Regression

That two values move together is something correlation already tells you. Correlation is the degree to which two values move together, delivering direction and strength as one number.

The trouble is that most real questions can't be answered with that number. When Ravi reported a correlation of 0.8 between ad spend and new signups, what Grace asked was how much to spend next month. A 0.8 says they move together and carries nothing about how many signups per unit of spend.

There's also the problem that spend wasn't the only thing moving. A new release shipped in that period and the season changed. Correlation can't separate out how much of the signup increase belongs to ad spend. If the month spend went up happens to be the month the release landed, both correlations come back high.

Regression exists to handle both at once. You build an equation containing several variables and extract a number for each one's contribution to the outcome. That number becomes the answer for signups per unit of spend, with a condition attached: it holds while the other variables stay where they are. The condition is easy to forget and it is where most misreadings begin.

Before writing any equation, write down in one sentence what you are trying to explain.

1. Decide the value being explained. New signups, monthly revenue, seconds to save a document — something that comes out as a number. It goes on the left side of the equation.
2. Choose the variables that will explain it. Ad spend, season, whether the plan changed. What to avoid here is including a value that only came into existence after the outcome. Explain signups using first logins after signup and the R-squared reads 0.99 while the equation is worthless.
3. Clean the data. Decide how to handle blanks, whether to keep or drop conspicuous outliers, and whether units need aligning. This step consumes half the total time.
4. Fit the equation. The output is one coefficient per variable. A spend coefficient of 3.2 reads as: holding the other variables where they are, raising spend by one unit adds 3.2 signups.
5. Check how much gets explained. R-squared is the share of the outcome's variation the equation accounts for, and it isn't read alone. Plot the differences between predicted and actual values and look for a shape that leans one way.
6. Check whether each coefficient could be chance. That's what a p-value is for — the probability of a result like this arising from chance alone. Read the confidence interval alongside it. A coefficient of 3.2 with an interval from 0.4 to 6.0 is not a state in which you can quote a size.

Three situations suit it: predicting a numeric outcome, learning which of several factors contributes how much, and isolating one factor's effect while holding others fixed. The third is where the practical value sits, because it is the only one correlation cannot approximate.

Five limits.

1. It doesn't establish causation. A coefficient computed from observed data is still a measure of moving together. Claiming causation requires a randomized experiment. Causation is one thing actually producing another.
2. A missing variable distorts the result. Leave the real cause out of the equation and its share gets absorbed into the coefficients that remain. Omit the product launch and its effect rides on the spend coefficient, making advertising look stronger than it is.
3. It assumes a straight line. The basic form treats the outcome as rising at a constant rate as a variable rises. Where the relationship curves — spend ten times as much and the effect flattens — the equation stops matching.
4. Variables that move together make coefficients unstable. Include both ad spend and impressions, which barely move independently, and coefficients swing or flip sign.
5. It collapses outside the range it has seen. An equation built on monthly spend between two numbers has no basis for a figure thirty times larger, however confidently the arithmetic produces one.

Some cases don't suit it at all. When the outcome is a category rather than a number, the method changes: predicting whether someone will churn belongs to classification, not here.

"The spend coefficient is 3.2 with a confidence interval from 0.4 to 6.0. That's too wide to quote."

A coefficient exists but its size can't be asserted.

"You left the product launch out as a variable. That inflates the advertising effect."

A missing variable is distorting the other coefficients.

"R-squared is 0.97 and you used post-signup logins as a variable."

A post-outcome value went in, so the equation means nothing.

Pagedeck uses this when setting the ad budget. Ravi's first equation contained spend alone, with a coefficient of 4.1. Mia asked for season and product launch dates to go in as variables, and rerunning it brought the spend coefficient down to 2.6.

The number didn't get worse; until then spend had been absorbing other factors' shares. When Grace wanted to use the value directly as budget justification, Ravi added one caveat: the equation can only speak within the range of spend seen so far, and the effect of tripling the budget isn't in this data. Extending the line past the observations is arithmetic, not evidence.

The most common mistake is reading a coefficient as the size of a cause. Spend went into the equation and a coefficient came out, so it looks like the amount spend produced. But a coefficient from observed data is still a measure of moving together, and the phrase holding the others fixed applies only to variables you actually included. If a factor you left out is the real cause, the coefficient is standing in for that factor's share. Where causation has to be claimed, run a randomized experiment rather than resting on a regression.

Treating a high R-squared as a good equation is also frequent. A large explained share looks like good work. But the figure rises simply from adding variables, and including a post-outcome value pushes it toward 0.99. An equation built that way predicts the past well and can't be used for next month. If prediction is the purpose, the only way to know is applying it to a period that wasn't used in building it, and a drop in accuracy there is the signal that matters.