# Effect of treatment of subjects with continuous traits

How do I find the effect of a treatment on a group with continuous traits?

Basically, I have a treatment that is given to a group of children who watch differing hours/kinds of TV programming. A child may watch $V$ hours of violent tv programming and $P$ hours of peaceful programming. I want to find out how the treatment effects my dependent variable, given how much violent/peaceful programming they watch.

I want to capture the effect of the hours of tv watched AND the kind of tv watched.

How do I specify a model like this?

I'm considering something like (VTV is violent tv hours, PTV is peaceful tv hours),

$$Y_i = \alpha_0 + \beta_0 TREAT +\beta_1\frac{VTV-PTV}{TVHOURS},$$ should I include an interaction term maybe? I am really struggling to understand the interpretation here.

If I include $VTV$ and $PTV$ and $TVHOURS$ separately, I run into some multicollinearity issues.

Are there any models that I can look into that might help me here?

• What kind of outcome variable do you have? Mar 18, 2018 at 23:10
• the outcome variable is $ln(y)$ where $y$ is continuous. Mar 18, 2018 at 23:18

Is it true that VTV + PTV = TVHOURS?

If yes, then PTV/TVHOURS = 1 - VTV/HOURS, so you could presumably include only the proportion of violent TV hours (out of all hours of TV watched) as a predictor variable in your model: VTV/TVHOURS.

Your model can then include TREAT, VTV/TVHOURS and their interaction. The model would enable you to determine whether there is a difference between treatment groups in the effect of the proportion of violent TV hours on your dependent variable.

• Yes, $VTV+PTV + TVHOURS$. If I include only $VTV/TVHOURS$, how do we differentiate between say someone who watches no tv vs someone who watches only peaceful tv? Assuming we set $VTV/TVHOURS$ to zero when they have watched no tv, wouldn't the model give the same interpretation in both cases? Mar 18, 2018 at 21:13
• Good point! Do you think you should restrict your model to those who watched some TV? It only makes sense to define that proportion for those subjects. If we define the proportion as zero for subjects who didn't watch any TV, we introduce confounding: we won't know whether the proportion is zero because they watched no TV or because they watched some TV but none of it was violent TV. Mar 18, 2018 at 22:55
• I guess the other thing you could do is to keep all subjects, define the proportion as zero in cases where the subject watched no TV and then include the total number of TV hours (or a binary indicator like "number of TV hours watched" with value 0 for 0 hours and 1 for more than 0 hours) in the model. Mar 18, 2018 at 23:01
• But the interpretation might be tricky still if you keep all subjects. If you include the binary indicator as suggested above, you would risk saying nonsensical things like: for those who watched no tv, the proportion of violent TV affects the dependent variable. Mar 18, 2018 at 23:05
• Yes, this is the problem I am running into as well. I would really like to keep all subjects, is there really no other way? Mar 18, 2018 at 23:06

Perhaps one way to keep all subjects in your model AND get a sensible interpretation would be to create a categorical variable called TV, which would have the following categories:

0 --> if subject watched no TV

1 --> if subject watched TV and a "small" proportion (or portion) of that watching was allocated to violent programs

2 --> if subject watched TV and a "moderate" proportion (or portion) of that watching was allocated to violent programs

3 --> if subject watched TV and a "large" proportion (or portion) of that watching was allocated to violent programs

Then you would throw TREAT, TV and their interaction in your model for ALL subjects.

You would have to decide how to define when the proportion VTV/TVHOURS is "small", "moderate" or "large" (basically, take the range 0 to 1 for a proportion and divide it into 3 segments).

This would enable you to say things like: among those who watched no TV, there was no evidence that treatment A was better than treatment B; among those who watched TV but only allocated a "small" portion of their viewing to violent programs, treatment A was significantly better than treatment B, etc. These types of interpretation would be meaningful.

• Right. This is exactly what we have been working with for now. We were hoping that there was some way to include the continuous variable in the model, and to avoid the somewhat arbitrary categorization. Mar 18, 2018 at 23:50
• I can't think of a way of doing that which would not run into the confounding issue mentioned earlier. Maybe others on this forum may have better ideas. Ayou can include the TV variable, as well as the VTV and TVHOURS into your model. Or exclude TV but include VTV and TVHOURS. The problem is with including the actual proportion of violent hours in the model - but the amount of watching should be fine. Mar 18, 2018 at 23:53