How would an individual decide on treating Non-normal data for, e.g. control charts; data transformation vs Non-normal distributions such as Weibull? I have access to Minitab but the decision point on data treatment evades me. Johnson transformations seem useful to establish, for example, process stability, but I've read negative connotations about such an approach and using non- normal distributions instead but I don't know how, meaningfully - signed, a process monitor
If you want to use parametric statistics, you assume some underlying distribution. You should therefore always try to use a model with a distribution that makes sense given the process that you are studying. If you don't have reason to assume any underlying structure, then perhaps you should use a model that does not assume some a priori probability distribution (e.g. semi- or non-parametric).
The probable reason you came across negative connotations of using transformations is that these can be easily misused to avoid having to think about the underlying process. For example, a log-transformation may help a skewed probability distribution 'seem more normal', but if the data you are considering are independent counts, then you would have a much nicer solution using Poisson regression (after all, counts are discrete and should follow a Poisson distribution if independent).
You mention you are a process monitor, so let me try to give some relatable examples:
- Milk cartons are approximately filled to 1L. Some slightly less, some slightly more, but they contain 1L milk on average, with further deviations being exceedingly rare. Slightly fuller or emptier is considered to be equally likely. This could very well be approximately normally distributed.
- You observe the time until failure of certain components. The failure is assumed to be solely due to the lifetime of a component expiring. All lifetimes are non-negative and one component may take considerably longer than another to fail. This behaviour is described by the Weibull distribution or related distributions (exponential, log-normal).
- You count the number of independent failures of components. This should follow a Poisson distribution.
- You want to determine the ratio of failure of components using one machine vs. another machine. This is a binomial (or logistic) regression problem.
A transformation can be useful if there is a justification for its use. It can also be used to simplify a problem. See for example this question on when to take the logarithm. It is not bad to use transformations, but the reason you should do so with caution is because it tends to take away the thinking part and reduces it to "find the transformation that makes my data best fit the model", rather than "find the model that best fits my data".
In short, different probability distributions or transformations could both be the answer to your modeling problem. The important part is that you can theoretically justify your model choice and/or validate it externally using some measure of predictive accuracy. If you can't substantiate your model choice, why use a parametric model in the first place?