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Yoonsang KimDulal K. BhaumikDulal K. Bhaumik
Robert D. Gibbons
Confidence and Prediction Intervals for Small Sample Asbestos Fiber Counts
based on Lognormal and Gamma Distributions
Asbestos Data
• For testing and laboratory assessment, the New York State Department of Health uses various types of airborne asbestos data measured by transmission electron microscopy (TEM).
• Collected as a part of the New York State Environmental Laboratory Approval Program.
• All fiber counts are expressed in structures/mm2. • Amosite type used in this study. • Asbestos samples were taken from 14 different locations, which are
denoted by samples, known to be contaminated • Sent to 35 laboratories to measure the fiber counts. Each asbestos
sample was measured by 25 to 29 laboratories.
2
Confidence Interval
• Confidence intervals for the mean asbestos fiber count are used for 1. if the confidence interval includes zero we do not have
evidence of asbestos contamination.
2. if this is a health based standard or a cleanup standard, the confidence interval can be used to determine if the data are either consistent with the standard, exceed the standard, or are below the standard, and the corresponding environmental impact decision can be made.
3
Prediction Interval
• Prediction intervals are useful for 1. testing individual new samples when a larger set of
background samples are available.
2. Interest is in determining the probability that the new count was drawn from the distribution of the background data.
3. In areas where low level asbestos counts are routinely observed, prediction limits are useful for identifying new areas of possible environmental concern.
4
5
Introduction
• For monitoring purposes, the most common measure of the asbestos concentration is the average number of counts and its upper confidence limit (UCL).
• Determining the fiber count distribution is a fundamental first step in deriving a test and constructing UCL of the mean.
• Asbestos fiber count (and size) data are usually right skewed. Often use the lognormal distribution to analyze.• H-statistic (Land 1973) is used to compute the UCL of the lognormal
mean (EPA guidance 2002).• Finkelstein (2008) used the t-test on log-transformed asbestos fiber
counts.
Introduction
• What is the problem?1. H-statistic is upward biased and difficult to implement in
practice (US EPA Singh et al. 1997, Krishnamoorthy et al. 2003)
2. The normal distribution based approach on log-transformed data requires large sample size. (e.g. t-test)
3. Such data sets are typically not large enough to provide adequate power for hypothesis testing and interval estimation.
6
Introduction
• The gamma distribution is widely used to analyze right-skewed data in various environmental monitoring applications.• Prediction and tolerance intervals to analyze alkalinity
concentration of groundwater (Krishnamoorthy et al. 2008, Aryal et al. 2008)
• US EPA proposed the use of a gamma distribution (2002) because the mean concentration may be overestimated based on the lognormal distribution (leading to an upward biased UCL).
• The gamma distribution has not been used properly to characterize asbestos concentration.
7
Introduction
• We propose methods for both lognormal and gamma distributions to analyze asbestos fiber counts.
• Lognormal based method: the generalized confidence interval (GCI) method (Krishnamoorthy and Mathew 2003).
• Gamma based method: Bhaumik et al. (2009), Krishnamoorthy et al. (2008), Aryal et al. (2008) are considered.
• Compute confidence intervals and prediction intervals for data collected from the New York State Dept of Health.
• Bayesian approach is also explored.
8
Lognormal distribution
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• Suppose X has a lognormal distribution log(X)~N(µ,σ2)
• Mean of X
• It is common to construct CI for E(X) based on Wald type statistic, provided the sample size is large.
• Cannot obtain the correct coverage probability without large sample size.
• In order to provide results for small samples, we explore the idea of generalized confidence intervals.
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22
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Lognormal distribution
• Check if asbestos data fit to lognormal distributions.• Anderson-Darling goodness-of-fit tests for log-transformed data:
normal distributions fit moderately well to 9 samples (p-values>0.09) out of a total 14 asbestos samples.
• Normal quantile-quatile plot (next slide).
10
Normal Quantile-Quatile Plot for log-transformed asbestos fiber counts data
11
Lognormal distribution doesn’t fit to samples 187Q, 7420, 5209
Generalized Confidence Interval
• Generalized confidence interval for the mean asbestos fiber count is • Useful to construct an interval for a function of
parameters of lognormal distribution, i.e.,
• Applicable to small sample sizes.
• Easy to compute.
12
)2/exp()( 2 XE
Generalized Confidence Interval
• The coverage probabilities of GCI are very close to the nominal levels.
• Comparison to Augus’s parametric bootstrap (1994) and Land’s (1973) methods.1. Confidence limits of GCI were very close to Land’s limit.
2. Parametric bootstrap results were unsatisfactory for small sample size and large variance of log(X).
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Generalized Confidence Interval
• Algorithm to construct GCI for the lognormal mean.1. Compute mean and variance of Y=log(X); denote them by .
2. Generate standard normal random variate Z and Chi-square random variate U2 with df=n-1.
3. Compute the generalized pivot statistic for µ+σ2/2.
4. Repeat steps 2 & 3 m times to obtain T1, …, Tm.
5. Compute 100(α/2) and 100(1–α/2) percentiles of T; denote them by Tα/2 and T1–α/2.→ CI for µ+σ2/2.
6. (exp(Tα/2 ), exp(T1–α/2)) is 100(1–α)% confidence interval for E(X).
14
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15
Generalized Confidence I ntervals
Asbestos Sample
2.5% 97.5% length coverage
2778 133.814 198.423 64.609 0.946 6001 347.088 446.871 99.783 0.950 3739 685.520 886.422 200.902 0.957 187Q* 696.021 937.035 241.014 0.943 4915 718.348 913.321 194.972 0.955 7420* 752.728 1038.391 285.663 0.945 5284 874.881 1048.481 173.600 0.947 8306* 1059.756 1515.303 455.547 0.943 6482* 1140.639 1675.967 535.328 0.956 5099 1505.401 1829.945 324.544 0.946 8214 1594.489 1902.151 307.662 0.949 879Q 1968.742 2342.955 374.213 0.954 5209* 5814.953 14488.370 8673.417 0.944 1987 7211.358 9521.924 2310.566 0.945
*; Lognormal distributions do not fi t these asbestos samples (p<0.05)
Coverage probability for GCI
i. Assume the true value of .
ii. Generate lognormal distribution with parameters .
iii. Compute confidence interval for η following the algorithm for GCI (steps 1 to 5).
iv. Assign 1 if the interval contains the true value of η, otherwise 0.
v. Repeat (ii – iv) many times (e.g., 5000 times).
vi. Proportion of 1s is the simulated coverage probability.
16
2ˆˆ 2
)ˆ,ˆ( 2
Gamma distribution
• Suppose X has a gamma distribution. Probability density function of gamma distribution is
• Mean of X is
• Check if asbestos data fit to lognormal distributions.• Based on Anderson-Darling goodness-of-fit tests, the gamma
distribution fit 13 out 14 asbestos samples (p-values>0.15). • Gamma quantile-quantile plots (next slide).
17
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Gamma Quantile-Quatile Plot for each of asbestos samples
18
Gamma distribution does not fit to the sample 5209.
Gamma distribution
• The confidence Interval by Bhaumik et al. (2009).
• Advantages: 1. Type 1 error rate is better than tests previously developed for a
gamma mean.
2. Does not depend on any unknown parameters under the null hypothesis.
• Disadvantage: Slightly conservative when the true shape parameter κ is smaller
than 0.5. However, for our asbestos data, the estimated shape parameter was larger than 2.
19
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20
Confidence I ntervals by
Bhaumik, Kapur, and Gibbons (2009)
Asbestos
Sample 2.5% 97.5% length coverage
2778 128.958 199.146 70.188 0.949 6001 339.484 445.861 106.376 0.953
3739 669.629 887.249 217.620 0.949
187Q 678.224 923.298 245.073 0.955
4915 701.560 923.905 222.345 0.954
7420 733.617 992.186 258.569 0.950
5284 859.905 1057.269 197.363 0.950
8306 1026.725 1458.669 431.944 0.951
6482 1100.815 1615.013 514.198 0.952
5099 1476.980 1846.848 369.868 0.953
8214 1567.768 1919.825 352.056 0.948
879Q 1933.656 2371.538 437.882 0.952
5209* 5118.761 9645.213 4526.451 0.954
1987 7031.485 9477.144 2445.659 0.948
*; Gamma distribution does not fi t this asbestos samples (p<0.05)
Coverage probability
• The confidence interval was obtained by inverting the test statistic T2 for H0: E(X)=µ0 .
• T2 has an approximate F distribution with df 1 and (n-1)
• Coverage probability can be computed based on T2.
1. Generate gamma random variables with parameters .
2. Compute T2. If T2 falls in Fα/2 and F1–α/2 , assign 1, otherwise 0.
3. Repeat steps 1 and 2 many times.
4. Proportion of 1s is the simulated coverage proability.
21
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2xxn
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ˆ,ˆ
Gamma distribution
• Prediction Intervals for a single new asbestos fiber.1. Krishnamoorthy, Mathew, and Mukherjee (2008)
2. Aryal, Bhaumik, Mathew, and Gibbons (2008)
22
Gamma distribution
• Krishnamoorthy et al. (2008)1. Wilson-Hilferty normal approximation used; the cubed root of a
gamma random variable has an approximate normal distribution.
2. Suppose X is an asbestos fiber measurement and Z=X1/3. The 100(1–α)% prediction interval for X is
3. Advantages: • Easy to compute.• No need to estimate parameters κ and θ.
4. Drawback: • The lower limit can be negative (commonly set to 0).
23
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StZ Zn
Gamma distribution
• Aryal et al. (2008)1. A log-transformed gamma random variable has approximately
a normal distribution when the shape parameter κ is large (κ>7).
2. The approximation normal distribution has a mean and variance,
where ψ( ) is a digamma function and ψ’( )is a trigamma function.
3. The 100(1–α)% prediction interval for X is
are maximum likelihood estimators.
24
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t n1
1ˆˆexp 1,2/1
ˆ,ˆ
Gamma distribution
• Aryal et al. (2008) (cont.)1. Advantage:
• The lower limit is never negative.• Can be used even when the shape parameter κ is smaller
than 7. We obtained coverage probabilities close to the nominal level when κ estimates are small too (see the table).
2. Drawback:• Parameters should be estimated. However, still not difficult to
compute.
25
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Prediction I nterval
by Krishnamoorthy et al. (2008)
AsbestosSample
2.5% 97.5% length coverage
2778 45.512 345.304 299.792 0.948 6001 187.446 665.016 477.570 0.950 3739 360.880 1337.184 976.304 0.949 187Q 353.847 1408.010 1054.163 0.956 4915 383.953 1381.438 997.485 0.943 7420 372.842 1538.780 1165.938 0.949 5284 572.305 1432.671 860.366 0.954 8306 459.982 2367.617 1907.635 0.952 6482 448.018 2704.219 2256.201 0.955 5099 945.542 2555.547 1610.005 0.951 8214 1030.173 2623.781 1593.608 0.947 879Q 1294.393 3201.672 1907.279 0.950 5209* 1095.369 18640.198 17544.829 0.946 1987 3498.026 14814.550 11316.524 0.951
*; Gamma distribution does not fi t the data (p<0.05).
27
Prediction I nterval
by Aryal, Bhaumik, Mathew and Gibbons (2008)
Asbestos Sample
2.5% 97.5% length coverage
2778 53.391 385.036 331.645 0.949 6001 199.969 690.802 490.833 0.949
3739 386.929 1392.959 1006.030 0.946
187Q 381.533 1473.813 1092.280 0.947
4915 410.808 1437.324 1026.516 0.946
7420 398.791 1624.280 1225.489 0.945
5284 594.231 1457.777 863.546 0.945
8306 506.886 2539.025 2032.139 0.951
6482 503.747 2946.278 2442.531 0.946
5099 987.659 2610.042 1622.383 0.946
8214 1070.138 2673.999 1603.861 0.950
879Q 1342.704 3257.939 1915.235 0.947
5209* 1364.464 23495.108 22130.644 0.945
1987 3793.839 15603.382 11809.543 0.952
*; Gamma distribution does not fi t the data (p<0.05).
Bayesian Intervals
• Specify prior knowledge about parameters or the quantity of interest as a prior density.
• Miller’s (1980) conjugate prior density for the gamma parameters;• Denote X~Gamma(к,β) for 1/θ=β. The joint conjugate prior
density for (к,β) with hyperparameters (p,q,r,s) is
where C is a normalizing constant.• Indicates past data or hypothetical experiment with a sample
size r (=s), a sum of observations q, a product of observations p.• Non-informative prior density was used.
28
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1),( 1
1
qpC r
s
1),(
Bayesian Intervals
• The posterior density of (к,β) is
• How to construct Bayesian confidence interval?1. Simulate 10,000 draws (кℓ,βℓ) from the posterior distribution
above for ℓ=1, …,10,000.
2. Compute кℓ/βℓ for all ℓs. These are treated as random draws from the posterior density of a gamma mean; no need to derive a posterior density of the mean.
3. Compute the highest posterior density (HPD) interval based on the draws к1/β1,…, к10000/β10000. This interval corresponds to the confidence interval for the mean of gamma distribution.
29
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Bayesian Intervals
• Highest Posterior Density (HPD) interval• Calculated in a way that values within the interval have higher
probability than values outside the interval.• i.e., the HPD interval (L,U) satisfies the equation,
• Used Chen and Shao’s algorithm (1999) to compute (L,U) ; built-in function in the R package boa (Smith 1997)
30
),|(),|( 11 nn xxUpxxLp
Bayesian Intervals
• How to construct Bayesian prediction interval?1. Simulate 10,000 draws (кℓ,βℓ) from the posterior distribution of
(к,β) for ℓ=1, …,10,000.
2. Simulate 10,000 draws x* from then gamma density with drawn values in step 1 as parameters (i.e., from the posterior predictive density).
3. Compute HPD interval based on drawn values of x*. This corresponds to the prediction interval.
31
Bayesian Intervals
32
• Asbestos sample 2778:(a) Contour plot of the joint posterior distribution of the (к,β).
(b) Histogram of marginal posterior distribution of к.
(c) Histogram of marginal posterior distribution of β.
Bayesian Intervals
• Asbestos sample 2778:(d) Histogram of posterior distribution of the mean к/β; 95% HPD interval is
(131.5,189.5).
(e) Histogram of posterior predictive distribution for a new observation; 95% interval is (31.4, 307.1).
33
34
Bayesian 95% HPD interval for the mean asbestos fiber count
Asbestos Sample
Lower Upper length coverage
2778 131.561 189.529 57.967 0.949 6001 343.811 437.293 93.482 0.948
3739 676.982 868.047 191.065 0.960
187Q 688.741 901.491 212.750 0.956
4915 709.600 905.910 196.309 0.957
7420 742.551 969.267 226.715 0.958
5284 861.079 1043.956 182.877 0.966
8306 1044.585 1418.159 373.574 0.948
6482 1120.526 1555.327 434.801 0.954
5099 1484.859 1824.875 340.016 0.963
8214 1573.404 1898.674 325.270 0.963
879Q 1941.713 2350.662 408.949 0.963
5209* 5268.752 8913.364 3644.612 0.948 1987 7126.943 9254.154 2127.211 0.956
35
Bayesian 95% HPD interval for a new
single asbestos observation Asbestos Sample
Lower Upper length coverage
2778 34.642 309.058 274.416 0.945 6001 174.617 623.352 448.735 0.944
3739 326.944 1247.083 920.139 0.944
187Q 316.091 1302.812 986.721 0.939
4915 357.181 1298.953 941.772 0.941
7420 332.407 1437.482 1105.075 0.941
5284 561.918 1382.222 820.304 0.944
8306 396.128 2180.309 1784.181 0.944
6482 359.784 2448.767 2088.983 0.941
5099 930.007 2453.735 1523.728 0.945
8214 1006.871 2536.357 1529.487 0.944
879Q 1258.247 3067.029 1808.782 0.946
5209* 356.444 15699.855 15343.410 0.945 1987 3173.504 13823.135 10649.631 0.944
Coverage probability for Bayesian Intervals
1. Generate gamma data with fixed parameters (к,β) Note: this is not exactly the Bayesian idea. But this was done to
compare its performance to other interval’s.
• Bayesian confidence interval:2. Compute HPD interval for the mean. If it contains the true value
of к/β, assign 1, otherwise 0.
3. Repeat this many times, and compute proportion of 1s.
• Bayesian prediction interval.2. Compute HPD interval for a single gamma variable.
3. Generate a single value from gamma distribution with (к,β).
4. If this new single value falls in the HPD interval, assign 1.
5. Repeat this many times, and compute proportion of 1s.
36
Conclusion
• We explored o several approaches for obtaining confidence intervals for
average asbestos fiber count and o prediction intervals for a single new asbestos count o based on lognormal and gamma distributions for small samples.
• Overall, Bayesian approach provides shorter length than non-Bayesian approaches, except that the length of GCI for a lognormal mean is as short as its corresponding HPD interval.
• The GCI and HPD interval can be used as good alternative methods to Land’s (1973) H-statistic based confidence limit.
• All methods we considered have good coverage probability.• The methods we explored can be used to characterize asbestos
fiber size measurements (length and diameter).
37
Appendix 1
• AHERA: Asbestos Hazard Emergency Response Act (1986); legislation requiring the cataloging of asbestos containing building materials in schools.
• Structures/mm2: asbestos structures, as defined by AHERA (fiber, bundle, matrix or cluster), per square millimeter of filter; reporting units for AHERA TEM analyses.
• http://www.eia-usa.org/fact-sheets/asbestos/
Appendix 2
• Type of asbestos:
The term “asbestos” refers to six fibrous minerals that have been commercially exploited and occur naturally in the environment. The U.S. Bureau of Mines has names more than 100 mineral fibers as “asbestos-like” fibers, yet only six are recognized regulated by the U.S. government. The six asbestiform minerals recognized by the government include, tremolite asbestos, actinolite asbestos, anthophyllite asbestos, chrysotile asbestos, amosite asbestos, and crocidolite asbestos.
• Amosite asbestos is identified by its straight, brittle fibers that are light gray to brown in color. Amosite is also referred to as brown asbestos and its name is derived from the asbestos mines located in South Africa. In years past, amosite was often used as an insulating material and at one time it was the second-most commonly used type of asbestos. Throughout recent decades, commercial production of amosite has decreased and its use as an insulating material has been banned in many countries.
• http://www.asbestos.com/asbestos/types.php
• Amosite asbestos is identified by its straight, brittle fibers that are light gray to brown in color. Amosite is also referred to as brown asbestos and its name is derived from the asbestos mines located in South Africa. In years past, amosite was often used as an insulating material and at one time it was the second-most commonly used type of asbestos. Throughout recent decades, commercial production of amosite has decreased and its use as an insulating material has been banned in many countries.
