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I'm using data (secondary analysis) to conduct an exploratory analysis. I have a sample size of $n = 67$, and the exploratory analysis includes two independent variables and four dependent variables. I am assessing how each of the two independent variables correlates (partial correlations) with the four dependent variables with and without control variables (two variables are controlled). Next, I am conducting linear regression on the relations that change with and without controls. Since my sample size is small, what assumptions am I violating by conducting this exploratory analysis? I'm asking so I can mention these in my paper as limitations.

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  • $\begingroup$ I think we'd be generally better off talking ideal conditions rather than assumptions. Whether ideal conditions hold is often a matter of degree not kind. The biggest danger with a sample of this size is overfitting I am not clear whether you have controlled as well as (other) independent variables, but I would be wary of even two-predictor models with a sample this small, and I do appreciate that you are limited by what is available. $\endgroup$
    – Nick Cox
    Commented Apr 4, 2023 at 10:53

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You can only know which assumptions are violated by your model by, first, fitting your linear models, and then, checking whether the model assumptions of linear regression are satisfied for each of your models. The common mistake is that some researchers check only the distribution of single variables (whether the relevant variables are normally distributed or not). However, what is essential is to check whether the relationships between the dependent and independent variables, on which the model is constructed, are linear in nature. In other words, you should check the MODEL assumptions, not the variable assumptions. You can look at these two sources for detailed explanations:

Pek, J., Wong, O., & Wong, A. C. M. (2018). How to address non-normality: A taxonomy of approaches, reviewed, and illustrated. Frontiers in Psychology, 9, 1–17. https://doi.org/10.3389/fpsyg.2018.02104

Osborne, J. W., & Waters, E. (2002). Four assumptions of multiple regression that researchers should always test. Practical Assessment, Research and Evaluation, 8(2), 1–5. https://doi.org/doi.org/10.7275/r222-hv23

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