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

Series — page 2

A field says what an essay is about. A series follows one idea essay by essay — from the question that introduces it to the one that assumes all the others.
The weight on the population, σ = 3. Each curve is one population spread τ. A group's estimate moves B = se²/(se² + τ²) of the way to the population mean, where se = σ/√n is what the group's own mean does not know. At τ = 1 a group of 9 observations sits halfway.

Shrinkage

  1. 2 The weight that decides
  2. 3 What the plug-in forgets
  3. 4 A group from the population's own tail
  4. 5 Estimates that are too alike
  5. 6 A league table of a hundred
  6. +5 more
10 essays · hierarchical Four datasets, slope 0.50, R² 0.67. Every one of these fits reports the same slope to two decimals and the same R². Only the first is a linear relationship with noise: the second is a curve, the third is a line with one outlier, and the fourth has its slope set by a single point.

Summary

  1. 1 Four datasets, one summary
  2. 2 R² is not a measure of fit
  3. 3 R² is a property of the design
  4. 4 The t statistic wearing different clothes
  5. 5 The summary that was meant to work
  6. +5 more
10 essays · regression The normal density at sigma = 1.00. The bands hold 68.27%, 95.45%, 99.73% of the mass. Those figures are integrals of the curve drawn, not the memorised 68-95-99.7.

Bands

  1. 1 The shape, and where its mass is
  2. 2 Two standard deviations of what
  3. 3 A tenth as wide, and both of them right
  4. 4 Ninety-three observations, and nothing assumed
  5. 5 All of the next ten
  6. +4 more
9 essays · normal The false-positive rate against the number of analyses, on pure noise. The data has no effect in it. Each analysis is correct and each p-value is honest. With 20 correlated outcomes available, something reaches p below 0.05 58% of the time.

Forking

  1. 3 Twenty analyses of nothing
  2. 4 How many analyses there really were
  3. 5 What naming it in advance costs
  4. 6 The correction that makes the estimate worse
  5. 7 Naming a handful in advance
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9 essays · testing Twenty walks up the same hill, σ = 2. Each walk fits a plane to the same four-corner factorial, takes its gradient as a direction, and steps along it until a run comes in below the one before. The true optimum is the cross. 80% of the walks stop before the best point on their own path — not because the direction was wrong, but because one noisy run is enough to stop them, and the direction error costs only 3.9% of the available gain.

Optimum

  1. 1 Walking up the gradient
  2. 2 The optimum is a ratio, and its interval is sometimes the whole line
  3. 3 Where the enumeration stops
  4. 4 The sign the curvature has
  5. 4 A basis drawn rather than chosen
  6. +4 more
9 essays · surface Twenty 95% intervals for a proportion that really is 0.35. 2 of the twenty miss the true value. The 95% is a property of the procedure across repetitions — no single interval has a 95% chance of anything, because it either contains 0.35 or it does not.

Repetition

  1. 1 Twenty intervals and one expected miss
  2. 2 Five times in six
  3. 3 Two intervals that overlap
  4. 4 Significant in one, not in the other
  5. 5 The verdicts watch the threshold, not the gap
  6. +4 more
9 essays · intervals Which allocations reverse the overall comparison. The per-group success rates are held fixed; only the split of each group between treatment and control changes. 32% of the allocations reverse, and the worst reverses by 13.1 percentage points.

Simpson

  1. 1 Simpson's reversal is a region, not a table
  2. 2 The reversal a coin cannot prevent
  3. 3 The change that is not confounding
  4. 4 Conditioning on what the treatment caused
  5. 5 Every patient at one half
  6. +4 more
9 essays · paradox Student's t on 5 degrees of freedom, against the normal. The two-sided 95% critical value is 2.571 for t(5) and 1.960 for the normal — 31% wider. Using the normal at this sample size makes every interval too short by that much.

Student

  1. 2 The correction for not knowing the spread
  2. 3 Where the two tails disagree
  3. 4 A degrees of freedom that is not a count
  4. 5 The skewness of a difference
  5. 6 The side a bound is read from
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9 essays · intervals Expected width against coverage, n = 30, p = 0.15. The Wald interval is the shortest and covers 94.2%. Clopper–Pearson covers 98.3% and is 13% wider. Shortness is not a virtue on its own — an interval of zero width is the shortest of all.

Width

  1. 3 The shortest interval is the one that misses
  2. 4 Two degrees of freedom, one total
  3. 5 A width promised for a difference
  4. 6 Blinded, and still exact
  5. 7 The bias that lands in the slope
  6. +4 more
9 essays · intervals What each forecaster says, and what is true. The true probability of the event given a forecaster's signal, and what three forecasters report. The truth is Φ(-0.5 + 1.2w) and the honest forecaster reports it, so its curve and the truth are the same line. The loud forecaster pushes every probability towards the ends and the hedged one pulls every probability towards the middle; both have their mean report held at the base rate of 0.3744, so each crosses the truth exactly once and neither can be caught by checking its average. Their reliability terms are 0.008500 and 0.013025 against the honest forecaster's zero, and all three have the same area under the ROC curve, 0.868312.

Calibration

  1. 1 An identity in three terms
  2. 2 A curve that is a binning
  3. 3 Calibrated and useless
  4. 4 A score that rewards lying
  5. 5 The miscalibration a perfect forecaster shows
  6. +3 more
8 essays · calibrate The same study read three ways. At time 2 the truth is 0.497. Kaplan–Meier gives 0.532; dropping the censored subjects gives 0.180; treating the censoring time as the event time gives 0.392. Both naive readings understate survival, because the subjects they mishandle are the ones doing well.

Censoring

  1. 1 The data that stops early
  2. 2 The curve that survives censoring
  3. 3 The interval at the end of the curve
  4. 4 A dropout the data cannot see
  5. 5 One minus Kaplan–Meier is not a risk
  6. +3 more
8 essays · survival The same covariate, three ways round. Three worlds over a treatment, an outcome and a covariate, joined by the same three edges at the same three strengths — 0.90, 0.50 and 0.70 — differing only in which way the two edges touching the covariate point. In the first the covariate causes both and adjusting for it recovers the effect of 0.50 exactly. In the second the treatment causes the covariate, the effect is 1.13, and adjusting returns 0.50 — the direct edge alone, with the part that travels through the covariate deleted. In the third the treatment and the outcome both cause the covariate, the effect is 0.50, and adjusting returns -0.087. The regression that produces those three numbers is one formula, and nothing in the data says which panel it is being run in.

Conditioning

  1. 1 One arithmetic, three decisions
  2. 2 The two worlds that look the same
  3. 3 Adjusting for everything
  4. 4 A collider before the treatment
  5. 5 The sample is a condition
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8 essays · collider The coverage is exact and it is not the nominal rate. ⌈(m+1)(1−α)⌉/(m+1) against m, the number of calibration points, at α = 0.05. It is a closed form and needs no data. It never falls below 95.0% and never reaches 1−α+1/(m+1), the two bounds the rank argument gives. It equals 95.0% exactly at 10 of the 182 sizes drawn — the sizes where (m+1)α is a whole number, which are 20 apart — and sits above it everywhere else, worst at 38 points where it is 97.4359%, or 2.4359% of coverage nobody asked for. Below 19 points there is no such order statistic and the interval is the whole line, which is where the curve starts.

Exchangeability

  1. 1 Coverage from exchangeability alone
  2. 2 What the split costs
  3. 3 Marginal is not conditional
  4. 4 The score is the modelling
  5. 5 When the order matters
  6. +3 more
8 essays · conformal The first stage an instrument needs is set by the violation nobody can see. The error each estimator converges on when the instrument has a direct effect of 0.05 on the outcome — a path the exclusion restriction asserts is zero and no sample can check. The instrument's error is δ/π exactly, so it is the reciprocal of the very quantity that made the method work: 1.0000 at a first stage of 0.05 and 0.0833 at 0.60. Least squares carries the confounding instead, at 0.3440 at a first stage of 0.30. The two cross at π = 0.1389, and the crossing is exactly δ times 2.7778 — the first stage an instrument needs is proportional to the violation it is assumed not to have, and below that line the method being corrected is the better estimator.

Exclusion

  1. 1 The assumption nothing tests
  2. 2 Weak, and back where it started
  3. 3 What the first stage does not know
  4. 4 Whose effect it is
  5. 5 Two instruments that disagree
  6. +3 more
8 essays · instrument More blocks, and the light tail is called wrong more often. The share of records whose three-way family call is right, over 400 records at each block count, at 100 readings a block. A 95% interval for the shape is formed and the record is recorded as calling Fréchet, Weibull or Gumbel according to whether that interval sits above zero, below it, or straddles it. The two signed parents go from 51.5% and 71.0% at 20 blocks to certainty by 100. The light-tailed one goes the other way — 83.0%, 75.5%, 55.0%, 36.5%, 4.8% — because its estimate sits at about -0.1157 whatever the record length, and a longer record only shrinks the interval onto that number.

Extremes

  1. 1 Three shapes, one limit
  2. 2 The maximum converges slowly
  3. 3 The threshold is a dial
  4. 4 A level with no data in it
  5. 5 Two intervals for one return level
  6. +3 more
8 essays · extreme Six groups of 10, each fitting its own slope, then borrowing. Each faint line is one group's own least-squares slope through its own centre; each solid line is that slope after pooling towards the population slope of 0.79. Every group has the same 10 observations. The group whose x values span 0.4 has a slope standard error of 1.86 and moves 91% of the way in; the group spanning 2.0 has a standard error of 0.37 and moves 28%.

Levels

  1. 1 The slope that borrows
  2. 2 Pooling a proportion
  3. 3 Borrowing towards a line
  4. 4 Two levels at once
  5. 5 Two groupings that cross
  6. +3 more
8 essays · multilevel Three mechanisms leave the slope alone; one does not. The bias of the complete-case slope under each of four missingness rules, counted over 4000 studies of 200 rows at 35.0% missing, with the closed form printed beside each count. Missingness that depends on nothing, on the regressor, or on the second covariate leaves the slope exactly where it was — the closed forms are zero to machine precision and the counts are -0.0005, -0.0005 and -0.0011 against standard errors of about 0.0018. Missingness that depends on the outcome moves it by -0.1635, which is 27.3% of the slope being estimated. The same share of rows is lost in every case.

Missingness

  1. 1 Three mechanisms and one dataset
  2. 2 Dropping the incomplete rows
  3. 3 One imputation is not an observation
  4. 4 The variance between imputations
  5. 5 An imputation model the analysis does not contain
  6. +3 more
8 essays · missing One wrong model, four designs, four slopes. The slope a straight line converges to when the truth is a quadratic, under four covariate distributions, by two routes: the population projection in closed form, and the mean of 2500 fitted slopes at 200 rows apiece. The even spread over [0, 2] gives 1.6000 and the same spread moved to [1, 3] gives 2.6000, while widening it to [0, 4] gives 2.6000 — the same number as the shifted one, because a symmetric design's target is the truth's tangent slope at the design's own mean and does not read the spread at all. An exponential spread with the SAME mean as the first gives 2.6000. So two studies of one world, each fitting the same wrong model, honestly report slopes 1.0000 apart, and neither is making an error.

Misspecification

  1. 1 What a wrong model estimates
  2. 2 The bread and the filling
  3. 3 Robust is not free
  4. 4 Three corrections and a leverage
  5. 5 Right for the wrong reason
  6. +3 more
8 essays · sandwich Three series and one relation between them. Above, three series generated from Δy = Πy₋₁ + ε with Π of rank 1. Below, the combination y1 −y2. It stays inside a band of 9.5 while the series themselves travel 28.4. The count of combinations that behave this way is the rank of Π, and it is what every method in the field sets out to estimate.

Rank

  1. 1 Three series and a count
  2. 2 Which series goes on the left
  3. 3 Counting what is still wandering
  4. 4 The weight that is a vector
  5. 5 The rank is a decision
  6. +3 more
8 essays · systems Where the normal approximation converges, and where it does not. Relative error against the exact binomial. At n = 1280 the error at the median is 0.96% and three sigma out it is 25.7% — a factor of 27. The tail is where the approximation is used.

Rate

  1. 2 The tail converges last
  2. 3 A bound written for a coin
  3. 4 A correction that goes below zero
  4. 5 An approximation built at the threshold
  5. 6 A tail the sample never saw
  6. +3 more
8 essays · normal

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