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

Series — page 3

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.
A weight that balances, and one that unbalances. The standardised difference between the arms on each covariate, integrated over the population rather than counted in a sample. Unweighted, the arms differ by 0.8310 on the first covariate and 0.6015 on the second, which is what makes the raw difference of arm means 2.7102 against a true average effect of 1.0000. Weighting each unit by one over its own assignment probability removes both differences exactly — -2.78e-17 and -5.69e-19, which is machine precision and not a small number — because the weighted density of the treated arm is the population's own whatever the propensity is. Weighting by a score fitted without the second covariate balances the first to 0.0035 and pushes the second out to 0.7057, further apart than doing nothing.

Weighting

  1. 1 A score that balances
  2. 2 How many observations a weight leaves
  3. 3 The region with no comparison
  4. 4 Either model, but not neither
  5. 5 The estimated weight is the better one
  6. +3 more
8 essays · weights What a 95% credible interval covers, n = 20. Computed by summing over all 21 possible counts rather than by simulating them. Jeffreys' prior covers close to 95% across the range; a confident prior centred in the wrong place covers almost nothing where the truth is far from it.

Credible

  1. 2 What a credible interval covers
  2. 3 Where the two schools agree
  3. 4 The interval that integrates
  4. 5 The shortest interval, and the one that does not move
  5. 6 An interval for something else
  6. +2 more
7 essays · bayes Two factors that interact — where each design looks. The true response at the four corners is -10, 6, 4, 0. One factor at a time visits three of them, sees that raising either factor alone helps, and recommends raising both — a corner it never ran, and one that is worse than either single change. It picks the best corner 0.0% of the time against the factorial design's 99.4%, on the same number of runs.

Factorial

  1. 1 One factor at a time
  2. 2 The design that cannot see a curve
  3. 3 Three levels, and the ring where the design says the same thing
  4. 4 The word a fraction costs
  5. 5 The design that refuses the corners
  6. +2 more
7 essays · design Coverage against sample size, true proportion 0.15. Coverage does not improve monotonically. n = 19 covers 93.8% while the larger n = 20 covers 81.9%. The sample space is discrete, so the endpoints jump as n changes.

Oscillation

  1. 2 More data is not monotonically better
  2. 3 A hole no sample size fills
  3. 4 What a guaranteed minimum costs
  4. 5 The coin that makes it exact
  5. 6 The coin that is already there
  6. +2 more
7 essays · intervals Eight groups, τ = 1 against a within-group spread of 3. Each row is a group. The hollow circle is the group's own mean, the filled one is the estimate after pooling, and the small mark is the truth the data was generated from. The group of 3 moves 75% of the way to the population mean of 0.10; the group of 40 moves 18%.

Pooling

  1. 1 Eight groups, one population
  2. 3 When borrowing goes wrong
  3. 4 When the spread estimates to zero
  4. 5 The fewest groups that can borrow
  5. 6 Where the borrowing goes
  6. +2 more
7 essays · hierarchical weakly informative — Beta(2, 2), updated by 5 of 20. The prior is worth 4 observations. With 20 observations the posterior mean is 0.292, against a data proportion of 0.250 and a prior mean of 0.500.

Prior

  1. 1 What a prior is worth
  2. 2 The prior the data estimates
  3. 3 A prior on the spread
  4. 4 A control borrowed from the last trial
  5. 5 A history that agrees with itself
  6. +2 more
7 essays · bayes Coverage of four nominal 95% intervals, n = 30. Computed exactly by summing over all 31 possible counts, not simulated. The Wald interval drops to 26.0% and is jagged everywhere; Clopper–Pearson never falls below 95% and pays for it in width.

Routes

  1. 2 Two routes to every number
  2. 3 The arcsine that closes it, and the error that was overstated
  3. 4 The check worth more than the check
  4. 5 A rate needs a rate beside it
  5. 6 A companion cut into many
  6. +2 more
7 essays · method Least squares estimates persistence low, by an amount with a formula. 3000 series of 50 observations at each persistence. The lower curve is the counted bias of the least-squares estimate of φ, and the open marks on it are −(1 + 3φ)/n, computed rather than fitted. The upper curve is the bias left after adding that quantity back, evaluated at the estimate rather than at the truth nobody has: -0.0020 at φ = 0.3, -0.0023 at φ = 0.5, -0.0039 at φ = 0.7, -0.0059 at φ = 0.8, -0.0108 at φ = 0.9, -0.0165 at φ = 0.95. The formula is a leading-order expression and it understates the bias where the persistence is nearest one — -0.0882 counted against -0.0770 predicted at φ = 0.95, which is the corner of the parameter space every one of these approximations is worst in.

Bias

  1. 1 Correcting the persistence
  2. 2 The repair that moves the wrong number
  3. 3 Correcting the forecast instead
  4. 4 The correction that leaves the region
  5. 5 What the interval is short by
  6. +1 more
6 essays · evaluation 12,000 studies of a real effect of 0.3, n = 16. Power is 21%. The studies that reached significance report a mean effect of 0.621 — 2.07 times the truth. Every one of them is honest; the selection did the inflating.

Curse

  1. 3 The winner's curse
  2. 4 The estimate after the choice
  3. 5 Shrinking the arm that was chosen
  4. 6 A prior that lets one arm stand out
  5. 7 A second stage sized by the first
  6. +1 more
6 essays · testing Two measurements of the same thing, correlated 0.60. Pick the worst 15% on the first measurement and their average rises by 0.78 on the second. Pick the best and theirs falls by 0.48. No treatment was given to anybody.

Rtm

  1. 2 Regression to the mean
  2. 3 Two analyses of one baseline
  3. 4 The measurement that got them enrolled
  4. 5 A lead that a heavy tail keeps
  5. 6 The slope of a density nobody can see
  6. +1 more
6 essays · paradox Twenty samples of 40, every one of them genuinely normal. Each panel is a quantile-quantile plot of 40 draws from a normal distribution. The worst point in the worst panel sits 0.87 standard deviations off the line. Anything a reader would reject here would be a false alarm.

Seeds

  1. 1 The seed is part of the figure
  2. 2 A coverage table with its own error
  3. 3 The same draws for both methods
  4. 4 A simulation that stops when it looks settled
  5. 5 The draws aimed at the tail
  6. +1 more
6 essays · method 8 exponential draws, standardised, against the normal. The source is one-sided and skewed. At n = 8 the standardised sum has skew 0.695, and the theory says 2/sqrt(n) = 0.707 — so the convergence is visible AND its rate is predicted.

Clt

  1. 1 Sums of almost anything
  2. 2 Where the derivative is zero
  3. 3 A ratio whose interval has to be the whole line
  4. 4 A flat point with more than one direction
  5. 5 Testing for a flat point first
5 essays · normal The variance touches p and never crosses it. d(x) = f(x)′M⁻¹f(x) along the diagonal of a square region, for the D-optimal measure. The line at 6 is the number of parameters in the model. Kiefer and Wolfowitz's theorem says a design is D-optimal exactly when the largest d anywhere in the region is p — not approximately, equals — so the optimal curve is tangent to that line at its support points and below it everywhere else. Here the largest value anywhere on a 41×41 grid is 6.000000000.

Equivalence

  1. 1 The theorem that says when to stop
  2. 2 How many places a design goes
  3. 3 Augmenting a design that has already run
  4. 4 The criterion with no derivative
  5. 5 A certificate for the worst direction
5 essays · optimality Twenty points and one more, at leverage 0.74. Without the distant point the slope is 0.495; with it the slope is -0.389. Its leverage is 0.737 and its Cook's distance is 24.1, against a conventional threshold of 1.

Leverage

  1. 1 The line that one point drew
  2. 2 Two points that hide each other
  3. 3 A robust loss and a far x
  4. 4 The start an efficient robust line inherits
  5. 5 The other half a trimmed fit rejected
5 essays · regression 20,000 p-values from a true null, n = 12. Flat, as it must be: under the null a p-value is uniform on (0,1). The Kolmogorov–Smirnov distance from uniform is 0.0090 (p = 0.81). That flatness is the check that catches an error a single rejection rate would miss.

Uniformity

  1. 1 A p-value that is not flat is not a p-value
  2. 2 The p-value a replication gets
  3. 3 Two ways to combine p-values
  4. 4 The smallest of three combinations
  5. 5 Studies that share a control
5 essays · testing Coverage of four nominal 95% intervals, n = 20. Computed exactly by summing over all 21 possible counts, not simulated. The Wald interval drops to 18.2% and is jagged everywhere; Clopper–Pearson never falls below 95% and pays for it in width.

Coverage

  1. 1 What the 95% refers to
  2. 2 An interval that covers and says nothing
  3. 3 A count that bets against its interval
3 essays · intervals

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