```Date: Mon, 22 Mar 2010 16:41:37 -0700 Reply-To: John Uebersax Sender: "SAS(r) Discussion" From: John Uebersax Organization: http://groups.google.com Subject: Re: df for confidence interval with a random effect (maybe Satterthwaite) Comments: To: sas-l@uga.edu Content-Type: text/plain; charset=ISO-8859-1 Thanks Ray, Okay, so if one uses the variance formula (= v) and the Welch- Satterthwaite degrees-of-freedom formula (= W-Sdf) you gave, then to estimate the 95% confidence of future observations, each being a random observation from a random batch, would that be: LL = mean - t_critical * sqrt(v) UL = mean + t_critical * sqrt(v) where t_critical = t_inv(.05/2, W-Sdf) and t_inv is the inverse t statistic cdf with W-Sdf degrees of freedom? (and, if so, could anyone suggest an example of estimating a CI this way in the 'quality' literature?) John On Mar 10, 11:48 pm, Ray Koopman wrote: > On Mar 10, 2:45 pm, JohnUebersax wrote: > > > > > Hello Group, > > > Suppose I want to construct a confidence interval for a variable's > > value (not mean) based on 8 samples, each with 200 observations. > > There is a random effect across samples, such that: > > > observed value = true score + sample effect (r) + error > > > or simply > > > y = x + r + e > > > So r is like a per-sample bias, normally distributed across samples, > > and independent of x. Simply treating the data as one large sample > > (N=1600) and computing a classic confidence interval: > > > LL = grand mean + t_inv(alpha, 1599) * std dev > > UL = grand mean - t_inv(alpha, 1599) * std dev > > > where std dev is computed from total variance and t_inv is the > > inverse t function, won't work, because this doesn't fully recognize > > uncertainty associated with r, which has only 8 - 1 = 7 df. So the > > true df for t_inv is between 7 and 1599. But what is the value? > > > I seem to recall that in this case one may apply a simple formula > > (Sattherwaite?) to get the correct df. > > > Would that work here, and if so could anyone suggest an informative > > source that gives the correct formula and maybe an example? > > > (p.s. I suppose this can be approached in a more complex way, e.g., > > hierarchical modeling, but want to pursue the simpler method before > > considering any alternatives.) > > > Thanks in advance. > > > JohnUebersaxPhD > > If all the usual random-model assumptions hold then the estimate of > the overall variance is > > v = MSB/n + MSW*(1-1/n) > > and the Welch-Satterthwaite df approximation is > > v^2 > f' = ---------------------------, > (MSB/n)^2 (MSW*(1-1/n))^2 > --------- + --------------- > k-1 k(n-1) > > where MSB and MSW are the between-group and within-group mean squares > from a one-way anova, k is the number of groups (in your case, 8), and > n is the number of observations in each sample (in your case, 200). > > This is discussed in many texts in the sections on random models > and variance component estimation, such as sections 12.5-12.7 in > Mendenhall's (1968) Introduction to Linear Models and The Design > and Analysis of Experiments. ```

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