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Okay, I'm a stats newbie so I'll try to be as specific and clear as possible.

I have a set of predictor variables (2 predictor variables) and a set of response variables (7 response variables). I am thus looking to conduct a canonical correlation analysis to investigate the amount of variance in the response variables that is explained by the predictor variables (please let me know if this is not described accurately!).

First off, I am assessing if any of my variables should be transformed/standardised in some way.

Three of my response variables are percentages, the rest of the variables (response and predictors) are continuous.

I am guessing I need to transform the percentage data. However, I am aware that arcsine transformations and such like are now not considered useful. If so, what is the best way to deal with these?

Also, my data appears severely non-normal (assessed using qq plots and shapiro-wilk tests). However, since I have a sample of more than 300 independent observations, can I assume normality via the Central Limit Theorem?

Please let me know if there's anything I can provide to make things clearer in answering my question.

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As far as I know, data transformation depends on the underlying distributions rather than on the data types. Plotting variables versus each other would tell you whether the relationships are approximately linear and whether the variance is constant throughout the range of the variables.

In this situation, I would give it a try without transformation and use diagnostics to know if a transformation is needed. Plotting the canonical variables versus each other will already tell you a lot about the potential problems. You can also have a look at this question and Regression Modeling Strategies for more guidelines.

About the last point, the Central Limit Theorem applies to the mean of a sample. Sampling deeper and deeper does not make the distribution more Gaussian. However, the estimated mean will be approximately Gaussian, centered around the true mean of the distribution. In short, having a large sample does not solve the issue.

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