Confidence intervals on maximum likelihoods of observed dataI observed 400 episodes of nursing care in a hospital. I tracked the movement of the nurses between 5 rooms A − E. The maximum likelihood of them moving from room i → j is given by: P i j = # of times from room i → j Total # of transitions to any room - Is there a way of defining a confidence interval on this maximum likelihood estimate P i j ?- And for all maximum likelihood estimates of all possible room combinations?

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Confidence intervals on maximum likelihoods of observed dataI observed 400 episodes of nursing care in a hospital. I tracked the movement of the nurses between 5 rooms   A  −  E. The maximum likelihood of them moving from room   i  →  j is given by:      P          i      j        =                              # of times from room                       i            →            j                                    Total # of transitions to any room                    - Is there a way of defining a confidence interval on this maximum likelihood estimate       P          i      j      ?- And for all maximum likelihood estimates of all possible room combinations?

ignaciopastorp6 · Accepted Answer

Step 1One could probably think about the nature of the dependence among the various random variables that would have been observed, but for now I&#039;ll do something simpler:You have n independent Bernoulli trials; in this case   n  =  400. You have x successes; in this case, x is the numerator in the fraction. So the number of successes is binomially distributed with an unobservable parameter p. Theory tells us the expected value of the number of successes is 400p and the variance of the number of successes is   400  p  (  1  −  p  ). That means the expected proportion of successes is p and the variance of the proportion is   p  (  1  −  p  )      /    400. So we can cite the central limit theorem and we have an approximately normal distribution; thus about a 0.95 chance of being between -1.96 and +1.96. We use x/400 as an estimate of p.Step 2Our 95% confidence interval therefore has endpoints      x    400    ±  1.96                    (        x                  /                400        )        (        1        −        (        x                  /                400        )        )            400

Confidence intervals on maximum likelihoods of observed data. I observed 400 episodes of nursing care in a hospital. I tracked the movement of the nurses between 5 rooms A-E.

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