This library is a port of Larry Hunter's Lisp statistics library to chicken scheme.
The library provides a number of formulae and methods taken from the book "Fundamentals of Biostatistics" by Bernard Rosner (5th edition).
To use this library, you need to understand the underlying statistics. In brief:
The Binomial distribution is used when counting discrete events in a series of trials, each of which events has a probability p of producing a positive outcome. An example would be tossing a coin n times: the probability of a head is p, and the distribution gives the expected number of heads in the n trials. The binomial distribution is defined as B(n, p).
The Poisson distribution is used to count discrete events which occur with a known average rate. A typical example is the decay of radioactive elements. A poisson distribution is defined Pois(mu).
The Normal distribution is used for real-valued events which cluster around a specific mean with a symmetric variance. A typical example would be the distribution of people's heights. A normal distribution is defined N(mean, variance).
- average-rank value sorted-valuesprocedure
returns the average position of given value in the list of sorted values: the rank is based from 1.
> (average-rank 2 '(1 2 2 3 4)) 5/2
- beta-incomplete x a bprocedure
- bin-and-count items nprocedure
Divides the range of the list of items into n bins, and returns a vector of the number of items which fall into each bin.
> (bin-and-count '(1 1 2 3 3 4 5) 5) #(2 1 2 1 1)
- combinations n kprocedure
returns the number of ways to select k items from n, where the order does not matter.
- factorial nprocedure
returns the factorial of n.
- (find-critical-value p-function p-value #:increasing?)procedure
given 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).
- fisher-z-transform rprocedure
returns the transformation of a correlation coefficient r into an approximately normal distribution.
- gamma-incomplete a xprocedure
- gamma-ln xprocedure
- permutations n kprocedure
returns the number of ways to select k items from n, where the order does matter.
- random-normal mean sdprocedure
returns a random number distributed with specified mean and standard deviation.
- random-pick itemsprocedure
returns a random item from the given list of items.
- random-sample n itemsprocedure
returns a random sample from the list of items without replacement of size n.
- random-weighted-sample n items weightsprocedure
returns a random sample from the list of items without replacement of size n, where each sample has a defined probability of selection (weight).
- sign nprocedure
returns 0, 1 or -1 according to if n is zero, positive or negative.
- square nprocedure
- cumsum sequencesprocedure
returns the cumulative sum of a sequence.
These functions provide information on a given list of numbers, the items. Note, the list does not have to be sorted.
- mean itemsprocedure
returns the arithmetic mean of the items (the sum of the numbers divided by the number of numbers).
(mean '(1 2 3 4 5)) => 3
- median itemsprocedure
returns the value which separates the upper and lower halves of the list of numbers.
(median '(1 2 3 4)) => 5/2
- mode itemsprocedure
returns 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.)
> (mode '(1 2 3 4)) (1 2 3 4) 1 > (mode '(1 2 2 3 4)) (2) 2 > (mode '(1 2 2 3 3 4)) (2 3) 2
- geometric-mean itemsprocedure
returns 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).
(geometric-mean '(1 2 3 4 5)) => 2.60517108469735
- range itemsprocedure
returns the difference between the biggest and the smallest value from the list of items.
(range '(5 1 2 3 4)) => 4
- percentile items percentprocedure
returns 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.
(percentile '(1 2 3 4) 50) => 5/2 (percentile '(1 2 3 4) 67) => 3
- variance itemsprocedure
- standard-deviation itemsprocedure
- coefficient-of-variation itemsprocedure
returns 100 * (std-dev / mean) of the items.
(coefficient-of-variation '(1 2 3 4)) => 51.6397779494322
- standard-error-of-the-mean itemsprocedure
returns std-dev / sqrt(length items).
(standard-error-of-the-mean '(1 2 3 4)) => 0.645497224367903
- mean-sd-n itemsprocedure
returns three values, one for the mean, one for the standard deviation, and one for the length of the list.
> (mean-sd-n '(1 2 3 4)) 5/2 1.29099444873581 4
- binomial-probability n k pprocedure
returns the probability that the number of positive outcomes for a binomial distribution B(n, p) is k.
> (do-ec (: i 0 11) (format #t "i = ~d P = ~f~&" i (binomial-probability 10 i 0.5))) i = 0 P = 0.0009765625 i = 1 P = 0.009765625 i = 2 P = 0.0439453125 i = 3 P = 0.1171875 i = 4 P = 0.205078125 i = 5 P = 0.24609375 i = 6 P = 0.205078125 i = 7 P = 0.1171875 i = 8 P = 0.0439453125 i = 9 P = 0.009765625 i = 10 P = 0.0009765625
- binomial-cumulative-probability n k pprocedure
returns the probability that less than k positive outcomes occur for a binomial distribution B(n, p).
> (do-ec (: i 0 11) (format #t "i = ~d P = ~f~&" i (binomial-cumulative-probability 10 i 0.5))) i = 0 P = 0.0 i = 1 P = 0.0009765625 i = 2 P = 0.0107421875 i = 3 P = 0.0546875 i = 4 P = 0.171875 i = 5 P = 0.376953125 i = 6 P = 0.623046875 i = 7 P = 0.828125 i = 8 P = 0.9453125 i = 9 P = 0.9892578125 i = 10 P = 0.9990234375
- binomial-ge-probability n k pprocedure
returns the probability of k or more positive outcomes for a binomial distribution B(n, p).
- binomial-le-probability n k pprocedure
returns the probability k or fewer positive outcomes for a binomial distribution B(n, p).
- poisson-probability mu kprocedure
returns the probability of k events occurring when the average is mu.
> (do-ec (: i 0 20) (format #t "P(X=~2d) = ~,4f~&" i (poisson-probability 10 i))) P(X= 0) = 0.0000 P(X= 1) = 0.0005 P(X= 2) = 0.0023 P(X= 3) = 0.0076 P(X= 4) = 0.0189 P(X= 5) = 0.0378 P(X= 6) = 0.0631 P(X= 7) = 0.0901 P(X= 8) = 0.1126 P(X= 9) = 0.1251 P(X=10) = 0.1251 P(X=11) = 0.1137 P(X=12) = 0.0948 P(X=13) = 0.0729 P(X=14) = 0.0521 P(X=15) = 0.0347 P(X=16) = 0.0217 P(X=17) = 0.0128 P(X=18) = 0.0071 P(X=19) = 0.0037
- poisson-cumulative-probability mu kprocedure
returns the probability of less than k events occurring when the average is mu.
> (do-ec (: i 0 20) (format #t "P(X=~2d) = ~,4f~&" i (poisson-cumulative-probability 10 i))) P(X= 0) = 0.0000 P(X= 1) = 0.0000 P(X= 2) = 0.0005 P(X= 3) = 0.0028 P(X= 4) = 0.0103 P(X= 5) = 0.0293 P(X= 6) = 0.0671 P(X= 7) = 0.1301 P(X= 8) = 0.2202 P(X= 9) = 0.3328 P(X=10) = 0.4579 P(X=11) = 0.5830 P(X=12) = 0.6968 P(X=13) = 0.7916 P(X=14) = 0.8645 P(X=15) = 0.9165 P(X=16) = 0.9513 P(X=17) = 0.9730 P(X=18) = 0.9857 P(X=19) = 0.9928
- poisson-ge-probability mu kprocedure
returns the probability of k or more events occurring when the average is mu.
- normal-pdf x mean varianceprocedure
returns the likelihood of x given a normal distribution with stated mean and variance.
> (do-ec (: i 0 11) (format #t "~3d ~,4f~&" i (normal-pdf i 5 4))) 0 0.0088 1 0.0270 2 0.0648 3 0.1210 4 0.1760 5 0.1995 6 0.1760 7 0.1210 8 0.0648 9 0.0270 10 0.0088
- convert-to-standard-normal x mean varianceprocedure
returns a value for x rescaling the given normal distribution to a standard N(0, 1).
> (convert-to-standard-normal 5 6 2) -1/2
- phi xprocedure
returns the cumulative distribution function (CDF) of the standard normal distribution.
> (do-ec (: x -2 2 0.4) (format #t "~4,1f ~,4f~&" x (phi x))) -2.0 0.0228 -1.6 0.0548 -1.2 0.1151 -0.8 0.2119 -0.4 0.3446 0.0 0.5000 0.4 0.6554 0.8 0.7881 1.2 0.8849 1.6 0.9452
- z percentileprocedure
returns the inverse of the standard normal distribution. Input is a percentile, between 0.0 and 1.0.
- t-distribution degrees-of-freedom percentileprocedure
returns 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.
- chi-square degrees-of-freedom percentileprocedure
returns the point at which chi-square distribution has percentile to its left, using given degrees-of-freedom.
- chi-square-cdf x degrees-of-freedomprocedure
returns the probability that a random variable is to the left of x using the chi-square distribution with given degrees-of-freedom.
These 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.
- binomial-probability-ci n p alphaprocedure
returns two values, the upper and lower bounds on an observed probability p from n trials with confidence (1-alpha).
> (binomial-probability-ci 10 0.8 0.9) 0.724273681640625 0.851547241210938 ; 2 values
- poisson-mu-ci k alphaprocedure
returns two values, the upper and lower bounds on the poisson parameter if k events are observed; the bound is for confidence (1-alpha).
> (poisson-mu-ci 10 0.9) 8.305419921875 10.0635986328125 ; 2 values
- normal-mean-ci mean standard-deviation k alphaprocedure
returns 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).
> (normal-mean-ci 0.5 0.1 10 0.8) 0.491747852700165 0.508252147299835 ; 2 values
- normal-mean-ci-on-sequence items alphaprocedure
returns 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).
> (normal-mean-ci-on-sequence '(1 2 3 4 5) 0.9) 2.40860081649174 3.59139918350826 ; 2 values
- normal-variance-ci standard-deviation k alphaprocedure
returns 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).
- normal-variance-ci-on-sequence items alphaprocedure
returns 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).
returns 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).
- normal-sd-ci-on-sequence sequence itemsprocedure
returns 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).
These functions report on the significance of an observed sample against a given distribution.
- (z-test x-bar n #:mu #:sigma #:tails)procedure
Given 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:
- ':both, the probability of the difference between x-bar and #:mu
- ':positive, the probability that observation is >= x-bar
- ':negative, the probability that observation is <= x-bar
e.g. given a distribution with mean 50 and standard deviation 10
; probability that a single observation is <= 40 > (z-test 40 1 #:mu 50 #:sigma 10 #:tails ':negative) 0.158655 ; probability that 10 observations are <= 40 > (z-test 40 10 #:mu 50 #:sigma 10 #:tails ':negative) 0.000783 ; probability that 5 observations give a mean of 40 > (z-test 40 5 #:mu 50 #:sigma 10) 0.025347
- (z-test-on-sequence observations #:mu #:sigma #:tails)procedure
As for z-test except x-bar and n are computed from given observations.
- (t-test-one-sample x-bar sd n mu #:tails)procedure
Given 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:
- ':both two-sided (default)
- ':positive one-sided, x-bar >= mu
- ':negative one-sided, x-bar <= mu
- (t-test-one-sample-on-sequence observations mu #:tails)procedure
As for t-test-one-sample except x-bar, sd and n are computed from given observations.
- (t-test-paired t-bar sd n #:tails)procedure
Computes 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.
- (t-test-paired-on-sequences before after #:tails)procedure
Computes the significance of the difference between two sequences of data: one before an experimental change and one after. #:tails is as for t-significance.
> (t-test-paired-on-sequences '(4 3 5) '(1 1 3)) 0.0198039411803931
- (t-test-two-sample mean-1 sd-1 n-1 mean-2 sd-2 n-2 #:variances-equal? #:variance-significance-cutoff #:tails)procedure
Computes the significance of the difference of two means given the sample standard deviations and sizes.
- (t-test-two-sample-on-sequences sequence-1 sequence-2 #:tails)procedure
Significance of difference of two sequences of observations.
- (f-test variance-1 n1 variance-2 n2 #:tails)procedure
Tests for the equality of two variances.
- (chi-square-test-one-sample observed-variance sample-size test-variance #:tails)procedure
Tests for significance of difference between an observed and a test variance.
- (binomial-test-one-sample p-hat n p #:tails #:exact?)procedure
Returns the significance of a one sample test with n observations, observed probability p-hat and expected probability p.
- (binomial-test-two-sample p-hat-1 n-1 p-hat-2 n-2 #:tails #:exact?)procedure
Returns the significance of a two sample test.
- (fisher-exact-test a b c d #:tails)procedure
Given 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.
- (mcnemars-test a-discordant-count b-discordant-count #:exact?)procedure
For 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.
- (poisson-test-one-sample observed mu #:tails #:approximate?)procedure
Computes significance of the number of observed events under a Poisson distribution against mu expected events.
- (sign-test-on-sequence sequence-1 sequence-2 #:exact? #:tails)procedure
Takes two equal-sized sequences of observations, and reports if the entries of one are different to those in the other.
- (wilcoxon-signed-rank-test differences #:tails)procedure
Given at least 16 differences, reports if the positive differences are significantly larger or smaller than the negative differences.
- (wilcoxon-signed-rank-test-on-sequences sequence-1 sequence-2 #:tails)procedure
Given two sequences of at least 16 observations, computes wilcoxon-signed-rank-test on the differences.
- chi-square-test-rxc contingency-tableprocedure
Given a contingency table (a SRFI-63 array), returns significance of relation between row and column variable.
- chi-square-test-for-trend row1-counts row2-countsprocedure
Returns p significance of trend, and prints a string to show if increasing or decreasing.
Sample size estimates
- (t-test-one-sample-sse mean-1 mean-2 sigma-1 #:alpha #:1-beta #:tails)procedure
Returns 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.
> (t-test-one-sample-sse 5.0 5.2 0.5) 163
- (t-test-two-sample-sse mean-1 sigma-1 mean-2 sigma-2 #:alpha #:1-beta #:tails #:sample-ratio)procedure
Returns 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.
- (t-test-paired-sse difference-mean difference-sigma #:alpha #:1-beta #:tails)procedure
Returns the size of sample to produce a given mean and standard deviation in the differences of two samples.
- (binomial-test-one-sample-sse p-estimated p-null #:alpha #:1-beta #:tails)procedure
Returns 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.
- (binomial-test-two-sample-sse p-one p-two #:alpha #:1-beta #:tails #:sample-ratio)procedure
Returns 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.
- (binomial-test-paired-sse pd pa #:alpha #:1-beta #:tails)procedure
Sample size estimate for McNemar's discordant pairs test.
- (correlation-sse rho #:alpha #:1-beta)procedure
Returns 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).
> (correlation-sse 0.80 #:alpha 0.05 #:1-beta 0.9) 11
Correlation and regression
- linear-regression xs ysprocedure
Given a line definition as lists of point coordinates, first prints to the terminal and then returns 5 values for the best fitting line through the points:
- the y-intercept
- the slope
- the correlation coefficient, r
- the square of the correlation coefficient, r^2
- the significance of the difference of the slope from zero, p
(This is also called the Pearson correlation; used when relation expected to be linear. Also see spearman-rank-correlation.)
> (linear-regression '(1.0 2.0 3.0) '(0.1 0.3 0.8)) Intercept = -0.3, slope = 0.35, r = 0.970725343394151, R^2 = 0.942307692307692, p = 0.154420958311267 -0.3 0.35 0.970725343394151 0.942307692307692 0.154420958311267 ; 5 values
- correlation-coefficient xs ysprocedure
As above, but only returns the value of r:
> (correlation-coefficient '(1.0 2.0 3.0) '(0.1 0.3 0.8)) 0.970725343394151
- (correlation-test-two-sample r1 n1 r2 n2 #:tails)procedure
Returns 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.
- (correlation-test-two-sample-on-sequences points-1 points-2 #:tails)procedure
As above, but computes the correlations from given lists of points.
- spearman-rank-correlation xs ysprocedure
Returns 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.)
Significance test functions
- (t-significance t-value degrees-of-freedom #:tails)procedure
returns 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.
> (t-significance 0.2 5) 0.849360513995829 > (t-significance 0.2 5 #:tails ':positive) 0.424680256997915 > (t-significance 0.2 5 #:tails ':negative) 0.575319743002086
- (f-significance f-value numerator-dof denominator-dof #:one-tailed?)procedure
returns 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.
> (f-significance 1.5 8 2) 0.920449812578091 > (f-significance 1.5 8 2 #:one-tailed? #t) 0.460224906289046
Peter Lane wrote the Scheme version of this library. The original Lisp version was written by Larry Hunter.
This egg is hosted on the CHICKEN Subversion repository:
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GPL version 3.0.
- 0.12: removed GSL dependency
- 0.11: refactoring correlation and regression interface to take two separate dataset arguments
- 0.9: ported to CHICKEN 5
- 0.8: added cumsum and random-weighted-sample
- 0.5: fixed warning in compilation (thanks to Felix for pointing it out)
- 0.4: all functions should now be working
- 0.3: some error fixes and addition of tests for majority of functions
- 0.2: fixed some errors in keywords and find-critical-value
- 0.1: initial package