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Dose-response relationships

Dose-response relationships. Tjalling Jager Theoretical Biology. Dose-response analysis. This morning: Introduction in effects assessment Analysis of survival data Analysis of continuous data Problems with these methods An alternative approach. Why effects assessment?.

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Dose-response relationships

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  1. Dose-response relationships Tjalling Jager Theoretical Biology

  2. Dose-response analysis This morning: • Introduction in effects assessment • Analysis of survival data • Analysis of continuous data • Problems with these methods • An alternative approach

  3. 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

  4. Test organisms (aquatic)

  5. Standardisation Toxicity tests are highly standardised (OECD, ISO, etc.): • species • exposure time • endpoints • test medium, temperature etc.

  6. 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

  7. Standard test set-up

  8. Survival test

  9. Survival test

  10. After 2 days …

  11. Reproduction test

  12. Reproduction test

  13. After 21 days …

  14. Range of Concentrations

  15. Plot response vs. dose What pattern to expect? Response log concentration

  16. Linear? Response log concentration

  17. Threshold, linear? Response log concentration

  18. Threshold, curve? Response log concentration

  19. S-shape? Response log concentration

  20. Hormesis? Response log concentration

  21. Essential chemical? Response log concentration

  22. Contr. NOEC * LOEC Standard approaches 1. Statistical testing 2. Curve fitting Response assumes threshold log concentration

  23. EC50 Standard approaches 1. Statistical testing 2. Curve fitting Response usually no threshold log concentration

  24. 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

  25. Dose-response analysis This morning: • Introduction in effects assessment • Analysis of survival data • Analysis of continuous data • Problems with these methods • An alternative approach

  26. Available data • Number of live animals after fixed exposure period • Example: Daphnia exposed to nonylphenol

  27. 100 80 60 survival (%) 40 20 0 0.001 0.01 0.1 1 concentration (mg/L) Plot dose-response curve Procedure • plot fraction survival after 48 h • concentration on log scale Objective • derive LC50 • (seldom NOEC) first: parametric analysis

  28. 100 80 60 survival (%) 40 20 0 0.001 0.01 0.1 1 concentration (mg/L) What model? Requirements • start at 100% and decrease to zero • inverse cumulative distribution?

  29. 1 cumulative density probability density Cumulative distributions E.g. the normal distribution …

  30. Distribution of what? Assumptions • animal dies instantly when exposure exceeds ‘threshold’ • threshold varies between individuals • spread of distribution indicates individual variation

  31. 1 1 20% mortality cumulative density cumulative density 20% mortality Concept of “tolerance”

  32. 1 1 50% mortality cumulative density cumulative density 50% mortality What is the LC50? ?

  33. std. normal distribution + 5 100 100 80 80 60 60 mortality (%) 40 40 20 20 data 0 0 2 3 4 5 6 7 8 9 probits 0.001 0.001 0.01 0.01 0.1 0.1 1 1 concentration (mg/L) Graphical method Probit transformation Linear regression on probits versus log concentration

  34. 100 80 60 survival (%) 40 20 0 0.001 0.01 0.1 1 concentration (mg/L) Fit model, least squares? Error is not normal: • discrete numbers of survivors • response must be between 0-100%

  35. 1 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 …

  36. 100 80 60 survival (%) 40 20 0 0.001 0.01 0.1 1 concentration (mg/L) Fit model, least squares?

  37. 100 80 60 survival (%) 40 20 0 0.001 0.01 0.1 1 concentration (mg/L) Max. likelihood estimation

  38. 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.

  39. Resulting fits: close-up 1 0.9 0.8 0.7 0.6 fraction surviving 0.5 0.4 data 0.3 log-logistic log-normal 0.2 Weibull gamma 0.1 0 -1 10 concentration

  40. 100 80 60 survival (%) 40 20 0 0.001 0.01 0.1 1 log concentration (mg/L) Non-parametric analysis Spearman-Kärber: wted. average of midpoints • weights is number of deaths in interval • only for symmetrical distributions

  41. Interpolate at 95% Interpolate at 5% “Trimmed” Spearman-Kärber 100 80 60 survival (%) 40 20 0 0.001 0.01 0.1 1 log concentration (mg/L)

  42. 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’

  43. Dose-response analysis This morning: • Introduction in effects assessment • Analysis of survival data • Analysis of continuous data • Problems with these methods • An alternative approach

  44. 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

  45. Analysis of continuous data Endpoints • In ecotoxicology, usually growth (fish) and reproduction (Daphnia) Two approaches • NOEC and LOEC (statistical testing) • ECx (regression modelling)

  46. NOEC * Contr. LOEC Derive NOEC Response log concentration

  47. 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

  48. 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! See e.g., Laskowski (1995), Crane & Newman (2000)

  49. Regression modelling Select model • log-logistic (ecotoxicology) • anything that fits (mainly toxicology) • straight line • exponential curve • polynomial

  50. Least-squares estimation 100 80 60 reproduction (#eggs) 40 Note: lsq is equivalent to max. likelihood, assuming normally-distributed errors 20 0 0.001 0.01 0.1 1 concentration (mg/L)

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