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where

k | is the contamination coefficient, where . |

s | is an independent estimate of , where . |

df | is the number of degrees of freedom for s, where
. |

is the prior probability of contamination for the ith
observation in the sample, where i = 1, ... ,n and n
is the number of observations in the sample. Note that
. | |

y_{i} | is the ith observation in the sample, where
i = 1, ... ,n and n is the number of observations in
the sample. When the BAYESACT call is used
to perform a Bayes
analysis of designs (see "Description" below),
the y_{i}s are estimates for effects. |

is the variable that contains the returned posterior
probability of contamination for the ith observation
in the sample, where i = 1, ... ,n and n is the
number of observations in the sample. | |

p_{0} | is the variable that contains the posterior probability that the sample is uncontaminated. |

Specifically, the BAYESACT call assumes a normal random sample of *n*
independent observations, with a mean of 0 (a centered sample) where
some of the observations may have a larger variance than others:

where *i* = 1, ... ,*n*. The parameter *k* is called the
*contamination coefficient*. The value of is the
*prior probability* of contamination for the *i*th observation.
Based on the prior probability of contamination for each observation,
the call gives the posterior probability of contamination for
each observation and the posterior probability that the
entire sample is uncontaminated.

Box and Meyer (1986) suggest computing posterior probabilities
of contamination for the analysis of saturated orthogonal factorial
designs. Although these designs give uncorrelated estimates for
effects, the significance of effects cannot be tested in an analysis
of variance since there are no degrees of freedom for error. Box
and Meyer suggest computing posterior probabilities of contamination
for the effect estimates. The prior probabilities
() give the likelihood that an effect
will be significant, and the contamination coefficient (*k*)
gives a measure of how large the significant effect will be. Box
and Meyer recommend using and
*k*=10, implying that about 1 in 5 effects will be about
10 times larger than the remaining effects. To adequately
explore posterior probabilities, examine them over a range of
values for prior probabilities and a range of contamination
coefficients.

If an independent estimate of is unavailable (as is
the case when the *y*_{i}s are effects from a saturated
orthogonal design), use 0 for *s* and *df* in the
BAYESACT call. Otherwise, the call assumes *s* is
proportional to the square root of a random
variable with *df* degrees of freedom. For example, if
the *y*_{i}s are estimated effects from an orthogonal
design that is not saturated, then use the BAYESACT call
with *s* equal to the estimated standard error of the estimates
and *df* equal to the degrees of freedom for error.

From Bayes' theorem, the posterior probability that *y*_{i} is
contaminated is

for a given value of , where is the density of a normal distribution with mean and variance .

The probability that the sample is uncontaminated is

Posterior probabilities that are independent of are
derived by integrating and *p* over a
noninformative prior for . If an estimate of is
available (when *df*>0), it is appropriately incorporated.
Refer to Box and Meyer (1986) for details.

data; retain post1-post7 postnone; call bayesact(10,0,0, 0.2, 0.2, 0.2, 0.2, 0.2, 0.2, 0.2, -5.4375,1.3875,8.2875,0.2625,1.7125,-11.4125,1.5875, post1, post2, post3, post4, post5, post6, post7, postnone); run;

return the following posterior probabilities:

POST1 0.42108 POST2 0.037412 POST3 0.53438 POST4 0.024679 POST5 0.050294 POST6 0.64329 POST7 0.044408 POSTNONE 0.28621

The probability that the sample is uncontaminated is 0.28621. A
situation where this BAYESACT call would be appropriate is a saturated
2^{7} design in 8 runs, where the estimates for main effects are
as shown in the function above (-5.4375, 1.3875, . . . , 1.5875).

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