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1[[tags: egg]]
2[[toc:]]
3
4== Statistics
5
6This library is a port of [[http://compbio.ucdenver.edu/hunter/|Larry Hunter]]'s Lisp statistics library to chicken scheme.
7
8The library provides a number of formulae and methods taken from the book "Fundamentals of Biostatistics" by Bernard Rosner (5th edition).
9
10=== Statistical Distributions
11
12To use this library, you need to understand the underlying statistics.  In brief:
13
14The [[http://en.wikipedia.org/wiki/Binomial_distribution|Binomial
15distribution]] is used when counting discrete events in a series of
16trials, each of which events has a probability p of producing a
17positive outcome.  An example would be tossing a coin {{n}} times: the
18probability of a head is {{p}}, and the distribution gives the
19expected number of heads in the {{n}} trials.  The binomial
20distribution is defined as B(n, p).
21
22The [[http://en.wikipedia.org/wiki/Poisson_distribution|Poisson
23distribution]] is used to count discrete events which occur with a
24known average rate.  A typical example is the decay of radioactive
25elements.  A poisson distribution is defined Pois(mu).
26
27The [[http://en.wikipedia.org/wiki/Normal_distribution|Normal
28distribution]] is used for real-valued events which cluster around a
29specific mean with a symmetric variance.  A typical example would be
30the distribution of people's heights.  A normal distribution is
31defined N(mean, variance).
32
33=== Provided Functions
34
35==== Utilities
36
37<procedure>(average-rank value sorted-values)</procedure>
38returns the average position of given value in the list of sorted values: the rank is based from 1.
39 > (average-rank 2 '(1 2 2 3 4))
40 5/2
41
42<procedure>(beta-incomplete x a b)</procedure>
43
44<procedure>(bin-and-count items n)</procedure>
45Divides the range of the list of {{items}} into {{n}} bins, and returns a vector of the number of items which fall into each bin.
46 > (bin-and-count '(1 1 2 3 3 4 5) 5)
47 #(2 1 2 1 1)
48
49<procedure>(combinations n k)</procedure>
50returns the number of ways to select {{k}} items from {{n}}, where the order does not matter.
51
52<procedure>(factorial n)</procedure>
53returns the factorial of {{n}}.
54
55<procedure>(find-critical-value p-function p-value #:increasing?)</procedure>
56given a monotonic function {{p-function}} taking a single value {{x}} to {{y}}, returns the value of {{x}} which makes {{(p-function x)}} closest to {{p-value}}.  A boolean keyword parameter {{#:increasing?}} determines if function should be increasing or decreasing (the default).
57
58<procedure>(fisher-z-transform r)</procedure>
59returns the transformation of a correlation coefficient {{r}} into an approximately normal distribution.
60
61<procedure>(gamma-incomplete a x)</procedure>
62
63<procedure>(gamma-ln x)</procedure>
64
65<procedure>(permutations n k)</procedure>
66returns the number of ways to select {{k}} items from {{n}}, where the order does matter.
67
68<procedure>(random-normal mean sd)</procedure>
69returns a random number distributed with specified mean and standard deviation.
70
71<procedure>(random-pick items)</procedure>
72returns a random item from the given list of items.
73
74<procedure>(random-sample n items)</procedure>
75returns a random sample from the list of items without replacement of size {{n}}.
76
77<procedure>(random-weighted-sample n items weights)</procedure>
78returns a random sample from the list of items without replacement of size {{n}}, where each sample has a defined probability of selection (weight).
79
80<procedure>(sign n)</procedure>
81returns 0, 1 or -1 according to if {{n}} is zero, positive or negative.
82
83<procedure>(square n)</procedure>
84
85<procedure>(cumsum sequences)</procedure>
86returns the cumulative sum of a sequence.
87
88==== Descriptive statistics
89
90These functions provide information on a given list of numbers, the {{items}}.  Note, the list does not have to be sorted.
91
92<procedure>(mean items)</procedure>
93returns the arithmetic mean of the {{items}} (the sum of the numbers divided by the number of numbers).
94 (mean '(1 2 3 4 5)) => 3
95
96<procedure>(median items)</procedure>
97returns the value which separates the upper and lower halves of the list of numbers.
98 (median '(1 2 3 4)) => 5/2
99
100<procedure>(mode items)</procedure>
101returns two '''values'''.  The first is a list of the ''modes'' and the second is the frequency.  (A mode of a list of numbers is the most frequently occurring value.)
102 > (mode '(1 2 3 4))
103 (1 2 3 4)
104 1
105 > (mode '(1 2 2 3 4))
106 (2)
107 2
108 > (mode '(1 2 2 3 3 4))
109 (2 3)
110 2
111
112<procedure>(geometric-mean items)</procedure>
113returns the geometric mean of the {{items}} (the result of multiplying the items together and then taking the nth root, where n is the number of items).
114 (geometric-mean '(1 2 3 4 5)) => 2.60517108469735
115
116<procedure>(range items)</procedure>
117returns the difference between the biggest and the smallest value from the list of {{items}}.
118 (range '(5 1 2 3 4)) => 4
119
120<procedure>(percentile items percent)</procedure>
121returns the item closest to the {{percent}} value if the {{items}} are sorted into order; the returned item may be in the list, or the average of adjacent values.
122 (percentile '(1 2 3 4) 50) => 5/2
123 (percentile '(1 2 3 4) 67) => 3
124
125<procedure>(variance items)</procedure>
126
127<procedure>(standard-deviation items)</procedure>
128
129<procedure>(coefficient-of-variation items)</procedure>
130returns 100 * (std-dev / mean) of the {{items}}.
131 (coefficient-of-variation '(1 2 3 4)) => 51.6397779494322
132
133<procedure>(standard-error-of-the-mean items)</procedure>
134returns std-dev / sqrt(length items).
135  (standard-error-of-the-mean '(1 2 3 4)) => 0.645497224367903
136
137<procedure>(mean-sd-n items)</procedure>
138returns three '''values''', one for the mean, one for the standard deviation, and one for the length of the list.
139 > (mean-sd-n '(1 2 3 4))
140 5/2
141 1.29099444873581
142 4
143
144==== Distributional functions
145
146<procedure>(binomial-probability n k p)</procedure>
147returns the probability that the number of positive outcomes for a binomial distribution B(n, p) is k.
148 > (do-ec (: i 0 11)
149          (format #t "i = ~d P = ~f~&" i (binomial-probability 10 i 0.5)))
150 i = 0 P = 0.0009765625
151 i = 1 P = 0.009765625
152 i = 2 P = 0.0439453125
153 i = 3 P = 0.1171875
154 i = 4 P = 0.205078125
155 i = 5 P = 0.24609375
156 i = 6 P = 0.205078125
157 i = 7 P = 0.1171875
158 i = 8 P = 0.0439453125
159 i = 9 P = 0.009765625
160 i = 10 P = 0.0009765625
161
162<procedure>(binomial-cumulative-probability n k p)</procedure>
163returns the probability that less than {{k}} positive outcomes occur for a binomial distribution B(n, p).
164 > (do-ec (: i 0 11)
165          (format #t "i = ~d P = ~f~&" i (binomial-cumulative-probability 10 i 0.5)))
166 i = 0 P = 0.0
167 i = 1 P = 0.0009765625
168 i = 2 P = 0.0107421875
169 i = 3 P = 0.0546875
170 i = 4 P = 0.171875
171 i = 5 P = 0.376953125
172 i = 6 P = 0.623046875
173 i = 7 P = 0.828125
174 i = 8 P = 0.9453125
175 i = 9 P = 0.9892578125
176 i = 10 P = 0.9990234375
177
178<procedure>(binomial-ge-probability n k p)</procedure>
179returns the probability of {{k}} or more positive outcomes for a binomial distribution B(n, p).
180
181<procedure>(binomial-le-probability n k p)</procedure>
182returns the probability {{k}} or fewer positive outcomes for a binomial distribution B(n, p).
183
184<procedure>(poisson-probability mu k)</procedure>
185returns the probability of {{k}} events occurring when the average is {{mu}}.
186 > (do-ec (: i 0 20)
187          (format #t "P(X=~2d) = ~,4f~&" i (poisson-probability 10 i)))
188 P(X= 0) = 0.0000
189 P(X= 1) = 0.0005
190 P(X= 2) = 0.0023
191 P(X= 3) = 0.0076
192 P(X= 4) = 0.0189
193 P(X= 5) = 0.0378
194 P(X= 6) = 0.0631
195 P(X= 7) = 0.0901
196 P(X= 8) = 0.1126
197 P(X= 9) = 0.1251
198 P(X=10) = 0.1251
199 P(X=11) = 0.1137
200 P(X=12) = 0.0948
201 P(X=13) = 0.0729
202 P(X=14) = 0.0521
203 P(X=15) = 0.0347
204 P(X=16) = 0.0217
205 P(X=17) = 0.0128
206 P(X=18) = 0.0071
207 P(X=19) = 0.0037
208
209<procedure>(poisson-cumulative-probability mu k)</procedure>
210returns the probability of less than {{k}} events occurring when the average is {{mu}}.
211 > (do-ec (: i 0 20)
212          (format #t "P(X=~2d) = ~,4f~&" i (poisson-cumulative-probability 10 i)))
213 P(X= 0) = 0.0000
214 P(X= 1) = 0.0000
215 P(X= 2) = 0.0005
216 P(X= 3) = 0.0028
217 P(X= 4) = 0.0103
218 P(X= 5) = 0.0293
219 P(X= 6) = 0.0671
220 P(X= 7) = 0.1301
221 P(X= 8) = 0.2202
222 P(X= 9) = 0.3328
223 P(X=10) = 0.4579
224 P(X=11) = 0.5830
225 P(X=12) = 0.6968
226 P(X=13) = 0.7916
227 P(X=14) = 0.8645
228 P(X=15) = 0.9165
229 P(X=16) = 0.9513
230 P(X=17) = 0.9730
231 P(X=18) = 0.9857
232 P(X=19) = 0.9928
233
234<procedure>(poisson-ge-probability mu k)</procedure>
235returns the probability of {{k}} or more events occurring when the average is {{mu}}.
236
237<procedure>(normal-pdf x mean variance)</procedure>
238returns the likelihood of {{x}} given a normal distribution with stated mean and variance.
239 > (do-ec (: i 0 11)
240          (format #t "~3d ~,4f~&" i (normal-pdf i 5 4)))
241  0 0.0088
242  1 0.0270
243  2 0.0648
244  3 0.1210
245  4 0.1760
246  5 0.1995
247  6 0.1760
248  7 0.1210
249  8 0.0648
250  9 0.0270
251 10 0.0088
252
253<procedure>(convert-to-standard-normal x mean variance)</procedure>
254returns a value for {{x}} rescaling the given normal distribution to a standard N(0, 1).
255 > (convert-to-standard-normal 5 6 2)
256 -1/2
257
258<procedure>(phi x)</procedure>
259returns the cumulative distribution function (CDF) of the standard normal distribution.
260 > (do-ec (: x -2 2 0.4)
261          (format #t "~4,1f ~,4f~&" x (phi x)))
262 -2.0 0.0228
263 -1.6 0.0548
264 -1.2 0.1151
265 -0.8 0.2119
266 -0.4 0.3446
267  0.0 0.5000
268  0.4 0.6554
269  0.8 0.7881
270  1.2 0.8849
271  1.6 0.9452
272
273<procedure>(z percentile)</procedure>
274returns the inverse of the standard normal distribution.  Input is a percentile, between 0.0 and 1.0.
275
276<procedure>( t-distribution degrees-of-freedom percentile)</procedure>
277returns the point in the t-distribution given the {{degrees-of-freedom}} and {{percentile}}.  {{degrees-of-freedom}} must be a positive integer, and {{percentile}} a value between 0.0 and 1.0.
278
279<procedure>(chi-square degrees-of-freedom percentile)</procedure>
280returns the point at which chi-square distribution has {{percentile}} to its '''left''', using given {{degrees-of-freedom}}.
281
282<procedure>(chi-square-cdf x degrees-of-freedom)</procedure>
283returns the probability that a random variable is to the '''left''' of {{x}} using the chi-square distribution with given {{degrees-of-freedom}}.
284
285====  Confidence intervals
286
287These functions report bounds for an observed property of a distribution: the bounds are tighter as the confidence level, alpha, varies from 0.0 to 1.0.
288
289<procedure>(binomial-probability-ci n p alpha)</procedure>
290returns two values, the upper and lower bounds on an observed probability {{p}} from {{n}} trials with confidence {{(1-alpha)}}.
291 > (binomial-probability-ci 10 0.8 0.9)
292 0.724273681640625
293 0.851547241210938
294 ; 2 values
295
296<procedure>(poisson-mu-ci k alpha)</procedure>
297returns two values, the upper and lower bounds on the poisson parameter if {{k}} events are observed; the bound is for confidence {{(1-alpha)}}.
298 > (poisson-mu-ci 10 0.9)
299 8.305419921875
300 10.0635986328125
301 ; 2 values
302
303<procedure>(normal-mean-ci mean standard-deviation k alpha)</procedure>
304returns two values, the upper and lower bounds on the mean of the normal distibution of {{k}} events are observed; the bound is for confidence {{(1-alpha)}}.
305 > (normal-mean-ci 0.5 0.1 10 0.8)
306 0.491747852700165
307 0.508252147299835
308 ; 2 values
309
310<procedure>(normal-mean-ci-on-sequence items alpha)</procedure>
311returns two values, the upper and lower bounds on the mean of the given {{items}}, assuming they are normally distributed; the bound is for confidence {{(1-alpha)}}.
312 > (normal-mean-ci-on-sequence '(1 2 3 4 5) 0.9)
313 2.40860081649174
314 3.59139918350826
315 ; 2 values
316
317<procedure>(normal-variance-ci standard-deviation k alpha)</procedure>
318returns two values, the upper and lower bounds on the variance of the normal distibution of {{k}} events are observed; the bound is for confidence {{(1-alpha)}}.
319
320<procedure>(normal-variance-ci-on-sequence items alpha)</procedure>
321returns two values, the upper and lower bounds on the variance of the given {{items}}, assuming they are normally distributed; the bound is for confidence {{(1-alpha)}}.
322
323<procedure>normal-sd-ci standard-deviation k alpha)</procedure>
324returns two values, the upper and lower bounds on the standard deviation of the normal distibution of {{k}} events are observed; the bound is for confidence {{(1-alpha)}}.
325
326<procedure>(normal-sd-ci-on-sequence sequence items)</procedure>
327returns two values, the upper and lower bounds on the standard deviation of the given {{items}}, assuming they are normally distributed; the bound is for confidence {{(1-alpha)}}.
328
329==== Hypothesis testing
330
331These functions report on the significance of an observed sample against a given distribution.
332
333=====  (parametric)
334
335<procedure>(z-test x-bar n #:mu #:sigma #:tails)</procedure>
336Given {{x-bar}} the sample mean, {{n}} the number in the sample, {{#:mu}} the distribution mean (defaults to 0), {{#:sigma}} the distribution standard deviation (defaults to 1), and {{#:tails}} the significance to report on:
337
338* {{':both}}, the probability of the difference between {{x-bar}} and {{#:mu}}
339* {{':positive}}, the probability that observation is {{>= x-bar}}
340* {{':negative}}, the probability that observation is {{<= x-bar}}
341
342e.g. given a distribution with mean 50 and standard deviation 10
343
344 ; probability that a single observation is <= 40
345 > (z-test 40 1 #:mu 50 #:sigma 10 #:tails ':negative)
346 0.158655
347 ; probability that 10 observations are <= 40
348 > (z-test 40 10 #:mu 50 #:sigma 10 #:tails ':negative)
349 0.000783
350 ; probability that 5 observations give a mean of 40
351 > (z-test 40 5 #:mu 50 #:sigma 10)
352 0.025347
353
354<procedure>(z-test-on-sequence observations #:mu #:sigma #:tails)</procedure>
355As for {{z-test}} except {{x-bar}} and {{n}} are computed from given {{observations}}.
356
357<procedure>(t-test-one-sample x-bar sd n mu #:tails)</procedure>
358Given observed data with mean {{x-bar}}, standard devation {{sd}} and number of observations {{n}} ({{n < 30}}), return the significance of the sample compared with the population mean {{mu}}.  {{#:tails}} is one of:
359
360* {{':both}} two-sided (default)
361* {{':positive}} one-sided, {{x-bar >= mu}}
362* {{':negative}} one-sided, {{x-bar <= mu}}
363
364<procedure>(t-test-one-sample-on-sequence observations mu #:tails)</procedure>
365As for {{t-test-one-sample}} except {{x-bar}}, {{sd}} and {{n}} are computed from given {{observations}}.
366
367<procedure>(t-test-paired t-bar sd n #:tails)</procedure>
368Computes the significance of the differences between two sequences of data: the differences are given as their mean, {{t-bar}}, standard deviation, {{sd}}, and number of measurements, {{n}}.
369
370<procedure>(t-test-paired-on-sequences before after #:tails)</procedure>
371Computes the significance of the difference between two sequences of data: one before an experimental change and one after.  {{#:tails}} is as for {{t-significance}}.
372
373 > (t-test-paired-on-sequences '(4 3 5) '(1 1 3))
374 0.0198039411803931
375
376<procedure>(t-test-two-sample mean-1 sd-1 n-1 mean-2 sd-2 n-2 #:variances-equal? #:variance-significance-cutoff #:tails)</procedure>
377Computes the significance of the difference of two means given the sample standard deviations and sizes.
378
379<procedure>(t-test-two-sample-on-sequences sequence-1 sequence-2 #:tails)</procedure>
380Significance of difference of two sequences of observations.
381
382<procedure>(f-test variance-1 n1 variance-2 n2 #:tails)</procedure>
383Tests for the equality of two variances.
384
385<procedure>(chi-square-test-one-sample observed-variance sample-size test-variance #:tails)</procedure>
386Tests for significance of difference between an observed and a test variance.
387
388<procedure>(binomial-test-one-sample p-hat n p #:tails #:exact?)</procedure>
389Returns the significance of a one sample test with {{n}} observations, observed probability {{p-hat}} and expected probability {{p}}.
390
391<procedure>(binomial-test-two-sample p-hat-1 n-1 p-hat-2 n-2 #:tails #:exact?)</procedure>
392Returns the significance of a two sample test.
393
394<procedure>(fisher-exact-test a b c d #:tails)</procedure>
395Given a 2x2 contingency table, returns a p value using Fisher's exact test.  {{a}} and {{b}} form the first row of the contingency table, {{c}} and {{d}} the second row.
396
397<procedure>(mcnemars-test a-discordant-count b-discordant-count #:exact?)</procedure>
398For measuring effectiveness of, e.g., one treatment over another.  {{a-discordant-count}} is the number of times when A worked, {{b-discordant-count}} the number of times B worked.
399
400<procedure>(poisson-test-one-sample observed mu #:tails #:approximate?)</procedure>
401Computes significance of the number of observed events under a Poisson distribution against {{mu}} expected events.
402
403===== (non parametric)
404
405<procedure>(sign-test plus-count minus-count #:exact? #:tails)</procedure>
406
407<procedure>(sign-test-on-sequence sequence-1 sequence-2 #:exact? #:tails)</procedure>
408Takes two equal-sized sequences of observations, and reports if the entries of one are different to those in the other.
409
410<procedure>(wilcoxon-signed-rank-test differences #:tails)</procedure>
411Given at least 16 differences, reports if the positive differences are significantly larger or smaller than the negative differences.
412
413<procedure>(wilcoxon-signed-rank-test-on-sequences sequence-1 sequence-2 #:tails)</procedure>
414Given two sequences of at least 16 observations, computes {{wilcoxon-signed-rank-test}} on the differences.
415
416<procedure>(chi-square-test-rxc contingency-table)</procedure>
417Given a contingency table (a SRFI-63 array), returns significance of relation between row and column variable.
418
419<procedure>(chi-square-test-for-trend row1-counts row2-counts)</procedure>
420Returns p significance of trend, and prints a string to show if increasing or decreasing.
421
422==== Sample size estimates
423
424<procedure>(t-test-one-sample-sse mean-1 mean-2 sigma-1 #:alpha #:1-beta #:tails)</procedure>
425Returns the size of sample necessary to distinguish a normally distributed sample with {{mean-2}} from a population {{mean-1}} standard deviation {{sigma-1}}.  The significance {{#:alpha}} (defaults to 0.05), power {{#:1-beta}} (0.95) and sides {{#:tails}} (':both) may be altered.
426 > (t-test-one-sample-sse 5.0 5.2 0.5)
427 163
428
429<procedure>(t-test-two-sample-sse mean-1 sigma-1 mean-2 sigma-2 #:alpha #:1-beta #:tails #:sample-ratio)</procedure>
430Returns the size of sample necessary to distinguish a normally distributed sample N(mean-1, sigma-1) from a normally distributed sample N(mean-2, sigma-2).  The significance {{#:alpha}} (defaults to 0.05), power {{#:1-beta}} (0.95), sides {{#:tails}} (':both) and sample-ratio {{#:sample-ratio}} (1) may be altered.
431
432<procedure>(t-test-paired-sse difference-mean difference-sigma #:alpha #:1-beta #:tails)</procedure>
433Returns the size of sample to produce a given mean and standard deviation in the differences of two samples.
434
435<procedure>(binomial-test-one-sample-sse p-estimated p-null #:alpha #:1-beta #:tails)</procedure>
436Returns the size of sample needed to test whether an observed probability is significantly different from a particular binomial null hypothesis with a significance alpha and a power 1-beta.
437
438<procedure>(binomial-test-two-sample-sse p-one p-two #:alpha #:1-beta #:tails #:sample-ratio)</procedure>
439Returns the size of sample needed to test if given two binomial probabilities are significantly different.  {{#:sample-ratio}} can be given if the two samples differ in size.
440
441<procedure>(binomial-test-paired-sse pd pa #:alpha #:1-beta #:tails)</procedure>
442Sample size estimate for McNemar's discordant pairs test.
443
444<procedure>(correlation-sse rho #:alpha #:1-beta)</procedure>
445Returns the size of sample necessary to find a correlation of value {{rho}} with significance {{#:alpha}} (defaults to 0.05) and power {{#:1-beta}} (defaults to 0.95).
446 > (correlation-sse 0.80 #:alpha 0.05 #:1-beta 0.9)
447 11
448
449==== Correlation and regression
450
451<procedure>(linear-regression xs ys)</procedure>
452
453Given a line definition as lists of point coordinates, first prints to
454the terminal and then returns 5 '''values''' for the best fitting line
455through the points:
456
457* the y-intercept
458* the slope
459* the correlation coefficient, r
460* the square of the correlation coefficient, r^2
461* the significance of the difference of the slope from zero, p
462
463(This is also called the Pearson correlation; used when relation expected to be linear.  Also see {{spearman-rank-correlation}}.)
464
465 > (linear-regression '(1.0 2.0 3.0) '(0.1 0.3 0.8))
466 Intercept = -0.3, slope = 0.35, r = 0.970725343394151, R^2 = 0.942307692307692, p = 0.154420958311267
467 -0.3
468 0.35
469 0.970725343394151
470 0.942307692307692
471 0.154420958311267
472 ; 5 values
473
474<procedure>(correlation-coefficient xs ys)</procedure>
475As above, but only returns the value of ''r'':
476
477 > (correlation-coefficient '(1.0 2.0 3.0) '(0.1 0.3 0.8))
478 0.970725343394151
479
480<procedure>(correlation-test-two-sample r1 n1 r2 n2 #:tails)</procedure>
481Returns the significance of the similarity between two correlations.  {{#:tails}} determines how the comparison is made: {{':both}} measures the difference, {{':negative}} if {{r1 < r2}} and {{#':positive}} if {{r2 > r1}}.
482
483<procedure>(correlation-test-two-sample-on-sequences points-1 points-2 #:tails)</procedure>
484As above, but computes the correlations from given lists of points.
485
486<procedure>(spearman-rank-correlation xs ys)</procedure>
487Returns two '''values''', the Spearman Rank measure of correlation between the given lists of point coordinates, and the p-significance of the correlation.  (This correlation is used for non-linear relations; compare with {{linear-regression}}.)
488
489==== Significance test functions
490
491<procedure>(t-significance t-value degrees-of-freedom #:tails)</procedure>
492returns the probability of {{t-value}} for given {{degrees-of-freedom}}.  The keyword {{#:tails}} modifies the calculation to be two-sided (the default) with {{':both}}, or one-sided, {{':positive}} or {{':negative}}.
493
494 > (t-significance 0.2 5)
495 0.849360513995829
496 > (t-significance 0.2 5 #:tails ':positive)
497 0.424680256997915
498 > (t-significance 0.2 5 #:tails ':negative)
499 0.575319743002086
500
501<procedure>(f-significance f-value numerator-dof denominator-dof #:one-tailed?)</procedure>
502returns the probability of {{f-value}} for given {{numerator-dof}} and {{denominator-dof}}.  The boolean keyword {{#:one-tailed?}} indicates if calculation is two-sided (the default) or not.
503
504 > (f-significance 1.5 8 2)
505 0.920449812578091
506 > (f-significance 1.5 8 2 #:one-tailed? #t)
507 0.460224906289046
508
509
510=== Authors
511
512[[/users/peter-lane|Peter Lane]] wrote the Scheme version of this library.  The original Lisp version was written by [[http://compbio.ucdenver.edu/hunter/|Larry Hunter]].
513
514=== License
515
516GPL version 3.0.
517
518=== Version History
519
520* 0.12: removed GSL dependency
521* 0.11: refactoring correlation and regression interface to take two separate dataset arguments
522* 0.9: ported to CHICKEN 5
523* 0.8: added cumsum and random-weighted-sample
524* 0.5: fixed warning in compilation (thanks to Felix for pointing it out)
525* 0.4: all functions should now be working
526* 0.3: some error fixes and addition of tests for majority of functions
527* 0.2: fixed some errors in keywords and find-critical-value
528* 0.1: initial package
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