Point Estimation using Least Squares
SWoP uses least squares estimation for point estimation. Least Squares Estimation uses Calculus and not probability. Methods like maximum likelihood use Probability, but the least squares method does not.
This method of point estimation does not depend on the probability distribution of the outcome.
Because we are not concerned with the errors, or the distribution of the errors, or the variance, or the standard error or even the P value: The assumption that independence of error is not relevant.
By the above logic: The assumption that errors are independent of the predictor variable is not relevant.
Also by the above logic: The assumption that the errors are normally distributed does not hold. As we are not concerned with the errors because we are not calculating a P value.
Multicollinearity only effects the precision of the estimator. Precision again is concerned with the variance. If we do not need a variance as we do not need P values then: Multicollinearity or X having full column rank (i.e. any dependent variable must not be a linear combination of other dependent variables) is not relevant.
The assumption of linearity does hold if you use $y=mx+b$. If this linear relation did not hold you can always use a more appropriate equation to model the relationship as long as the least squares estimator is still valid. The least squares estimator is quite flexible.
This means that point estimation can be used for data that is correlated without any adjustment of the error terms because we are not calculating P values or 95\% Confidence Intervals.
Therefore least squares method also works under the paradigm of “Statistics without Probability” for correlated data.
Similarly, logistic regression can be done by transforming the binary outcome using a logit transformation and reverting back to least squares regression.
To replace Poisson regression the data can be transformed into a dataset suitable for logistic regression with a binary outcome instead of a rate outcome. This also circumvents the need for probability.
Also for survival analysis; an application of multivariate least squares estimation with 2 outcome variables (follow-up time and occurrence of an event) can be used. A more simple method would be just to transform the survival analysis data set into a longitudinal dataset and analyzed using logistic regression with the event as a binary outcome.
Longitudinal data can be analyzed without worrying about correlated error terms as interval estimation in SWOP does not use standard error as it has its own unique method of interval estimation and hypothesis testing.
In conclusion besides the linearity assumption, there are no other assumptions for using least squares in the paradigm of “Statistics without probability”. Note that the linearity assumption can be overcome by using a more appropriate equation to model the relationship between the predictor X and the outcome Y.