The Inverse Gaussian Model Sample Clauses

The Inverse Gaussian Model. Two hypothetical time paths of depression (one solid and one dashed) showing actual and linear smoothed depression scores. The participant’s depression score under the inverse Gaussian model evolves as Brownian motion with a trend but with random local changes. Reaching the threshold for remission is 1. For nonoverlapping time intervals (𝑡1, 𝑡2) and (𝑡3, 𝑡4) the random variables 𝑋(𝑡2) − 𝑋(𝑡1) and 𝑋(𝑡4) − 𝑋(𝑡3) must be statistically independent. 2. For any time interval (𝑡1, 𝑡2) , 𝑋(𝑡2) − 𝑋(𝑡1) must be normally distributed with mean and variance both proportional to (𝑡1, 𝑡2). It seems reasonable to believe that the two conditions specified above for the IG distribution may hold in this illustration. The independent increments are a key assumption of defining the stochastic process a Weiner process. The level of depression 𝑋(𝑡) is a random variable which has been assumed here to undergo independent changes over time. The property implies that a change in a patient’s depression status depends only on her current level of depression and not on its historical pattern. This property seems defensible as a first approximation to the true character of the process. The requirement for normally distributed depression level changes might seem restrictive but the depression scale on which 𝑋(𝑡) will be defined, should be an appropriately constructed measure of depression, and further could be transformed to meet this requirement providing it does not alter its ability to order depression status; rank preserving.