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I want to demonstrate a possible association between a dichotomous independent variable and a continuous dependent variable. Therefore, I wanted to use a linear regression analysis. However, the dependent variable is not normally distributed, while normality is an assumption of linear regression analysis. The other assumptions are met. How can I solve this problem or which other test can I use for this?

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You may transform the variable in several ways, in order to reduce skewness. For instance, you may take the log, square- or cube root of the variable. Which transformation will yield a most normal-like distribution depends on the nature of your data.

This is a useful article that explains some different approaches: https://medium.com/@TheDataGyan/day-8-data-transformation-skewness-normalization-and-much-more-4c144d370e55

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Linear regression works just fine for this. Inferences are valid as long as the sample size is large enough, regardless of the distribution of the variable (or its residual), due to the Central Limit Theorem. n>30 is one rule of thumb I am familiar with wrt what is large enough. In this instance, the analysis amounts to a difference in means so it is not necessary to frame it as linear regression, although it may be convenient to run it as such.

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I meant the assumption of normal distribution of residuals, not of the dependent variable. The residuals are not normally distributed. How can I solve this? Can I still use the linear regression analysis if I want to predict a possible association between alcoholism (yes/no) and duration of hospitalization (in days) after surgery?

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