Users Guide to the QDE Toolkit Pro

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Users Guide to the QDE Toolkit Pro
Ch 6 83
May 2, 2002
National Research Conseil national Council
Canada de recherches
Excel Tools for Presenting Metrological
Comparisons byB.M. Wood, R.J. Douglas A.G.
Steele
Chapter 6. Degrees of Freedom and Student
Distributions
This chapter reviews some basic information about
Student distributions and effective degrees of
freedom, and discusses how the default options
are set by the QDE Toolkit Pro, and how easily it
can cope with a participant who has chosen to
express an effective degrees of freedom as is
suggested in the ISO Guide to the expression of
Uncertainty in Measurement. It also presents an
improved method for determining the effective
degrees of freedom, aimed specifically at the
coverage factors.
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Ch 6 84
QDE Toolkit Pro - Probability Density Functions
(PDFs)
The PDF of the measurand underlies the ideas of
the standard uncertainty developed in the ISO
Guide to the Expression of Uncertainty in
Measurement. The QDE Toolkit Pro uses PDFs that
are symmetric about their mean value, so it is
straightforward to use the PDF of the measurand
as the PDF of the (repeated) measurement.
Left-right reflection of the PDF, and explicit
repeatability extrapolation of the uncertainty
are not addressed. QDE Toolkit Pro uses q Normal
(or Gaussian) PDFs q Student PDFs with n
effective degrees of freedom
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Ch 6 85
QDE Toolkit Pro - Normal and Student PDFs
q Normal (or Gaussian) PDFs PDF(x) aN
exp(-(x-x0)2/(2u2)), aN from ? PDF(x) dx 1 q
Student PDFs with n effective degrees of
freedom PDF(x) aS 1 ((x-x0)/u)2 / n-(n1)/2,
aS from ? PDF(x) dx 1 where n is used to
parameterize the excess tails of a PDF in the
limit as ???, the Student distribution converges
to the Normal distribution. Note that n does
not have to be an integer.
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Ch 6 86
QDE Toolkit Pro -Type B Estimates of n for
Student PDFs
n is used by QDE Toolkit Pro for its graphing of
PDFs, and for calculating coverage factors. The
basic thought is that u is not known perfectly,
but is only estimated by u0. For ngt10, the
traditional Type B method is based on the
variance of the chi distribution from its mean,
and can harness your expert opinion on how well
u is known - within Du ? ? 0.5u0/Du2 QDE
Toolkit Pro function nu_from_chi_variance. Howev
er, for smaller ns, the (reduced) chi
distribution is very asymmetric (either about its
mean or about 1 - which is how it is used). An
improved coupling of the Student distribution
tails to your experience can be made through the
inverse-chi distribution, using the related
question What Du would have u0Du larger than
84 of the possible us? (1-.84) (1-.68)/2If
you, as the responsible metrologist, have an
answer for this question, an improved Type B
estimate for the effective degrees of freedom,
aimed specifically at describing the tails of the
most appropriate Student distribution (good for
nS gt1), is ?S ? 0.5u0/Du2 1 3Du/u0
1.2Du/u02 QDE Toolkit Pro function
nu_sub_S.
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Ch 6 87
QDE Toolkit Pro - Improved Estimate for n the
details
The Student distribution is derived from the
chi-square distribution of estimates u02 for the
unknown variance of the underlying normal
curve.The variance of the reduced-chi
distribution is used in the ISO Guide to check
that ? ? 0.5u0/Du2
Tails here make broad tails in the Student
distribution
For nlt10, asymmetry in the distribution casts
doubt on this variance-based approach. To link to
expert opinion, we use the inverse-reduced-chi
distribution, and ask for expert insight about
the value of 1Du/u needed to have 84 of the
distribution below that value. Using inverse-chi
distributions, the derivation of the Student
distribution becomes intuitive and
straightforward.
Tails here make broad tails in the Student
distribution
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Ch 6 88
QDE Toolkit Pro - Improved Estimate for n the
details
The usual derivation of the Student distribution
(independent normal and chi-square distributions)
can be transformed, as shown here for ?4, into a
bivariate distribution of normal curves (width ?
u) having a marginal distribution that is an
inverse-reduced-chi distribution.To choose the
best degrees of freedom ? from this family of
Student distributions, based on an experts
opinion about how well u is known, he or she
needs only to express that opinion as a Du/u0,
such that u is expected to exceed (u0Du) 16
(1/6) of the time. (Easier than mental variance
calculations!!)
Inverse-chi Distribution
Student Distribution
Normal Curves
Du/u0
16
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Ch 6 89
QDE Toolkit Pro - Improved Estimate for n more
details
This graph shows the variance-based link between
expert opinion and the effective degrees of
freedom, referred to in the ISO Guide ? ?
0.5u0/Du2 1It also shows the corresponding
limits of the proper inverse-chi distribution,
for an interval with the same 68 confidence,
where the low-u and high-u tails are each
expected 16 of the time these limits are
labeled as CDF0.16 and CDF0.84.
The upper curve also shows excellent agreement
with the numerical fit ?S ? 0.5u0/Du2 1
3Du/u0 1.2Du/u02 2 that is used
in the QDE Toolkit Pro function nu_sub_S, and
its inverse Toolkit Pro function
delta_u_by_u_from_nu_sub_S, to link to expert
opinion about the limit on u that is expected to
be exceeded only 16 of the time.
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Ch 6 90
QDE Toolkit Pro - two definitions for n???
Do these Two Equations Imply Two Definitions for
Degrees of Freedom?No. In the context of
determining coverage factors, the variance-based
method is simply a bad approximation for degrees
of freedom lt 10, if the degrees of freedom is to
be used for evaluating coverage factors from
Student distributions. Nonetheless, we suggest
using the symbol nS for a degrees of freedom
aimed at describing the tails of the Student
distribution. Fortunately, in precision
metrology, usually n gt 10 and there is no
difficulty. Should we ever translate from one
Equation to the Other?No, we think that in
high-level Metrology it is almost never
appropriate to correct another Labs opinion in
this way. If they have used this bad
approximation for n, it is rather like them
having over-estimated their uncertainty, and in
the context of Key Comparisons they should expect
to live with the consequences derived from any
inaccurate estimates that they make.
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Ch 6 91
QDE Toolkit Pro - Getting the Effective Degrees
of Freedom
If you are convinced by the foregoing discussion,
and want to apply the improved estimator for the
degrees of freedom, then often only the
evaluation of your own Labs degrees of freedom
would be affected, and only occasionally would
any changes be really significant. If you are the
Pilot Lab, it could also affect your evaluation
of the effective degrees of freedom for the
travel uncertainty assigned to each Lab, and for
other effects being evaluated by the Pilot
Lab. Most often, the effective degrees of
freedom will simply be an input parameter
provided, along with their uncertainty, by some
of the Labs. Sometimes these will have to be
combined with other uncertainties, and their
effective degrees of freedom, as decided by the
Pilot Lab, adding the uncertainties in quadrature
and combining the degrees of freedom using the
Welch-Satterthwaite formula.
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Ch 6 92
QDE Toolkit Pro - Using the Effective Degrees of
Freedom
From this point on, the QDE Toolkit Pro can
really help to simplify the use of the effective
degrees of freedom. There are two equivalent
paths (red and blue) ni can be edited. After
the first Toolkit macro has run to completion,
the ni have the normal default, but can be
edited for re-running a Toolkit macro. ni can
be added initially, before any Toolkit macro is
run. After the macro has run to completion, the
input ni are used with the default correlation
coefficients.
Run
Edit
Edit
Ready for editing default correlation
coefficients and running any Toolkit macros.
Run
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Ch 6 93
QDE Toolkit Pro - infinite n??
Normal Distributions - infinite n The normal or
gaussian distribution is formally a Student
distribution with infinite degrees of freedom.
This is the default option in the ISO Guide to
the Expression of Uncertainty in Measurement if
someone has not told you about any departures
from the normal distribution, then they have
implicitly told you that it is normal. If there
are significant departures from the normal
distribution, this is their mistake for not
telling you so! Uniform Distributions -
infinite n The ISO Guide to the Expression of
Uncertainty in Measurement argues that the
degrees of freedom for a uniform distribution
should be infinite. In the context of a
variance-based understanding of the degrees of
freedom, this seems wrong the uniform
distribution is sometimes used to represent a
possible range of values that might include a
delta-function distribution as well as a
double-delta-function of the maximum width (a
goalpost distribution). With this variance a
variance-based degrees of freedom could be
determined. However, on our context of honing the
degrees of freedom to help determine the most
appropriate coverage factor, the bounded uniform
distribution cannot produce large tails, and an
infinite degrees of freedom seem more appropriate.