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The assumption of linearity simplifies the regression process and allows the calculation of the change in log-odds for any increase other than a unit increment. This provides a useful interpretation for independent variables measured on a continuous scale - a feature of this particular data set. By fitting all the independent variables, particularly the deprivation variables, as continuous it is being assumed that this relationship is linear.
Based on the previous model results, there is reason to question the assumption of linearity for Group_0, the educational attainment variable, and income and access, the sub-indices of deprivation. The results from the preliminary analysis suggest that the relationship between these variables and the imputation probability is non-linear. If the variables are fitted in their present format as continuous, the multivariable model results would not yield useful conclusions. Therefore, based on a quartile design variable analysis, Group_0, income and access are fitted as categorical variables.
Quartile design analysis creates categorical variables with 4 levels using three cutpoints based on the distributional quartiles. Thus the continuous variables are replaced by these (new) 4-level categorical variables. The lowest quartile serves as the reference group; this category is used as a comparison group for all the other groups, and hence results can be interpreted relative to this reference group. For the deprivation model, there is justification in using the sub-indices of multiple deprivation because it shows a more careful pattern of differences that have a better ability to predict non-response. Therefore the Access and Income singular deprivation indices are fitted as categorical variables (the remaining domains are left as continuous). In doing so, it is possible to ascertain which deprivation sub-indices have a comparatively better predictive ability.
The Type III Analysis of Effects (shown above in Table 4.4.1(a) (11 Kb PDF file and Table 4.4.1(b)(11 Kb PDF file)) tests all factors in the model as if each one were the last one included in the model (i.e. this controls for all other effects). This indicates that, with the other indices in the model, the employment and education indices are rendered redundant.
The tables show that the employment and education sub-indices have the least significance, after controlling for other effects. While it is widely acknowledged that unemployment is a good predictor of under-enumeration, a person with a low income is much more likely to be unemployed (although the reverse is not necessarily true). In this respect, employment deprivation is said to be confounded as it is associated with both the outcome variable (imputation probability) and a primary independent variable (low income).
Table 4.4.2 (11 Kb PDF file) shows the how the Other Census model changes when Group_0 is fitted as a categorical variable.
When fitted as a class variable, Group_0 becomes non-significant. Also notice how house_assoc – the variable looking at social tenures not administered by the council – is absent from the model. The reason for this is that the odds ratio of the covariate in this model ends up being very close to 1 which is indicative of no effect[Footnote 1]. Hence the final model is fitted with sin_par_f, NS_SeC_3, no_car and single.
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The odds ratio estimate of 1.035, in the univariate model, drops down to 1.005 in the multivariate model - i.e. with the addition of other variables in the model, the differences between areas with a high number of housing association tenants and those without becomes non-significant.
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