Nked together, providing a partial ranking. The rank-ordered logit accommodates tied

Nked together, providing a partial ranking. The rank-ordered logit accommodates tied rankings (Allison and Christakis 1994:206-8). The likelihood function is an extension of the simpler discrete choice likelihood (equation 3.5), except that Yij is a rank rather than a 0/1 indicator for the chosen alternative, and the model includes an additional term ijk which equals 1 if the ranking of the kth choice is greater than or equal to the ranking of the jth choice, and is zero otherwise. That is,NIH-PA Author Manuscript NIH-PA Author Manuscript NIH-PA Author Manuscript(3.10)In the case where one alternative is ranked “first,” and all others are tied for “last,” the rankordered logit model simplifies to the discrete choice model for a single choice.4. COMPLICATIONS FOR CPI-455 biological activity Thonzonium (bromide)MedChemExpress Thonzonium (bromide) actual CHOICE DATAIn this section we discuss features of residential choice data that require modifications of standard discrete choice models. These include the aggregation of alternatives, violations of the independence from irrelevant alternatives assumption, unfeasibly large choice sets, choice based sampling, and the treatment of a respondent’s current place of residence. We discuss how each of these problems can be handled within the choice model. Aggregation of Alternatives In actual residential choice, individuals select among houses units, apartments, or even rooms. Typically, however, we observe choices of aggregate units such as Census tracts. When the units that individuals actually choose are not the ones that we observe, it is necessary to modify the choice model to take account of the differential size and variability of the aggregate units (Ben-Akiva and Lerman 1985, Chapter 9). Denote by L the actual choice set (e.g., housing units). Pi(l) is the probability that the ith decision-maker choosing the lth housing unit (where lL). The L housing units are partitioned into J non-overlapping aggregates (e.g., Census tracts denoted as Cj) such that the total number of units in the jth aggregate, . The probability of choosing the jth tract is equal to the sum of the probabilities that the respondent chooses each of the tract’s constituent housing units. Thus, the probability that the chooser selects a housing unit located in the jth parcel is , and the utility associated with the jth aggregate is the average utility of all its housing units:(4.1)An implication of this result is that, all else equal, aggregate utilities and choice probabilities vary with the size of the aggregate units. Census tracts with more housing units will, ceteris paribus, be chosen more often than those with fewer. Further, within tracts, individual dwelling units may be heterogeneous in their desirability. Thus the estimated effects of other measured characteristics of tracts may be distorted by their correlations with tract size and variability. To take these complications into account we modify the general choice model in Equation 3.4 as follows:Sociol Methodol. Author manuscript; available in PMC 2013 March 08.Bruch and MarePage(4.2)NIH-PA Author Manuscript NIH-PA Author Manuscript NIH-PA Author Manuscriptwhere Uij is the average utility of the housing units within the jth Census tract, Mj is the number of housing units in the jth Census tract, Bj measures the variation in the utilities of housing units within the jth Census tract, and 1,2 are positive scaling coefficients (BenAkiva and Lerman 1993). Estimates of the Mj are typically available from census data and thus can be straightforwardly.Nked together, providing a partial ranking. The rank-ordered logit accommodates tied rankings (Allison and Christakis 1994:206-8). The likelihood function is an extension of the simpler discrete choice likelihood (equation 3.5), except that Yij is a rank rather than a 0/1 indicator for the chosen alternative, and the model includes an additional term ijk which equals 1 if the ranking of the kth choice is greater than or equal to the ranking of the jth choice, and is zero otherwise. That is,NIH-PA Author Manuscript NIH-PA Author Manuscript NIH-PA Author Manuscript(3.10)In the case where one alternative is ranked “first,” and all others are tied for “last,” the rankordered logit model simplifies to the discrete choice model for a single choice.4. COMPLICATIONS FOR ACTUAL CHOICE DATAIn this section we discuss features of residential choice data that require modifications of standard discrete choice models. These include the aggregation of alternatives, violations of the independence from irrelevant alternatives assumption, unfeasibly large choice sets, choice based sampling, and the treatment of a respondent’s current place of residence. We discuss how each of these problems can be handled within the choice model. Aggregation of Alternatives In actual residential choice, individuals select among houses units, apartments, or even rooms. Typically, however, we observe choices of aggregate units such as Census tracts. When the units that individuals actually choose are not the ones that we observe, it is necessary to modify the choice model to take account of the differential size and variability of the aggregate units (Ben-Akiva and Lerman 1985, Chapter 9). Denote by L the actual choice set (e.g., housing units). Pi(l) is the probability that the ith decision-maker choosing the lth housing unit (where lL). The L housing units are partitioned into J non-overlapping aggregates (e.g., Census tracts denoted as Cj) such that the total number of units in the jth aggregate, . The probability of choosing the jth tract is equal to the sum of the probabilities that the respondent chooses each of the tract’s constituent housing units. Thus, the probability that the chooser selects a housing unit located in the jth parcel is , and the utility associated with the jth aggregate is the average utility of all its housing units:(4.1)An implication of this result is that, all else equal, aggregate utilities and choice probabilities vary with the size of the aggregate units. Census tracts with more housing units will, ceteris paribus, be chosen more often than those with fewer. Further, within tracts, individual dwelling units may be heterogeneous in their desirability. Thus the estimated effects of other measured characteristics of tracts may be distorted by their correlations with tract size and variability. To take these complications into account we modify the general choice model in Equation 3.4 as follows:Sociol Methodol. Author manuscript; available in PMC 2013 March 08.Bruch and MarePage(4.2)NIH-PA Author Manuscript NIH-PA Author Manuscript NIH-PA Author Manuscriptwhere Uij is the average utility of the housing units within the jth Census tract, Mj is the number of housing units in the jth Census tract, Bj measures the variation in the utilities of housing units within the jth Census tract, and 1,2 are positive scaling coefficients (BenAkiva and Lerman 1993). Estimates of the Mj are typically available from census data and thus can be straightforwardly.