Comparison of dose-response curves

A superior method for detecting interactions can also be detected by comparing the complete dose-response curves of an agent in the presence and absence of a second agent. This approach allows complete characterization of the dose-response curve, including slope, intercept and LD50 or LC50 (Johnson et al., 2006, 2009).

1) Preliminary toxicity bioassays are performed and the “non-killing” dose of the first agent is determined (steps 1-2 in the section

2) The dose-response of the second “killing” agent is determined by treating bees as recommended for oral, topical or foliage exposure (sections 3.2.1.-3.2.3.), with the exception that all bees are treated with a uniform dose of the “non-killing” agent before, or simultaneous with, administration of a the recommended series of doses of the “killing” agent. A control dose-response series, in which bees are not exposed to the “non-killing” agent at all, is also performed for comparison.

3) Each dose-response series should be repeated at least 3 times.

4) For analysis, the doses are transformed on a log scale and the mortality is transformed on a probit scale, and a dose-response line is fit (Fig. 10). Comparison of the dose-response curves can be performed using commercially available software such as PoloPC (Robertson et al., 2007) or using ‘glm’ in the R statistical package (R Development Core Team, 2010) (see section 7.3. for a sample script).

5) Three different tests are available to determine the presence of a significant interaction between agents by comparing dose-response curves.

- Comparison of the overlap of 95% confidence intervals around the calculated the LD50 or LC50.  The LD50 or LC50 values, and accompanying 95% confidence intervals, are calculated from the log-probit lines using Fieller's method, with correction for heterogeneity where appropriate (Finney, 1971). If the confidence intervals do not overlap, then the treatments are deemed significantly different. However, this test has been criticized for being overly conservative (Payton et al., 2003), it does not generate p-values and there is no method for correcting for multiple comparisons.

- A ratio test comparing the ratio of the LD50 or LC50 derived from the pair of dose-response curves can be performed. This test will produce the synergism or antagonism ratio and the associated 95% confidence interval. If the confidence intervals do not overlap “1”, then the treatments are deemed significantly different (Robertson et al., 2007). The ratio test does not generate a p-value and there is no method to correct for multiple comparisons.

- Interactions can be determined by comparing the dose-response lines using a test analogous to ANCOVA (Johnson et al., 2013). Models are fit using ‘glm’ in R with all data from both dose-response curves.  For the full model, the second “killing” agent serves as the covariate, and the presence or absence of the “non-killing” agent serves as a categorical factor. The interaction between the “killing” agent dose and “non-killing” agent is then compared using two simplified models with the explanatory power of the terms in the models assessed through a process of model simplification in reference to the likelihood ratio (Savin et al., 1977).  The first simplified model leaves out the interaction term and, when compared with the full model, tests for differences in slope between the dose-response lines.  The second simplified model leaves out the “non-killing” factor entirely and tests for evidence of an agonistic or antagonistic interaction between the two agents.  Model comparison using the likelihood ratio generates a p-value which may be adjusted for multiple comparisons using the Bonferroni correction for multiple comparisons.

Fig. 10. Test for synergistic interaction between thymol (an acaricide) and chlorothalonil (a fungicide) in bees. Symbols indicate raw mortality data for groups of bees treated with acetone (“*”, control, N=864) or chlorothalonil (“*”, N=467). Solid black and red lines are fit independently to data for acetone and chlorothalonil treatments, respectively. Curved dotted lines correspond to 95% confidence intervals. Dashed green lines were generated using a model where the slope is identical for both lines. The “Test of Parallelism” is a likelihood ratio test between the green lines and the red and black lines (deviance = 0.035, df=1,17, p-value= 1). The single dashed blue line represents a model fit to pooled data for both treatment groups. The “Test of Equality” is a likelihood ratio test between the blue line and the red and black lines (deviance = 10.449, df= 2,18, p-value < 0.0001).1298JDE revised Fig10