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I a total of 8 Independent Variables (4 continuous - Scales outcomes - and 4 categorical - Demographic and other personality questions) and 2 Dependent Variables (1 continuous and 1 count). The DVs involve data from an Iterated Prisoners' Dilemma - 1)participants mean consumption per game (continuous variable) and 2) the number of cooperations participants played during each game (count variable).

I have tested the DVs for Poisson distributions, but none of them is. Residuals of the continuous variables are not normally distributed.

My main aim is to analyse the main effect of the IVs on the 2 DVs. I am also interested in testing the possible interactions between two of the IVs and their subsequent effects on the outcome variables. What is the best statistical model to use considering that all the variables are not normally distributed? Or at least, what are the first steps I should take knowing that the data is not normally distributed?

I have been looking into Generalized linear models, but how can I run any model if I don't know the exact distribution of my data? Should I try to normalize my data?

Count DV: enter image description here

Continuous DV: enter image description here Thank you

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  • $\begingroup$ It would be great if you could expand on the problem. What are your DVs and IVs? Can you provide scatter plots so that others can see how the DVs are related to IVs? $\endgroup$ – jcken Jun 12 '20 at 10:57
  • $\begingroup$ @jcken thanks for the answer, I edited the question by adding more details regarding the experiment. I'm not sure what are the scatterplots you are interested in but I added a couple of Histograms to show the distribution of the DVs. $\endgroup$ – John Walk Jun 12 '20 at 11:13
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    $\begingroup$ The responses (you say DVs) don't have to follow reference distributions closely for associated models to be useful. Thus I would certainly start with a Poisson model for # of cooperations and possibly start with a gamma model and/or a logarithmic link for mean consumption. A shared flavour for both example responses is that values can't be negative and that mean responses given predictors will be positive. Such features lend themselves to generalized linear models. $\endgroup$ – Nick Cox Jun 12 '20 at 11:19
  • $\begingroup$ @JohnWalk scatter plots show the relation between the IV and DV; if you want to model this relationship it would be useful to understand what it looks like! E.g. it could show an approximately linear or polynomial relationship between IV and DV $\endgroup$ – jcken Jun 12 '20 at 11:36
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Ok let us try to answer the question depending on the nature of your dependent/independent variables:

  • For continuous data: unless you're fitting a classifier such as LDA or QDA, do not be worried about normality. Logistic regression works well with non-normal data and as far as I know, SVM is not affected by it either. The only problem would be if you are doing linear regression, but even then, normality rules only apply to the model residuals, not to your dependent/independent variables. Lack of normality becomes an issue when you want to get CI for your beta estimates or hypothesis testing (which rely on normality of residuals).

  • For count data. I don't think there is a problem using any of the above models either. I routinely use count data in hierarchical clustering for my RNA-expression analysis, t-tests etc., but there may be models that are affected by this. Good luck!

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  • $\begingroup$ Thank you for your insights. A couple of questions: 1) isn't Logistic regression used only for categorical response variables with only 2 values? and 2) what you do mean when you say that the lack of normality becomes an issue when you do hypothesis testing? Thanks again. $\endgroup$ – John Walk Jun 12 '20 at 13:10
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    $\begingroup$ Sure, logistic regression is commonly used for binary responses. That means that you would have to encode your response as 0 or 1 depending on the target value. For instance, I work with drugs of which target responses are continuous, but you can set a cutoff of activity and declare active (1) or inactive (0) depending on whether your activity is above or below this threshold. Normality becomes an issue because in linear regression because the t-statistic used to get the p-value relies on your betas being normal. If your sample size is big, this is less of an issue $\endgroup$ – Ramon Soto Garcia Jun 12 '20 at 13:32
  • $\begingroup$ Thanks for the clarification. That sounds like a valuable process, although, given the nature of my continuous variable I cannot really encode it into a binary variable. $\endgroup$ – John Walk Jun 12 '20 at 15:36
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    $\begingroup$ It's fine, use linear regression or any other model that uses a quantitative output, just remember to check model assumptions and compare performance metrics with test data. Good luck! $\endgroup$ – Ramon Soto Garcia Jun 12 '20 at 15:41

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