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I was wondering what (if any) the benefit is of reporting effect sizes (e.g., eta-squared) after running a regression model in which the dependent variable is z-transformed.

In such a model, the beta estimates of your predictor variables indicate change in z-score per unit of the predictor. This is pretty intuitive in terms of understanding the magnitude of the effect.

Effect sizes are also helpful in determining the magnitude of the effect of a certain variable. For example in terms of explained variance in the model.

I am unsure if it is informative to report standardized effect sizes in addition to reporting z-scores (with standard error).

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This depends to a large extend on your intended audience.

If your intended audience is very familiar with eta square, then by all means report it. As you mentioned this is looking at the same object from a slightly different angle, but sometimes such multiple angles help. I find it often helpful to look up or derive the exact relation between the two. Even if this will never show up in the final article and nobody ever asks about it at a conference, knowing such relations makes me much more comfortable with the results and often leads to much clearer/preciser way of writing down the results.

If your intended audience if not familiar with eta square, then it would probably just add confusion. So in that case I would not mention eta square.

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