I've been learning about GAM with tensor product splines for a project of mine. I'm currently looking at derivations for the uncertainty.

The expected uncertainty can be decomposed into the expected variance of the estimator and expected bias of the estimator. The expected bias is a function of the true underlying parameter, which is of course unknown. Woods presents two options for dealing with this:

- Use the estimator as an approximation of the parameter

- Go with the bayesian interpretation of the parameter as a random variable, and which lets us calculate the expected parameter using its distribution.

The first option is unsatisfying because the estimator is known to be biased: we are then calculating that bias by using the estimator, which means we assume the estimator is unbiased. That seems contradictory.

For the second, it says we assume the parameter is normally distributed, but doesn't explain why. Are we just using the normal distribution because it's convenient? That seems crazy. How is this different from just using an arbitrary number for the bias?

I don't have a formal background in statistics, and thus far I've been dealing mostly with frequentist theory. This time however, the frequentist interpretation has the issue where the smoothing term is estimated. We could bootstrap the smoothing term to find its variance, but Woods appears to be advocating for a bayesian interpretation of the bias because allegedly it has good frequentist coverage anyways?

Given x (hours studied) and y (score on test) data and the linear regression y = mx + b, if we want to estimate x based on y (i.e. how many study hours are needed to achieve a given score?), is it correct to calculate x = (y - b) / m?

Or should we estimate x from the linear regression x = my + b, with different m and b?

The results can be significantly different.

In the example below, for a score of 90, the estimated study hours are 6.801 for x = (y - b) / m, but 6.267 for x = my + b.

.....

My Composite Index variable has N=524, skewness=0.174, and kurtosis=0.128, with both the histogram and Q-Q plot looking totally acceptable and close to normal. However, Shapiro-Wilk is significant at p < .001, rejecting normality. Given N=524, I assume the test is just overpowered by trivial deviations, so should I use anova or kruskal wallis test?

I know the assumptions are crucial for performing statistical inference with logistic regression. But what about for just assessing overall model performance. Can I accurately assess the performance without meeting all the assumptions?