# How to fit a model that forces all points through zero while allowing for interaction effects

I'm trying to build a model to predict the percentage of a target audience reached as a function of the amount spent on several media channels (e.g. TV and radio) and the type of campaign. The fitted values of the model should look something like the plot below; i.e. they should have following properties:

• a 'diminishing returns' relationship between reach and spend
• the slope of this relationship depends on the categorical 'campaign' variable
• reach should be zero when spend on all media channels is zero. (Note that while only one media channel is shown on the graph there will actually be two or three that I'm modelling).

I've been using linear regression - I'm open to nonlinear methods though. The diminishing returns property is simple (modelling the log of spend), but I'm having trouble building a model that includes the other two properties. Initially I thought that I could just include an interaction term between campaign and each media spend (to force the slope of the media spend effect to depend on the campaign type) and exclude an intercept (to force points through zero). However, the main effects of campaign type then cause nonzero reach at zero spend for the levels of campaign other than the baseline level. I can fit a model that specifically excludes the main effects of campaign while still including the interaction effects, but I understand that this is bad practice (this model also gives predictions that don't pass a sense-check - although I'm not sure if this is just because I'm fitting an overly flexible model to relatively little data).

Can someone please advise on methods I could try to build a model with the above properties?

• Can you elaborate what you mean by the other media channels? Then I will include them in my answer. Commented Jul 31, 2018 at 1:54
• Also, is it important to force the model to go through the origin? If you fit the model you describe, does it result in a model with a large intercept? Commented Jul 31, 2018 at 2:02
• 'Other media channels' means one predictor for radio spend and one predictor for tv spend. Both should have a diminishing returns relationship with reach, and reach should be zero when both tv spend and radio spend are zero. The aim is to be able to predict reach for a given tv spend and radio spend.
– jay
Commented Jul 31, 2018 at 23:35
• If the goal is prediction, you can compare the performance of forcing all curves to go through zero and not, since the former violates marginality. Can you include an example of what the data looks like? Commented Jul 31, 2018 at 23:40
• As for whether the model needs to be forced through the origin, not 100% necessary but strongly preferred. Some of the fitted values from the model will be presented to clients and they will (correctly) think it's strange if there is nonzero reach when there is zero media spend. As for a large intercept, it's relatively large for some of the levels of campaign if I exclude the interaction term.
– jay
Commented Jul 31, 2018 at 23:41