# How do I test that there's a main effect and no interaction with one contrast?

Suppose there's a treatment applied to a cell culture to see if there's an effect on gene expression. The measurements are performed at two time points: 10 and 20 days after treatment. For the sake of simplicity, suppose that I work with the following model:

$$log y_{s, g} = \beta_{0} + \beta_{t}x_{t} + \beta_{20d}x_{20d} + \beta_{int}x_{t}x_{20d}$$

where $$x_{t}$$ is 0 for control and 1 for treatment and $$x_{20d}$$ is 0 for 10 days after treatment and 1 for 20 days. $$s$$ and $$g$$ are sample and gene index, respectively.

I want to select all genes where effect of treatment is the same between two time points. I can do it like so:

1. Use the full model and select genes where $$\beta_{int}$$ is not significant
2. For those genes I use the simpler model $$log y_{s, g} = \beta_{0} + \beta_{t}x_{t}$$ to select genes for which treatment $$\beta_{t}$$.

However, is there a way to write a single contrast that I can use to select genes with stable treatment effect over time and avoid doing multiple tests for a gene?

Thanks!

There are some misconceptions here. If you want to select genes where the effect is the same, then your step

1. Use the full model and select genes where $$\beta_{int}$$ is not significant

is not that. A non-significant test does not mean that the effect is the same. For example, consider a case where there is a difference but you have very little data and thus not enough power to detect it: the test will be not significant but there still is a difference.

Perhaps a better question to ask is whether the difference in gene expression after the 10th and the 20th day is significantly affected by treatment? To this end you could compute a new variable, which for each gene is the difference between the effect on the 10th and 20th day, i.e.

$$y_{new} = y_{10d} - y_{20d}$$,

and then check whether there is a significant effect of treatment:

$$y_{new} = \beta_0 + \beta_t x_t + \epsilon$$,

• "...the test will be not significant but there still is a difference." - well, that's conceptually true. But I'm ok with having some false negatives. What you're proposing seems to be a different hypothesis. I want to specifically select genes where treatment effect is the same for time points. Why isn't the interaction term a good choice? Commented Dec 24, 2020 at 19:58
• in that case why don't you just test whether $\beta_{int}$ is not significant and $\beta_t$ is significant in your full model?
– Jan
Commented Dec 24, 2020 at 22:54