model and the same vector V of covariates. It is this assumption of identical γ's that identifies the model and allows the vignettes to be used to correct DIF at the individual level. The CHOPIT model, once specified, is then estimated via standard maximum-likelihood methods, where the likelihood function maximized is

L(β, σ2, θ, γ | y, z, V, X)

In the King et al. (2003) article, a more general model is presented that includes an individual-level random effect in the mean equation. This is strongly identified only in the context of a latent variable analysis of several simultaneous self-assessment questions. Although it is possible to estimate this variant of the model in which the three dependent variables discussed here are treated as multiple measures of a latent school-community strength factor, we choose not to here because we believe that the

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