Date: Mon, 22 Mar 2010 16:41:37 -0700
Reply-To: John Uebersax <jsuebersax@GMAIL.COM>
Sender: "SAS(r) Discussion" <SAS-L@LISTSERV.UGA.EDU>
From: John Uebersax <jsuebersax@GMAIL.COM>
Subject: Re: df for confidence interval with a random effect (maybe
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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
(and, if so, could anyone suggest an example of estimating a CI this
way in the 'quality' literature?)
On Mar 10, 11:48 pm, Ray Koopman <koop...@sfu.ca> wrote:
> On Mar 10, 2:45 pm, JohnUebersax<jsueber...@gmail.com> 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
> 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.