Sensitivity analysis on a growth model

WHAT THIS TESTS This evaluates whether you can quantify leverage in an interconnected model correctly, normalizing across scales and accounting for interactions rather than eyeballing.

A GOOD ANSWER COVERS Local analysis: perturb one input at a time by a normalized amount, say plus and minus 10 percent, hold others at baseline, and record the resulting percent change in the long-term output to compute an elasticity, percent output change per percent input change. Normalizing to percentages makes inputs on different scales comparable. Because acquisition, engagement, and monetization interact, one-at-a-time can miss interactions and nonlinear regions, so complement with global sensitivity analysis: Monte Carlo sample the joint input space and use variance-based methods, such as Sobol indices, to attribute output variance to each input and to interactions. Rank inputs by elasticity or variance contribution, and tornado-chart the result. Validate around the operating region you actually care about.

COMMON WRONG ANSWERS Comparing absolute changes across metrics with different units, so the biggest-numbered metric looks most important by artifact of scale. Pure one-at-a-time analysis on a strongly interacting model, missing interactions. Perturbing inputs by inconsistent amounts. Analyzing far from the realistic operating range.

LIKELY FOLLOW-UPS Why normalize to elasticities rather than absolute deltas? When does one-at-a-time fail and global analysis become necessary? What does a Sobol index tell you that a local derivative does not?

ONE CONCRETE EXAMPLE You bump Day-7 retention by 10 percent and the 12-month user forecast rises 18 percent, an elasticity of 1.8; a 10 percent bump in ad-spend conversion rate raises it 6 percent, elasticity 0.6. Retention is the higher-leverage lever. A follow-up Monte Carlo with Sobol indices confirms retention dominates output variance and reveals a meaningful retention-by-monetization interaction.