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The function names appear below. IMPORTANT: There are really *3* sets of functions. The first set has an 'l' prefix, which can be used with list or tuple arguments. The second set has an 'a' prefix, which can accept NumPy array arguments. These latter functions are defined only when NumPy is available on the system. The third type has NO prefix (i.e., has the name that appears below). Functions of this set are members of a "Dispatch" class, c/o David Ascher. This class allows different functions to be called depending on the type of the passed arguments. Thus, stats.mean is a member of the Dispatch class and stats.mean(range(20)) will call stats.lmean(range(20)) while stats.mean(Numeric.arange(20)) will call stats.amean(Numeric.arange(20)). This is a handy way to keep consistent function names when different argument types require different functions to be called. Having implementated the Dispatch class, however, means that to get info on a given function, you must use the REAL function name ... that is "print stats.lmean.__doc__" or "print stats.amean.__doc__" work fine, while "print stats.mean.__doc__" will print the doc for the Dispatch class. NUMPY FUNCTIONS ('a' prefix) generally have more argument options but should otherwise be consistent with the corresponding list functions. Disclaimers: The function list is obviously incomplete and, worse, the functions are not optimized. All functions have been tested (some more so than others), but they are far from bulletproof. Thus, as with any free software, no warranty or guarantee is expressed or implied. :-) A few extra functions that don't appear in the list below can be found by interested treasure-hunters. These functions don't necessarily have both list and array versions but were deemed useful CENTRAL TENDENCY: geometricmean harmonicmean mean median medianscore mode MOMENTS: moment variation skew kurtosis skewtest (for Numpy arrays only) kurtosistest (for Numpy arrays only) normaltest (for Numpy arrays only) ALTERED VERSIONS: tmean (for Numpy arrays only) tvar (for Numpy arrays only) tmin (for Numpy arrays only) tmax (for Numpy arrays only) tstdev (for Numpy arrays only) tsem (for Numpy arrays only) describe FREQUENCY STATS: itemfreq scoreatpercentile percentileofscore histogram cumfreq relfreq VARIABILITY: obrientransform samplevar samplestdev signaltonoise (for Numpy arrays only) var stdev sterr sem z zs zmap (for Numpy arrays only) TRIMMING FCNS: threshold (for Numpy arrays only) trimboth trim1 round (round all vals to 'n' decimals; Numpy only) CORRELATION FCNS: covariance (for Numpy arrays only) correlation (for Numpy arrays only) paired pearsonr spearmanr pointbiserialr kendalltau linregress INFERENTIAL STATS: ttest_1samp ttest_ind ttest_rel chisquare ks_2samp mannwhitneyu ranksums wilcoxont kruskalwallish friedmanchisquare PROBABILITY CALCS: chisqprob erfcc zprob ksprob fprob betacf gammln betai ANOVA FUNCTIONS: F_oneway F_value SUPPORT FUNCTIONS: writecc incr sign (for Numpy arrays only) sum cumsum ss summult sumdiffsquared square_of_sums shellsort rankdata outputpairedstats findwithin N(s*f0.59999999999999998sDispatchcBs tZdZd„Zd„ZRS(s‹ The Dispatch class, care of David Ascher, allows different functions to be called depending on the argument types. This way, there can be one function name regardless of the argument type. To access function doc in stats.py module, prefix the function with an 'l' or 'a' for list or array arguments, respectively. That is, print stats.lmean.__doc__ or print stats.amean.__doc__ or whatever. cGsh|_x_|D]W\}}xH|D]@}||iiƒjotdt|ƒ‚n||i|1 by 100 in lscoreatpercentile(). f100.0N(spercentslensinliststargetcfs histogramshslrlsbinsizesextrasscumsumscopysdeepcopyscumhistsrangesisfloatsscore( sinlistspercentsscoresishscumhistslrlsbinsizestargetcfsextras((s5/afs/cs.wisc.edu/u/n/a/naze/gametheory/sexes/stats.pyslscoreatpercentileËs  0i c Cs™t|||ƒ\}}} } tt i |ƒƒ}t ||t| ƒƒ}||d||| |t| ƒ||tt|ƒƒd}|SdS(sß Returns the percentile value of a score relative to the distribution given by inlist. Formula depends on the values used to histogram the data(!). Usage: lpercentileofscore(inlist,score,histbins=10,defaultlimits=None) iidN(s histogramsinlistshistbinss defaultlimitsshslrlsbinsizesextrasscumsumscopysdeepcopyscumhistsintsscoresfloatsislenspct( sinlistsscoreshistbinss defaultlimitssishscumhistslrlspctsbinsizesextras((s5/afs/cs.wisc.edu/u/n/a/naze/gametheory/sexes/stats.pyslpercentileofscoreßs Dic Cs|tjort|ƒttgjpt|ƒdjo|} dt|ƒ} n|d} |d} | | t |ƒ}n]t|ƒt |ƒt |ƒd}t|ƒt |ƒ|t |ƒ}t |ƒ|d} dg|} d}xq|D]i}yO|| djo|d}n-t|| t |ƒƒ}| |d| |Wt|ƒt |dƒSdS(s˜ Returns the variance of the values in the passed list using N-1 for the denominator (i.e., for estimating population variance). Usage: lvar(inlist) iiN( slensinlistsnsmeansmns deviationssrangesissssfloat(sinlists deviationssismnsn((s5/afs/cs.wisc.edu/u/n/a/naze/gametheory/sexes/stats.pyslvaros  cCstit|ƒƒSdS(s Returns the standard deviation of the values in the passed list using N-1 in the denominator (i.e., to estimate population stdev). Usage: lstdev(inlist) N(smathssqrtsvarsinlist(sinlist((s5/afs/cs.wisc.edu/u/n/a/naze/gametheory/sexes/stats.pyslstdev~scCs't|ƒttit|ƒƒƒSdS(s¢ Returns the standard error of the values in the passed list using N-1 in the denominator (i.e., to estimate population standard error). Usage: lsterr(inlist) N(sstdevsinlistsfloatsmathssqrtslen(sinlist((s5/afs/cs.wisc.edu/u/n/a/naze/gametheory/sexes/stats.pyslsterrˆscCs-t|ƒ}t|ƒ}|ti|ƒSdS(s‹ Returns the estimated standard error of the mean (sx-bar) of the values in the passed list. sem = stdev / sqrt(n) Usage: lsem(inlist) N(sstdevsinlistssdslensnsmathssqrt(sinlistsnssd((s5/afs/cs.wisc.edu/u/n/a/naze/gametheory/sexes/stats.pyslsem’s  cCs"|t|ƒt|ƒ}|SdS(s² Returns the z-score for a given input score, given that score and the list from which that score came. Not appropriate for population calculations. Usage: lz(inlist, score) N(sscoresmeansinlists samplestdevsz(sinlistsscoresz((s5/afs/cs.wisc.edu/u/n/a/naze/gametheory/sexes/stats.pyslzžscCs5g}x$|D]}|it||ƒƒq W|SdS(sZ Returns a list of z-scores, one for each score in the passed list. Usage: lzs(inlist) N(szscoressinlistsitemsappendsz(sinlistszscoressitem((s5/afs/cs.wisc.edu/u/n/a/naze/gametheory/sexes/stats.pyslzs©s cCs5t|t|ƒƒ}t|ƒ|}|||!SdS(s‹ Slices off the passed proportion of items from BOTH ends of the passed list (i.e., with proportiontocut=0.1, slices 'leftmost' 10% AND 'rightmost' 10% of scores. Assumes list is sorted by magnitude. Slices off LESS if proportion results in a non-integer slice index (i.e., conservatively slices off proportiontocut). Usage: ltrimboth (l,proportiontocut) Returns: trimmed version of list l N(sintsproportiontocutslenslslowercutsuppercut(slsproportiontocutsuppercutslowercut((s5/afs/cs.wisc.edu/u/n/a/naze/gametheory/sexes/stats.pys ltrimboth¹s srightcCsy|djo*d}t|ƒt|t|ƒƒ}n4|djo&t|t|ƒƒ}t|ƒ}n|||!SdS(s Slices off the passed proportion of items from ONE end of the passed list (i.e., if proportiontocut=0.1, slices off 'leftmost' or 'rightmost' 10% of scores). Slices off LESS if proportion results in a non-integer slice index (i.e., conservatively slices off proportiontocut). Usage: ltrim1 (l,proportiontocut,tail='right') or set tail='left' Returns: trimmed version of list l srightisleftN(stailslowercutslenslsintsproportiontocutsuppercut(slsproportiontocutstailsuppercutslowercut((s5/afs/cs.wisc.edu/u/n/a/naze/gametheory/sexes/stats.pysltrim1És  $ cCs¯d}x1|ddddddgjodGtƒ}q W|ddddgjoîd Gt||ƒ}tti|d ƒti|d ƒƒ\} } | d jod t t | dƒƒ}nd}|GH|ddgjoâ|d djo;t||d ƒ\} } dGt | dƒGt | dƒGHq=t|ƒdjpt|ƒdjo8t||ƒ\}} dGt |dƒGt | dƒGHq=t||ƒ\} } dGt | dƒGt | dƒGHq¢|d djo;t||d ƒ\} } dGt | dƒGt | dƒGHq¢t||ƒ\} } dGt | dƒGt | dƒGHnbd}x1|ddddddgjodGtƒ}qJW|ddgjo‹t||ƒ\} }}} }dGHdddddgt | dƒt |dƒt |dƒt | dƒt |dƒgg}ti|ƒnŠ|ddgjo=t||ƒ\}} dGHd Gt |dƒGt | dƒGHn:t||ƒ\}} d!GHd"Gt |dƒGt | dƒGHd#GHtSd$S(%s¾ Interactively determines the type of data and then runs the appropriated statistic for paired group data. Usage: lpaired(x,y) Returns: appropriate statistic name, value, and probability ssisrsIsRscsCs9 Independent or related samples, or correlation (i,r,c): s Comparing variances ...iif0.050000000000000003s unequal, p=isequalses Independent samples t-test: is( Rank Sums test (NONparametric, n>20): s. Mann-Whitney U-test (NONparametric, ns<20): s Related samples t-test: s# Wilcoxon T-test (NONparametric): sdsDs9 Is the data Continuous, Ranked, or Dichotomous (c,r,d): s/ Linear regression for continuous variables ...sSlopes InterceptsProbs SEestimates% Correlation for ranked variables ...sSpearman's r: s/ Assuming x contains a dichotomous variable ...sPoint Biserial r: s N( ssampless raw_inputsobrientransformsxsysrsF_onewayspstatscolexsfspsstrsroundsvartypes ttest_indstslensranksumsszs mannwhitneyusus ttest_relscorrtypes linregresssmsbsseeslolsprintccs spearmanrspointbiserialrsNone(sxsysseesvartypeslolsrssamplesscorrtypesbsfsmspsustsz((s5/afs/cs.wisc.edu/u/n/a/naze/gametheory/sexes/stats.pyslpairedàs^- #&####W#c CsEd}t|ƒt|ƒjo td‚nt|ƒ}tt|ƒ}tt|ƒ}t|ƒ} t|ƒ}|t ||ƒt |ƒt |ƒ}ti|t|ƒt|ƒ|t|ƒt|ƒƒ}||} |d}| ti|d| |d| |ƒ} td|d|t|| | ƒƒ}| |fSdS(s  Calculates a Pearson correlation coefficient and the associated probability value. Taken from Heiman's Basic Statistics for the Behav. Sci (2nd), p.195. Usage: lpearsonr(x,y) where x and y are equal-length lists Returns: Pearson's r value, two-tailed p-value f1.0000000000000001e-30s/Input values not paired in pearsonr. Aborting.if1.0f0.5N(sTINYslensxsys ValueErrorsnsmapsfloatsmeansxmeansymeanssummultssumsr_numsmathssqrtsssssquare_of_sumssr_densrsdfstsbetaisprob( sxsysr_numsr_densTINYsprobsdfsymeansnsrstsxmean((s5/afs/cs.wisc.edu/u/n/a/naze/gametheory/sexes/stats.pys lpearsonr s     ';  +(c Csäd}t|ƒt|ƒjo td‚nt|ƒ}t|ƒ}t|ƒ}t ||ƒ} dd| t ||ddƒ}|t i|d|dd|ƒ} |d}td|d||| | ƒ}||fSdS( sð Calculates a Spearman rank-order correlation coefficient. Taken from Heiman's Basic Statistics for the Behav. Sci (1st), p.192. Usage: lspearmanr(x,y) where x and y are equal-length lists Returns: Spearman's r, two-tailed p-value f1.0000000000000001e-30s0Input values not paired in spearmanr. Aborting.iiif1.0f0.5N(sTINYslensxsys ValueErrorsnsrankdatasrankxsrankyssumdiffsquaredsdsqsfloatsrssmathssqrtstsdfsbetaisprobrs( sxsysrankxsrssdfsrankysnsprobrssTINYsdsqst((s5/afs/cs.wisc.edu/u/n/a/naze/gametheory/sexes/stats.pys lspearmanr:s    $' "cCsÄd}t|ƒt|ƒjo td‚nti||ƒ}ti|ƒ}t|ƒdjo td‚nTti|t dƒƒ} ti || dƒ}ti|d|dƒ}ti|d|dƒ}tti|dƒƒ}tti|dƒƒ} t|ƒ} tit|ƒt| ƒt|ƒt| ƒƒ} | |tti|dƒƒ| }| d}|ti|d||d||ƒ} td|d||| | ƒ}||fSd S( s Calculates a point-biserial correlation coefficient and the associated probability value. Taken from Heiman's Basic Statistics for the Behav. Sci (1st), p.194. Usage: lpointbiserialr(x,y) where x,y are equal-length lists Returns: Point-biserial r, two-tailed p-value f1.0000000000000001e-30s5INPUT VALUES NOT PAIRED IN pointbiserialr. ABORTING.is3Exactly 2 categories required for pointbiserialr().iif1.0f0.5N(sTINYslensxsys ValueErrorspstatsabutsdatasuniques categoriessrangescodemapsrecodesrecodedslinexandsmeanscolexsxmeansymeansnsmathssqrtsfloatsadjusts samplestdevsrpbsdfstsbetaisprob(sxsysrpbsTINYsdfsrecodedsdatas categoriessprobsymeanscodemapsnsadjuststsxmean((s5/afs/cs.wisc.edu/u/n/a/naze/gametheory/sexes/stats.pyslpointbiserialrRs*   3$ +"cCsbd} d} d}xÏtt|ƒdƒD]·}x®t|t|ƒƒD]—}||||}||||} || }|o=| d} | d} |djo|d}qÜ|d}qE|o| d} qE| d} qEWq)W|t i | | ƒ}dt|ƒddt|ƒt|ƒd}|t i |ƒ} tt| ƒdƒ} || fSdS(sÓ Calculates Kendall's tau ... correlation of ordinal data. Adapted from function kendl1 in Numerical Recipies. Needs good test-routine.@@@ Usage: lkendalltau(x,y) Returns: Kendall's tau, two-tailed p-value iif4.0f10.0f9.0f1.4142136000000001N(sn1sn2sisssrangeslensxsjsysksa1sa2saasmathssqrtstaussvarszserfccsabssprob(sxsysaastausisssksjssvarsa1sa2sn1sn2szsprob((s5/afs/cs.wisc.edu/u/n/a/naze/gametheory/sexes/stats.pys lkendalltaurs2    0cCsÈd}t|ƒt|ƒjo td‚nt|ƒ} tt|ƒ}tt|ƒ}t|ƒ}t|ƒ} t| t ||ƒt |ƒt |ƒƒ}ti| t|ƒt|ƒ| t|ƒt|ƒƒ}||} dtid| |d| |ƒ}| d}| ti|d| |d| |ƒ} td|d||| | ƒ}|t| t|ƒt|ƒƒ}| ||}tid| | ƒt|ƒ} ||| || fSdS(s½ Calculates a regression line on x,y pairs. Usage: llinregress(x,y) x,y are equal-length lists of x-y coordinates Returns: slope, intercept, r, two-tailed prob, sterr-of-estimate f9.9999999999999995e-21s1Input values not paired in linregress. Aborting.f0.5f1.0iiN(sTINYslensxsys ValueErrorsnsmapsfloatsmeansxmeansymeanssummultssumsr_numsmathssqrtsssssquare_of_sumssr_densrslogszsdfstsbetaisprobsslopes intercepts samplestdevssterrest(sxsysslopesr_numsr_densTINYsdfs interceptsprobssterrestsymeansnsrstsxmeansz((s5/afs/cs.wisc.edu/u/n/a/naze/gametheory/sexes/stats.pys llinregress•s(    -; ' +"$!sSamplesac Csøt|ƒ} t|ƒ} t|ƒ} | d}| d| t|ƒ}| |t i |d| ƒ} td|dt|ƒ|| | ƒ}|djoPd}t||dd|ddd|| | | t|ƒt|ƒ|| |ƒn| |fSdS( s’ Calculates the t-obtained for the independent samples T-test on ONE group of scores a, given a population mean. If printit=1, results are printed to the screen. If printit='filename', the results are output to 'filename' using the given writemode (default=append). Returns t-value, and prob. Usage: lttest_1samp(a,popmean,Name='Sample',printit=0,writemode='a') Returns: t-value, two-tailed prob if1.0f0.5isSingle-sample T-test.s Populations--N(smeansasxsvarsvslensnsdfsfloatssvarspopmeansmathssqrtstsbetaisprobsprintitsstatnamesoutputpairedstatss writemodesnamesminsmax( saspopmeansprintitsnames writemodesstatnamessvarsprobsdfsnstsvsx((s5/afs/cs.wisc.edu/u/n/a/naze/gametheory/sexes/stats.pys lttest_1sampµs     (  sSamp1sSamp2cCsBt|ƒ} t|ƒ} t|ƒd} t|ƒd} t|ƒ}t|ƒ}||d} |d| |d| t | ƒ}| | ti|d|d|ƒ}td| d| | ||ƒ}|djo\d}t||||| | t|ƒt|ƒ||| | t|ƒt|ƒ|||ƒn||fSdS(s™ Calculates the t-obtained T-test on TWO INDEPENDENT samples of scores a, and b. From Numerical Recipies, p.483. If printit=1, results are printed to the screen. If printit='filename', the results are output to 'filename' using the given writemode (default=append). Returns t-value, and prob. Usage: lttest_ind(a,b,printit=0,name1='Samp1',name2='Samp2',writemode='a') Returns: t-value, two-tailed prob iif1.0f0.5isIndependent samples T-test.N(smeansasx1sbsx2sstdevsv1sv2slensn1sn2sdfsfloatssvarsmathssqrtstsbetaisprobsprintitsstatnamesoutputpairedstatss writemodesname1sminsmaxsname2(sasbsprintitsname1sname2s writemodesstatnamessvarsprobsdfsv1sv2sx2sx1stsn1sn2((s5/afs/cs.wisc.edu/u/n/a/naze/gametheory/sexes/stats.pys lttest_indÐs$     $'"  sSample1sSample2cCs‰t|ƒt|ƒjo td‚nt|ƒ} t|ƒ} t|ƒ}t|ƒ} t|ƒ}d} x8t t|ƒƒD]$}| ||| ||| } q{W|d}| t|ƒ} ti|| d| t|ƒƒ}| | |}td|d||||ƒ} |djo\d}t||||| |t|ƒt|ƒ||| | t|ƒt|ƒ||| ƒn|| fSdS(s™ Calculates the t-obtained T-test on TWO RELATED samples of scores, a and b. From Numerical Recipies, p.483. If printit=1, results are printed to the screen. If printit='filename', the results are output to 'filename' using the given writemode (default=append). Returns t-value, and prob. Usage: lttest_rel(a,b,printit=0,name1='Sample1',name2='Sample2',writemode='a') Returns: t-value, two-tailed prob s"Unequal length lists in ttest_rel.iif2.0f0.5sRelated samples T-test.N(slensasbs ValueErrorsmeansx1sx2svarsv1sv2snscovsrangesisdfsfloatsmathssqrtssdstsbetaisprobsprintitsstatnamesoutputpairedstatss writemodesname1sminsmaxsname2(sasbsprintitsname1sname2s writemodesstatnamesdfsv1sv2sx2sx1sprobscovsisnstssd((s5/afs/cs.wisc.edu/u/n/a/naze/gametheory/sexes/stats.pys lttest_relïs0       " %"  cCs¦t|ƒ}|tjo't|ƒt|ƒgt|ƒ}nd}xBtt|ƒƒD].}|||||dt||ƒ}qYW|t ||dƒfSdS(sD Calculates a one-way chi square for list of observed frequencies and returns the result. If no expected frequencies are given, the total N is assumed to be equally distributed across all groups. Usage: lchisquare(f_obs, f_exp=None) f_obs = list of observed cell freq. Returns: chisquare-statistic, associated p-value iiiN( slensf_obssksf_expsNonessumsfloatschisqsrangesis chisqprob(sf_obssf_expsischisqsk((s5/afs/cs.wisc.edu/u/n/a/naze/gametheory/sexes/stats.pys lchisquares  ',cCszd}d}d}d}t|ƒ} t|ƒ} | } | } d} |i ƒ|i ƒx¼|| jo || jo¡||}||}||jo|t| ƒ}|d}n||jo|t| ƒ}|d}n||}ti|ƒti| ƒjo |} qYqYWyGti| | t| | ƒƒ}t|dd|t| ƒƒ}Wn d}nX| |fSdS(sÚ Computes the Kolmogorov-Smirnof statistic on 2 samples. From Numerical Recipies in C, page 493. Usage: lks_2samp(data1,data2) data1&2 are lists of values for 2 conditions Returns: KS D-value, associated p-value if0.0if0.12f0.11f1.0N(sj1sj2sfn1sfn2slensdata1sn1sdata2sn2sen1sen2sdssortsd1sd2sfloatsdtsmathsfabsssqrtsensksprobsabssprob(sdata1sdata2sprobsensfn2sfn1sd2sj1sj2sdsen1sen2sn1sn2sd1sdt((s5/afs/cs.wisc.edu/u/n/a/naze/gametheory/sexes/stats.pys lks_2samp%s>         !& cCst|ƒ} t|ƒ} t||ƒ}|d| !} || }| | | | ddt | ƒ}| | |}t ||ƒ}t||ƒ}tit|ƒƒ}|djo td‚nti|| | | | ddƒ} t|| | d| ƒ} |dt| ƒfSdS(sº Calculates a Mann-Whitney U statistic on the provided scores and returns the result. Use only when the n in each condition is < 20 and you have 2 independent samples of ranks. NOTE: Mann-Whitney U is significant if the u-obtained is LESS THAN or equal to the critical value of U found in the tables. Equivalent to Kruskal-Wallis H with just 2 groups. Usage: lmannwhitneyu(data) Returns: u-statistic, one-tailed p-value (i.e., p(z(U))) iif2.0s*All numbers are identical in lmannwhitneyuf12.0f1.0N(slensxsn1sysn2srankdatasrankedsrankxsrankyssumsu1su2smaxsbigusminssmallusmathssqrts tiecorrectsTs ValueErrorssdsabsszszprob(sxsysbigusu1su2ssmallusrankysrankedsTsrankxsn1sn2szssd((s5/afs/cs.wisc.edu/u/n/a/naze/gametheory/sexes/stats.pys lmannwhitneyuLs     $  'cCsìt|ƒ\}}t|ƒ}d}d}x›||djo‰||||djobd}xC||djo||||djo|d}|d}q`W||d|}n|d}q-W|t |d|ƒ}d|SdS(s Corrects for ties in Mann Whitney U and Kruskal Wallis H tests. See Siegel, S. (1956) Nonparametric Statistics for the Behavioral Sciences. New York: McGraw-Hill. Code adapted from |Stat rankind.c code. Usage: ltiecorrect(rankvals) Returns: T correction factor for U or H f0.0iiif1.0N( s shellsortsrankvalsssortedsposnslensnsTsisntiessfloat(srankvalssposnsntiessTsisnssorted((s5/afs/cs.wisc.edu/u/n/a/naze/gametheory/sexes/stats.pys ltiecorrectis" * c Cs·t|ƒ}t|ƒ}||} t| ƒ}|| }||}t|ƒ}|||dd}||t i ||||ddƒ}ddtt|ƒƒ}||fSdS(só Calculates the rank sums statistic on the provided scores and returns the result. Use only when the n in each condition is > 20 and you have 2 independent samples of ranks. Usage: lranksums(x,y) Returns: a z-statistic, two-tailed p-value if2.0f12.0if1.0N(slensxsn1sysn2salldatasrankdatasrankedssumsssexpectedsmathssqrtszszprobsabssprob( sxsysexpectedsssrankedsn1sn2szsprobsalldata((s5/afs/cs.wisc.edu/u/n/a/naze/gametheory/sexes/stats.pys lranksums‚s       +cCst|ƒt|ƒjo td‚ng}xJtt|ƒƒD]6}||||}|djo|i|ƒq?q?Wt|ƒ}t t |ƒ} t | ƒ} d}d} xKtt| ƒƒD]7}||djo| | |} q¿|| |}q¿Wt|| ƒ}||dd}ti||dd|ddƒ}ti||ƒ|} d dtt | ƒƒ} || fSd S( s¶ Calculates the Wilcoxon T-test for related samples and returns the result. A non-parametric T-test. Usage: lwilcoxont(x,y) Returns: a t-statistic, two-tail probability estimate s"Unequal N in wilcoxont. Aborting.if0.0if0.25f2.0f1.0f24.0iN(slensxsys ValueErrorsdsrangesisdiffsappendscountsmapsabssabsdsrankdatas absrankedsr_plussr_minussminswtsmnsmathssqrtssesfabsszszprobsprob(sxsysdiffswtscountsr_plussdsismns absrankedsprobsabsdszsr_minussse((s5/afs/cs.wisc.edu/u/n/a/naze/gametheory/sexes/stats.pys lwilcoxont˜s2    'c Gsœt|ƒ}dgt|ƒ}g}tt|ƒ}x(tt|ƒƒD]}|||}qGWt|ƒ}t |ƒ}x=tt|ƒƒD])}|d||!||<|d||5qŠWg} xQtt|ƒƒD]=}| i t||ƒdƒ| |t||ƒ| |0, 1.0-zprob(z) = 1-tail probability for any z, 2.0*(1.0-zprob(abs(z))) = 2-tail probability Adapted from z.c in Gary Perlman's |Stat. Usage: lzprob(z) f6.0f0.0f0.5f1.0f0.000124818987f0.001075204047f0.0051987750189999996f0.019198292004f0.059054035642f0.15196875136400001f0.31915293269400002f0.53192300729999997f0.79788456059299995f2.0f4.5255659000000002e-05f 0.00015252929f1.9538131999999999e-05f0.00067690498600000005f0.0013906042840000001f0.00079462081999999998f0.0020342548740000001f0.0065497912140000001f0.010557625006f0.011630447319f0.0092794533410000008f0.0053535791080000002f0.0021412687410000001f0.00053531084899999996f0.99993665752399996N(sZ_MAXszsxsmathsfabssyswsprob(szsZ_MAXswsysxsprob((s5/afs/cs.wisc.edu/u/n/a/naze/gametheory/sexes/stats.pyslzprobFs      R w cCs¹d}d}d}d||}xŽtddƒD]}}|ti|||ƒ}||}ti |ƒd|jpti |ƒd|jo|Sn| }ti |ƒ}q0WdSd S( su Computes a Kolmolgorov-Smirnov t-test significance level. Adapted from Numerical Recipies. Usage: lksprob(alam) f2.0f0.0f-2.0iiÉf0.001f1e-08f1.0N( sfacssumstermbfsalamsa2srangesjsmathsexpstermsfabs(salamstermssumsjstermbfsa2sfac((s5/afs/cs.wisc.edu/u/n/a/naze/gametheory/sexes/stats.pyslksprobps 4cCs4td|d||t|||ƒƒ}|SdS(sÿ Returns the (1-tailed) significance level (p-value) of an F statistic given the degrees of freedom for the numerator (dfR-dfF) and the degrees of freedom for the denominator (dfF). Usage: lfprob(dfnum, dfden, F) where usually dfnum=dfbn, dfden=dfwn f0.5N(sbetaisdfdensdfnumsfloatsFsp(sdfnumsdfdensFsp((s5/afs/cs.wisc.edu/u/n/a/naze/gametheory/sexes/stats.pyslfprob…s,cCs\d} d}d}} }||} |d} |d} d| || }xt | dƒD]ô}t|dƒ}||}||||| |||}| ||}|||}|| | ||| |||}||| }|||}| }||}||}||} d}t| |ƒ|t| ƒjo| Sq[q[WdGHdS(s  This function evaluates the continued fraction form of the incomplete Beta function, betai. (Adapted from: Numerical Recipies in C.) Usage: lbetacf(a,b,x) iÈf2.9999999999999999e-07f1.0is-a or b too big, or ITMAX too small in Betacf.N(sITMAXsEPSsbmsazsamsasbsqabsqapsqamsxsbzsrangesisfloatsemstemsdsapsbpsappsbppsaoldsabs(sasbsxsemstemsappsamsEPSsapsazsqapsqabsqamsITMAXsbmsaoldsbpsbzsdsisbpp((s5/afs/cs.wisc.edu/u/n/a/naze/gametheory/sexes/stats.pyslbetacf‘s4    "'   ! cCs¡dddddd g}|d}|d}||d ti|ƒ}d}x6tt|ƒƒD]"}|d }||||}qaW| tid |ƒSd S( sœ Returns the gamma function of xx. Gamma(z) = Integral(0,infinity) of t^(z-1)exp(-t) dt. (Adapted from: Numerical Recipies in C.) Usage: lgammln(xx) f76.180091730000001f-86.505320330000004f24.014098220000001f -1.231739516f 0.00120858003f5.3638199999999999e-06f1.0f5.5f0.5if2.5066282746500002N( scoeffsxxsxstmpsmathslogssersrangeslensj(sxxstmpsserscoeffsjsx((s5/afs/cs.wisc.edu/u/n/a/naze/gametheory/sexes/stats.pyslgammln³s   cCs|djp |djo td‚n|djp |djo d}nTtit||ƒt|ƒt|ƒ|ti|ƒ|tid|ƒƒ}||d||djo"|t |||ƒt |ƒSn'd|t ||d|ƒt |ƒSdS(sW Returns the incomplete beta function: I-sub-x(a,b) = 1/B(a,b)*(Integral(0,x) of t^(a-1)(1-t)^(b-1) dt) where a,b>0 and B(a,b) = G(a)*G(b)/(G(a+b)) where G(a) is the gamma function of a. The continued fraction formulation is implemented here, using the betacf function. (Adapted from: Numerical Recipies in C.) Usage: lbetai(a,b,x) f0.0f1.0sBad x in lbetaif2.0N( sxs ValueErrorsbtsmathsexpsgammlnsasbslogsbetacfsfloat(sasbsxsbt((s5/afs/cs.wisc.edu/u/n/a/naze/gametheory/sexes/stats.pyslbetaiÈs   S"cGs t|ƒ} dg| }dg| }dg| }g}tti |ƒ}tt |ƒ}tt |ƒ}tt|ƒ}x(t t|ƒƒD]} ||| }q‹Wti |ƒ}t|ƒ}t|ƒt|ƒt|ƒ}d}x7|D]/}|tti |ƒƒtt|ƒƒ}qëW|t|ƒt|ƒ}||}| d} || } |t| ƒ}|t| ƒ}||} t| | | ƒ}| |fSdS(s Performs a 1-way ANOVA, returning an F-value and probability given any number of groups. From Heiman, pp.394-7. Usage: F_oneway(*lists) where *lists is any number of lists, one per treatment group Returns: F value, one-tailed p-value iiN(slenslistssasmeanssvarssnssalldatasmapsNsarraystmpsameansavarsrangesisbignsasssasquare_of_sumssfloatssstotsssbnslistssswnsdfbnsdfwnsmsbsmswsfsfprobsprob(slistssvarsstmpssstotsmswsbignsnssmsbsmeanssdfwnsdfbnsasfsislistsprobssswnsalldatasssbn((s5/afs/cs.wisc.edu/u/n/a/naze/gametheory/sexes/stats.pys lF_onewayås:      -    cCs$||t|ƒ|t|ƒSdS(sr Returns an F-statistic given the following: ER = error associated with the null hypothesis (the Restricted model) EF = error associated with the alternate hypothesis (the Full model) dfR-dfF = degrees of freedom of the numerator dfF = degrees of freedom associated with the denominator/Full model Usage: lF_value(ER,EF,dfnum,dfden) N(sERsEFsfloatsdfnumsdfden(sERsEFsdfnumsdfden((s5/afs/cs.wisc.edu/u/n/a/naze/gametheory/sexes/stats.pyslF_value s swicCsDt|dƒttgjo |g}nt||ƒ}g}t i |ƒ} xat t |ƒƒD]M} || dgjp|| djp|| djo|| g}qaqaW|iƒx|D] }| |=qÃWdgt | dƒ}x_t t | dƒƒD]G} ti| | ƒ} tti| ƒ} ttt | ƒƒ||| }q*q*W|SdS(s  Returns an integer representing a binary vector, where 1=within- subject factor, 0=between. Input equals the entire data 2D list (i.e., column 0=random factor, column -1=measured values (those two are skipped). Note: input data is in |Stat format ... a list of lists ("2D list") with one row per measured value, first column=subject identifier, last column= score, one in-between column per factor (these columns contain level designations on each factor). See also stats.anova.__doc__. Usage: lfindwithin(data) data in |Stat format iiN(slensdatasnumfacts withinvecsrangescolspstatsuniquescolexs examplelevelslinexandsrowss factsubjssallsubjs(sdatasrowss factsubjss withinvecsallsubjssnumfacts examplelevelscol((s5/afs/cs.wisc.edu/u/n/a/naze/gametheory/sexes/stats.pys lfindwithins c CsÛti|tiƒ}|tjoGti|ƒ}t|ƒ}ti |d|ƒ}ti i |ƒ}nkt |ƒttgjox|i|}ti |d|ƒ}ti i ||ƒ}|djo/t|iƒ}d|| 0.0; >0 means extra weight in left tail). Use askewtest() to see if it's close enough. Dimension can equal None (ravel array first), an integer (the dimension over which to operate), or a sequence (operate over multiple dimensions). Usage: askew(a, dimension=None) Returns: skew of vals in a along dimension, returning ZERO where all vals equal if1.5isNumber of zeros in askew: iN( sNspowersamomentsas dimensionsdenomsequalszerostypes ArrayTypesasumswhere(sas dimensionszerosdenom((s5/afs/cs.wisc.edu/u/n/a/naze/gametheory/sexes/stats.pysaskew= s ) cCstit|d|ƒdƒ}ti|dƒ}t|ƒti jot |ƒdjodGt |ƒGHn||}ti |dt|d|ƒ|ƒSdS(s­ Returns the kurtosis of a distribution (normal ==> 3.0; >3 means heavier in the tails, and usually more peaked). Use akurtosistest() to see if it's close enough. Dimension can equal None (ravel array first), an integer (the dimension over which to operate), or a sequence (operate over multiple dimensions). Usage: akurtosis(a,dimension=None) Returns: kurtosis of values in a along dimension, and ZERO where all vals equal iisNumber of zeros in akurtosis: iN( sNspowersamomentsas dimensionsdenomsequalszerostypes ArrayTypesasumswhere(sas dimensionszerosdenom((s5/afs/cs.wisc.edu/u/n/a/naze/gametheory/sexes/stats.pys akurtosisP s ) cCs­|tjoti|ƒ}d}n|i|}tii|ƒti i|ƒf}t ||ƒ}t ||ƒ}t||ƒ}t||ƒ}||||||fSdS(s< Returns several descriptive statistics of the passed array. Dimension can equal None (ravel array first), an integer (the dimension over which to operate), or a sequence (operate over multiple dimensions). Usage: adescribe(inarray,dimension=None) Returns: n, (min,max), mean, standard deviation, skew, kurtosis iN(s dimensionsNonesNsravelsinarraysshapesnsminimumsreducesmaximumsmmsameansmsastdevssdsaskewsskews akurtosisskurt(sinarrays dimensionsmmsskewsmsnskurtssd((s5/afs/cs.wisc.edu/u/n/a/naze/gametheory/sexes/stats.pys adescribec s   $c Cs|tjoti|ƒ}d}nt||ƒ}t|i|ƒ}|ti |d|dd|dƒ}d||d|d|d|d|d |d |d |d }d ti d|dƒ} dti titi | ƒƒƒ}ti d| dƒ}titi|dƒd|ƒ}|ti||ti ||ddƒƒ}|dt|ƒdfSdS(s2 Tests whether the skew is significantly different from a normal distribution. Dimension can equal None (ravel array first), an integer (the dimension over which to operate), or a sequence (operate over multiple dimensions). Usage: askewtest(a,dimension=None) Returns: z-score and 2-tail z-probability iiif6.0if3.0iiFf2.0iii iÿÿÿÿf1.0N(s dimensionsNonesNsravelsasaskewsb2sfloatsshapesnssqrtsysbeta2sW2slogsdeltasalphaswheresequalsZszprob( sas dimensionsZsbeta2sb2sdeltasysalphasnsW2((s5/afs/cs.wisc.edu/u/n/a/naze/gametheory/sexes/stats.pys askewtest| s   +J%!0c Cs>|tjoti|ƒ}d}nt|i|ƒ}|djo dG|GHnt||ƒ} d|d|d}d||d|d|d|d|d|d }| |ti |ƒ} d ||d |d|d |d ti d |d|d ||d|dƒ} d d | d| ti dd| dƒ}ddd|}d| ti d|dƒ} titi| dƒd| ƒ} titi| dƒ|tidd|| ddƒƒ}||ti dd|ƒ}titi| dƒd|ƒ}|dt|ƒdfSdS(sa Tests whether a dataset has normal kurtosis (i.e., kurtosis=3(n-1)/(n+1)) Valid only for n>20. Dimension can equal None (ravel array first), an integer (the dimension over which to operate), or a sequence (operate over multiple dimensions). Usage: akurtosistest(a,dimension=None) Returns: z-score and 2-tail z-probability, returns 0 for bad pixels iis<akurtosistest only valid for n>=20 ... continuing anyway, n=f3.0if24.0iiif6.0ii f8.0f2.0f4.0f9.0icf1.0N(s dimensionsNonesNsravelsasfloatsshapesns akurtosissb2sEsvarb2ssqrtsxs sqrtbeta1sAsterm1sdenomswhereslesssequalspowersterm2sZszprob( sas dimensionsAsZsEsterm2sterm1svarb2snsb2sdenomsxs sqrtbeta1((s5/afs/cs.wisc.edu/u/n/a/naze/gametheory/sexes/stats.pys akurtosistest• s(     :[/!=!cCs‰|tjoti|ƒ}d}nt||ƒ\}}t||ƒ\}}ti |dƒti |dƒ}|t |dƒfSdS(sT Tests whether skew and/OR kurtosis of dataset differs from normal curve. Can operate over multiple dimensions. Dimension can equal None (ravel array first), an integer (the dimension over which to operate), or a sequence (operate over multiple dimensions). Usage: anormaltest(a,dimension=None) Returns: z-score and 2-tail probability iiN( s dimensionsNonesNsravelsas askewtestsssps akurtosistestskspowersk2s achisqprob(sas dimensionsk2spsssk((s5/afs/cs.wisc.edu/u/n/a/naze/gametheory/sexes/stats.pys anormaltestµ s   "cCs“ti|ƒ}ti|ƒ}tit|ƒƒ}x@t t|ƒƒD],}ti i ti |||ƒƒ|| 1D. Usage: az(a, score) N(sscoresameansas asamplestdevsz(sasscoresz((s5/afs/cs.wisc.edu/u/n/a/naze/gametheory/sexes/stats.pysazí scCs>g}x$|D]}|it||ƒƒq Wti|ƒSdS(s Returns a 1D array of z-scores, one for each score in the passed array, computed relative to the passed array. Usage: azs(a) N(szscoressasitemsappendszsNsarray(saszscoressitem((s5/afs/cs.wisc.edu/u/n/a/naze/gametheory/sexes/stats.pysazsù s cCs.t||ƒ}t|dƒ}|||SdS(sÚ Returns an array of z-scores the shape of scores (e.g., [x,y]), compared to array passed to compare (e.g., [time,x,y]). Assumes collapsing over dim 0 of the compare array. Usage: azs(scores, compare, dimension=0) iN(sameanscompares dimensionsmnss asamplestdevssstdsscores(sscoresscompares dimensionsmnsssstd((s5/afs/cs.wisc.edu/u/n/a/naze/gametheory/sexes/stats.pysazmap scCsK|d„}t|ƒtijo3yti|ƒ}WqUti|dƒ}qUXn|i}|i ƒddddgjo4ti |ƒ}tit ||ƒƒ}||_n“|i ƒddgjooti |ƒd}xLt t|ƒƒD]8}t||ƒtjot|||ƒ||threshmax are replaced by newval instead of by threshmin/threshmax (respectively). Usage: athreshold(a,threshmin=None,threshmax=None,newval=0) Returns: a, with values threshmax replaced with newval iiN( sNszerossasshapesmasks threshminsNoneswhereslesss threshmaxsgreatersclipsnewval(sas threshmins threshmaxsnewvalsmask((s5/afs/cs.wisc.edu/u/n/a/naze/gametheory/sexes/stats.pys athreshold6 s ) )cCs5t|t|ƒƒ}t|ƒ|}|||!SdS(s£ Slices off the passed proportion of items from BOTH ends of the passed array (i.e., with proportiontocut=0.1, slices 'leftmost' 10% AND 'rightmost' 10% of scores. You must pre-sort the array if you want "proper" trimming. Slices off LESS if proportion results in a non-integer slice index (i.e., conservatively slices off proportiontocut). Usage: atrimboth (a,proportiontocut) Returns: trimmed version of array a N(sintsproportiontocutslensaslowercutsuppercut(sasproportiontocutsuppercutslowercut((s5/afs/cs.wisc.edu/u/n/a/naze/gametheory/sexes/stats.pys atrimbothG s cCs‹ti|ƒdjo*d}t|ƒt|t|ƒƒ}n=ti|ƒdjo&t|t|ƒƒ}t|ƒ}n|||!SdS(s€ Slices off the passed proportion of items from ONE end of the passed array (i.e., if proportiontocut=0.1, slices off 'leftmost' or 'rightmost' 10% of scores). Slices off LESS if proportion results in a non-integer slice index (i.e., conservatively slices off proportiontocut). Usage: atrim1(a,proportiontocut,tail='right') or set tail='left' Returns: trimmed version of array a srightisleftN( sstringslowerstailslowercutslensasintsproportiontocutsuppercut(sasproportiontocutstailsuppercutslowercut((s5/afs/cs.wisc.edu/u/n/a/naze/gametheory/sexes/stats.pysatrim1X s $cCsyt|iƒdjo td‚n|id}t|dƒ}titi |ƒ|ƒt |ƒti i ||ƒSdS(s… Computes the covariance matrix of a matrix X. Requires a 2D matrix input. Usage: acovariance(X) Returns: covariance matrix of X is acovariance requires 2D matricesiN( slensXsshapes TypeErrorsnsameansmXsNsdots transposesfloatsmultiplysouter(sXsnsmX((s5/afs/cs.wisc.edu/u/n/a/naze/gametheory/sexes/stats.pys acovarianceo s   cCs?t|ƒ}ti|ƒ}|titii||ƒƒSdS(sˆ Computes the correlation matrix of a matrix X. Requires a 2D matrix input. Usage: acorrelation(X) Returns: correlation matrix of X N( s acovariancesXsCsNsdiagonalsVssqrtsmultiplysouter(sXsCsV((s5/afs/cs.wisc.edu/u/n/a/naze/gametheory/sexes/stats.pys acorrelation} s cCs²d}x1|ddddddgjodGtƒ}q W|ddddgjoñd Gt||ƒ}tti|d ƒti|d ƒƒ\} } | d jod t t | dƒƒ}nd}|GH|ddgjoå|d djo>t||td ƒ\} } dGt | dƒGt | dƒGHq@t|ƒdjpt|ƒdjo8t||ƒ\}} dGt |dƒGt | dƒGHq@t||ƒ\} } dGt | dƒGt | dƒGHq¥|d djo;t||d ƒ\} } dGt | dƒGt | dƒGHq¥t||ƒ\} } dGt | dƒGt | dƒGHnbd}x1|ddddddgjodGtƒ}qMW|ddgjo‹t||ƒ\} }}} }dGHdddddgt | dƒt |dƒt |dƒt | dƒt |dƒgg}ti|ƒnŠ|ddgjo=t||ƒ\}} dGHd Gt |dƒGt | dƒGHn:t||ƒ\}} d!GHd"Gt |dƒGt | dƒGHd#GHtSd$S(%sü Interactively determines the type of data in x and y, and then runs the appropriated statistic for paired group data. Usage: apaired(x,y) x,y = the two arrays of values to be compared Returns: appropriate statistic name, value, and probability ssisrsIsRscsCs9 Independent or related samples, or correlation (i,r,c): s Comparing variances ...iif0.050000000000000003s unequal, p=isequalses Independent samples t-test: is( Rank Sums test (NONparametric, n>20): s. Mann-Whitney U-test (NONparametric, ns<20): s Related samples t-test: s# Wilcoxon T-test (NONparametric): sdsDs9 Is the data Continuous, Ranked, or Dichotomous (c,r,d): s/ Linear regression for continuous variables ...sSlopes InterceptsProbs SEestimates% Correlation for ranked variables ...sSpearman's r: s/ Assuming x contains a dichotomous variable ...sPoint Biserial r: s N( ssampless raw_inputsobrientransformsxsysrsF_onewayspstatscolexsfspsstrsroundsvartypes ttest_indsNonestslensranksumsszs mannwhitneyusus ttest_relscorrtypes linregresssmsbsseeslolsprintccs spearmanrspointbiserialr(sxsysseesvartypeslolsrssamplesscorrtypesbsfsmspsustsz((s5/afs/cs.wisc.edu/u/n/a/naze/gametheory/sexes/stats.pysapaired‰ s^- #&####W#c Csd}t|ƒ}t|ƒ} t|ƒ}|ti i ||ƒti i |ƒti i |ƒ} t i |t|ƒt|ƒ|t|ƒt|ƒƒ}| |}|d}|t i |d||d||ƒ} td|d||| | |ƒ} || fSdS(s÷ Calculates a Pearson correlation coefficient and returns p. Taken from Heiman's Basic Statistics for the Behav. Sci (2nd), p.195. Usage: apearsonr(x,y,verbose=1) where x,y are equal length arrays Returns: Pearson's r, two-tailed p-value f9.9999999999999995e-21if1.0f0.5N(sTINYslensxsnsameansxmeansysymeansNsaddsreducesr_numsmathssqrtsasssasquare_of_sumssr_densrsdfstsabetaisverbosesprob( sxsysverbosesr_densymeansnsdfsrsTINYsxmeanstsprobsr_num((s5/afs/cs.wisc.edu/u/n/a/naze/gametheory/sexes/stats.pys apearsonrÉ s   :;  +%c CsÉd}t|ƒ}t|ƒ}t|ƒ}ti i ||dƒ} dd| t ||ddƒ}|ti|d|dd|ƒ} |d}td|d||| | ƒ}||fSdS(sí Calculates a Spearman rank-order correlation coefficient. Taken from Heiman's Basic Statistics for the Behav. Sci (1st), p.192. Usage: aspearmanr(x,y) where x,y are equal-length arrays Returns: Spearman's r, two-tailed p-value f1.0000000000000001e-30iiif1.0f0.5N(sTINYslensxsnsrankdatasrankxsysrankysNsaddsreducesdsqsfloatsrssmathssqrtstsdfsabetaisprobrs( sxsysrankxsrssdfsrankysnsprobrssTINYsdsqst((s5/afs/cs.wisc.edu/u/n/a/naze/gametheory/sexes/stats.pys aspearmanrÞ s   $' "cCs¡d}ti|ƒ}ti||ƒ}t|ƒdjo t d‚nWti|t i dƒƒ} ti || dƒ}ti|d|dƒ}ti|d|dƒ}tti|dƒƒ}tti|dƒƒ} t|ƒ} tit|ƒt| ƒt|ƒt| ƒƒ} | |tti|dƒƒ| }| d}|ti|d||d||ƒ} td|d||| | ƒ}||fSdS( s Calculates a point-biserial correlation coefficient and the associated probability value. Taken from Heiman's Basic Statistics for the Behav. Sci (1st), p.194. Usage: apointbiserialr(x,y) where x,y are equal length arrays Returns: Point-biserial r, two-tailed p-value f1.0000000000000001e-30is:Exactly 2 categories required (in x) for pointbiserialr().iif1.0f0.5N(sTINYspstatsauniquesxs categoriessaabutsysdataslens ValueErrorsNsarangescodemapsarecodesrecodeds alinexandsameansacolexsxmeansymeansnsmathssqrtsfloatsadjusts asamplestdevsrpbsdfstsabetaisprob(sxsysrpbsTINYsdfsrecodedsdatas categoriessprobsymeanscodemapsnsadjuststsxmean((s5/afs/cs.wisc.edu/u/n/a/naze/gametheory/sexes/stats.pysapointbiserialrô s&  3$ +"cCsbd} d} d}xÏtt|ƒdƒD]·}x®t|t|ƒƒD]—}||||}||||} || }|o=| d} | d} |djo|d}qÜ|d}qE|o| d} qE| d} qEWq)W|t i | | ƒ}dt|ƒddt|ƒt|ƒd}|t i |ƒ} tt| ƒdƒ} || fSdS(sÑ Calculates Kendall's tau ... correlation of ordinal data. Adapted from function kendl1 in Numerical Recipies. Needs good test-cases.@@@ Usage: akendalltau(x,y) Returns: Kendall's tau, two-tailed p-value iif4.0f10.0f9.0f1.4142136000000001N(sn1sn2sisssrangeslensxsjsysksa1sa2saasmathssqrtstaussvarszserfccsabssprob(sxsysaastausisssksjssvarsa1sa2sn1sn2szsprob((s5/afs/cs.wisc.edu/u/n/a/naze/gametheory/sexes/stats.pys akendalltau s2    0cGsd}t|ƒdjoe|d}t|ƒdjo|d}|d}q’|dd…df}|dd…df}n|d}|d}t|ƒ} t|ƒ} t|ƒ} | t i i ||ƒt i i |ƒt i i |ƒ}t i| t|ƒt|ƒ| t|ƒt|ƒƒ}||} dt id| |d| |ƒ}| d}| t i|d| |d| |ƒ} td|d||| | ƒ}|t| ƒt|ƒt|ƒ}| || }t id| | ƒt|ƒ}||| ||fSdS(sW Calculates a regression line on two arrays, x and y, corresponding to x,y pairs. If a single 2D array is passed, alinregress finds dim with 2 levels and splits data into x,y pairs along that dim. Usage: alinregress(*args) args=2 equal-length arrays, or one 2D array Returns: slope, intercept, r, two-tailed prob, sterr-of-the-estimate f9.9999999999999995e-21iiiNf0.5f1.0(sTINYslensargssxsysnsameansxmeansymeansNsaddsreducesr_numsmathssqrtsasssasquare_of_sumssr_densrslogszsdfstsabetaisprobsfloatsslopes intercepts asamplestdevssterrest(sargssslopesr_numsr_densTINYsdfsprobs interceptssterrestsymeansnsrstsxmeansysxsz((s5/afs/cs.wisc.edu/u/n/a/naze/gametheory/sexes/stats.pys alinregress5 s2       :; ' +"$!c Cs9t|ƒtijoti|ƒ}nt|ƒ} t|ƒ} t |ƒ} | d}| d| t |ƒ}| |ti|d| ƒ} td|d||| | ƒ}|djond}t||dd|ddd|| | | tiiti|ƒƒtiiti|ƒƒ|| |ƒn| |fSdS( s’ Calculates the t-obtained for the independent samples T-test on ONE group of scores a, given a population mean. If printit=1, results are printed to the screen. If printit='filename', the results are output to 'filename' using the given writemode (default=append). Returns t-value, and prob. Usage: attest_1samp(a,popmean,Name='Sample',printit=0,writemode='a') Returns: t-value, two-tailed prob if1.0f0.5isSingle-sample T-test.s Populations--N(stypesasNs ArrayTypesarraysameansxsavarsvslensnsdfsfloatssvarspopmeansmathssqrtstsabetaisprobsprintitsstatnamesoutputpairedstatss writemodesnamesminimumsreducesravelsmaximum( saspopmeansprintitsnames writemodesstatnamessvarsprobsdfsnstsvsx((s5/afs/cs.wisc.edu/u/n/a/naze/gametheory/sexes/stats.pys attest_1samp^ s$     "  $cCs—|tjo(ti|ƒ}ti|ƒ}d}nt||ƒ}t||ƒ} t ||ƒ} t ||ƒ} |i |}|i |}||d} |d| |d| t| ƒ} ti| dƒ}ti|d| ƒ} || ti| d|d|ƒ}ti|d|ƒ}td| dt| ƒ| ||ƒ}t|ƒtijoti||i ƒ}nt|ƒdjo|d}n|djoät|ƒtijo|d}nt|ƒtijo|d}nd}t|||||| ti"i#ti|ƒƒti$i#ti|ƒƒ||| | ti"i#ti|ƒƒti$i#ti|ƒƒ|||ƒdSn||fSdS(s Calculates the t-obtained T-test on TWO INDEPENDENT samples of scores a, and b. From Numerical Recipies, p.483. If printit=1, results are printed to the screen. If printit='filename', the results are output to 'filename' using the given writemode (default=append). Dimension can equal None (ravel array first), or an integer (the dimension over which to operate on a and b). Usage: attest_ind (a,b,dimension=None,printit=0, Name1='Samp1',Name2='Samp2',writemode='a') Returns: t-value, two-tailed p-value iiif1.0f0.5sIndependent samples T-test.N(&s dimensionsNonesNsravelsasbsameansx1sx2savarsv1sv2sshapesn1sn2sdfsfloatssvarsequalszerodivproblemswheressqrtstsabetaisprobsstypes ArrayTypesreshapeslensprintitsstatnamesoutputpairedstatss writemodesname1sminimumsreducesmaximumsname2(sasbs dimensionsprintitsname1sname2s writemodesstatnamesprobsssvarsdfsv1sv2sx2szerodivproblemsx1stsn1sn2((s5/afs/cs.wisc.edu/u/n/a/naze/gametheory/sexes/stats.pys attest_ind| sH     $'(  $$ cCsŒ|tjo(ti|ƒ}ti|ƒ}d}nt|ƒt|ƒjo td‚nt||ƒ}t||ƒ} t ||ƒ} t ||ƒ} |i|}t|dƒ} ||idƒ}ti|tii|||ƒtii||ƒd| ƒ} ti| dƒ}ti|d| ƒ} tii||ƒ| }ti|d|ƒ}td| dt| ƒ| ||ƒ}t|ƒtijoti ||iƒ}nt|ƒdjo|d}n|djoœd}t#|||||| ti&iti|ƒƒti'iti|ƒƒ||| | ti&iti|ƒƒti'iti|ƒƒ|||ƒd Sn||fSd S( s Calculates the t-obtained T-test on TWO RELATED samples of scores, a and b. From Numerical Recipies, p.483. If printit=1, results are printed to the screen. If printit='filename', the results are output to 'filename' using the given writemode (default=append). Dimension can equal None (ravel array first), or an integer (the dimension over which to operate on a and b). Usage: attest_rel(a,b,dimension=None,printit=0, name1='Samp1',name2='Samp2',writemode='a') Returns: t-value, two-tailed p-value isUnequal length arrays.isdif1.0f0.5sRelated samples T-test.N()s dimensionsNonesNsravelsasbslens ValueErrorsameansx1sx2savarsv1sv2sshapesnsfloatsdfsastypesdssqrtsaddsreducesdenomsequalszerodivproblemswherestsabetaisprobsstypes ArrayTypesreshapesprintitsstatnamesoutputpairedstatss writemodesname1sminimumsmaximumsname2(sasbs dimensionsprintitsname1sname2s writemodesstatnamesprobssdenomsdfsv1sv2sx2szerodivproblemsx1sdsnst((s5/afs/cs.wisc.edu/u/n/a/naze/gametheory/sexes/stats.pys attest_rel° sD     A(  $$ cCsšt|ƒ}|tjo6tit|ƒt|ƒgt|ƒti ƒ}n|i ti ƒ}ti i ||d|ƒ}|t||dƒfSdS(sF Calculates a one-way chi square for array of observed frequencies and returns the result. If no expected frequencies are given, the total N is assumed to be equally distributed across all groups. Usage: achisquare(f_obs, f_exp=None) f_obs = array of observed cell freq. Returns: chisquare-statistic, associated p-value iiN(slensf_obssksf_expsNonesNsarrayssumsfloatsFloatsastypesaddsreduceschisqs chisqprob(sf_obssf_expskschisq((s5/afs/cs.wisc.edu/u/n/a/naze/gametheory/sexes/stats.pys achisquareâ s  6cCs§d}d} d}d}|id} |id}| d} |d} t i |idt i ƒ} t i|dƒ}t i|dƒ}x¶|| jo | |jo›||}|| }||jo|t| ƒ}|d}n||jo| t| ƒ}| d} n||}t|ƒt| ƒjo |} q‰q‰WyJti| | t| | ƒƒ}t|dd|t i| ƒƒ}Wn d}nX| |fSdS(sû Computes the Kolmogorov-Smirnof statistic on 2 samples. Modified from Numerical Recipies in C, page 493. Returns KS D-value, prob. Not ufunc- like. Usage: aks_2samp(data1,data2) where data1 and data2 are 1D arrays Returns: KS D-value, p-value if0.0if0.12f0.11f1.0N(sj1sj2sfn1sfn2sdata1sshapesn1sdata2sn2sen1sen2sNszerossFloatsdssortsd1sd2sfloatsdtsabssmathssqrtsensaksprobsfabssprob(sdata1sdata2sprobsensfn2sfn1sd1sd2sj1sj2sdsen1sen2sn1sn2sdt((s5/afs/cs.wisc.edu/u/n/a/naze/gametheory/sexes/stats.pys aks_2sampô s>         !) cCs$t|ƒ} t|ƒ} tti||fƒƒ}|d| !} || }| | | | ddt | ƒ}| | |}t||ƒ}t||ƒ}tit|ƒƒ}|djo td‚nti|| | | | ddƒ} t|| | d| ƒ} |dt| ƒfSdS(s© Calculates a Mann-Whitney U statistic on the provided scores and returns the result. Use only when the n in each condition is < 20 and you have 2 independent samples of ranks. REMEMBER: Mann-Whitney U is significant if the u-obtained is LESS THAN or equal to the critical value of U. Usage: amannwhitneyu(x,y) where x,y are arrays of values for 2 conditions Returns: u-statistic, one-tailed p-value (i.e., p(z(U))) iif2.0s*All numbers are identical in amannwhitneyuf12.0f1.0N(slensxsn1sysn2srankdatasNs concatenatesrankedsrankxsrankyssumsu1su2smaxsbigusminssmallusmathssqrts tiecorrectsTs ValueErrorssdsabsszszprob(sxsysu2su1sbigussmallusrankysrankedsTsrankxsn1sn2szssd((s5/afs/cs.wisc.edu/u/n/a/naze/gametheory/sexes/stats.pys amannwhitneyu s     $  'cCsõtti|ƒƒ\}}t|ƒ}d}d}x›||djo‰||||djobd}xC||djo||||djo|d}|d}qiW||d|}n|d}q6W|t |d|ƒ}d|SdS(s Tie-corrector for ties in Mann Whitney U and Kruskal Wallis H tests. See Siegel, S. (1956) Nonparametric Statistics for the Behavioral Sciences. New York: McGraw-Hill. Code adapted from |Stat rankind.c code. Usage: atiecorrect(rankvals) Returns: T correction factor for U or H f0.0iiif1.0N( s ashellsortsNsarraysrankvalsssortedsposnslensnsTsisntiessfloat(srankvalssposnsntiessTsisnssorted((s5/afs/cs.wisc.edu/u/n/a/naze/gametheory/sexes/stats.pys atiecorrect8 s"  * c CsÂt|ƒ}t|ƒ}ti||fƒ} t| ƒ}|| }||}t |ƒ}|||dd}||t i||||ddƒ}ddtt|ƒƒ}||fSdS(sÉ Calculates the rank sums statistic on the provided scores and returns the result. Usage: aranksums(x,y) where x,y are arrays of values for 2 conditions Returns: z-statistic, two-tailed p-value if2.0f12.0if1.0N(slensxsn1sysn2sNs concatenatesalldatas arankdatasrankedssumsssexpectedsmathssqrtszszprobsabssprob( sxsysprobsexpectedsssrankedsn1sn2szsalldata((s5/afs/cs.wisc.edu/u/n/a/naze/gametheory/sexes/stats.pys aranksumsR s      +cCsjt|ƒt|ƒjo td‚n||}titi|dƒ|ƒ}t|ƒ}t |ƒ} t | ƒ} d}d} xKtt| ƒƒD]7}||djo| | |} q‘|| |}q‘Wt|| ƒ}||dd}ti||dd|ddƒ} ti||ƒ| } ti||ƒ| } d dtt | ƒƒ}||fSd S( sà Calculates the Wilcoxon T-test for related samples and returns the result. A non-parametric T-test. Usage: awilcoxont(x,y) where x,y are equal-length arrays for 2 conditions Returns: t-statistic, two-tailed p-value s#Unequal N in awilcoxont. Aborting.if0.0if0.25f2.0f1.0f24.0iN(slensxsys ValueErrorsdsNscompresss not_equalscountsabssabsds arankdatas absrankedsr_plussr_minussrangesisminswtsmnsmathssqrtssesfabsszszprobsprob(sxsysprobswtscountsr_plussdsismns absrankedsabsdszsr_minussse((s5/afs/cs.wisc.edu/u/n/a/naze/gametheory/sexes/stats.pys awilcoxontg s,     'c Gs¿t|ƒdjp td‚t|ƒ}dgt|ƒ}tt|ƒ}g}x.tt|ƒƒD]}|||i ƒ}qdWt |ƒ}t |ƒ}x=tt|ƒƒD])}|d||!||<|d||5q­Wg} xQtt|ƒƒD]=}| it||ƒdƒ| |t||ƒ| |0, 1.0-zprob(z) = 1-tail probability for any z, 2.0*(1.0-zprob(abs(z))) = 2-tail probability Adapted from z.c in Gary Perlman's |Stat. Can handle multiple dimensions. Usage: azprob(z) where z is a z-value cCsd |d|d|d|d|d|d|d|d |d |d |d |d |d|d}|SdS(Nf4.5255659000000002e-05f 0.00015252929f1.9538131999999999e-05f0.00067690498600000005f0.0013906042840000001f0.00079462081999999998f0.0020342548740000001f0.0065497912140000001f0.010557625006f0.011630447319f0.0092794533410000008f0.0053535791080000002f0.0021412687410000001f0.00053531084899999996f0.99993665752399996(sysx(sysx((s5/afs/cs.wisc.edu/u/n/a/naze/gametheory/sexes/stats.pysyfunc2swcCs_d|d|d|d|d|d|d|d|d ti|ƒd }|SdS( Nf0.000124818987f0.001075204047f0.0051987750189999996f0.019198292004f0.059054035642f0.15196875136400001f0.31915293269400002f0.53192300729999997f0.79788456059299995f2.0(swsNssqrtsx(swsx((s5/afs/cs.wisc.edu/u/n/a/naze/gametheory/sexes/stats.pyswfunc=sWf6.0f0.5f1.0f2.0iiN(syfuncswfuncsZ_MAXsNszerosszsshapesFloatsxsfabssyswhereslesssgreatersprob(szswfuncsZ_MAXsyfuncsysxsprob((s5/afs/cs.wisc.edu/u/n/a/naze/gametheory/sexes/stats.pysazprob's  5%1cCs€t|ƒtijo8dti|itiƒ}|itiƒ}d}n%ti dƒ}ti |tiƒ}ti |iƒ}dti|iti ƒ} ti |iti ƒ}ti |iti ƒ} ti d||tiƒ} tiiti |iƒƒ}x!tddƒD]}t|ƒ|joPn| ||} ti| dƒ} ti| d|ƒ}|| }| ti| ƒ}||}titit|ƒd | ƒtit|ƒd |ƒddƒ}ti|ti!|dƒ||ƒ}ti"||ddƒ}| } t|ƒ} qW|o#titi!|dƒd |ƒSn$titi!|dƒd |ƒdSd S( s¤ Returns the probability value for a K-S statistic computed via ks_2samp. Adapted from Numerical Recipies. Can handle multiple dimensions. Usage: aksprob(alam) iÿÿÿÿif-1.0f2.0f-2.0iÉiýÿÿif0.001f1e-08f1.0N(#stypesalamsNs ArrayTypesonessshapesFloat64sfrozensastypes arrayflagsarrayszerossmasksFloatsfacssumstermbfsa2smultiplysreduces totalelementssrangesjsasums exponentsslesss overflowmaskswheresexpsterms less_equalsabssnewmasksequalsclip(salamsnewmasksterms totalelementssfrozensmaskssumsjs arrayflags overflowmaskstermbfsa2sfacs exponents((s5/afs/cs.wisc.edu/u/n/a/naze/gametheory/sexes/stats.pysaksprobNs@   9 %#cCspt|ƒtijo,td|d||d|||ƒSn+td|d||t|||ƒƒSdS(s Returns the 1-tailed significance level (p-value) of an F statistic given the degrees of freedom for the numerator (dfR-dfF) and the degrees of freedom for the denominator (dfF). Can handle multiple dims for F. Usage: afprob(dfnum, dfden, F) where usually dfnum=dfbn, dfden=dfwn f0.5f1.0N(stypesFsNs ArrayTypesabetaisdfdensdfnumsfloat(sdfnumsdfdensF((s5/afs/cs.wisc.edu/u/n/a/naze/gametheory/sexes/stats.pysafprobws,cCs’d}d}d}t|ƒtijo ti|iti ƒd}n+d}ti dgƒ}ti |gƒ}ti |iƒ}d}} }||}|d} |d}d||| }xut|dƒD]c}tititi|dƒƒƒdjoPnt|dƒ}||} |||||| || }| ||}|||}|| |||| | || }||| }|||} | d}|| }|| }|| } d}ti%t&| |ƒ|t&| ƒƒ}ti(|ti|dƒ| |ƒ}ti)||ddƒ}qÓWt*ti|dƒƒ} | djo|odG| GdGHn|o|Sn |dSd S( sà Evaluates the continued fraction form of the incomplete Beta function, betai. (Adapted from: Numerical Recipies in C.) Can handle multiple dimensions for x. Usage: abetacf(a,b,x,verbose=1) iÈf2.9999999999999999e-07iiÿÿÿÿif1.0s1a or b too big, or ITMAX too small in Betacf for s elementsN(-sITMAXsEPSs arrayflagstypesxsNs ArrayTypesonessshapesFloatsfrozensarrayszerossmasksbmsazsamsasbsqabsqapsqamsbzsrangesissumsravelsequalsfloatsemstemsdsapsbpsappsbppsaoldslesssabssnewmaskswheresclipsasums noconvergesverbose(sasbsxsverbosesemsappsamsEPSsaps noconvergesazstemsqapsbppsqabsqamsITMAXsbms arrayflagsnewmasksaoldsbpsbzsdsfrozensmasksi((s5/afs/cs.wisc.edu/u/n/a/naze/gametheory/sexes/stats.pysabetacf…sR    + "'    &%cCs¡dddddd g}|d}|d}||d ti|ƒ}d}x6tt|ƒƒD]"}|d }||||}qaW| tid |ƒSd S( sß Returns the gamma function of xx. Gamma(z) = Integral(0,infinity) of t^(z-1)exp(-t) dt. Adapted from: Numerical Recipies in C. Can handle multiple dims ... but probably doesn't normally have to. Usage: agammln(xx) f76.180091730000001f-86.505320330000004f24.014098220000001f -1.231739516f 0.00120858003f5.3638199999999999e-06f1.0f5.5f0.5if2.5066282746500002N( scoeffsxxsxstmpsNslogssersrangeslensj(sxxstmpsserscoeffsjsx((s5/afs/cs.wisc.edu/u/n/a/naze/gametheory/sexes/stats.pysagammln¹s   c CsQd}t|ƒtijo@tti|dƒti|dƒƒdjo t d‚q\nti ti |dƒ||ƒ}ti ti |dƒd||ƒ}ti ti |dƒti |dƒddƒ}t ||ƒt |ƒt |ƒ|ti|ƒ|tid|ƒ}ti ti|dƒd|ƒ}ti|ƒ}t|ƒtijowti ti||d||dƒ|t||||ƒt|ƒd|t||d||ƒt|ƒƒ}np||d||djo'|t||||ƒt|ƒ}n,d|t||d||ƒt|ƒ}|Sd S( s‚ Returns the incomplete beta function: I-sub-x(a,b) = 1/B(a,b)*(Integral(0,x) of t^(a-1)(1-t)^(b-1) dt) where a,b>0 and B(a,b) = G(a)*G(b)/(G(a+b)) where G(a) is the gamma function of a. The continued fraction formulation is implemented here, using the betacf function. (Adapted from: Numerical Recipies in C.) Can handle multiple dimensions. Usage: abetai(a,b,x,verbose=1) f1.0000000000000001e-15iisBad x in abetaif1.0iÿÿÿÿiýÿÿf2.0N(sTINYstypesasNs ArrayTypesasumslesssxsgreaters ValueErrorswheresequalsbtsgammlnsbslogs exponentssexpsabetacfsverbosesfloatsans(sasbsxsverbosesTINYsbtsanss exponents((s5/afs/cs.wisc.edu/u/n/a/naze/gametheory/sexes/stats.pysabetaiÎs& /!%1J!% 2'+cCsÏt|ƒt|ƒjo dGHdSnt|ƒ} ti|ƒ} ti| t| ƒfƒ} x@t t| ƒƒD],}ti || |ƒ| dd…|f=0 f1.3999999999999999iiN(sNsasarraysastypeslesssgreaterszerossshape(sa((s5/afs/cs.wisc.edu/u/n/a/naze/gametheory/sexes/stats.pysasignss 2,cCs„t|ƒtijo|iƒdddgjo|itiƒ}n|tjoti ti |ƒƒ}n t|ƒt t gjoUtii||ƒ}|djo/t|iƒ}d|||tjoti|ƒ}d}nt||||ƒSdS(sÑ Squares each value in the passed array, adds these squares & returns the result. Unfortunate function name. :-) Defaults to ALL values in the array. Dimension can equal None (ravel array first), an integer (the dimension over which to operate), or a sequence (operate over multiple dimensions). Set keepdims=1 to maintain the original number of dimensions. Usage: ass(inarray, dimension=None, keepdims=0) Returns: sum-along-'dimension' for (inarray*inarray) iN(s dimensionsNonesNsravelsinarraysasumskeepdims(sinarrays dimensionskeepdims((s5/afs/cs.wisc.edu/u/n/a/naze/gametheory/sexes/stats.pysass¼s   cCsM|tjo(ti|ƒ}ti|ƒ}d}nt||||ƒSdS(s‹ Multiplies elements in array1 and array2, element by element, and returns the sum (along 'dimension') of all resulting multiplications. Dimension can equal None (ravel array first), an integer (the dimension over which to operate), or a sequence (operate over multiple dimensions). A trivial function, but included for completeness. Usage: asummult(array1,array2,dimension=None,keepdims=0) iN(s dimensionsNonesNsravelsarray1sarray2sasumskeepdims(sarray1sarray2s dimensionskeepdims((s5/afs/cs.wisc.edu/u/n/a/naze/gametheory/sexes/stats.pysasummultÎs   cCsx|tjoti|ƒ}d}nt|||ƒ}t|ƒti jo|i ti ƒ|Snt |ƒ|SdS(s¶ Adds the values in the passed array, squares that sum, and returns the result. Dimension can equal None (ravel array first), an integer (the dimension over which to operate), or a sequence (operate over multiple dimensions). If keepdims=1, the returned array will have the same NUMBER of dimensions as the original. Usage: asquare_of_sums(inarray, dimension=None, keepdims=0) Returns: the square of the sum over dim(s) in dimension iN( s dimensionsNonesNsravelsinarraysasumskeepdimsssstypes ArrayTypesastypesFloatsfloat(sinarrays dimensionskeepdimsss((s5/afs/cs.wisc.edu/u/n/a/naze/gametheory/sexes/stats.pysasquare_of_sumsßs   cCsB|tjoti|ƒ}d}nt||d||ƒSdS(sŒ Takes pairwise differences of the values in arrays a and b, squares these differences, and returns the sum of these squares. Dimension can equal None (ravel array first), an integer (the dimension over which to operate), or a sequence (operate over multiple dimensions). keepdims=1 means the return shape = len(a.shape) = len(b.shape) Usage: asumdiffsquared(a,b) Returns: sum[ravel(a-b)**2] iiN( s dimensionsNonesNsravelsasinarraysasumsbskeepdims(sasbs dimensionskeepdimssinarray((s5/afs/cs.wisc.edu/u/n/a/naze/gametheory/sexes/stats.pysasumdiffsquaredôs   c Cst|ƒ}|d}t|ƒ}|d}xÚ|djoÌx»t||ƒD]ª}x¡t||d| ƒD]ˆ}x|djo|||||joX||}|||||<||||<||}|||||<||||