# Test to determine whether a variable changes more in one group than another

I'm trying to determine an appropriate statistical test for a study I'm undertaking. Below, I've tried to present a simplified version of the problem, using the terms dose and response which hopefully will allow the answer will be generalisable to other people reading this question.

I have two groups: one affected by a condition, and a control group.

Each group is given a treatment at a range of different doses. The response is measured at each dose:

As can be seen, the affected group always shows a higher response than the control group, and there is more variability between individuals in the affected group (error bars show standard error).

Note that at higher doses, a change in dose doesn't cause much change in the response. The interesting part is what happens at lower doses. I hypothesise that the rate of change (between dose and response) will be higher in the affected group at lower doses than the control group.

I'm looking for tests to determine:

• Given that dose clearly has an effect on the response in both groups, are the affected group significantly more affected than controls by a change in dose - particularly at the lower doses?

I am at a complete loss as to how I can achieve this. Any suggestions are welcome. I've tried constructing an ANOVA and looking at post-hoc pairwise comparisons for each group. However, since the dose levels are on a continuous scale, I wonder whether I should really be modelling these data first.

Is the affected group significantly more affected than controls by a change in dose - particularly at lower doses?

What you are describing is equivalent to assessing the significance of an interaction in a regression model. Namely, the interaction between group and dose in the model:

response ~ group * dose

Your question doesn't state explicitly what the response is, but judging by the higher variance with higher values and what looks like a plateau, you may want to perform a transformation of the response. The right transformation depends on the process generating the data, so I can't tell for certain unless you include more information on this response.

• +1 for the answer. Do take care that the dose-response relation does not look linear (the plateau mentioned in this answer). Transformation is indeed one option, but there are more techniques available (such as splines).
– IWS
Nov 1, 2017 at 9:56
• Thank you. Given that the data plateau, I'll have a look into log transformation first. The data are actually from a vision experiment. 'dose' is the duration of a light being on, and 'response' is likelihood of seeing the light. The scale is upside-down; i.e. a higher response actually means it's harder to see. That's why my 'affected' group have a higher response. Lower 'dose's are harder to see. I'll try a log transformation at first. Nov 2, 2017 at 18:00

It looks like you have repeated measures for each individual across time which need to be accounted for. The most appropriate model is probably a linear mixed model (or growth model) with individual as your random factor and time and time squared as your random effects, and time and time squared as your fixed effects. Then you can add in a dummy variable for Affected/Control as a fixed effect to see if the curve for your affected group is higher than your curve for your control group.