# 7.1.4.2. 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 LD_{50} or LC_{50}
(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 7.1.4.1.).

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 LD_{50} or LC_{50}. The LD_{50} or LC_{50 }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 LD_{50}
or LC_{50} 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).