STAT430 : Questions?

Referers: Fall2007 :: (Remote :: Orphans :: Tree )

Dorman Wiki
Dorman Lab Wiki
This is an old revision of Questions from 2007-11-06 00:05:08.
  - After going through the notes again I am still a little fuzzy on the difference between confidence interval and prediction interval.  Is the confidence interval the 1- α range of E[y] while the prediction interval is the 1- α range for y itself?

  - So, a residual is the observed value minus the predicted value, but is this the only definition of residual?  If it is, it then seems like SST should be the sum of squares of residuals.  Is this so?  By the notation we have used I could also justify it being SSR.  I'm just a little turned around about this at the moment...  This could clarify the meaning of MSR if it is them SSR/k.  Then it's the mean squared residual?

  - In the MLR assumptions we state that cov( ε l o n i , ε l o n j ) = 0 for all i not equal to j and that this holds for random samples, but not necessarily for time series data or repeated measures on an individual.  However, it seems like the majority of studies are one of the latter two.  Is our work in experimental design teaching us how to satisfy this assumption by blocking and such so that our data is of a proper form for MLR analysis?  Is it that we simply violate this assumption some times?  Or, am I thinking of studies in the wrong way; possibly the same mistake of language as thinking of a random variable in the same way as a random number generator?

Yes, there is a typo in the formula for E [ X ] .  Here is corrected formula plus detailed derivation.

Agreed.  My notes for derivation of E [ X ] read simpler.

It is not an iff statement.  One must have all moments external link match (when they exist) with those of a known distribution to conclude that a random variable has this distribution.  See moment generating function external link.  Thus, we would also have to check higher moments, like E [ X 3 ] , match those of a Poisson random variable to conclude that X Poisson.

Yes, this is abusive notation.  So, Ω is the sample space consisting of all possible outcomes of a random experiment.  A random variable maps Ω to some subspace of R .  If we sort of forget about the random experiment and outcomes, and treat the random variable as the outcome, then we can call this R subspace Ω X .  Proper, careful notation would probably use something other than Ω for this purpose.

As per our discussion about goodness-of-fit tests, the degrees of freedom should be m -1 less the number of parameters estimated, where m is the number of categories.  In the test of independence, the number of categories is n r n c .  Under independence, there are n r -1 parameters to estimate for the marginal pmf on rows, one for each category minus the constraint that the pmf i p i = 1 sums to one.  Similarly, there are n c -1 additional parameters to estimate for the pmf on columns.  Therefore, the number of degrees of freedom is n r n c - ( n r -1 ) - ( n c -1 ) -1 = ( n r -1 ) ( n c -1 ) , in agreement with the rule for tests of independence.  In conclusion, the test of independence can be viewed as a special type of goodness-of-fit test.

There is only a vague course schedule.  We will cover rudimentary experimental design, multiple linear regression, general linear models, logistic regression, poisson regression, stochastic processes (Bernoulli, Poisson, Brownian, discrete time Markov chain), simulation, including random number generation, Monte Carlo integration, and MCMC.
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