Transcript Reference Interval Statistics

Topics
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Reference interval studies
The importance of seeing
Parametric, Non-parametric and other
Estimating the error of a reference interval study
Sample size for a reference interval study
Outlier exclusion
Partitioning for age, sex or other
Data mining techniques
Requirements for reference interval sharing
The importance of seeing
• This is a workshop on statistical techniques
• The human brain is a very powerful
mathematical engine
• The best inputs are graphical not numerical
• ALWAYS graph your data
• ALWAYS think about your data
The importance of numbers
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is Gaussian?
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Both!
Parametric statistics
• Can be used on parametric distributions
• Parametric distributions are those which can be
described by parameters
• Gaussian Distribution defined by 2 parameters:
• Mean (average) – indication of the center
• Standard deviation – indication of scatter
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Symmetrical distribution (not skewed)
68.3% within +/- 1SD
95.4% within +/- 2SD
99.7% within +/- 3SD
Non-parametric statistics
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No assumptions about distribution
Percentiles determined by ranking
Measure of centre is median (50th percentile)
Measure of scatter is percentiles (eg 2.5th and
97.5th)
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Non-parametric statistics
• xth percentile is X × (n+1)th lowest sample
• Example:
75th centile, n=138
75th = 0.75 x 139 th lowest sample
= 104th lowest
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Non-Parametric - numbers
• To determine two percentiles P% apart
• Need at least (100/P)-1 observations
• Examples
– 95th Centile (separate from 90th)
– Need (100/5)-1 = 19 observations
– 97.5th Centile (separate from 95th)
– Need (100/2.5)-1 = 39 observations
– 99th Centile (separate from 98th)
– Need (100/1)-1 = 99 observations
Robust Techniques
• Methods giving more weight to the more
common (central) values than to the peripheral
results
• Described by Amadeo Pesce
– Estimating reference intervals with n=20!
– Horne PS, Pesce AJ, Copeland BE. Clin Chem
1998;44:622-631.
• Techniques not readily available ***
• Data-mining techniques may be considered
“robust”
Confidence Intervals
• Reference interval studies are experiments
• There is “Experimental error”
• This is revealed when more than one reference
interval study is performed.
• Even if every other factor is the same, a different
sampling of a population will produce a different
result
• The confidence interval of the Upper and Lower
reference intervals describe this error
Statistical Imprecision of RI study
• Estimates of reference limits has limitations
• Expressed as the confidence interval of the
Reference Limits, eg 90% CI of the upper and
lower reference limits
• Confidence intervals decrease as the number
of people in the study increases.
Large n
Small n
CI - Parametric
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Mean +{z1s +/- z2*SQRT[s2/n + (z12*s2)/2n]}
s = SD
n=sample size
z1 = probit value related to percentile
(=1.96 for 97.5th percentile)
• z2 = covering factor for confidence level
(= 1.64 for 90%)
CI - Parametric
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• Mean = 20, SD = 10
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Confidence interval error
Reference Limits
• +/-2SD +/- 1.64*SQRT[s2/n + (1.962*s2)/2n]}
EXAMPLE
2.5th Centile, n=250
97.5th = 0.025 x (n-1) = 0.025 x 249 = 6th lowest sample
90% confidence interval is 3rd to 12th lowest samples
Outlier exclusion
• “some observation whose discordancy from the
majority of the sample is excessive in relation to the
assumed distribution model for the sample, thereby
leading to the suspicion that it is not generated by
this model.”
• A vital part of a reference interval study using
parametric or non-parametric statistics
• Particularly difficult with “logarithmic” data
– (BNP data)
Outlier exclusion
• Dixons criteria
• If D (distance of outlier from next sample) is > 1/3 x
R (range of entire data set): exclude
• For groups of outliers treat each individually
– NCCLS, Horn and Pesce
• Other: remove any data outside +/- 4SD
• “Reliable statistical detection of outliers in reference
interval data remains a challenge”
– Solberg and Lahti, Clin Chem 2005;51:2326-2332
Distributions
• Commonly “assumed” distributions
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Gaussian
Square root
Logarithmic
More skewed
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Box - Cox Transformations
• A family of transformations
y = (xλ - 1)/λ
y = ln(x+c)
λ=0
• Covers many forms of transformation
λ=1
linear transformation (unchanged)
λ = 0.5
square root transformation
λ = 0.2
skewed right (less skewed than log)
λ = 0 (or close to zero) logarithmic transformation
λ = -0.2 Heavily skewed right (more than log)
λ < 0 “Over-log” transformation
• Normalises data more skewed than log distribution
Transformations of ALT
Raw data
Lambda=1
“Over-Log”
Lambda=-0.5
Logarithmic
Lambda=0
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NHANES III: ALT, male, age 20 to 80, n=6423
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NORIP STUDY
ALT (U/L)
Female ALT (n=1220)
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Percent:
1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0
+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
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+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
Female URL: 45.6 (90% CI 42.5 – 49.3, n=1220)
Male URL: 68 (90% CI 63.4 – 73.6, n=1080)
Partitioning
• Provision of separate reference intervals for
subgroups
• Sex and age (paediatric & geriatric) most
common
• Others may include race, menopausal status,
stage of gestation or menstrual cycle.
• Historically Harris and Boyd has been
recommended.
• New theories
– Lahti A et al. Clin Chem 2002;48:338-352
Lahti et al
• Criteria depends on asterisk rate of subgroups
when common intervals are applied.
• <3.2% asterisk rate of either subgroup: NO
• >4.1% asterisk rate of either subgroup: YES
• In-between: consider other factors
• Note: non-parametric approach also described
– Very complex
– Clin. Chem., May 2004; 50: 891 - 900.
Data Mining
• Bhattacharya, LG. Journal of the Biometric Society.
1967;23:115-135.
• Example data: Frequency Distribution of the forkal length of
the Porgy caught by pair-trawl fishery in the East China Sea.
Bhattacharya
• Assumptions
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Gaussian or Log Gaussian distributions
Most results unaffected by reason for testing blood
Ideal for “profiles”
No systematic effect of source on results. Eg
• Inpatients with low sodium and albumin
• Outpatients with delayed separation
• Beware
– No confidence limits for results
– User-influence on results
GJ - Excel Bhattacharya
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Input Data
Bhattacharya fit
Battacharya
function
Included data
Zero line
Linear (Included
data)