Sraman Dasgupta Hi; first of all, you need to provide some information on the original variables of the dataset. Was the response variable continuous and unbounded (i.e., includes positive and negative values)? Why did you need to do the Z-transformation or Standardize the dataset?

Shapiro-Wilk's test is NOT a definite answer to decide on the Normality of a Response Variable.

For GLM, the Mean-Variance Relationship is the first thing to look at, which serves as a rule-of-thumb for assessing which distribution and link function to use.

Assuming that there are non-positive response values in the original dataset, you may not use the Box-Cox transformation to normalize the Response variable, but you may use the Johnson transformation to normalize it, but that transformation is a bit complicated.

If you are inclined to use GLM only, one way is to square the Original Response Values so that it will be between ZERO and Infinity. Then, you may apply the Gamma Distribution. And, which Link function to apply depends on the Mean-Variance Relationship. But remember that, on squaring up, the response variable unit will also be squared.

Sraman Dasgupta there is no assumption that the dependent variables needs to be normally distributed. It is about the errors, which may be assessed with the residuals of the models.

As Samipan Majumder already mentioned, please give more information about your dependent variable, what you are trying to measure and how you determined normality (not only which test you used).

In order to check for normality of the dependent variable, I conducted Shapiro-Wilk's test in Rstudio. The response (i.e. dependent) variable has only positive integer values and is unbounded. Since the different independent variables had different units, I standardized the dataset.

Sraman Dasgupta By different units of independent variables, I believe that you mean that the range of values of such variables varies widely, correct? Nevertheless, I understood from your latest reply that your response variable follows a Non-Normal Right-Skewed Distribution. You may check the distribution/histogram to conclude its skewness if required.

Now, if you are interested in applying the GLM model, the Random Component that seems applicable here is that of the Gamma Distribution Random Component because it allows values from ZERO to positive INFINITY. The link function can either be the Canonical Link (i.e., Inverse Link function) or can be another link function, such as the Power Link function (g(mu_i) = mu_i^lambda, when lambda != 0) OR, (g(mu_i) = ln(mu_i), when lambda = 0). Here, mu_i = E(Y_i), where Y_i is the ith Response.

If required, you may compare the model results by using the original independent variables and, later, by standardizing the independent variables.

as mentioned above, it does not make any sense to test for normality of the dependent variable, but for the conditional distributions. In case of metric predictors, this can be assessed with the residuals.

Just to give you an example: imagine you have measured the height of people in centimeter across male and female participants. See the figure attached for population and sample distributions (simulated data just for illustration). We know that the male and female population each normally distributed. Therefore, samples from this population should also follow a normal distributions (with sampling error of course).

Now, if you would test and inspect the normality of your dependent variable "height", you would clearly see that the "data" is not normally distributed, since it is a bimodal distribution. The Anderson Darling test would give a significant result for our sample.

On the other hand, if you test each sample individually, they would be normal. Accordingly, if you use a regression with height as dependent variable and group/sex as predictor, the residuals would also follow a normal distribution. mod1