Normal Distribution
nothing is called 95% until it has been counted
Concept

R² — where it appears

The share of an outcome's sample variance that a fitted model accounts for. It is a function of the predictor's spread as well as of the relationship, and for a simple regression it determines the t statistic exactly given the sample size.

Named by 5 essays across 3 fields — each of them below, with the objects they name alongside it.

One relationship at five designs, residual spread 1.00. Every panel has the same slope of 1, the same intercept of 0 and the same residual standard deviation of 1.00. Only the range of x differs. R-squared runs from 0.021 to 0.849, and the estimated residual spread is 0.9932 in all five.

R² is a property of the design

One line, one residual spread, five studies that differ only in how far apart they placed their x values. R² runs from 0.021 to 0.849 and the estimated residual spread is 0.993 in every one of them. Nothing about the relationship changed.

spread · Summary Two studies with an R-squared of 0.85. The left study's points sit 0.50 from the line and the right study's 2.00 — a factor of 4.0. Both report an R-squared of 0.85 and, at the same sample size, the same standard error for the slope. The design was chosen to make it so, and it can always be chosen.

The t statistic wearing different clothes

For a simple regression, t² = (n − 2)R²/(1 − R²), exactly, on every dataset — checked to sixteen significant figures over five hundred fits. So a paper reporting R² and a p-value has reported one number twice, and two studies with the same R² have points four times further from the line.

spread · Summary R² against the number of useless predictors, n = 30. The response is pure noise and so is every predictor, so the true relationship is nothing at all. R² rises from 0.000 to 0.648 anyway, following k/(n − 1) — which is what a criterion that rewards higher R² is actually rewarding.

R² is not a measure of fit

Adding a predictor with no relationship to anything cannot reduce R², and in expectation raises it by 1/(n − 1). Twenty useless predictors on thirty points give an R² of 0.69 from pure noise.

regression · Summary The damage and the warning, against the same dial. Two readings at each persistence. In the darker colour, how often a regression between two independent series of 200 steps is called significant at 5%: 4.9% at φ = 0, 34.2% at 0.8, 52.4% at 0.9, 83.4% at a unit root. In the lighter, how often the standard unit-root test refuses a unit root on one of those series — the chance the analyst is told the series is stationary and may be regressed: 87.2% at φ = 0.9 and 31.9% at 0.95. At φ = 0.9 both are high at once, which is a correct diagnostic licensing a regression that is wrong half the time.

The cliff that is a slope

A regression between two independent series is called significant 4.9% of the time at no persistence, 52.4% at a lag-one correlation of 0.9, and 83.4% at a unit root. The rule the field offers asks whether the last of those holds, and at 0.9 the unit-root test correctly refuses one 87.2% of the time.

timeseries · Spurious Six cells, and 5% is the right answer in all of them. How often a regression between two independently generated series is called significant at the 5% level, for two worlds and three treatments, at 200 observations. Every pair is independent by construction, so 5% is correct everywhere and every other reading is a failure. Untreated: 82.9% and 100.0%. With a fitted line removed: 74.2% and 33.5%. Differenced: 5.0% and 5.2%. The treatment that controls the rate in both worlds is the one that discards the level and the trend, which is the quantity a study of trending series was about.

The repair that keeps the question

A regression between two independent trending series is significant 82.9% of the time on random walks and 100.0% on trend-stationary ones. Subtracting a fitted line leaves 74.2% and 33.5%; differencing leaves 5.0% and 5.2% and throws away the trend the study was about.

timeseries · Spurious

Named alongside it

The objects these essays reach for when they reach for this one.

Augmented dickey fullerAutocorrelationClosed formDeterministic trendEstimated varianceFalse positiveNear-unit rootRandom walkSample sizeSpurious regressionStationarityStatistical power

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