Topic 2 Nuclear Counting Statistics

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Topic 2 Nuclear Counting Statistics

• Random vs Systemic Errors
• Nuclear Counting Statistics
• Propagation of Errors
• Application of Statistical Analysis
• Statistical Analysis

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Random vs Systemic Errors

• Types of measurement errors Blunders, Systemic
  and Random Errors
• Blunders are gross errors such as incorrect
  instrument setting and wrong injection of
  radiopharmaceuticals.
• Systemic errors are results differing
  consistently from the correct one such as the
  length measurement by warped ruler.
• Random errors are variations in results from one
  measurement to another (physical limitation or
  variation of the quantity) such as the rate of
  the radiation emission.

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Accuracy and Precision

• Measurement results having systemic errors are
  said to be inaccurate
• Measurements that are very reproducible (same
  result for repeated measurements) is said to be
  precise.
• It is possible that result is precise but
  inaccurate and vice versa

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Nuclear Counting Statistics

• The Poisson Distribution
• Standard Deviation
• The Gaussian Distribution

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Poisson Distribution

• Defined only for non-negative integer values
• The probability of getting a certain result N
  when the true value is m P(Nm)e-mmN/N!
• Variance, ?2, is defined as such that 68.3 of
  the measurement results fall within ? ? of the
  true value m.
• For Poisson distribution, ?2 m.

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Nuclear Medicine Counting

• Nuclear Medicine radionuclide decay counting
  follows Poisson distribution.
• Nuclear Medicine question is that how good is the
  result N from a single measurement?
• The assumption is that N?m so that there is 68.3
  chance that m is within the range N??N. ??N is
  uncertainty in N.
• Percentage uncertainty is defined as V (?N/N) x
  100.

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Confidence Intervals
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Standard Deviation

• Standard deviation is calculated from the series
  of measurements where the number of measurements,
  n, and the mean value, are known

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Standard Deviation, Variance and Nuclear Counting

• Standard Deviation, SD, is an estimation of
  Variance ?.
• In nuclear counting, SD??N

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Gaussian Distribution

• Gaussian distribution is also called Normal
  Distribution.
• The Equation describe Gaussian distribution is

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Gaussian Distribution (continued)

• For large value m, Poisson distribution can be
  approximated by Gaussian distribution
• Gaussian distribution is defined for any value of
  x (positive integer for Poisson)
• Variance ?2 can have any value (mean value m for
  Poisson).
• If there is additional random errors in nuclear
  counting apart from the radionuclide decay, the
  results are described by Gaussian distribution
  with variance, ?2m(?N)2

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Propagation of Errors

• Sums and Differences ?(N1?N2?N3?)?N1N2N3
• Constant Multiplier ?(kN)k?Nk?N and
  percentage uncertainty
  V(kN) ?(kN)/kNx100100 /?N
• Products and Radios
  V(N1??N2??N3??)
  ?1/N11/N21/N3

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Exercises 1

• Given a formula in nuclear medicine
  Yk(N1-N2)/(N3-N4) Show
  that ?YVY?Y/100
  where
  VY is
  the percentage uncertainty of Y

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Applications of Statistical Analysis

• Effects of averaging
• Counting rates
• Significance of difference between counting
  measurements
• Effects of background
• Minimum detectable activity (MDA)
• Comparing counting systems
• Estimating required counting times
• Optimal division of counting times
• Statistics of ratemeters.

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Effects of Averaging

• If n counting measurements are used to compute an
  average result, then, the uncertainty in
  average is

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Counting Rates

• If N counts are recorded during time t, then the
  counting rate is RN/t. The uncertainty in
  counting rate is then given by

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Significance of Difference Between Counting
Measurements

• Suppose two measurements, N1 and N2. The
  difference ?N1-N2 could be true difference
  between these two counts or just random variation
  of one.
• If ?lt2?? is considered marginal (5 chance within
  random error)
• ?gt3?? is considered significant (1 chance) and
  2?? lt?lt3?? is questionable.

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Effects of Background Counting
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Effects of Background (continued)
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Effects of Background (continued)
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Minimum Detectable Activity
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Comparing Counting Systems
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Estimating Required Counting Times
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Optimal Division of Counting Times
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Statistical Tests

• The Chi-Square Test
• The t-Test
• The Treatment of Outliers
• Linear Regression.

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The Chi-Square Test

• The ?2 (chi-square) test is used to test whether
  random variations in a set of measurements are in
  fact consistent with Poisson distribution.
• The formula for calculating ?2 is

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The chi-square test (continued)

• The calculated ?2 value is then compared with the
  table (degree of freedom dfn-1)
• P is the probability that the observed variation
  in a series of n measurements from a Poisson
  distribution would equal or exceed the calculated
  ?2
• 0.02ltPlt0.98 are acceptable with P0.5 a perfect.
  Plt0.01 the variation is too large and Pgt0.99 is
  too good to be true. 0.01ltPlt0.02 and 0.98ltPlt0.99
  are suspicious.

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The t-test

• T-test is used to test the significance of the
  difference between the means from two sets of
  data that whether they are in fact from the same
  Guassian distribution.
• Different formulas are used for independent and
  paired data sets
• Independent data are obtained from two different
  groups whilst paired data have some kind of
  correlation between the two measurements.

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T-test (continued)

• The formula for the independent data sets is
  (with mean values X1 and X2, standard deviation
  SD1 andSD2, nn1n2-2 and dfn)

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T-test (continued)

• The formula for the paired data set is (with
  the difference average, Standard deviation of the
  pair differences SD, number of paired
  measurements n and dfn-1)

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T-test (continued)

• The calculated t (from independent and paired
  data) is then compared with the table.
• The probability of the data sets are in fact from
  the same Gaussian distribution is less than the
  tabled probability if the calculated t is larger
  than the tabled value. Less than 5 is considered
  to be different.

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Treatment of Outliers

• The following formula is used for the
  determination of whether or not a data value X is
  an outlier

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Treatment of Outliers (continued)

• If the calculated T is less than the tabled
  value, the probability of the data value to be an
  outlier is less than the tabled probability.
• Rejection of data must be done with caution.

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Linear Regression

• If we suspect that a parameter X is correlated
  with a measured quantity Y such as YabX

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Linear Regression (continued)

• we can use the following formula to determine a,b

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Linear Regression (continued)

• And carry out statistical analysis (r?1, strong
  correlation)

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Linear Regression (continued)

• A preferred test is t-test using the following
  formula (with number of X,Y pair, n and degree of
  freedom, dfn-2)