Transcript Regressie en Correlatie

Dose-response relationships
Tjalling Jager
Theoretical Biology
Dose-response analysis
This morning:
1. Introduction in effects assessment
2. Analysis of survival data
3. Analysis of continuous data
4. Problems with these methods
5. An alternative approach
Why effects assessment?
How toxic is chemical X?
– for RA of the production or use of X
– for ranking chemicals (compare X to Y)
– for environmental quality standards
Need measure of toxicity that is:
– good indicator for environment
– comparable between chemicals
Test organisms (aquatic)
Standardisation
Toxicity tests are highly standardised
(OECD, ISO, etc.):
–
–
–
–
species
exposure time
endpoints
test medium, temperature etc.
Types of tests
‘Acute’
– short-term
– usually mortality or immobility
– quantal or discrete response
‘Chronic’
– long-term
– usually sub-lethal endpoint
– graded or continuous response
Standard test set-up
Survival test
Survival test
After 2 days …
Reproduction test
Reproduction test
After 21 days …
Range of Concentrations
Plot response vs. dose
Response
What pattern to expect?
log concentration
Response
Linear?
log concentration
Response
Threshold, linear?
log concentration
Response
Threshold, curve?
log concentration
Response
S-shape?
log concentration
Response
Hormesis?
log concentration
Response
Essential chemical?
log concentration
Standard approaches
1. Statistical testing
2. Curve fitting
Contr.
Response
NOEC
*
LOEC
assumes threshold
log concentration
Standard approaches
Response
1. Statistical testing
2. Curve fitting
EC50
usually no threshold
log concentration
Standard summary statistics
NOEC
 highest tested concentration where effect is
not significantly different from control
EC50 or LC50
 the estimated concentration for 50% effect,
compared to control
Dose-response analysis
This morning:
1. Introduction in effects assessment
2. Analysis of survival data
3. Analysis of continuous data
4. Problems with these methods
5. An alternative approach
Available data
 Number of live animals after fixed exposure period
 Example: Daphnia exposed to nonylphenol
mg/L
0h
24 h
48 h
0.004
20
20
20
0.032
20
20
20
0.056
20
20
20
0.100
20
20
20
0.180
20
20
16
0.320
20
13
2
0.560
20
2
0
Plot dose-response curve
Procedure
– plot fraction survival after 48 h
– concentration on log scale
Objective
first: parametric analysis
100
survival (%)
– derive LC50
– (seldom NOEC)
80
60
40
20
0
0.001
0.01
0.1
concentration (mg/L)
1
What model?
Requirements
– start at 100% and decrease to zero
– inverse cumulative distribution?
survival (%)
100
80
60
40
20
0
0.001
0.01
0.1
concentration (mg/L)
1
Cumulative distributions
E.g. the normal distribution …
cumulative density
probability density
1
Distribution of what?
Assumptions
– animal dies instantly when exposure exceeds ‘threshold’
– threshold varies between individuals
– spread of distribution indicates individual variation
cumulative density
probability density
1
Concept of “tolerance”
cumulative density
1
80
60
20% mortality
40
20
0
0.001
0.01
0.1
1
concentration (mg/L)
probability density
survival (%)
100
20% mortality
What is the LC50?
cumulative density
1
80
60
?
40
20
0
0.001
50% mortality
0.01
0.1
1
concentration (mg/L)
probability density
survival (%)
100
50% mortality
Graphical method
Probit transformation
std. normal distribution + 5
mortality (%)
100
80
60
40
20
data
0
0.001
0.01
0.1
concentration (mg/L)
1
2 3 4 5 6 7 8 9
probits
Linear regression on probits versus log concentration
Fit model, least squares?
survival (%)
100
80
60
Error is not normal:
– discrete numbers of survivors
– response must be between 0-100%
40
20
0
0.001
0.01
0.1
concentration (mg/L)
1
How to fit the model
 Result at each concentration as binomial trial
 Probability to survive is p, to die 1-p
 Predicted p = f(c)
 Estimate parameters of the model f
– maximum likelihood estimation
– weighted least-squares …
– chi-square for goodness of fit …
1
Fit model, least squares?
survival (%)
100
80
60
40
20
0
0.001
0.01
0.1
concentration (mg/L)
1
Max. likelihood estimation
survival (%)
100
80
60
40
20
0
0.001
0.01
0.1
concentration (mg/L)
1
Which distribution?
Popular distributions
– log-normal (probit)
– log-logistic (logit)
– Weibull
ISO/OECD guidance document
A statistical regression model itself does not
have any meaning, and the choice of the
model is largely arbitrary.
Resulting fits: close-up
LC50
-log lik.
fraction surviving
1
0.9
Log-logistic
0.225
16.681
0.8
Log-normal
0.226
16.541
0.7
Weibull
0.242
16.876
0.6
Gamma
0.230
16.582
0.5
0.4
0.3
0.2
0.1
0
data
log-logistic
log-normal
Weibull
gamma
-1
10
concentration
Non-parametric analysis
Spearman-Kärber: wted. average of midpoints
survival (%)
100
 weights is number
of deaths in interval
 only for symmetrical
distributions
80
60
40
20
0
0.001
0.01
0.1
log concentration (mg/L)
1
“Trimmed” Spearman-Kärber
100
survival (%)
Interpolate at 95%
80
60
40
20
0
0.001
Interpolate at 5%
0.01
0.1
log concentration (mg/L)
1
Summary: survival
 Survival data are quantal data, reported as
fraction responding individuals
 Analysis types
– parametric (tolerance distribution)
– non-parametric (trimmed Spearman-Kärber)
 Model hardly affects LC50
 Error is ‘multinomial’
Dose-response analysis
This morning:
1. Introduction in effects assessment
2. Analysis of survival data
3. Analysis of continuous data
4. Problems with these methods
5. An alternative approach
Difference graded-quantal
Quantal: fraction of animals responding
– e.g. 8 out of 20 = 0.4
– always between 0% and 100%
– no standard deviations
Graded: degree of response of the animal
– e.g. 85 eggs or body weight of 23 g
– usually between 0 and infinite
– standard deviations when >1 animal
Analysis of continuous data
Endpoints
– In ecotoxicology, usually growth (fish) and
reproduction (Daphnia)
Two approaches
– NOEC and LOEC (statistical testing)
– ECx (regression modelling)
Derive NOEC
Contr.
Response
NOEC
*
LOEC
log concentration
Derivation NOEC
ANOVA: are responses in all groups equal?
H0: R(1) = R(2) = R(3) …
Post test: multiple comparisons to control, e.g.:
–
–
–
–
t-test with e.g. Bonferroni correction
Dunnett’s test
Fisher’s exact test with correction
Mann-Whitney test with correction
Trend tests
– stepwise: remove highest dose until no sign. trend
is left
What’s wrong?
 Inefficient use of data (most data are ignored)
 No statistically significant effect does not
mean no effect
– large effects (>50%) may occur at the NOEC
– large variability leads to high NOECs
 However, NOEC is still used!
Contr.
Response
NOEC
*
LOEC
See e.g., Laskowski (1995), Crane & Newman (2000)
log concentration
Regression modelling
Select model
– log-logistic (ecotoxicology)
– anything that fits (mainly toxicology)
Response
• straight line
• exponential curve
• polynomial
log concentration
Least-squares estimation
reproduction (#eggs)
100
80
60
n
SSQ   Ri (m eas.)  Ri (est.)
i 1
40
Note: lsq is equivalent to max.
likelihood, assuming normallydistributed errors
20
0
0.001
2
0.01
0.1
concentration (mg/L)
1
Example: Daphnia repro test
Standard protocol
– take juveniles <24 h old
– expose to chemical for 21 days
– count number of offspring daily
– use total number of offspring after 21 days
– calculate NOEC and EC50
Example: Daphnia and Cd
NOEC is (probably) zero
100
90
# juv./female
80
70
60
50
40
30
20
10
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
concentration
1.6
1.8
2
Example: Daphnia repro
Put data on log-scale and fit sigmoid curve
100
EC10
0.13 mM
(0.077-0.19)
90
# juv./female
80
70
60
EC50
0.41 mM
(0.33-0.49)
50
40
30
20
10
0
-2
10
-1
10
0
10
concentration
1
10
Regression modelling
Advantage
– use more of the data
– ECx is estimated with confidence interval
– poor data lead to large confidence intervals
Model is purely empirical
100
EC10
0.13 mM
(0.077-0.19)
90
80
# juv./female
– no understanding of the process
– extrapolation is dangerous!
70
60
EC50
0.41 mM
(0.33-0.49)
50
40
30
20
10
0
-2
10
-1
10
0
10
concentration
1
10
Summary: continuous data
Repro/growth data are ‘graded’ responses
– look at average response of animals
– not fraction of animals responding!
Thus: no ‘tolerance distribution’!
Analysis types
– statistical testing (e.g., ANOVA)  NOEC
– regression (e.g., log-logistic)  ECx
Dose-response analysis
This morning:
1. Introduction in effects assessment
2. Analysis of survival data
3. Analysis of continuous data
4. Problems with these methods
5. An alternative approach
Problems
Dilemma of risk assessment
Available data
Protection goal
• different exposure time
• different temperature
• different species
• time-variable conditions
• limiting food supplies
• interactions between species
•…
single time point
single endpoint
Response
Extrapolation?
LC50
ECx
NOEC
log concentration
Available data
Assessment
factor
Three LC50s
1000
One NOEC
100
Two NOECs
50
Three NOECs
10
‘Safe’ level for
field system
Where’s the science?
No attempt to understand process of toxicity
Response
 Dose-response approaches are descriptive
 Extrapolation through arbitrary ‘assessment factors’
 Ignores that LC50/ECx/NOEC change in time
LC50
ECx
NOEC
log concentration
Available data
Assessment
factor
Three LC50s
1000
One NOEC
100
Two NOECs
50
Three NOECs
10
Effects change in time
1
LC50
s.d.
tolerance
24 hours
0.370
0.306
48 hours
0.226
0.267
0.9
fraction surviving
0.8
0.7
0.6
0.5
24 hours
0.4
0.3
48 hours
0.2
0.1
0
0
0.1
0.2
0.3
0.4
concentration
0.5
0.6
0.7
Toxicokinetics
Why does LC50 decrease in time? Partly:
internal concentration
Change in time
depends on
1. chemical
2. test species
internal concentration
– effects are related to internal concentrations
– accumulation takes time
Daphnia
chemical B
chemical
small fishA
large
fish C
chemical
time
time
Chronic tests
With time, control response increases and all
parameters may change …
100
increasing time (t = 9-21d)
90
# juv./female
80
70
60
50
40
30
20
10
0
-2
10
-1
10
0
10
concentration
1
10
EC10 in time
survival
Alda Álvarez et al. (2006)
body length
cumul. reproduction
carbendazim
2.5
pentachlorobenzene
140
120
2
100
1.5
80
60
1
40
0.5
20
0
0
5
10
time (days)
15
20
0
0
2
4
6
8
10
time (days)
12
14
16
Toxicity is a process in time
 Effects change in time, how depends on:
– endpoint chosen
– species tested
– chemical tested
 Ignored by standardising exposure time
 No such thing as the ECx/LC50/NOEC
– difficult to compare chemicals, species, endpoints
Dose-response analysis
This morning:
1. Introduction in effects assessment
2. Analysis of survival data
3. Analysis of continuous data
4. Problems with these methods
5. An alternative approach
Biology-based modelling
Make explicit (but simple) assumptions on
mechanisms of toxicity
internal
concentration
in time
external
concentration
(in time)
toxicokinetics
toxicodynamics
effects
in time
Toxicokinetics
internal concentration
 Simplest form: 1-compartment model
 More detail in Module 2 …
time
Why do animals die?
Instant death at certain threshold?
lethal
exposure
lethal
exposure
?
?
Newman & McCloskey (2000)
Hazard modelling
 Chemical increases probability to die
 Effect depends on internal concentration
hazard rate
1 comp.
kinetics
NEC
hazard rate
blank value
internal concentration
survival in time
Example DEBtox
Results
Parameters are
• time-independent
• comparable between species and
chemicals
Use parameters to predict effects
• on different time-scale
• of time-varying exposure
• of different size animals
• of different chemicals
•…
Sub-lethal effects
Sub-lethal effects
Sub-lethal effects
toxicant
Sub-lethal effects
Dynamic Energy Budgets
assimilation
reproduction
maintenance
growth
DEBtox basics
 Effect depends on internal concentration
 Chemical changes parameter in DEB model
DEB parameter
toxicokinetics
NEC
DEB
blank value
internal concentration
growth and repro in time
Example DEBtox
Results
Parameters are
• time-independent
• comparable between species and
chemicals
Use parameters to predict effects
• on different time-scale
• of time-varying exposure
• of different size animals
• at population level
•…
Life-cycle data
 Follow growth/repro/survival over large part of
the life cycle
 Example:
– nematode Acrobeloides nanus
– exposed to cadmium in agar for
35 days
– body size, eggs and survival
determined regularly
Alda Álvarez et al. (2006)
Example: A. nanus and Cd
300
cumulative offspring
body length
60
50
40
30
200
100
20
0
5
10
15
20
25
30
35
time
0
0
5
10
15
20
25
30
35
time
fraction surviving
1
0.8
Mode of action: costs for growth
Parameters:
7 for basic life history
7 for chemical behaviour
0.6
0.4
0.2
0
0
5
10
15
20
time
25
30
35
Alda Álvarez et al. (2006)
Alternative approach
 Biology-based methods (DEBtox)
–
–
–
–
make explicit assumptions on processes
analyse all data in time
parameters do not change in time
basis for extrapolations
internal
concentration
in time
external
concentration
(in time)
toxicokinetics
toxicodynamics
effects
in time
Summary
Remember
Survival
 Usually acute
Growth / repro
 Usually (sub)chronic
Remember
Survival
 Usually acute
 Quantal response (dead
or alive)
Growth / repro
 Usually (sub)chronic
 Graded response
(#eggs, size)
Remember
Survival
 Usually acute
 Quantal response (dead
or alive)
 Needs at least 10
animals per dose
Growth / repro
 Usually (sub)chronic
 Graded response
(#eggs, size)
 Needs 1 animal per
dose (more for NOEC)
Remember
Survival
 Usually acute
 Quantal response (dead
or alive)
 Needs at least 10
animals per dose
 Analyse by finding
tolerance distribution or
non-parametric
Growth / repro
 Usually (sub)chronic
 Graded response
(#eggs, size)
 Needs 1 animal per
dose (more for NOEC)
 Analyse by standard
regression techniques
(curve fitting)
Remember
Survival
 Usually acute
 Quantal response (dead
or alive)
 Needs at least 10
animals per dose
 Analyse by finding
tolerance distribution or
non-parametric
 LC50, EC50 …
Growth / repro
 Usually (sub)chronic
 Graded response
(#eggs, size)
 Needs 1 animal per
dose (more for NOEC)
 Analyse by standard
regression techniques
(curve fitting)
 NOEC, EC50, EC10 …
Watch out!
Problems with standard analyses
– descriptive, no understanding of process
– statistics depend on exposure time
Alternative: biology-based
– make assumptions on mechanisms
– analyse effects data in time
Standard analysis may have role in risk
assessment but …
Science needs BB methods
Does food limitation increase effect of cadmium?
total juveniles after 15d
100
high food
80
low food
EC50
60
40
20
0
0
0.05
0.1
0.15
0.2
Cd concentration (mg/L)
0.25
Data Heugens et al. (2006)
Food limitation
assimilation
ad libitum
5%
reproduction
maintenance
growth
Food limitation
assimilation
limiting
50%
reproduction
maintenance
growth
Electronic DEB laboratory
Free downloads from
http://www.bio.vu.nl/thb/deb/deblab/
DEBtox
– Windows version 2.0.2. (2007)
– data from standard tests
DEBtool
– open source (Octave, MatLab)
– full range of DEB research
– advanced DEBtox applications