In media industry, "Engagement" word has been systematically diluted in its meaning by overusing it across several contexts without defining concretely what it means exactly. To someone it's Average Page Views / Visit
and to some it is defined as Total Engaged Minutes
(whatever that means). When I started my role as Product Data Analyst, success metrics of various A/B or Multivariate tests would involve vague things like  "increase engagement by 5%". Frustrated by absence of any robust definition, have embarked upon a project to answer this simplelooking question  "How do you define engagement for media or publishing sector ?"
Given a set of independent (can turn out to be correlated as we keep exploring) variables or predictors (ex  Pageviews/time, Frequency: sessions/time, Recency: time since last visit etc.), we may partition the problem at two levels  individual user level and aggregatelevel (Please refer to the The Brief, esp. the Normalization section).
Focusing our attention to userlevel scoring, we need to identify a subset (or all) of these variables with sufficient regression power to the target variables (ARPU, conversions). Given appropriate data (scaled, deduped, userlevel data; this is probably the hardest part of the project), a series of exploratory scatter plots of various pairs of (ith predictor, target variable), the correlation matrix of predictors and a simple multivariate linear regression (MVR) should be enough to propel us to the desired outcome. For example, let’s take the muchdiscussed engagement metric of Financial Times 
This can be expressed as a simple multivariate regression problem by taking logarithms on both sides
With right kind of data, this can easily be modeled with MVR. Now the “trick” of taking logarithm and subsequent modeling via MVR would be obvious when one explores the relationship between Volume and target (dependent) variables through the process of exploratory analysis.
Outline of Steps

Gather userlevel data. (let’s worry about aggregatelevel scoring later). This would involve merging multiple datasets, deduping and fair bit of datawrangling and dataenrichment.

Study exploratory scatter plots of various predictors and target variables. R (statistical software) has a special command for this  plotmatrix. Also study correlation matrix of various predictors.

Perform suitable data transformations (ex. log), scaling or normalization (zscoring etc.) etc.

Loosely regress target variable(s) into predictors (fancy of saying: run MVR). Various candidate predictors being considered are 
 Page views/time
 Average active time on article pages
 Article views/time
 Frequency: sessions/time
 Recency: time since last visit
 Referral channel (SEO, direct etc)
 Interactions (commenting, sharing, flagship product feature bespoke measures e.g. adding articles
to shortlist)
 User type/status: subscriber, registered or visitor
 Paywall hits/timeDependent variables  Value (ARPU, subs revenue, advertising revenue)  Conversions (visitor>member, member>subs, visitor>subs)

Using a combination of tstatistic(s) and Fstatistic (and some qualitative judgement) select a subset of “strong” predictors leaving out weaker ones. Fix any issues with scaling if we haven’t done it appropriately earlier.

Regress again with the reduced subset of predictors. Capture the coefficients. In the FT example above, this would approximately be (1,0.5,1).

Testing: Engagement scoring model needs ongoing management / improvement as our understanding about our audience improves. But some basic testing needs to be carried out 
 **Predictive Accuracy**:
Using Crossvalidation ensure predictive accuracy of the scoring model. A standard quantitative measure like MSE would be good.
 **Descriptive Accuracy**:
For example, assess questions like  What is the differential improvement of the engagement score with a small increase in  let’s say  commenting ?
Rationale: Why this would be a preferred approach ?
We can fit various exotic statistical models to the data. However simple MVR is preferred for couple of reasons 

Simplicity: MVR is simple and various nonlinear processes can be modeled by linear regression with sufficient accuracy.

Descriptive Power: Engagement scoring is not a predictive modelling task. We are not just concerned about predicting value given the predictors (in that case, Support Vector Machine would be a good starting point). Instead, we are deeply concerned about how each of the predictors influence (through direct and interaction effects) the overall score. Additionally, the score has to make sense qualitatively. The plot below explains it beautifully 
FYI, MVR falls under Least Squares in the plot. The key attribute we are looking here is  Interpretability. As the model complexity (flexibility) increases, its interpretability reduces commensurate with an increase in predictive power.