Limited Dependent Variables in Management Research

A limited dependent variable (LDV) is an outcome or response variable whose value is either restricted to a small number of (usually discrete) values or limited in its range of values. The first type of LDV is commonly called a categorical variable; its value indicates the group or category to which an observation belongs (e.g., male or female). Such categories often represent different choice outcomes, where interest centers on modeling the probability each outcome is selected. An LDV of the second type arises when observations are drawn about a variable whose distribution is truncated, or when some values of a variable are censored, implying that some values are wholly or partially unobserved. Methods such as linear regression are inadequate for obtaining statistically valid inferences in models that involve an LDV. Instead, different methods are needed that can account for the unique statistical characteristics of a given LDV.

Mapping to Quality of Life and Capability Measures in Cataract Surgery Patients: From Cat-PROM5 to EQ-5D-3L, EQ-5D-5L, and ICECAP-O Using Mixture Modelling

Objectives. Cataract is a prevalent and potentially blinding eye condition. Cataract surgery, the only proven treatment for this condition, is a very frequently undertaken procedure. The objective of this analysis was to develop a mapping algorithm that could be used to predict quality of life and capability scores from the Cat-PROM5, a newly developed, validated patient-reported outcome measure for patients undergoing cataract surgery. Methods. We estimated linear models and adjusted limited dependent variable mixture models. Data were taken from the Predict-CAT cohort of up to 1181 patients undergoing cataract surgery at two sites in England. The Cat-PROM5 was mapped to two quality of life measures (EQ-5D-3L and EQ-5D-5L) and one capability measure (ICECAP-O). All patients reported ICECAP-O and one or other of the EQ-5D measures both before and after cataract surgery. Model performance was assessed using likelihood statistics, graphical inspections of model fit, and error measurements. Results. Adjusted limited dependent variable mixture models dominated linear models on all performance criteria. Mixture models offered very good fit. Three component models that allowed component membership to be a function of covariates (age, sex, and diabetic status depending on specification and outcome measure) and which conditioned on covariates offered the best performance in almost all cases. An exception was the EQ-5D-5L post-surgery for which a two-component model was selected. Conclusions. Mapping from Cat-PROM5 to quality of life and capability measures using adjusted limited dependent variable mixture models is feasible, and the estimates can be used to support cost-effectiveness analysis in relation to cataract care. Mixture models performed strongly for both quality of life outcomes and capability outcomes.

The relationship between early life urbanicity and depression in late adulthood: evidence from the Survey of Health, Ageing and Retirement in Europe

ObjectivesWe aimed to study the association of childhood urbanicity with depressive symptoms in late adulthood.Design, setting and participantsWe used linear and logistic regressions to analyse data drawn from 20 400 respondents from the Survey of Health, Ageing and Retirement in Europe, a panel dataset incorporating a representative sample of the 50+ population in 13 European countries.Outcomes and analysisChildhood urbanicity was determined using self-reports of the respondents’ circumstances at age 10, and late-adulthood depression using the EURO-D scale. We conditioned on circumstances early in life as well as later in life, most importantly late-adulthood urbanicity. We estimated the associations using linear regression models and limited dependent variable models.ResultsA pooled regression of both men and women suggested that childhood urbanicity is associated non-monotonically with depression in late adulthood and is particularly apparent for those spending their childhoods in suburban settings. We found that individuals who spend the longest time in their childhood in a suburban home exhibit an average increase in probability of 3.4 (CI 1.1 to 5.7) percentage points in reporting four or more depressive symptoms. The association was robust to the inclusion of a host of household characteristics associated with childhood urbanicity and was independent of current urbanicity and current income. When broken down by gender, we found some evidence of associations between depressive outcomes and urban living for men, and stronger evidence of such associations with urban and suburban living for women who exhibit an increase of 5.6 (CI 2.2 to 9.0) percentage points in reporting four or more depressive symptoms.ConclusionsOur analysis reveals a relationship between childhood urbanicity and depression in late adulthood. The evidence presented on the nature of this relationship is not straightforward but is broadly suggestive of a link, differing by gender, between greater urbanicity and higher levels of depressive symptoms. The life-long nature of this association may potentially inform policy agendas aimed at improving urban and suburban living conditions.