1. PARAMETRIC ANALYSIS: Use one of several linearizing functions (dose metamers) to provide data in a form that can be modeled statistically using the exponential family of functions (eg., probit, logit, weibull) and likelihood estimation techniques (eg., Generalized Linear Models --GLiMs). The three transformations shown below are essentially equivalent within the range of effect between 16% and 84%, which constitutes the linear portion of the cumulative frequency response.
"Probits" are Probability Units, based on the gaussian (bell) shape of most population-based dose response relationships. A probit = NED+5, where "NED" is a Normal Equivalent Deviate for a Normal distribution with mean of zero and standard deviation (NED) of one. Typically regress probit value against the log(10) of concentration, dose, or time.
Logits represent the log-odds of experiencing a quantal response. A logit is defined as: LOGIT = ln(P/1-P), where "P" is the cumulative frequency or proportion (0<P<1) of respondants within a population. (eg. Logit(0.25)= ln(0.25/0.75)=-1.10; Logit(0.5) = ln(0.5/0.5)=0.0). Logits have biological meaning, whereas probits do not. Shape of distribution between probit and logit are similar except at extreme values of P (eg., 0.8<P<0.2). Typically regress logit value against the natural log of concentration, dose, or time.
A flexible generalization of the exponential model, the Weibull Cumulative Distribution Function is F(x)= 1 - e^{ -(ax)l}, where a and l are greater than 0, and are referred to as the scale and shape parameters, respectively. The Weibull transformation (U=ln(-ln(1-P))) is often useful when modeling multistage processes (eg., carcinogenic action) and time-to-death processes (eg., survival or failure time analysis). Typically regress Weibull value against the natural log of concentration, dose, or time. The Weibull distribution is essentially equivalent to the logistic at values of P<0.50, but diverges greatly from the logistic and lesser so from the probit at values of P>0.80.
Comparison of Different Transforming Functions:
NED | PROBIT | LOGIT | Weibull | EFFECT LEVEL (%) |
---|---|---|---|---|
-5 | 0 | -9.21 | -9.210 | 0.01 |
-2 | 3 | -3.79 | -3.806 | 2.20 |
-1 | 4 | -1.66 | -1.754 | 15.90 |
0 | 5 | 0.00 | -0.367 | 50.00 |
+1 | 6 | 1.66 | 0.609 | 84.10 |
+2 | 7 | 3.79 | 1.339 | 97.80 |
+5 | 10 | 9.21 | 2.220 | 99.99 |
Once a linearizing function is selected, endpoint analysis can proceed in a variety of ways.
a. Litchfield-Wilcoxon Method -- A graphical method of probit analysis. Plot data on probability or probit paper. Calculate average slope between 16% and 84% effect level and interpolate to the LC50. Confidence intervals approximated as LC50*(2.77/slope^N^{0.5}) and LC50/(2.77/slope^N^{0.5}) for the upper and lower 95% bounds, respectively.
b. Maximum Likelihood Method -- Computationally intense technique that iteratively solves for parameters in a function, maximizing the probability that the observed values are predicted by the chosen model. Generalized Linear Models (GLiMs) utiltize maximum likelihood estimation and are distinct from General Linear Models (GLMs) which use least squares regression and form the basis of Analysis of Variance (ANOVA). GLiMs include two ingredients: the distribution of the response variable (eg., Binomial, Poisson, Gamma, Normal) and a function (eg., f(m_{C})= Probit, Logit, log, Weibull, or Identity) that links the mean response at each observed treatment level to a linear predictor such as f(m_{C}) = b_{0} + b_{1}C. Methods must be done by computer, and several software packages contain GLiM routines (eg., SAS, S-Plus, GLIM). The Probit, Logist, and Lifereg procedures in SAS make specific assumptions concerning probability distributions and link functions. The SAS Genmod procedure allows for complete flexibility in defining the models. ANOVA models are actually special cases of GLiMs, with the assumption of a Normal distribution and an Identity link function.
(no assumptions on specific models of dose response):
a. Trimmed Spearman-Karber Method -- Only assumptions made are that (1) the population response (eg., mortality) increases monotonically with dose, and (2) the response is symmetrical about the mean. Can be done by hand (whew), but more practical to use software packages (eg., ToxStat).
Data are adjusted so that dose response is monotonic.
The cumulative distribution is then trimmed at either end (usually the top and bottom 10%), and the scale is readjusted to fit within 0<p<1.
LC50 is calculated as the mean of the trimmed and readjusted cumulative distribution.
b. Moving Averages-- Uses angles of proportion responding-- uses Arcsin transformation
(sin^{-1}(P^{0.5})
Angles range between 0 and 90^{o} (Arcsin(0.5) = 45^{o})
Use a "3-point" Moving Average to Calculate Estimate
Log(conc): | P(mort): | Angle: | Average: |
---|---|---|---|
1 | 0.20 | 26.5 | |
1.5 | 0.30 | 33.3 | 34.0 |
1.6 | 0.45 | 42.1 | 42.1 |
1.7 | 0.60 | 50.8 | 56.7 |
1.8 | 0.95 | 77.1 |
Interpolate between concentrations that span 45^{o} (log(LC50)= 1.62)
C. Binomial-- If dose(A) yields P(0) and dose(B) yields P(1) then can use binomial distribution to calculate LC50:
log(LC50)= (log(A)*log(B))^{0.5}
(Can use values of P other than 0 and 1)
Confidence limits: lower-- dose(A), upper-- dose(B)
Coefficent of confidence= 1 - (0.5)^{n1} - (0.5)^{n2}
(eg. if n1 and n2 are equal and are 10, then the range: dose(A) to dose(B) represents a (1- (0.5)^{10} - (0.5)^{10}) = 99.8% CI)
If both n1 and n2 equal 6 or more, the confidence coefficient will always exceed 95%.
Often it is desirable to measure a toxicity endpoint other than some type of quantal response. Common endpoints in ecotoxicology are growth and reproduction. Since these are continuous responses, appropriate statistical modeling of these data has historically been approached in a different manner compared to quantal responses. Three approaches are illustrated below.
Example Data Set:
Compound A | Compound B | |||
Conc (mg/L) | Survival: | Reproduction: | Survival: | Reproduction: |
0 | 100 | 20 | 100 | 20 |
1 | 100 | 19 | 100 | 18 |
3 | 90 | 20 | 90 | 22 |
6 | 80 | 18 | 90 | 8 |
12 | 50 | 15 | 90 | 3 |
Compound A--
Survival Effect : NOEC = 6 LOEC = 12 ChV = 8.48
Compound B --
Reproduction Effect: NOEC = 3 LOEC = 6 ChV = 4.24
Problems with the "Hypothesis Testing" Approach:
1. Level of effect must be one of concentration levels tested
2. Currently, NOEC considered an "endpoint" -- Problem: statistically invalid to use failure to reject null hypothesis as "truth" without knowledge of Type II error (not usually determined). Failure to reject null hypothesis is highly dependent on experimental design (e.g., sample size, variance both influence "power").
3. No way to estimate confidence intervals around endpoint (does not use all data).
B.2. Interpolation method: Icp (non parametric)
B.3. Relative Inhibition Estimation (RIp): If the toxicity response is described as an inhibition relative to the control (i.e., zero dose group) treatment, one can develop a series of appropriate parametric statistical models for nearly any type of continuous (or quantal) response. If define response scale such that 1-p= m_{C}/m_{0}, where m_{C} is the response at concentration "c" and m_{0} is the control response, models can be developed that describe the relative inhibition of fecundity, biomass, growth, or mortality. All models take the same general form: f(m_{C}) = b_{0} + b_{1}C + b_{2}C^{2}, and are analyzed using GLiMs, but it is required that link functions and error distributions be selected depending upon the response type.
Response: | Link: | Link Function: | Probability Distribution: |
Mortality (p) | Logit | ln(p/1-p) | Binomial |
Probit | F^{-1}(p) | Binomial | |
Fecundity (m) | Log | log(m_{C}) | Poisson |
Biomass or Growth (m) | Log | log(m_{C}) | Gamma |
Identity | m_{C} | Normal |