Estimating Intakes from Measurements

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Estimating Intakes from Measurements
Lecture 4
Prof Alan Birchall Internal Dosimetry Course,
SSI, Sweden Nov 16-17 (2006)
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Estimating Intakes from Measurements
Structure

1. Introduction
2. Estimating intakes from a single measurement
3. Estimating intakes from multiple measurements
4. Conclusions
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Estimating Intakes from Measurements
Structure

1. Introduction
2. Estimating intakes from a single measurement
3. Estimating intakes from multiple measurements
4. Conclusions
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1. Introduction
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2. Estimating an Intake from a Single
Measurement
Personal Air Sample
Sample 5 Bq
Sampling rate 10x Breathing rate
Therefore, Intake 50 Bq
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2. Estimating an Intake from a Single
Measurement
Bioassay sample
Let the measurement made at time t, be m
Calculate the bioassay function from 1 Bq
intake, f(t)
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2. Estimating an Intake from a Single
Measurement
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3. Estimating an Intake from multiple
measurements
Multiple estimate method
This is really an extension of the single
measurement method
Here, the best estimate is defined as the mean of
the estimates
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3. Estimating an Intake from multiple
measurements
Least Squares Method
Here, the best estimate is defined as the value
of I which minimises the sum of the squares of
the deviations
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3. Estimating an Intake from multiple
measurements
Least Squares Method
Note that this estimate is different that that of
the multiple estimate method
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3. Estimating an Intake from multiple
measurements
Weighted Least Squares Method
Here, the best estimate is defined as the value
of I which minimises the sum of the weighted
squares of the deviations
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3. Estimating an Intake from multiple
measurements
Weighted Least Squares Method
Here, the best estimate is defined as the value
of I which minimises the sum of the weighted
squares of the deviations
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3. Estimating an Intake from multiple
measurements
Weighted Least Squares Method
The question now is what values of ? do we use?
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3. Estimating an Intake from multiple
measurements
Weighted Least Squares Method
Need to look at these assumptions in a little
more detail
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3. Estimating an Intake from multiple
measurements
Weighted Least Squares Method (uniform absolute)
e.g. ? 3
This is the usual default (no error assumption)
Fit is independent of K
Sensible assumption when all data is of similar
magnitude
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3. Estimating an Intake from multiple
measurements
Weighted Least Squares Method (uniform relative)
e.g. ? 10
Fit is independent of K
Fit is different from uniform absolute
Usual assumption when data varies a lot
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3. Estimating an Intake from multiple
measurements
Weighted Least Squares Method (uniform relative)
e.g. ? 10
Fit is independent of K
Fit is different from uniform absolute
Usual assumption when data varies a lot
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3. Estimating an Intake from multiple
measurements
Weighted Least Squares Method (uniform relative)
e.g. ? 10
Multiple estimate method !!
Fit is independent of K
Fit is different from uniform absolute
Usual assumption when data varies a lot
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3. Estimating an Intake from multiple
measurements
Weighted Least Squares Method (inverse square
errors)
This can occur when errors are dominated by
counting statistics
Fit is independent of K
Usually gives an answer between uniform abs rel
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3. Estimating an Intake from multiple
measurements
Weighted Least Squares Method (explicit errors)
The best !!
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3. Estimating an Intake from multiple
measurements
Weighted Least Squares Method (explicit errors)
IMBA allows the user to enter errors on each
measurement value individually
IMBA also allows different error assumptions to
be made e.g. uniform absolute, uniform relative
e.t.c.
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3. Estimating an Intake from multiple
measurements
Weighted Least Squares Method (logarithmic errors)
So far, it has been assumed that the measurement
data is normally distributed.
Sometimes, measurement data can be lognormally
distributed.
This can be dealt with by first taking the logs
of the measurement data, and then doing a least
squares fit.
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3. Estimating an Intake from multiple
measurements
Weighted Least Squares Method (logarithmic errors)
In IMBA, each measurement value can be specified
as either normally or lognormally distributed.
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3. Estimating an Intake from multiple
measurements
Weighted Least Squares Method (examples)
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3. Estimating an Intake from multiple
measurements
Weighted Least Squares Method (examples)
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3. Estimating an Intake from multiple
measurements
Weighted Least Squares Method (examples)
Beware !
This is the same graph !!!
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3. Estimating an Intake from multiple
measurements
Weighted Least Squares Method (examples)
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3. Estimating an Intake from multiple
measurements
Weighted Least Squares Method (animal experiment)
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3. Estimating an Intake from multiple
measurements
Weighted Least Squares Method
Why minimise the squares of the residuals?
Why not minimise the residuals themselves, or the
cube of the residuals?
?
The answer lies in a method known as the Maximum
Likelihood Method
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3. Estimating an Intake from multiple
measurements
Maximum Likelihood method
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3. Estimating an Intake from multiple
measurements
Maximum Likelihood method
The least squares method follows directly from
the maximum likelihood method !
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3. Estimating an Intake from multiple
measurements
Advantages of the Least Squares Method
Simple to apply
Can propagate errors
Advantages of the Maximum Likelihood Method
Can deal with LOD data
In IMBA, the user can choose the fitting method
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4. Conclusions
It has been shown that the Weighted Least Squares
Method gives the best estimate of Intake.
When the errors on each data are not given, a
sensible choice must be made.
The Maximum Likelihood Method (MLM) gives the
same fit as the Weighted Least Squares Method
when all the data is normally distributed and
lies above the limit of detection.
THE END
Care must be taken when deciding whether a fit is
good (remember the pig !)