Dear experts and colleagues,
Hello, I received a review asking that my analysis is somewhat suspicious/incorrect.
Here is a brief explanation of my analysis.
I am comparing the disease group and the control group.
And my data are matrices of subjects constructed from brain MR data.
Since my subjects in each group show different demographic characteristics and behavioral tests, I ran ANCOVA as nuisance variables to remove their effects on variance in my findings.
The reviewer said General Linear Model assumes normal distribution among participants, but the included covariates in my ANCOVA show non-linear relationships.
However, from my knowledge, regression analysis does not assume normality for either predictors (IV) or an outcome (DV), and even multiple regression (ANCOVA) does not make any assumptions about the distributions of the explanatory variables or covariates.
Do I miss any point here? Or how can I answer the reviewer?
All the discussions and suggestions are welcomed!
The reviewer mentioned "GLM" what usually stands for generalized linear model. And one feature of GLMs is that they are not limited to the assumption of a normal distribution. So that's kind of a strange comment.
However, the ANCOVA is "just" a general (not genralized) linear model, usually just abbreviated LM. This is a special case of a GLM that is based on the normal distribution assumption. This distribution is conditional on the structural model µ(xi), that is, on all the predictor values in the model. Under the additional assumption that the variance (s²) is independent of the predictors, the conditional normal distribution is N(µ(xi), s²). The mean function is estimated from the data as m(xi), and thus, if the assumption is correct, the residulas yi - m(xi) should be relaizations of random variables that have all the same distribution N(0, s²). This is what can be checked.
If the assumption is reasonable at all depends on the nature of your response variable. This requires understanding of your experiment and what presisely is measured and how it is measured. For instance, if these are count values, or proportions, or intensities, then the assumption is not neccesarily reasonable or acceptable, just on theoretical grounds (like measuring velocity in kg is just not reasonable on theoretical grounds, to give a stupid analogy).
Do agree with the above statement, moreover nature of the experiment needs to be elaborated fully.
Your post says that the referee says that GLM assumes Normal distribution (of the outcome, or more exactly of the outcome conditionnally on the predictors), which is true if G is for general [errors are Gaussian with any kind of covariance matrix] but not if it stands for generalized [errors are still assumed iid, but Gaussian assumption is removed thanks to the link function formalism].
But it also says, as your write it, that he has concerns with non-linearity of the relation between X and Y, which has nothing to do with the distribution of the X data. So if he really speaks about non-linearity, your interpretation of his concern as a problem of normality of X is wrong. It says, basically, that you cannot model the relationship between X and Y as a line, which is another assumption of the ANCOVA, completly different and much more problematic than any kind of distribution assumption problem.
Best regards,
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