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>
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 <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
>
>                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.