# Supra-additive relations among timeseries (separate ratings of a single stimulus)

I conducted an online behavioural study (N=20) where people listened to a song with lyrics, and rated (in three counter-balanced conditions) the tension they perceived in the music alone (M), the lyrics alone (L), and in the song as a whole (M&L).

Subjects indicated their rating by moving the mouse to control a slider. They were encouraged to move the slider continuously so as to closely track their subjective tension rating, directing their attention to one or both of the M&L dimensions, as per the current condition. The slider returned data points no less than 50-100 ms apart.

One subject's ratings are shown below as an illustration.

I'd like to find out to what degree the "holistic" (M&L) rating contains "signal" that is above & beyond that found in the individual (M, L) ratings. This is admittedly quite vaguely construed, and so I'm not sure how to best operationalise this in my data analyses. Just computing correlations (or dynamic time warping) between the 3 signals pair-wise seems simplistic. Perhaps Markov-modelling can be of help, but I'm not sure how.

For more context, this is a pilot for a later study with brain measurements. The specific hypothesis I'm hoping these behavioural data weigh evidence for (or against), is that the brain processes that track tension in music and language integrate supra-additively, i.e. that "M&L" > "M"+"L", with the "+" loosely understood to be interaction rather than simple addition.

Why not model it as a linear regression? If you believe you have an interaction you can add it to the model. Then you can model it as well.

$$ML = aM+bL$$

You can then do hypothesis testing on the coefficients found for every subject.

Alternatively if you assume you have a single function describing all the subjects you can model it as a single regression with (or without) a random effect for the subject and get uncertainty estimates for the coefficients.

• By "hypothesis testing on the coefficients found for every subject" you mean for instance a one-sample t-test against zero for the coefficient of the interaction term? Aug 19, 2022 at 13:45
• I have high inter-subject agreement (ICC) in terms of all 3 ratings, so instead of the fine-grained option (RFX and FFX), I ran the regression on the averaged rating: ML ~ 1 + M*L. This returned significant coefficients for both predictors (M and L), each estimated at ~0.7; and also significant for their interaction term (M x L), estimated at ~0. My question is: how should I interpret the interaction term's close-to-zero (yet significant) slope, in the light of the supraadditivity hypothesis mentioned above? Aug 20, 2022 at 17:48
• Since you added the interaction then yes, I'd like to test the coefficient of the interaction for significance. As for the value being close to zero, I'd report it pretty much as you stated. It's approximately zero but the value we calculated is (what you found) and it's statistically significant. I don't know ilwho you need to report to, but assuming it's a workplace I'd report it as honestly as possible and discuss further actions. If it's a homework assignment I'd go for the grade and repot what the prof. Wants ;). Aug 20, 2022 at 20:29
• Also note that the significance of the result is dependent on your measurement frequency. If you reduce the sampling frequency, you have less samples and therefore your confidence intervals will become wider. So you need to discuss that as well. Say you measured in 1ms intervals instead of 100ms, your data will stay the same (people don't react that fast) but your sample size wil synthetically increase 100 fold and every result will become statistically significant. I think it's more important in this case that the value is close to zero than that it's significant. It's discussion worthy. Aug 20, 2022 at 20:34
• In other words, that the holistic condition can be obtained as a linear combination of the individual conditions (M and L), with no need to assume other types of interaction. To me, this all suggests there is no supra-additive interaction in the sense construed in my question!? Aug 21, 2022 at 13:05