# What's the proper way to determine whether dose response curves +/- another variable (besides dose) are statistically different?

I have an experiment wherein I measured uptake of a certain molecule into cells when delivered using Carrier A vs Carrier B. In other words, for example, I delivered 1, 2, 3, 4, 8, and 10 nmoles of DrugX to 6 wells of cells per dosage, HOWEVER in 3 wells DrugX was delivered using Carrier A and in the other 3 wells DrugX was delivered using Carrier B. I would like to statistically determine whether the dose-response curve I get when I used Carrier A for delivery is any different than the dose-response curve I get when I used Carrier B for delivery.

My initial thought was to run a paired T-test, but then I realized I would have to report results at each point and I would like to report on the curve as a whole. My next thought was to use a two-factor ANOVA, however from what it looks like, when it determines whether or not there is variance due to the change in carrier it looks at the average and variance of all the responses. Another thought I had would be to calculate area under the curve, but I'm not sure how. I finally settled on running a nonlinear regression on the Carrier A data, then on the Carrier B data, then as the dataset as a whole and compared the resulting fit curves. I got a nice p-value, however the $R^2$ values are lower than I'd like (about 0.9) because the curves didn't model especially well.

What would you do? What is the industry standard?

• For the ANOVA you would want to focus on the interaction terms, those would tell you if there is a difference in the effect of drug depending on carrier for each level of the drug. Same thing for a regression, you should be comparing models that allow for interactions with carrier, not just a main effect of carrier (it's possible you did this, I can't really tell from your brief description). If the drug effect follows some expected pattern, a generalized linear model might be better than a nonlinear regression. I can't speak to industry standards, however. – Bryan Krause Apr 6 '17 at 23:00
• Might be worth asking this on stats.stackexchange, too? I have seen paired t-tests with results at each point reported in the literature, but I dislike this quite a bit. I suspect the answer depends on what aspect of the dose-response curve you care about: is it the saturation value? The 50% peak response? – AJK Apr 7 '17 at 0:25
• What is the expected shape of the dose-response curve in this case? Or if there is no expected shape, what do the curves look like? If there's a transformation of the data that makes it look linear, I would also suggest fitting a generalized linear model (i.e., with a non-linear link function) and looking for a significant main and/or interaction effect of Carrier B, as Bryan Krause suggested. If there isn't such a transformation, there are likelihood-ratio tests you can do using non-parametric regression, but you might not have enough data (see e.g. EDGE from the Storey lab). – Patrick B. Apr 7 '17 at 19:17

Good call on not using multiple t-tests. The reason being, assuming a 95% confidence interval, each time you run the test you are "allowing" a 5% possibility of making a Type I error - this means that you incorrectly reject the null hypothesis in favour of the alternative hypothesis. If you run one t-test, there's a five percent chance of making an error. If you run two t-tests, there's a ten percent chance of making an error.

Between a regression model, or ANOVA, it does not matter. As the mathematics behind each are pretty much equivalent.

What you really want to look for, is the slopes of both models. If the two models have a statistically significant difference between their slopes, then this is indicating an interaction effect.

Two examples of (some, but not all) possible interaction effects:

1. Carrier A & B have a similar effect on dose-response at 2mmol. But as the dosage approaches 10mmol, Carrier B causes a dose-response that is nearly double that of Carrier A.

2. Above 5mmol, Carrier B creates a greater dose-response. Below 5mmol Carrier A creates a greater dose response. At 5mmol the dose-response is nearly equivalent.

I agree, t-tests are not appropriate. I don't think an ANOVA is great here either because you are not concerned with an average. With dose response curves, you typically are more concerned with an effective dose or a lethal dose derived from a model fit to the data.

There are well-developed standards for this type of analysis as it is common in pharmaceutical development, ecology, and enzyme kinetics. The most commonly used and acceptable models used to analyze dose response data are 3 or 4 parameter logistic models that end up looking s-shaped. You can find a brief overview of this here.

If you are familiar with R, there is an entire statistical package developed to compare dose response curves called drc. This is a general description of the package and its use. This paper describes more concisely the use of the package and the theory involved.