I am investigating the effect of certain cognitions/attitudes on sexual recidivism.
Sexual recidivism is the binary outcome variable (0 = no / 1 = yes). Originally the sex offenders could score 0 or 1 on eight different attitudes (file-information was scored). Based on theory I divided the attitudes in two categories and made new variables which were the sum of the corresponding attitudes. Therefore my predictors are now 'offense supportive attitudes' (OSA) and ' post-hoc excuse making' (EM). OSA can be 0-1-2-3 and EM can be 0-1-2-3-4-5, so I figured they are now ordinal predictors.
I have checked for multicollinearity and linearity of the logit; both assumptions have been met. Unfortunately, I don't know how to check the assumption of independence of errors (overdispersion). I think the durbin-watson test only analyses linear regression?
You can check graphically, which is perhaps the easiest and the most intuitive. Compute residual for each observation, i.e. y_i - inv_logit(X_i B) where y_i is your observation (0 or 1), and X_i is the covariate vector for the ith observation, and B is the coefficient vector. Now, your covariate is multidimensional (at least you have two). Check the density of the residual at each level of each covariate. In other words, check marginally. Now to check jointly is a little tricky. Lucky for you, you only have two covariates. You can plot the density of the residual on a 2 dimensional grid. Visually , you can check if the distribution of the residual is independent of the covariates very easily.
If you estimate your model via GENLIN (with logit link and binomial error distribution), you can use the SAVE sub-command to save a couple different types of residuals to the working data file. See the link below for details. HTH.
https://www.ibm.com/support/knowledgecenter/en/SSLVMB_25.0.0/statistics_reference_project_ddita/spss/advanced/syn_genlin_save.html
Apparent non - independence can be produced by several things.
1: non linearity of the relation between the logit and the predictor- you have ruled this out.
2 Missing important predictor variables - the only real way around this is good theory for including what is important.
3 Missing important 'structure' eg the observations are dependent over time (that is what a DW test is about) ; dependent over space (there is a Cliif-Ord test for logit models) ; and some sort of context (eg prison , or neighbourhood) which can be handled by multilevel model- your question does not describe this as an issue.
I do not think that residual analysis is really going to help with non-independence in your case. What it does help with is identifying outliers and overly influential observations - see for example:
Article Death and deprivation: an exploratory analysis of deaths in ...
Thank you all!
I have found underdispersion in all of my data, which is probably the result of what Kelvyn Jones said in #2. My independent variables are not good predictors, so the log reg model predicts all offenders to not reoffend. Therefore everyone who does reoffend are the biggest errors, causing autocorrelation between my errors.
Anyhow I do find it difficult to decide in which section of my thesis I should describe this problem. The assumption checks are usually in the methods section, but the explanation in this case is a little bit more complicated and inevitably includes the results (OSA en EM are not predictive, it is not a good model, the theory fails).
Underdispersion , that is less variation than expected from binomial random variable, is usually due to some underlying structure that is not being modelled properly (time , context ....) whereas apparent over dispersion can result from omitted variables.
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