Hi,
Some background to my question:
I have fitted 1st order multinomial Markov models with covariates.
Since all probabiilties (for each row of the transition matrix) sum to 1, I estimate only the probabilities and respective se's and CI's for the first k-1 of K states in each row of the transition matrix.
Therefore, It is easy to calculate the last transition probability estimate from the other k-1 estimates (for each row of the transition matrix).
However, it is unclear to me how I would obtain the confidence interval for the last category.
Is there a trick how to do that?
Or is there a way how I can approximate the se of the probability of the last category from the estimates of the se from the first k-1 categories?
I hope that my explanation of what my question is, is clear enough.
Thanks for anyone who can provide hints how to do it!
I suppose one possibility would be to redo the calculations, and using another state as the reference category.
However, if I don't want to do that, is there an analytical solution?
About the last variable, use binomial random variable. Since the estimations of variance and standard deviation can be calculated based on estimation of the probability, the problem is solved.
Maybe you want to check out the emmeans package, provided that you have done your modelling in R.
Actually I now think, it is more tricky then anticipated, and how I calculate the CI's wasn't correct.
You can see in the below, what I estimate is the beta's, and for the beta's I obtain se's and CI's
You can see from the formula, that when I backtransform the beta to the probabilities, the formula uses all the estimates beta_u . (for all s = 0,... k -1).
So, How would I then actually obtain backtransformed Confidence intervals?
Would I plug in the estimates of beta, and use the upper / lower CI value only for the specific beta_us_p
(with u = starting state, s = end state, p = for the respective covariate).
At the end, I may be better of using bootstrapping on the probabilities itself (instead of beta).
However, I then can't easily see how the CI's actually connect to the p-values.
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