Circular Data Analysis

Transcription

1 Chater 30 Circular Data Aalysis Itroductio This rocedure comutes summary statistics, eerates rose lots ad circular historams, comutes hyothesis tests aroriate for oe, two, ad several rous, ad comutes the circular correlatio coefficiet for circular data. Aular data, recorded i derees or radias, is eerated i a wide variety of scietific research areas. Examles of aular (ad cyclical) data iclude daily wid directios, ocea curret directios, dearture directios of aimals, directio of boe-fracture lae, ad orietatio of bees i a beehive after stimuli. The usual summary statistics, such as the samle mea ad stadard deviatio, caot be used with aular values. For examle, cosider the averae of the aular values ad 359. The simle averae is 80. But with a little thouht, we miht coclude that 0 is a better aswer. Because of this ad other roblems, a secial set of techiques have bee develoed for aalyzi aular data. This rocedure imlemets may of those techiques. 30-

2 Circular Data Aalysis Techical Details Suose a samle of ales a, a,..., a is to be summarized. It is assumed that these ales are i derees. Fisher (993) ad Mardia & Ju (000) cotai defiitios of various summary statistics that are used for aular data. These results will be reseted ext. Let C = cos ( a ) i, C = C, S = ( a ) si i, S = S, T = R = C + S, R = R S ta C S C > 0, > 0 S ta C C + π < 0 S ta S, C C + π < 0 > 0 To iterret these quatities it may be useful to imaie that each ale reresets a vector of leth oe i the directio of the ale. Suose these idividual vectors are arraed so that the beii of the first vector is at the orii, the beii of the secod vector is at the ed of the first, the beii of the third vector is at the ed of the secod, ad so o. We ca the imaie a sile vector a that will stretch from the orii to the ed of the last observatio. R, called the resultat leth, is the leth of a. R is the mea resultat leth of a. Note that R varies betwee zero ad oe ad that a value of R ear oe imlies that there was little variatio i values of the ales. The mea directio,θ, is a measure of the mea of the idividual ales. θ is estimated byt. The circular variace, V, measures the variatio i the ales about the mea directio. V varies from zero to oe. The formula for V is The circular stadard deviatio, v, is defied as v = V = R ( R ) l The circular disersio, used i the calculatio of cofidece itervals, is defied as The skewess is defied as T δ = R ( ) s = R T T si ( R ) 3/ 30-

3 The kurtosis is defied as Circular Data Aalysis ( ) k = R cos T T R 4 ( R ) Correctio for Groued Data Whe the ales are roued, a multilicative correctio for R may be ecessary. The corrected value is ive by where = * R = R π / J si / ( π J ) Here J is the umber of equi-sized arcs. Thus, for mothly data, J would be. Cofidece Iterval for the Mea Directio Uto & Fileto (989) ae 0 ive a cofidece iterval for the mea directio whe o distributioal assumtio is made as where Circular Uiform Distributio T ± σ = si ( z α / σ ) ( H) 4R H = cos T cos a si T si a ( ) ( i ) + ( ) ( i ) Uiformity refers to the situatio i which all values aroud the circle are equally likely. The robability distributio o a circle with this roerty is the circular uiform distributio, or simly, the uiform distributio. The robability desity fuctio is ive by The robability betwee ay two oits is ive by f ( a ) = 360 a a Pr ( a < a a a, a a + π ) =

4 Circular Data Aalysis Tests of Uiformity Uiformity refers to the situatio i which all values aroud the circle are equally likely. Occasioally, it is useful to erform a statistical test of whether a set of data do ot follow the uiform distributio. Several tests of uiformity have bee develoed. Note that whe ay of the followi tests are reected, we ca coclude that the data were ot uiform. However, whe the test is ot reected, we caot coclude that the data follow the uiform distributio. Rather, we do ot have eouh evidece to reect the ull hyothesis of uiformity. Rayleih Test The Rayleih test, discussed i Mardia & Ju (000) aes 94-95, is the score test ad the likelihood ratio test for uiformity withi the vo Mises distributio family. The Rayleih test statistic is R. For lare samles, the distributio of this statistic uder uiformity is a chi-square with two derees of freedom with a error of O. A closer aroximatio to the chi-square with two derees of freedom is achieved by aroximatio of ( ) the modified Rayleih test. This test, which has a error of O( ), is calculated as follows. R S * = R + Modified Kuier's Test The modified Kuier's test, Mardia & Ju (000) aes 99-03, was desied to test uiformity aaist ay alterative. It measures the distace betwee the cumulative uiform distributio fuctio ad the emirical distributio fuctio. It is accurate for samles as small as 8. The test statistic, V, is calculated as follows 4 where V 0. 4 V = V a( i) i a( i) i = max mi + to 360 to 360 Published critical values of V are V Alha This table was used to create a iterolatio formula from which the alha values are calculated. Watso Test The followi uiformity test is outlied i Mardia & Ju aes The test is coducted by calculati U ad comari it to a table of values. If the calculated value is reater tha the critical value, the ull hyothesis of uiformity is reected. Note that the test is oly valid for samles of at least eiht ales. The calculatio of U is as follows U i = u i u + i + ( ) = 30-4

5 where Circular Data Aalysis u( i) i u = =, u ( i) a( i) = 360 a () a ( ) a ( 3 ) a ( ) are the sorted ales. Note that maximum likelihood estimates of κ ad θ are used i the distributio fuctio. Mardia & Ju (000) reset a table of critical values that has bee etered ito NCSS. Whe a value of U is calculated, the table is iterolated to determie its siificace level. Published critical values of U are U Alha Vo Mises Distributios The Vo Mises distributio takes the role i circular statistics that is held by the ormal distributio i stadard liear statistics. I fact, it is shaed like the ormal distributio, excet that its tails are trucated. The robability desity fuctio is ive by where I ( x) f ( a; θ, κ ) = ex cos π I 0 ( κ ) [ κ ( a θ) ] (the modified Bessel fuctio of the first kid ad order ) is defied by I articular I ( x) r+ x =, = 0,,,! r! r= 0 ( r + ) I ( x) = r= 0 ( r! ) 0 = π π x cos The arameterθ is the mea directio ad the arameterκ is the cocetratio arameter. The distributio is uimodal. It is symmetric about A. It aears as a ormal distributio that is trucated at lus ad mius 80 derees. Wheκ is zero, the vo Mises distributio reduces to the uiform distributio. Asκ ets lare, the vo Mises distributio aroaches the ormal distributio. 0 e x ( θ ) r dθ 30-5

6 Circular Data Aalysis Poit Estimatio The maximum likelihood estimate of θ is the samle mea directio. That is, θ = T. The maximum likelihood of κ is the solutio to where That is, the MLE of κ is ive by ( ) A κ = R ( ) A x κ * = A ( ) ( x) I x =. I 0 ( R) This ca be aroximated by (see Fisher (993) ae 88 ad Mardia & Ju (000) aes 85-86) κ * = 5 3 5R R + R + R < R R < R R R 4R + R This estimate is very biased. This bias is corrected by usi the followi modified estimator. * * max κ, 0 κ * < κ * κ = 3 ( ) κ * κ ( + ) * κ 5 > 5 Test for a Secified Mea Directio of Vo Mises Data There are several differet hyothesis tests that have bee roosed for testi H 0 :θ = θ 0 versus H :θ θ 0, whereθ 0 is a secific value of the mea directio. The tests reseted here require the additioal assumtio that the data follow the Vo Mises distributio, at least aroximately. It will be useful to adot the followi otatio. * C = cos a i 0 θ ( ) * S = si a i 0 θ ( ) [ ] + [ ] * * * R = S C 30-6

7 Circular Data Aalysis Score Test The score test, ive by Mardia & Ju (000) ae 3, is comuted as χ S = A κ ( ) ( * S ) κ For lare, χ S follows the chi-square distributio with oe deree of freedom. Likelihood Ratio Test The likelihood ratio test, ive by Mardia & Ju (000) ae, is comuted as χ L * * [( ) ( C ) ] * ( C ) 4 R * if 5 ad C / 3 = 3 * ( C ) * lo if 5 ad C > / 3 * + ( C ) + 3 * ( R ) The test statistic, χ L, follows a chi-square distributio with oe deree of freedom. Watso & Williams Test The Watso ad Williams test, ive by Mardia & Ju (000) ae 3, is comuted as F = R C * * * ( R ) / ( ) if C * 5 / 6 The test statistic, F, follows a F distributio with oe ad - derees of freedom. Stehes Test This test, ive by Fisher (993) aes 93-94, is comuted as If κ, E follows the stadard ormal distributio. ( T θ ) si E = / 0 ( κr) Cofidece Iterval for Mea Directio assumi Vo Mises A eeral cofidece iterval for θ was ive above. Whe the data ca be assumed to follow a vo Mises distributio, a more aroriate iterval is ive by Mardia & Ju (000) ae 4 ad Uto & Fileto (989) ae 69. This cofidece iterval is ive by T [ α ] ( 4 α ) R z T ± cos R z if R / 3 ± z ( R ) α ex cos R if R > /

8 Circular Data Aalysis Test for a Secified Cocetratio of Vo Mises Data Suose you wat to test a oe-sided hyothesis coceri κ, ive that the data come from a Vo Mises distributio ad that the mea directio arameter is ukow. Fisher (993) ae 95 suests the followi rocedure whe κ. Whe testi κ = κ 0 versus κ < κ 0, reect the ull hyothesis if χ ;α 3 R < + κ 8κ 0 0 Whe testi κ = κ 0 versus κ > κ 0, reect the ull hyothesis if These tests are based o the result that χ ; α 3 R > + κ 8κ ( R) 3 + κ 8κ 0 0 ~ χ 0 0 Cofidece Iterval for Cocetratio of Vo Mises A aroximate cofidece iterval for κ whe κ > was ive by Mardia & Ju (000) aes 6-7 as where + + 3b + + 3d, 4b 4d b = χ d = χ ( R), α / ( R), α / Goodess of Fit Tests for the Vo Mises Distributio Stehes Modified Watso s Test The followi oodess-of-fit test, ublished by Lockhart & Stehes (985) as a modificatio of the Watso test for the circle, is outlied i Fisher (993) ae 84. The test is coducted by calculati U ad comari it to a table of values. If the calculated value is reater tha the critical value, the ull hyothesis of Vo Misesess is reected. Note that the test is oly valid for samles of at least 0 ales. The calculatio of U is as follows U i = i ( )

9 where Circular Data Aalysis ( i ) i = = ( i ) ( i ) = Fκ a ( ) T a () a ( ) a ( ) a ( ) Fκ a θ is the cumulative distributio fuctio of the vo Mises distributio. Note that maximum likelihood estimates of κ ad θ are used i the distributio fuctio. Lockhart & Stehes (985) reset a table of critical values that has bee etered ito NCSS. Whe a value of U is calculated, the table is iterolated to determie its siificace level. 3 are the sorted ales ad ( ) Cox Test Mardia & Ju (000) aes 4-43 reset a vo Mises oodess-of-fit test that was oriially ive by Cox (975). The test statistic, C, is distributed as a chi-squared variable with two derees of freedom uder the ull hyothesis that the data follow the vo Mises distributio. It is calculated as follows. where sc C = + v c s s ( κ ) vs( κ ) sc = cos a T ( i ) α ( κ ) ss = si ai T vc ( x) = + α4 α v ( x) S = ( ) [ α / + α / α α ] ( + α ) / α 3 ( ) α α α α 4 3 α Multi-Grou Tests Three multi-rou tests are available for testi hyotheses about two or more rous. The oarametric uiform-scores test tests whether the distributios of the rous are idetical. The Watso-Williams F test tests whether a set of mea directios are equal ive that the cocetratios are ukow, but equal, ive that the rous each follow a vo Mises distributio. The cocetratio homoeeity test tests whether the cocetratio arameters are equal, ive that the rous each follow a vo Mises distributio. 30-9

10 Circular Data Aalysis Mardia-Watso-Wheeler Uiform-Scores Test Suose you have oulatios followi ay commo distributio from which radom samles are take ad you wish to test whether these distributios are equal. Fisher (993) ae ad Mardia & Ju (000) aes reset a oarametric test that is calculated as follows i i where C Ri = cos( γ i ), S Ri = si( γ i ) = = ales. The circular raks are calculated usi W = ( CRi + S Ri ), = i where the r i are the raks of the corresodi a i. ri γ i = π i, ad γ i are the circular raks of the corresodi If all i are reater tha 0, the distributio of W is aroximately distributed as a chi-square with - derees of freedom. Sice raks are used i this test, ties become a issue. We have adoted the stratey of alyi averae raks. Note that little has bee doe to test the adotio of this stratey withi the realm of circular statistics. Watso-Williams Hih Cocetratio F Test Suose you have Vo Mises oulatios from which radom samles are take ad you wish to test whether their mea directios are equal. That is, you wish to test the ull hyothesis H : θ = θ =... = θ 0 Mardia & Ju (000) aes reset the Watso-William Hih-Cocetratio F Test that is calculated as follows F WW R R / 3 = = + 8 κ R / = ( ) ( ) where κ is the maximum likelihood estimate of the cocetratio based o R ad R = C = = + S, C = cos( a ). i, S = si( ai ), R = C S +, C = C, S = S, ad The distributio of F WW is aroximately distributed as a F with - ad - derees of freedom whe the assumtios that κ = κ =... = κ ad that the distributios are Vo Mises are made. The aroximatio also requires that κ. = = 30-0

11 Circular Data Aalysis Multi-Grou Cocetratio Homoeeity Test Suose you have rous from which radom samles are take ad you wish to test whether the cocetratios are equal. That is, you wish to test the ull hyothesis H : κ = κ =... = κ 0 Mardia & Ju (000) ae 39 resets such a test. It is divided ito three cases. Case I. R < U is aroximately distributed as a chi-square with - derees of freedom where w = ( ) U = = w f ad f = si ( R / ) 3 8 = w f = w Case II R U is aroximately distributed as a chi-square with - derees of freedom U = w h = = w h = w where w = ad h = R 089. sih. 58 Case III. R > U3 is aroximately distributed as a chi-square with - derees of freedom where ν U 3= =, ν =,ad R R = ν lo + d ν lo ν = ν d = 3 ( ) = ν ν. 30-

12 Circular Data Aalysis Circular Correlatio Measure This sectio discusses a measure of the correlatio betwee two circular variables reseted by Jammalamadaka ad SeGuta (00). Suose a samle of airs of ales ( a, a),( a, a ),...,( a, a ) is available. The circular correlatio coefficiet is calculated as r c = k = si si ( ak T, ) si( ak T, ) ( ak T, ) si ( ak T, ) k = k = wheret, is the mea directio of the first circular variable ad T, is the mea directio of secod. The siificace of this correlatio coefficiet ca be test usi the fact the z r is aroximately distributed as a stadard ormal, where z r = r c λ λ λ 0 0 ad i λ i = si k = ( ak T, ) si ( ak T, ) Data Structure The data cosist of oe or more variables. Each variable cotais a set of aular values. The rows may be searated ito rous usi the uique values of a otioal roui variable. A examle of a dataset cotaii circular data is Circular.S0. Missi values are etered as blaks (emty cells). Procedure Otios This sectio describes the otios available i this rocedure. Variables Tab These otios secify the variables that will be used i the aalysis. Data Variables Data Variables Secify oe or more variables that cotai the aular values. The values i these variables must be of the tye secified i 'Data Tye'. If more tha oe variable is secified, the format of the reorts deed o whether a 'Groui Variable' is used. If a 'Groui Variable' is secified, a searate set of reorts is eerated for each data variable. If o 'Groui Variable' is secified, each of these variables are treated as a differet rou i a sile set of reorts. 30-

13 Circular Data Aalysis Data Tye Secify the tye of circular data that is cotaied i the Data Variables. Note that all variables must be of the same data tye. The ossible data tyes are Ale (0 to 360) Data are i the rae 0 to 360 derees. Neative values are coverted to ositive values by subtracti them from 360 (e.. -0 becomes 340). Data outside 0 to 360 are coverted to this rae by subtracti (or addi) 360 util the value is i this rae. RADIAN (0 to i) Data are i the rae 0 to i radia. Neative values are coverted to ositive values by subtracti them from i. AXIAL (0 to 80) Data are bidirectioal. Axial data are coverted to aular data by multilyi by two. Axial data may be i the full rae. Comass Text data rereseti the 6 oits of the comass are etered. Values are coverted ito derees usi the recodes: N = 0, E = 90, S = 80, W = 70. Two ad three letters may be used. For examle, 'NNW' is orth by orth-west. Time (0-4) Time of day values betwee 0 ad 4 may be etered. Weekday Iteers rereseti the days of the week are etered. The relatioshi is = Moday, = Tuesday,..., 7 = Suday. The iteers are coverted to derees usi = 80/7, = 80/7+360/7, ad so o. Moth of Year Iteers rereseti the moths of the year are etered. The relatioshi is = Jauary, = February,..., = December. The iteers are coverted to derees usi = 80/, = 80/+360/, ad so o. Groui Variable Groui Variable This otioal variable searates the values of the Data Variables ito rous. A searate aalysis is the eerated for each rou. Note that whe a roui variable is secified, the correlatios are ot eerated. Groui Correctio Factor Whe the same data values occur reeatedly, a correctio factor is suested for the calculatio of R bar. This correctio factor deeds o the umber of uique values, which is etered here. If '0' is etered, o correctio factor is used. Otios Hyothesized Values Hyothesized Theta This otioal arameter secifies the hyothesized value of theta (mea directio) uder the ull hyothesis. A set of hyotheses tests are coducted to determie if the data suort this hyothesized value. 30-3

14 Circular Data Aalysis Note that this is a sile-rou test. If there are several rous, a searate test is rovided for each rou. Hyothesized Kaa This otioal arameter secifies the hyothesized value of kaa (cocetratio) uder the ull hyothesis. A hyothesis test is coducted to determie if the data suort this hyothesized value. Note that this is a sile-rou test. If there are several rous, a searate test is rovided for each rou. Otios Cofidece Coefficiet Cofidece Coefficiet Secify the value of cofidece coefficiet for the cofidece itervals. Reorts Tab The otios o this ael cotrol which reorts ad lots are dislayed. Select Reorts Summary Reorts... Correlatios Select these otios to dislay the idicated reorts. Reort Otios Show Notes This otio cotrols whether the available otes ad commets that are dislayed at the bottom of each reort. This otio lets you omit these otes to reduce the leth of the outut. Precisio Secify the recisio of umbers i the reort. A sile-recisio umber will show seve-lace accuracy, while a double-recisio umber will show thirtee-lace accuracy. Note that the reorts were formatted for sile recisio. If you select double recisio, some umbers may ru ito others. Also ote that all calculatios are erformed i double recisio reardless of which otio you select here. This is for reorti uroses oly. Variable Names This otio lets you select whether to dislay oly variable ames, variable labels, or both. Value Labels This otio alies to the Grou Variable(s). It lets you select whether to dislay data values, value labels, or both. Use this otio if you wat the outut to automatically attach labels to the values (like =Yes, =No, etc.). See the sectio o secifyi Value Labels elsewhere i this maual. Reort Otios Decimal Places Mea ad Probability Decimals Secify the umber of diits after the decimal oit to dislay o the outut of values of this tye. Note that this otio i o way iflueces the accuracy with which the calculatios are doe. Eter 'All' to dislay all diits available. The umber of diits dislayed by this otio is cotrolled by whether the PRECISION otio is SINGLE or DOUBLE. 30-4

15 Circular Data Aalysis Plots Tab The otios o this ael cotrol the aearace of the lots. Select Plots Rose Plot / Circular Historam (Combied) ad Rose Plots / Circular Historams (Idividual) Select these otios to dislay the idicated lots. Format Click the lot format butto to chae the lot settis (see the Widow Otios below). Edit Duri Ru Checki this otio will cause the bar chart format widow to aear whe the rocedure is ru. This allows you to modify the format of the rah with the actual data. Rose Plot / Circular Historam Widow Otios This sectio describes the secific otios available o the Rose Plot / Circular Historam Format widow, which is dislayed whe a Rose Plot / Circular Historam Format butto is clicked. Commo otios, such as axes, labels, leeds, ad titles are documeted i the Grahics Comoets chater. Rose Plot Tab Data Tye The data tye of the lot is secified ideedetly of the data tye secified o the Variables tab of the Circular Data Aalysis rocedure. 30-5

16 Circular Data Aalysis Directio This otio idicates whether the orietatio of the lot is i a 'Clockwise' or 'Couter-Clockwise' directio. 30-6

17 Circular Data Aalysis Referece Ale (Rotatio) This otio lets you idicate the ositio of 0 derees by eteri a offset ale. O the default circle, 0 derees is o the riht (east), 90 derees is at the to (orth), 80 derees is o the left (west), ad 70 derees is at the bottom (south). This otio lets you add a 'offset' to each ale which moves the ositio of 0 derees aroud the circle. The offset must be betwee 0 ad 360 derees. Iterior Obects The two choices for lot styles are Rose Plot ad Circular Historam. 30-7

18 Circular Data Aalysis Grou Dislay Whe the data is roued data, this otio determies whether the etals withi a bi are side-by-side, stacked uo each other, or overlaid. Side-by-Side The bi width is divided equally by the umber of rous ad the etals are laid out sequetially i the bi. Althouh the etals are arrower, they still ecomass the oits of the rou that withi the boudaries of the whole bi. Stacked A sile etal i each bi is divided by the umber of rous. Rose lots with the rou dislay set to Stacked may be misleadi because the roortioal area is larer for the outside rous. Overlaid Each etal for each rou is overlaid i each bi. Some deree of trasarecy is recommeded whe usi the Overlaid rou dislay. It is also difficult to distiuish rous whe there are more tha or 3 rous. Radius for Iterior Obects This otio secifies the distace to the outer ede of the bis ad etals of the rose lot or circular historam. Petal Width Secify the ercet of the total width of each bi that is to be used for each etal. Percet for Historam Base This is the ercet of Radius for Iterior Obects that is used for the base of the circular historam. Number of Bis Secify the umber of bis for the circle. 30-8

19 Circular Data Aalysis First Bi Start Ale This ermits the user to chae the ale at which the bii beis. This is useful, for examle whe the Data Tye is set to Comass, sice this otio ca be used to ceter the bis o the directios. Whe Data Tye is set to Comass, the recommeded First Bi Start Ale is 45 for 4 bis,.5 for 8 bis, ad.5 for 6 bis. Number of Radial Axes Select the umber of axes that o from the ceter to the outer ede of the iterior reio. If this is set to the same umber as the umber of bis, these axes show the edes of the bis. The radial axes also bei at the First Bi Start Ale. Alterate Bi Fill Check this box to show a backroud fill for each bi. The fills alterate beii with Fill. Whe the umber of bis is odd, the adacet first ad last bis will both have Fill. 30-9

20 Data & Meas Tab Circular Data Aalysis Raw Data Symbols Check this box to show the raw data symbols. Number of Raw Data Symbol Bis Secify the umber of bis for the raw data oits. To use o bii set this to 0 or All Uiques. First Bi Start Ale This ermits the user to chae the ale at which the bii beis. This is useful, for examle whe the Data Tye is set to Comass, sice this otio ca be used to ceter the bis o the directios. Whe Data Tye is set to Comass, the recommeded First Bi Start Ale is 45 for 4 bis,.5 for 8 bis, ad.5 for 6 bis. Radius for Raw Data Symbols This is the distace from the ceter at which the symbols are show. Width for Multile Symbols at Oe Locatio This secifies the width of the bad that cotais the symbols whe there is more tha oe value at some locatios. 30-0

21 Circular Data Aalysis Mea Symbols ad Lies Use these otios to set u the visual reresetatio of the circular meas for each rou. Refereces Tab Directio Refereces The otios i this sectio allow you to secify the tick marks ad refereces oi aroud the lot. Maitude Refereces The otios i this sectio allow you to secify the tick marks ad refereces oi from the ceter to the outside of the lot. 30-

22 Circular Data Aalysis Examle Aalysis of Circular Data This sectio resets a examle of how to ru this rocedure. The data are wid directios of two rous. The data are foud i the Circular dataset. You may follow alo here by maki the aroriate etries or load the comleted temlate Examle by clicki o Oe Examle Temlate from the File meu of the Circular Data Aalysis widow. Oe the Circular dataset. From the File meu of the NCSS Data widow, select Oe Examle Data. Click o the file Circular.NCSS. Click Oe. Oe the Circular Data widow. Usi the Aalysis meu or the Procedure Naviator, fid ad select the Circular Data Aalysis rocedure. O the meus, select File, the New Temlate. This will fill the rocedure with the default temlate. 3 Secify the variables. O the Circular Data widow, select the Variables tab. (This is the default.) Double-click i the Data Variables text box. This will bri u the variable selectio widow. Select Wid from the list of variables ad the click Ok. Wid will aear i the Data Variables box. Double-click i the Groui Variable text box. This will bri u the variable selectio widow. Select Grou from the list of variables ad the click Ok. Grou will aear i the Paired Variables box. Set Hyothesized Theta to 40. Set Hyothesized Kaa to. 4 Ru the rocedure. From the Ru meu, select Ru Procedure. Alteratively, ust click the ree Ru butto, The followi reorts ad charts will be dislayed i the Outut widow. Summary Statistics Sectio Mea Circular Samle Mea Resultat Circular Stadard Circular Vo Mises Size Directio Leth Variace Deviatio Disersio Cocetratio Grou (N) (Theta) (R bar) (V) (v) (Delta) (Kaa) Grou This is the rou (or variable) reseted o this lie. Samle Size This is the umber of omissi values i this rou. Mea Directio This is estimated mea directio, T. 30-

23 Circular Data Aalysis Mea Resultat Leth This is the estimated mea resultat leth, R. It is a measure of data cocetratio. A R close to zero imlies low data cocetratio. A R close to oe imlies hih data cocetratio. Circular Variace The circular variace, V, is a measure of variatio i the data. Note that V = R. Circular Stadard Deviatio The circular stadard deviatio is v = l ( R ) Circular Disersio T The circular disersio, δ =, is aother measure of variatio. R Vo Mises Cocetratio This is the estimated cocetratio arameter of the vo Mises distributio, κ.. Note that it is ot the square root of the circular variace. Mea Directio Sectio Lower 95.0% Uer 95.0% Stadard Samle Mea Cofidece Cofidece Error of Size Directio Limit Limit Mea Grou (N) (Theta) of Theta of Theta Directio This reort rovides the lare samle cofidece iterval for the mea directio as described by Uto & Fileto (989) ae 0. Note that this iterval does ot require the assumtio that the data come from the vo Mises distributio. Variatio Statistics Sectio Circular Samle Circular Stadard Circular Size Variace Deviatio Disersio Skewess Kurtosis Grou (N) (V) (v) (Delta) (s) (k) This reort rovides measures of data variatio ad disersio which were defied i the Statistical Summary Reort. It also rovides measures of the skewess ad kurtosis of the data. Skewess This is a measure of the skewess (lack of symmetry about the mea) i the data. Symmetric, uimodal datasets have a skewess value ear zero. Kurtosis This is a measure of the kurtosis (eakedess) i the data. Vo Mises datasets have a kurtosis ear zero. 30-3

24 Circular Data Aalysis Vo Mises Distributio Estimatio Sectio Lower 95.0% Uer 95.0% Lower 95.0% Uer 95.0% Samle Mea Cofidece Cofidece Vo Mises Cofidece Cofidece Size Directio Limit Limit Coc. Limit Limit Grou (N) (Theta) of Theta of Theta (Kaa) of Kaa of Kaa This reort rovides estimates ad cofidece itervals of the arameters (mea directio ad cocetratio) of the vo Mises distributio that best fits the data. Note that the vo Mises distributio is a symmetric, uimodal distributio. You should check the rose lot or circular historam to determie if the data are symmetric. The formulas used i the estimatio ad cofidece itervals were ive earlier i this chater. They come from Mardia & Ju (000). Trioometric Momets Sectio Mea Mea Mea Mea Grou N Cos(a) Si(a) Cos(a) Si(a) R bar R bar Theta Theta This reort rovides summary statistics that are used i other calculatios. Mea Cos(a) C = a i. This is cos( ) Mea Si(a) S = a i. This is si( ) Mea Cos(a) C = a i. This is cos( ) Mea Si(a) S = a i. This is si( ) R bar This is R = ( C S ) R bar +. This is R = ( C S )

25 Circular Data Aalysis Theta, Theta This is calculated usi the followi formula with set to ad the, resectively. T = S ta C S C > 0, > 0 S ta C C + π < 0 S ta S, C C + π < 0 > 0 Multile-Grou Hyothesis Tests Sectio Null Hyothesis Test Test Prob Reect H0 (H0) Name Statistic Level at 0.05 Level Equal Distributios Uiform Scores Test Yes Equal Directios Watso-Williams F Test No Equal Cocetratios Cocetratio Homoeeity Test No Notes: These statistics test various hyotheses about the arameters of vo Mises distributios. They require that each rou follow the vo Mises distributio. The Uiform Scores test requires samles of at least 0. The Watso-Williams F-test assumes that all kaa's are equal ad that their averae is >. This reort rovides tests for three hyotheses about the features of several vo Mises datasets. That is, it rovides a test of whether the distributios are idetical, whether the mea directios are idetical, ad whether the cocetratios are idetical. These tests are documeted i the Techical Details sectio of this chater. Two-Grou Hyothesis Tests Sectio Equal Distributios Equal Directios Equal Cocetratios First Secod Test Prob Test Prob Test Prob Grou Grou Statistic Level Statistic Level Statistic Level Notes: These statistics test various hyotheses about the arameters of vo Mises distributios. They require that each rou follow the vo Mises distributio. Equal distributios tested by the Mardia-Watso-Wheeler uiform scores test. Requires all Ni > 0. Equal directios tested by the Watso-Williams F test. Assumes Vo Mises data with equal kaa's, all >. Equal cocetratios tested by cocetratio homoeeity test. Assumes Vo Mises data. This reort rovides the same three tests as the Multile-Grou Hyothesis Tests Sectio, take two rous at a time. It allows you to ioit where differeces occur. 30-5

26 Circular Data Aalysis Tests for a Secified Mea Directio Assumi Vo Mises Data Test Statistic & Prob Levels Tests for a Secified Mea Directio Assumi Vo Mises Data - Test Statistics - Wid Actual H0 Score Likelihood Watso & Stehes Samle Mea Mea Test Ratio Williams Test Size Directio Directio Z CS F Z Grou (N) (Theta) (Theta0) Value Value Value Value Notes: These rocedures test whether the mea directio is equal to a secified value, whe kaa (cocetratio) is ukow. They assume that the data follow the vo Mises distributio. The Score Test requires a lare samle size. The Likelihood Ratio Test requires a samle size of at least 5. The Watso & Williams Test requires a lare value of kaa. The Stehes Test requires kaa to be reater tha. Tests for a Secified Mea Directio Assumi Vo Mises Data - Probability Levels - Wid Actual H0 Score Likelihood Watso & Stehes Samle Mea Mea Test Ratio Williams Test Size Directio Directio Prob Prob Prob Prob Grou (N) (Theta) (Theta0) Level Level Level Level Notes: This reort ives the robability levels of the test statistics dislayed i the revious reort. Althouh the robability levels of four tests are ive, you should use oly oe of these. This sectio reorts the results of four tests of the hyothesis that the mea directio of a articular rou is equal to a secific value. These are two-sided tests. They were documeted earlier i this chater. The first table ives the values of the test statistics. The secod table ives the robability levels. The ull hyothesis is reected whe the robability level is less tha 0.05 (or some other aroriate cutoff). Tests for a Secified Cocetratio Assumi Vo Mises Data Prob Prob Samle Actual H0 Level of Level of Size Cocetratio Cocetratio Chi-Square (H:Kaa (H:Kaa Grou (N) (Kaa) (Kaa0) Value < Kaa0) > Kaa0) Notes: These statistics test whether the kaa (cocetratio) arameter is equal to the secified value. The tests require that the estimated kaa is >. This sectio reorts the results of two, oe-sided tests of the hyothesis that the cocetratio arameter of each rou is equal to a secific value. They were documeted earlier i this chater. The first robability level is for testi the ull hyothesis that kaa is reater tha or equal to kaa0. The secod robability level is for test the ull hyothesis that kaa is less tha or equal to kaa

27 Circular Data Aalysis Uiform Distributio Goodess-of-Fit Tests Rayleih's Rayleih's Kuier's Kuier's Watso's Watso's Samle Test Test Test Test Test Test Size Statistic Prob Statistic Prob Statistic Prob Grou (N) (S*) Level (V) Level (U) Level Notes: The tests i this reort assess the oodess-of-fit of the uiform distributio. The Rayleih test requires samles of at least 0. The Kuier ad Watso tests require samles of at least 8. This sectio reorts the results of three oodess-of-fit tests for the uiform distributio. They were documeted earlier i this chater. These tests may be viewed as testi whether the data are distributed uiformly aroud the circle. Vo Mises Distributio Goodess-of-Fit Tests Watso's Watso's Cox's Cox's Samle Test Test Test Test Size Statistic Prob Statistic Prob Grou (N) (U) Level (S) Level Notes: The tests i this reort assess the oodess-of-fit of the vo Mises distributio. Both tests require samles of at least 0. This sectio reorts the results of two oodess-of-fit tests for the vo Mises distributio. They were documeted earlier i this chater. Several hyothesis tests assume that the data follow a vo Mises distributio. These tests allow you to check the accuracy of this assumtio. 30-7

28 Rose Plots Circular Data Aalysis These lots show the distributio of the data aroud the circle. 30-8

29 Circular Historams Circular Data Aalysis The circular historams are eerated by setti the Iterior Obects o Plot to Circular Historam. 30-9