# Analysis of Sensitivity Dis **tributions fo**  **r Es timation**  **of Acut e Haz**  **ard timation**  **of Acut Conc**  **en trations t**  **o Aquatic Animals**

Mat thew E tte rs on

**1** 

# Mat thew E tte rs on Office of R esear ch and Development Mid

- Con tinen t Eco logy Division, Duluth, MN

U. S.EP A Scien tific Advisory P anel R eview January 31, 2012 Arling ton, V

A

# Talk Outline

1. What is an SSD?

2. Definitions

3. Methodological background

4. Seven questions about SSD performance

5. Future directions

**2**0. 7 0. 8 0. 9 1

0. 7 0. 8 0. 9 1 

# 1. What is an SSD?

0. 7 0. 8 0. 9 1

0. 7 0. 8 0. 9 1

0. 7 0. 8 0. 9 1

0. 7 0. 8 0. 9 1

0. 7 0. 8 0. 9 1 

## tri an gu la r no rmal lo gistic gu mbel pareto we ibu ll bu rr 100



# 1. Do we ha ve enough data? 2. Is the dis tribution of samples adequate? 3. How did we fit the statistical dis tribution? 4. Is the dis tribution a good r epr esen tation of the

-1 0 1 2 3 4 5 6 7 8 0 0. 1 0. 2 0. 3 0. 4 0. 5 0. 6

Log 10 L C50CumulativePercentile

-1 0 1 2 3 4 5 6 7 8 0 0. 1 0. 2 0. 3 0. 4 0. 5 0. 6

Log 10 L C50CumulativePercentile



## 3

-1 0 1 2 3 4 5 6 7 8 0 0. 1 0. 2 0. 3 0. 4 0. 5 0. 6

Log 10 L C50CumulativePercentile

-1 0 1 2 3 4 5 6 7 8 0 0. 1 0. 2 0. 3 0. 4 0. 5 0. 6

Log 10 L C50CumulativePercentile



### p = 0.05

-1 0 1 2 3 4 5 6 7 8 0 0. 1 0. 2 0. 3 0. 4 0. 5 0. 6

Log 10 L C50CumulativePercentile

-1 0 1 2 3 4 5 6 7 8 0 0. 1 0. 2 0. 3 0. 4 0. 5 0. 6

Log 10 L C50CumulativePercentile



### HC5 -1 0 1 2 3 4 5 6 7 8 0 0. 1 0. 2 0. 3 0. 4 0. 5 0. 6

Log 10 L C50CumulativePercentile 

# 4. Is the dis tribution a good r epr esen tation of the data? 5. Can we pr edict the location of the true HC5 with a specified lev el of confidence? 6. Did we choose the right distribution? 7. Wh y not fit multiple dis tributions?

Malathion Less sensitive → ← More sensitive

# 2. Definitions

• Previous definitions: SMAV, GMAV, HC5, MDRs, typical 3 OPP species

• **θ** = set of parameters that define the specific form of a distribution (for example for the normal distribution: **θ** = [ μ , σ ] )

• Reference value = empirical HC5 calculated using all available GMAVs

• AChEI = acetylcholinesterase inhibition

**4**

# 3. **Methodological Background** 

• Data

- Acute LC50 or EC50 from 96 h exposures (48 h for Cladocerans )

• Statistical distributions:

• normal, logistic, triangular, Gumbel

• Pareto, Weibull, Burr



# • Fitting methods (parameter estimation)

• maximum likelihood

• moment estimators

• graphical methods

**5**

# Parameter **estimation (Maximum Likelihood)**

  |

n

n 

# pdf x  

#  **θ** 



#   

#   ln |

n

n 

# L p df x  

#  **θ** 

# (normal: **θ** = [ μ , σ ] ) 

# 3. **Methodological Background** 

**6** n 

#       2
2 1 

# 1 ln 2 l n 2 2 n

i i 

# n L n x

  



 

#    





#        2
2 1 

# 1 ln 2 l n 2 2 n

i i 

# n L n x

   

 

#    





#  (normal: **θ** = [

μ , σ ] )

Fo r these analyses, maximum lik elihood was a va ilable f or all seven distributions.

# sample mean: = theor etical mean: f ( **θ** ) sample va riance: s 2 

# = theor etical v ariance: g ( **θ** )

x

Dis tribution par ameter s mean va riance es timator 1 es timator 2 

# Parameter **estimation (Moment Estimators)** 

# 3. **Methodological Background** 

**7** 

# Dis tribution par ameter s mean va riance es timator 1 es timator 2

normal

logis tic

triangular 2 2 

# 3   

# ˆ s

  ˆ x  



# ˆ x   ˆ 3

s 
  

#   2 

#  

#  

,    **θ** 

#  

,    **θ**



# Fo r these analyses, moment es timator s were av ailable f or the normal, logis tic, triangular
, Gumbel , and P ar eto dis tributions 

#  

, a b  **θ** 2

a b  

#   2



## 24 b a

 

# ˆ ˆ 2 a x b   ˆ 6

b x s  

# 1. Calculat e the empirical per cen tile:  

1 i i 

# rank x p n   

# Parameter **estimation (Graphical Methods)** 

Ex ample below uses normal dis tribution with μ = 0 , σ = 1 

# 3. **Methodological Background** 

**8** 

# 2. Calculat e ‘ sta ndar d’ quan tile:

3. Set up a linear model in z and x :
4. ‘Solv e’ the linear model using leas t

- squar es 

#   1 

# | 0 ,1 i i 

# z F p

 

# 

i i 

# x z    

Fo r these analyses, gr aphical methods wer e a va ilable f or only the normal, logis tic, and triangular dis tributions.25



# ˆ ˆ ln p HC p    

    

# Sampling variance and the HC5 **3. Methodological**  **Background**

0 0. 1 0. 2 0. 3 0. 4 0. 5 0. 6 0. 7 0. 8 0. 9 1 0 5 10 15 20

p 

## 9



# ˆ ˆ ln 1 p HC p p    

   

  



#     

#   2



# ˆ ˆ va r v ar var ln 1 p HC p p    

 
   

 

     2 

# ln 1 p
p        

    

# 4.1. How does sample size influence accuracy

**(bias and variance) of the estimated HC 5**  

# ?

Rationale:

• sampling variance decreases with increasing sample size

• Previous researchers have reported substantial bias with some fitting methods (especially maximum likelihood) at sample sizes in the range of interest (3 ≤ n ≤ 8)

**10**

# Methods:

• Simulated 10,000 replicate sets of data at 5 sample sizes (n = [3, 8, 20, 50, 100]) under seven generating distributions ( **simulation data** ). (based on malathion) 

# 4.1. How does sample size influence accuracy **(bias and variance) of the estimated HC 5**  

# ?

on malathion)

• Fit all seven distributions (normal, triangular, etc.) to each replicate using three different fitting techniques (where possible)

• Calculated bias and sampling variance for all combinations of generating distribution and fitted distribution **11**RatiooftheaverageestimatedHC5tothetrueHC5 

# 4.1. How does sample size influence accuracy

**(bias and variance) of the estimated HC 5**  

# ?

10 100 Ge ne ra ti ng d is tri bu ti on = tri an gu la r

Es ti ma ti on Me th od = m ax im um l ik el ih ood tri an gu la r no rmal lo gistic gu mbel pareto we ibu ll bu rr 100

**12** 

# Sample Siz eRatiooftheaverageestimatedHC5tothetrueHC5



# 3 8 20 50 100 0.01 0.1 15

# 4.1. How does sample size influence accuracy

**(bias and variance) of the estimated HC 5**  

# ?

3 8 20 50 100 0.01 0.1 1 10 100 Ge ne ra ti ng d ist ri bu ti on = tri an gu la r

Es ti ma ti on Meth od = m ax imu m lik elih ood trian gu la r no rmal lo gistic gu mbel pareto we ibu ll bu rr

3 8 20 50 100 0.01 0.1 1 10 100 Ge ne ra ti ng d ist ri bu ti on = b ur r

Es ti ma ti on Meth od = m ax imum likelih ood

10 100RatiooftheaverageestimatedHC5tothetrueHC5

Ge ne ra ti ng d ist ri bu ti on = tri an gu la r

Es ti ma ti on Meth od = m omen t e st ima to rs 10 100 Ge ne ra ti ng d ist ri bu ti on = b ur r

Es ti ma ti on Meth od = m omen t e st imat or s



## 13 3 8 20 50 100 0.01 0.1 1 10RatiooftheaverageestimatedHC Es ti ma ti on Meth od = m omen t e st ima to rs

3 8 20 50 100 0.01 0.1 1 10 Es ti ma ti on Meth od = m omen t e st imat or s

3 8 20 50 100 0.01 0.1 1 10 100



### Samp le Si ze

Ge ne ra ti ng d ist ri bu ti on = tri an gu la r Es ti ma ti on Meth od = graph ical est ima to rs

3 8 20 50 100 0.01 0.1 1 10 100



### Samp le Si ze

Ge ne ra ti ng d ist ri bu ti on = b ur r Es ti ma ti on Meth od = graph ical est imat or s

# 1, 000 10, 000 Ge ne ra ti ng d is tri bu ti on = tri an gu la r

Es ti ma ti on Me th od = ma xi mu m lik el ih ood tri an gu la r no rmal lo gistic gu mbel pareto we ib ull bu rr 10,000Widthof95%confidenceintervalaroundtheHC5 

# 4.1. How does sample size influence accuracy **(bias and variance) of the estimated HC 5**  

# ?

**14** 

# 3 8 20 50 100 1 10 100

Sample Siz eWidthof95%confidenceintervalaroundtheHC53 8 20 50 100 1 10 100 1,000 10,000 Ge ne ra ti ng d ist ri bu ti on = tri an gu lar

Est imat io n Me th od = m ax imu m likelih ood tri an gu la r no rmal lo gi stic gu mbel pareto we ibu ll bu rr

3 8 20 50 100 1 10 100 1,000 10,000 Ge ne ra ti ng d ist ri bu ti on = b ur r

Es ti mat ion Meth od = m ax imu m likelih ood

10,000Widthof95%CIaroundtheHC5

Ge ne ra ti ng d ist ri bu ti on = tri an gu lar

Est imat io n Me th od = m om en t esti ma tors 10,000 Ge ne ra ti ng d ist ri bu ti on = b ur r Es ti mat ion Meth od = m omen t esti ma tors 

# 4.1. How does sample size influence accuracy **(bias and variance) of the estimated HC 5**  

# ?

**15** 3 8 20 50 100 1 10 100 1,000Widthof95%CIaroundtheHC

Ge ne ra ti ng d ist ri bu ti on = tri an gu lar

Est imat io n Me th od = m om en t esti ma tors

3 8 20 50 100 1 10 100 1,000 Ge ne ra ti ng d ist ri bu ti on = b ur r

Es ti mat ion Meth od = m omen t esti ma tors

3 8 20 50 100 1 10 100 1,000 10,000

Samp le Si ze

Ge ne ra ti ng d ist ri bu ti on = tri an gu lar Est imat io n Me th od = gra ph ic al est ima to rs

3 8 20 50 100 1 10 100 1,000 10,000

Samp le Si ze

Ge ne ra ti ng d ist ri bu ti on = b ur r Es ti mat ion Meth od = g ra ph ic al est ima to rs

# Conclusions:

• When (3 ≤ n ≤ 8), bias is likely to be present (but direction depends on fitting method) **4.1. How does sample size influence accuracy (bias and variance) of the estimated HC**  **5** 

# ?

• When (3 ≤ n ≤ 8), sampling variance is likely to be large

• To control bias and variance, additional constraints must be imposed

**16**

# 4.2. How do taxonomic **requirements influence**  **accuracy of the estimated HC**  **5** 

# ? **4.2. How do taxonomic requirements**  **influence accuracy**  **of the estimated HC 5**  

# ?

Rationale:

• Previous result showing large bias and variance at sample sizes of interest (3 ≤ n ≤ 8)

• Imposition of data requirements is intended to ensure the inclusion of results from both sensitive and insensitive taxa

**17**

# Methods:

• Used data on 12 AChEI pesticides (8 organophosphates and 4 carbamates)

• Drew random samples of size n = 8 that either 

# 4.2. How do taxonomic **requirements influence**  **accuracy of the estimated HC**  **5** 

# ? **4.2. How do taxonomic requirements**  **influence accuracy**  **of the estimated HC 5**  

# ?

• Drew random samples of size n = 8 that either met or did not necessarily meet the 8 minimum data requirements of the 1985 guidelines ( **empirical data** )

• True HC5 unknown so I substituted the reference HC5 (estimated using the same distribution & fitting method) **18**no rmal lo gistic trian gu la r Gu mbel Pa reto We ibu ll Bu rr

0.0001 0.001 0.01 0.1 1 10 100 a. 8 GM AVs ch os en at ran do m 8 GM AVs m eetin g ALC M DRs

10 100estimatedusing57GMAVs

b. 

# 4.2. How do taxonomic **requirements influence**  **accuracy of the estimated HC**  **5** 

# ? **4.2. How do taxonomic requirements**  **influence accuracy**  **of the estimated HC 5**  

# ?

Maximum Lik elihood

**19** no rmal lo gistic tri an gu la r Gu mbel Pa reto 0.0001 0.001 0.01 0.1 1 10RatioofHC5atn=8toHC5estimatedusing57GMAVs

no rmal lo gi stic tri an gu la r 0.0001 0.001 0.01 0.1 1 10 100



### Dist ri buti on c.



# malathion Momen t Es timator s

Gr aphical Methodsno rmal lo gi stic trian gu la r Gu mbel Pa reto We ibu ll Bu rr

0.01 0.1 1 10 100 1,000 a. 8 GM AVs ch os en at ran do m 8 GM AVs meetin g ALC M DRs

100 1,000Widthof95%CIaroundtheHC5

b. 

# 4.2. How do taxonomic **requirements influence**  **accuracy of the estimated HC**  **5** 

# ? **4.2. How do taxonomic requirements**  **influence accuracy**  **of the estimated HC 5**  

# ?

Maximum Lik elihood

**20** no rmal lo gistic trian gu la r Gu mbel Par eto 0.01 0.1 1 10 100Widthof95%CIaroundtheHC

no rmal lo gi stic trian gu la r 0.01 0.1 1 10 100 1,000



### Dist rib uti on c.



# malathion Momen t Es timator s

Gr aphical Methods

# Conclusions:

• Impositions of taxonomic requirements introduces a negative (conservative) bias relative to reference values (estimated **4.2. How do taxonomic requirements**  **influence accuracy**  **of the estimated HC 5**  

# ? **4.2. How do taxonomic requirements**  **influence accuracy**  **of the estimated HC 5**  

# ?

relative to reference values (estimated using all data)

• Taxonomic requirements reduce sampling variance

**21**

# Rationale:

• 1985 Guidelines describe the use of graphical methods **4.3. How do different estimation methods**  **influence accuracy**  **of the estimated HC 5**  

# ? **4.3. How do different estimation methods**  **influence accuracy**  **of the estimated HC 5**  

# ?

• Previous researchers have reported substantial bias with maximum likelihood

• Likely avenues for future work would employ maximum likelihood

**22**

# Methods:

• Used the **simulation data**  described under question 1

• Used **empirical data**  described under **4.3. How do different estimation methods**  **influence accuracy**  **of the estimated HC 5**  

# ? **4.3. How do different estimation methods**  **influence accuracy**  **of the estimated HC 5**  

# ?

• Used **empirical data**  described under question 2

• Fit seven distributions to all sets of data (both simulated and empirical) using three methods: **maximum likelihood** 
, **moment estimators** 
, **graphic methods**  **23**

# Gener ating Distribution = normal, n = 3

0.01 0.1 1 10 100 Gen er at in g dist ri bu ti on = n orm al

Est ima ti on Me th od = m ax imu m lik elih ood

100RatiooftheaverageestimatedHC5tothetrueHC5

Gen er at in g dist ri bu ti on = n orm al

Est ima ti on Me th od = m omen t esti ma to rs 

# 3. How do different estimation **methods influence**  **accuracy of the estimated HC**  **5** 

# ? **3. How do different estimation methods**  **influence accuracy**  **of the estimated HC 5**  

# ?

**24** 0.01 0.1 1 10RatiooftheaverageestimatedHC Gen er at in g dist ri bu ti on = n orm al

Est ima ti on Me th od = m omen t esti ma to rs

trian gu la r no rmal lo gi stic Gu mbel Pa reto We ibu ll Bu rr 0.01 0.1 1 10 100



### Dist rib uti on

Gen er at in g dist ri bu ti on = n orm al Est ima ti on Me th od = g raph ical met ho ds

# Gener ating Distribution = normal, n = 3

0 10 100 1,000 10,000 Est ima ti on Me th od = m ax imu m lik elih ood

10,000Widthofthe95%confidencerangeofHC5estimates

Est ima ti on Me th od = m omen t esti ma to rs 

# 4.3. How do different estimation **methods influence**  **accuracy of the estimated HC**  **5** 

# ? **4.3. How do different estimation methods**  **influence accuracy**  **of the estimated HC 5**  

# ?

**25** 0 10 100 1,000Widthofthe95%confidencerangeofHC

Est ima ti on Me th od = m omen t esti ma to rs

trian gu la r no rmal lo gi stic Gu mbel Pa reto We ibu ll Bu rr 0 10 100 1,000 10,000



### Dist rib uti on

Est ima ti on Me th od = g raph ical met ho ds

# Conclusions:

• At (3 ≤ n ≤ 8) **maximum likelihood**  was usually positively biased (liberal)

• **Moment estimators** were usually least biased 

# 4.3. How do different estimation **methods influence**  **accuracy of the estimated HC**  **5** 

# ? **4.3. How do different estimation methods**  **influence accuracy**  **of the estimated HC 5**  

# ?

• **Moment estimators** were usually least biased

• **Graphical methods** were usually negatively biased (conservative )

• Sampling variance did not differ much between methods, but was least with **graphical methods**

**26**

# 4.4. Are measures of fit (goodness

**- of** 

**- fit,**  **Akaike’s Information**  **Criterion) reliable indicators**  **of performance? 4.4. Are measures of fit (goodness** 

**- of** 

**- fit,**  **Akaike’s Information**  **Criterion) reliable indicators**  **of performance?**

Rationale:

• When we fit a distribution to empirical data we will always get an HC5 estimate. Can we diagnose how good it is?

• Model selection criteria (AIC) may help to choose among competing distributions

**27**

# Methods:

• Used the **simulation data**

• Assessed goodness of fit for each replicate using 

# 4.4. Are measures of fit (goodness

**- of** 

**- fit,**  **Akaike’s Information**  **Criterion) reliable indicators**  **of performance? 4.4. Are measures of fit (goodness** 

**- of** 

**- fit,**  **Akaike’s Information**  **Criterion) reliable indicators**  **of performance?**

• Assessed goodness of fit for each replicate using parametric bootstrap sampling and tallied the proportion of fits with significant (at α = 0.05) lack of fit

• Tallied AIC c 

# weight (among 7 distributions) assigned to each sample

**28**

## n = 8 normal logistic triangular Gumbel Pareto Weibull Burr

normal 

# 0.1 **1** 0.1 1 0.08 0.07 0.03 0.07 0.09 logistic 

# 0.03 **0.04** 0.03 0.06 0.07 0.04 0.03GeneratingDistribution 

# Fit te d Dis tribution 

# 4.4. Are measures of fit (goodness

**- of** 

**- fit,**  **Akaike’s Information**  **Criterion) reliable indicators**  **of performance? 4.4. Are measures of fit (goodness** 

**- of** 

**- fit,**  **Akaike’s Information**  **Criterion) reliable indicators**  **of performance?**

**29** triangular 

# 0.05 0.10 **0.07** 0.07 0.05 0.09 0.05

Gumbel 

# 0.07 0.06 0.10 **0.02** 0.03 0.09 0.05

Pareto 

# 0.30 0.19 0.31 0.02 **0.01** 0.12 0.15

Weibull 

# 0.13 0.08 0.13 0.09 0.20 **0.05** 0.13

Burr 

# 0.07 0.1 1 0.12 0.05 0.06 0.09 **0.06**GeneratingDistribution



# Fr equency with which lack of fit is concluded at α = 0.05 **simulation data** (n = 8)

## n = 100 normal logistic triangular Gumbel Pareto Weibull Burr

normal 

# 0.07 0.15 0.45 0.57 1.00 0.75 0.13 logistic 

# 0.17 **0.07** 0.86 0.55 1.00 0.82 0.12GeneratingDistribution 

# Fit te d Dis tribution 

# 4.4. Are measures of fit (goodness

**- of** 

**- fit,**  **Akaike’s Information**  **Criterion) reliable indicators**  **of performance? 4.4. Are measures of fit (goodness** 

**- of** 

**- fit,**  **Akaike’s Information**  **Criterion) reliable indicators**  **of performance?**

**30** triangular 

# 0.10 0.23 **0.09** 0.63 1.00 0.73 0.17

Gumbel 

# 0.81 0.63 1.00 **0.03** 1.00 0.61 0.07

Pareto 

# 1.00 1.00 1.00 0.1 1 **0.05** 0.26 0.10

Weibull 

# 0.78 0.60 1.00 0.22 1.00 **0.04** 0.85

Burr 

# 0.20 0.05 0.88 0.49 1.00 0.87 **0.05**GeneratingDistribution



# Fr equency with which lack of fit is concluded at α = 0.05 **simulation data** (n = 100)

# n normal logistic triangular Gumbel We ibull Pareto Burr 

# AIC c 

# we igh t assigned to 10,000 samples of **log**

**- logistic**  data 

# 4.4. Are measures of fit (goodness

**- of** 

**- fit,**  **Akaike’s Information**  **Criterion) reliable indicators**  **of performance? 4.4. Are measures of fit (goodness** 

**- of** 

**- fit,**  **Akaike’s Information**  **Criterion) reliable indicators**  **of performance?**

**31** 

# 8 

# 0.14 **0.13** 0.20 0.12 0.17 0.23 0.01

20 

# 0.18 **0.19** 0.24 0.12 0.16 0.06 0.06

50 

# 0.22 **0.32** 0.17 0.07 0.09 0.00 0.13

100 

# 0.24 **0.46** 0.06 0.02 0.03 0.00 0.20

# Conclusions:

• Neither AIC c 

# nor goodness

- of

- fit tests are likely to help diagnose fit or choose among **4.4. Are measures of fit (goodness**

**- of** 

**- fit,**  **Akaike’s Information**  **Criterion) reliable indicators**  **of performance? 4.4. Are measures of fit (goodness** 

**- of** 

**- fit,**  **Akaike’s Information**  **Criterion) reliable indicators**  **of performance?**

c 

# likely to help diagnose fit or choose among distributions at (3 ≤ n ≤ 8)

• Both would be quite useful at larger sample sizes

**32**

# 4.5. What are actual **coverage rates of 95%**

**confidence limits?** 

Rationale:

• We wish to be able to estimate:

• But we can actually estimate: “the pr obability that the true HC5 is higher than a giv en concentr ation. ”



# • But we can actually estimate:

**33** 

# “the pr obability that a calculated lower boundary
concentr ation is less than the true HC5. ”



#   Pr 5 0 .95

n 

# x k s H

C    Pr(lower boundary conc. ≤ HC5) ≈ 0.95

Aldenber g & Slob 1993 

#   Pr 5 0 .95

n 

# x s H k C   

# Methods:

• Used both **simulation data** and **empirical data**

• Used published extrapolation constants ( k
n 

# , Aldenberg & 

# 4.5. What are actual **coverage rates of 95%**

**confidence limits?** 

• Used published extrapolation constants ( k
n 

# , Aldenberg & Slob 1993, and others) to calculate lower 95% confidence bounds on the HC5

• Applied the methodology to distributions in the location

scale family only (normal, logistic, triangular)

• Tallied the frequency with which the actual (or reference) HC5 was greater than the estimated 95% boundary

**34**

# Fit ting Method Dis tribution normal logis tic triangular 

# 4.5. What are actual **coverage rates of 95%**

**confidence limits?** 

**35** 

# normal logis tic triangular Maximum Lik elihood 0.95 0.96 0.94
Momen t Es timator s 0.94 0.95 0.93
Gr aphical Methods 0.95 0.97 0.95

Fr equency with which the calculated lower boundary is lower than the true HC5 f or **simulation dat a** 

# Fit ting Method Dis tribution normal logis tic triangular 

# 4.5. What are actual **coverage rates of 95%**

**confidence limits?** 

**36** 

# normal logis tic triangular Maximum Lik elihood 0.70 0.73 0.66
Momen t Es timator s 0.82 0.85 0.78
Gr aphical Methods 0.70 0.65 0.64

Fr equency with which the calculated lower boundary is lower than the r ef er ence HC5 f or **empirical data and MDRs not**  **met (n = 8)**

# Fit ting Method Dis tribution normal logis tic triangular 

# 4.5. What are actual **coverage rates of 95%**

**confidence limits?** 

**37** 

# Fr equency with which the calculated lower boundary is lower than the r ef er ence HC5 f or **empirical data and MDRs met**  **n = 8)**

. 

# normal logis tic triangular Maximum Lik elihood 1.000 0.999 0.998
Momen t Es timator s 1.000 1.000 1.000
Gr aphical Methods 0.999 0.997 0.989

# Conclusions:

• For data simulated from a known distribution 95% confidence limits provide
coverage at the nominal rate **4.5. What are actual coverage rates of 95%** 

**confidence limits?** 

coverage at the nominal rate

• For empirical data 95% confidence limits provide less than nominal coverage without data requirements and greater than nominal coverage with data requirements **38**

# 4.6. How do different candidate **distributions**

**perform relative to each other?** 

Rationale:

• Choice of distribution has been described as both important and difficult in existing literature

• There are many distributions to choose from!

**39**

# Conclusions:

• In general, distributions on transformed data performed better than distributions on untransformed data

• Of the above, triangular performed best across all 

# 4.6. How do different candidate **distributions perform**  **relative to each other?**

• Of the above, triangular performed best across all AChE inhibitors

• Of distributions on untransformed data, Burr performed best

• Regardless of the above points, there remains considerable uncertainty about which distribution performs best **40**

# 4.7. Does model

**- averaging**  **across distributions**

**improve estimates of the HC**  **5** 

# ? **4.7. Does model**

**- averaging**  **across distributions improve**  **estimates of the HC 5**  

# ?

Rationale:

• If no one distribution is best, perhaps model

- averaging methods can be used to generate a weighted average HC5 estimate across all distributions?

**41**

# Conclusions:

• Calculating a model

- averaged HC5 did not help with simulated data

• In contrast, with empirical data, weighted **4.7. Does model**

**- averaging**  **across distributions improve**  **estimates of the HC 5**  

# ? **4.7. Does model**

**- averaging**  **across distributions improve**  **estimates of the HC 5**  

# ?

**42** 

# • In contrast, with empirical data, weighted HC5 estimates were far less biased than estimates from a single distribution

• The reason for the different performance for simulated versus empirical data is not clear and should be investigated further

# 5. What future research is suggested

**by these results?** 

• We need more data (information), but problem is defined for (3 ≤ n ≤ 8)

• “Borrow” information from other species & chemicals

• Use hierarchical modeling techniques

**43**

#   EC/L C50 ~ n or

mal
,   

#   l ogistic
,

  

# 5. What future research is suggested

**by these results?** 

**44** 

#   EC/L C50 0, MOA



#      de Zwart (2002) 

#   EC/L C50 0,

    



#  

, f t axonomy ch emical   We b ICE

## 45