Appendix F The models

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The logistic regression models estimate the odds ratio of event Y conditional on a given set of characteristics X as follows:

Odds ratio = prob(Y|X)/(1- prob(Y|X) = exp(Xβ) ,

where β is the vector of parameter of interest.

With a simple transformation, the probability of event Y given X can be recovered easily:

Prob(Y|X)= 1/(1+exp(Xβ))

The probability of spending 30% or more of household income on shelter for at least one year between 2002 and 2004 (model 1) and the probability of spending 30% or more of household income on shelter all three years (model 2) can be derived based on characteristics X. To look at the effect of a particular variable on these two probabilities, all other variables are set to their mean values. Since all explanatory variables are categorical, their mean values would be equivalent to their share in the weighted sample.

For instance, to hold the effect of disability constant when looking at city effect, X_disabled is set to 0.365 and X_non-disabled is set to 0.635 to reflect the disability composition of the sample. The same technique is applied to all other variables, except for X_city where the city of interest will be given a 100% weight and all other cities will not be given any weights. This provides the same basis for doing cross city comparisons.

Each model was run with a number of different combinations of variables and the estimates were relatively stable. The results of the most general models are presented in this report. Table 7 presents the estimated probabilities for these models.

The variable "total household income" was not included in the models because it is part of the calculation of the characteristic of interest i.e., the STIR.