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My data set includes 435 randomly sampled cases; on the DV 406 participants report "no likelihood" of the DV (one score) and 29 participants indicated "some likelihood" of the DV (range: 10 points). Non-parametric tests (chi squared tests) have already been performed on the data between genders related to our IVs.

I am interested in using 5 IVs to predict the presence of the DV (between genders, if possible); however, the sample and residuals are non-normally distributed, and log and square root transformations have not changed the distribution in any meaningful way. I've considered bootstrapping, logistic regression for rare events (the event occurs at a rate in our sample of 6.4%) and case-control design, though I've looked into none extensively.

It may be that multiple regression is possible with this data set (I have tested multicollinearity and it is not an issue), but I am not sure and need assurance. I am a novice and I would appreciate any leads on what types of analyses that will predict the DV.

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    $\begingroup$ It's inevitable that no transformation will change this distribution to normal. 406/435 give a massive spike and whatever you transform that spike to, it's still a spike. Multiple regression is completely inappropriate here given the nature of the response. You might get somewhere with logit regression but even fitting a model with 5 predictors could be a tough call when you have only 29 positive responses. You write as if you have tried multiple regression already, but consider that it will predict values out of range much of the time. $\endgroup$
    – Nick Cox
    Commented Mar 13, 2016 at 19:12

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It appears that you have an ordinal $Y$ with clumping at one value. This is an ideal candidate for semiparametric regression, e.g., proportional odds ordinal logistic model. Full-likelihood semiparametric models can handle extreme ties extremely well, and the inference is the same no matter how $Y$ is transformed as long as the transformation is monotonic.

This site has dealt with this issue a few times. See for example How to model distributions which are not normally distributed .

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If your DV is scored "none" and "some" (which it appears to be, unless I've misunderstood what you wrote) then it is a dichotomy and you should use logistic regression.

However, this will not predict the presence of the DV, it will be about the relationship between the IVs and "some" vs. "none".

Also, since you have only 29 people who said "some", a model with 5 IVs is likely to be overfit. You either need more data or a simpler model.

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