162 Chapter 5 different local groups (between term). We estimated these terms using a within-between random effects model (REWB), sometimes designated as an “hybrid” model [35, 36]. The within term uses the yearly variation in the number of psychologists to which the GP is exposed to through her local group to measure the effects on antidepressant prescription for this GP overtime. The within term is free from omitted time-invariant bias, as in the statistically equivalent fixed-effects model [35, 48]. Because the identification is based on observing the same GP under different levels of psychotherapy supply, the within term is purged of bias by GP characteristics that are likely constant during the 4-year study period (e.g. education and preferences). It is also free from the influence of time invariant local group characteristics that are eliminated as addictive effects captured by the nested structure of GPs into local groups [49]. In addition to the within term, the REWB model offers the flexibility to include random effects and other time-invariant predictors, as noted by Mundlak (1978) [50]. In a fixed effects model, time-invariant GP variables would be absorbed by the GP fixed-effect. This would prevent investigating how the variation in the average level of psychologists between GPs in local groups is related to their antidepressant prescription, which we can do with the REWB. Unlikely the within term, the between GPs term is not free from time-invariant bias, and it is likely confounded by GP/local groups unobserved variables linked to both psychologist supply and antidepressants prescription. The economic interpretation of the between term is interesting precisely because it proxies a range of unmeasured characteristics and processes shaping the association observed. By comparing the between and within terms we can conclude about the relevance of these unmeasured characteristics and processes. If the terms are similar in magnitude, then the variation within GP is the most relevant. On the other hand, a larger between term should flag the need to understand and target the unobserved GP and local groups’ determinants of depression treatment which are driving the association. Examples of factors explaining why certain local groups have lower(higher) psychologists supply and higher(lower) antidepressant therapy include the level of mental health services integration, local referral and treatment protocols, and GPs awareness and literacy on the topic. Statistical model The key aspect in estimating the REWB model lies in the inclusion of the mean of the variable of interest as an additional explanatory variable ( ¯Pl ). This is often described as the Mundlak approach [50] and is closely related to the correlated random effects model [49]. Following Bell and Jones (2019) [35] we further included in the model the demeaned explanatory variable (Plt − ¯Pl) instead of its raw time-varying form ( Plt ). While the models themselves would be equivalent, the use of the demeaned explanatory variable allows for the interpretation of the between term as not being controlled for the raw value of Plt . The REWB model is specified as:
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