Descriptive measures of association for bivariate distributions

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1 Chapter 4 Descriptive measures of associatio for bivariate distributios Now we come to describe ad characterise specific features of bivariate frequecy distributios, ie, itrisic structures of raw data sets {(x i, y i )},, obtaied from statistical samples S for 2-D variables (X, Y ) from some populatio of study objects Let us suppose that the spectrum of values resp categories of X is a 1, a 2,, a k, ad the spectrum of values resp categories of Y is b 1, b 2,, b l, where k, l N Hece, for the bivariate joit distributio there exists a total of k l possible combiatios {(a i, b j ),,k;j=1,,l of values resp categories for (X, Y ) I the followig we will deote associated absolute (observed) frequecies by o ij := o (a i, b j ), ad relative frequecies by h ij := h (a i, b j ) 41 (k l) cotigecy tables Cosider a raw data set {(x i, y i )},, for a 2-D variable (X, Y ) givig rise to k l combiatios of values resp categories {(a i, b j )},,k;j=1,,l The bivariate joit distributio of observed absolute frequecies o ij may be coveietly represeted i terms of a (k l) cotigecy table (or cross tabulatio) by o ij b 1 b 2 b j b l j a 1 o 11 o 12 o 1j o 1l o 1+ a 2 o 21 o 22 o 2j o 2l o 2+ a i o i1 o i2 o ij o il o i+ a k o k1 o k2 o kj o kl o k+ i o +1 o +2 o +j o +l (41) where it holds for all i = 1,, k ad j = 1,, l that 0 o ij ad k l o ij = (42) j=1 The correspodig margial absolute frequecies of X ad of Y are l o i+ := o i1 + o i2 + + o ij + + o il =: o ij (43) j=1 o +j := o 1j + o 2j + + o ij + + o kj =: k o ij (44)

2 20CHAPTER 4 DESCRIPTIVE MEASURES OF ASSOCIATION FOR BIVARIATE DISTRIBUTIONS Oe obtais the related bivariate joit distributio of observed relative frequecies h ij followig the systematics of Eq (??) to yield h ij b 1 b 2 b j b l j a 1 h 11 h 12 h 1j h 1l o 1+ a 2 h 21 h 22 h 2j h 2l o 2+ a i h i1 h i2 h ij h il h i+ a k h k1 h k2 h kj h kl o k+ i h +1 h +2 h +j h +l 1 Agai, it holds for all i = 1,, k ad j = 1,, l that (45) 0 h ij 1 ad k l h ij = 1 (46) j=1 while the margial relative frequecies of X ad of Y are h i+ := h i1 + h i2 + + h ij + + h il =: h +j := h 1j + h 2j + + h ij + + h kj =: l h ij (47) j=1 l h ij (48) O the basis of a (k l) cotigecy table displayig the relative frequecies of the bivariate joit distributio of some 2-D (X, Y ), oe may defie two kids of related coditioal relative frequecy distributios, amely (i) the coditioal distributio of X give Y by ad (ii) the coditioal distributio of Y give X by h(a i b j ) := h ij h +j (49) h(b j a i ) := h ij h i+ (410) The, by meas of these coditioal distributios, a otio of statistical idepedece of variables X ad Y is defied to correspod to the simultaeous properties h(a i b j ) = h(a i ) = h i+ ad h(b j a i ) = h(b j ) = h +j (411) Give these properties hold, it follows from Eqs (49) ad (410) that h ij = h i+ h +j (412) 42 Measures of associatio for the metrical scale level Next, specifically cosider a raw data set {(x i, y i )},, from a statistical sample S for some metrically scaled 2-D variable (X, Y ) The bivariate joit distributio of (X, Y ) i this sample ca be coveietly represeted graphically i terms of a scatter plot Let us ow itroduce two kids of measures for the descriptio of specific characteristic features of such distributios

3 42 MEASURES OF ASSOCIATION FOR THE METRICAL SCALE LEVEL Sample covariace The first stadard measure characterisig bivariate joit distributios of metrically scaled 2-D (X, Y ) descriptively is the dimesioful sample covariace s XY (metr), defied by (i) From raw data set: alteratively: (ii) From relative frequecy distributio: s XY := 1 [(x 1 x)(y 1 y) + + (x x)(y y)] (413) 1 =: (x i x)(y i y), (414) s XY = 1 [x 1y x y xy] (415) = 1 (x i y i xy) (416) s XY = [(a 1 x)(b 1 y)h (a k x)(b l y)h kl ] (417) = k l (a i x)(b j y)h ij, (418) j=1 alteratively: s XY = [a 1b 1 h a k b l h kl xy] (419) = k l a i b j h ij xy (420) j=1 Remark 421 The alterative formulae provided here prove computatioally more efficiet It is worthwhile to poit out that i the research literature it is stadard to defie for bivariate joit distributios of metrically scaled 2-D (X, Y ) a dimesioful symmetric (2 2) covariace matrix S accordig to ( ) s 2 S := X s XY (421) the compoets of which are defied by Eqs (??) ad (414) The determiat of S, give by s Y X s 2 Y det(s) = s 2 X s2 Y s2 XY, is positive as log as s 2 X s2 Y s2 XY > 0 which applies i most practical cases The S is regular, ad thus a correspodig iverse S 1 exists The cocept of a regular covariace matrix S ad its iverse S 1 geeralises i a straightforward fashio to the case of multivariate joit distributios of metrically scaled m D(X, Y,, Z), where S R m m is give by s 2 X s XY s XZ s Y X s 2 Y s Y Z S := (422) s ZX s ZY s 2 Z

4 22CHAPTER 4 DESCRIPTIVE MEASURES OF ASSOCIATION FOR BIVARIATE DISTRIBUTIONS Exercise 1 Fid the covariace of the eruptio duratio ad waitig time i the data set faithful Observe if there is ay liear relatioship betwee the two variables Exercise 2 Cosider the data show i the ext table, o cosumptio C ad icome Y for coutries A ad B (measured i dollars): Coutry A Coutry B Obs C Y C Y For coutry A, compute the sample meas ad stadard deviatios for both variables Compute the covariace ad the correlatio coefficiet based o the ext auxiliar table: Obs = i C i Y i (C i C) 2 (Y i Y ) 2 (C i C)(Y i Y ) 2 Costruct a scatter diagram for cosumptio ad icome (cosumptio i the Y axis ad icome i the X axis) 3 Repeat the exercise for coutry B Compare the meas ad stadard deviatios 4 Compare the covariaces for both coutries Commet ituitively 5 Suppose that the data o cosumptio for coutry A is altered i the followig way: (a) Observatios o cosumptio ad icome are measured i cets istead of dollars (b) Observatios o cosumptio are measured i cets ad observatios o icome i dollars (c) Observatios o cosumptio are measured i dollars, but icome is measured i Crows (1 dollar = 785 Crows) (d) To all the observatios o icome ad cosumptio the umber 10 is added (arbitrarily) (e) Oly to the observatios o icome the umber 10 is added For all the cases, compute the covariace ad commet your fidigs ituitively 422 Bravais ad Pearso s sample correlatio coefficiet The sample covariace s XY costitutes the basis for the secod stadard measure characterisig bivariate joit distributios of metrically scaled 2-D (X, Y ) descriptively, the ormalised dimesioless sample correlatio coefficiet r (metr) devised by the Frech physicist Auguste Bravais ( ) ad the Eglish mathematicia ad statisticia Karl Pearso ( ) for the purpose of aalysig correspodig raw data {(x i, y i )},, for the existece of liear statistical associatios It is defied i terms of the bivariate sample covariace s XY ad the uivariate sample stadard deviatios s X ad s Y by (cf Bravais (1846) ad Pearso (1901)) r := s XY (423) s X s Y

5 42 MEASURES OF ASSOCIATION FOR THE METRICAL SCALE LEVEL 23 Due to ormalisatio, the rage of the sample correlatio coefficiet is 1 r +1 The sig of r ecodes the directio of a correlatio As to iterpretig the stregth of a correlatio via the magitude r, i practice oe typically employs the followig qualitative Rule of thumb: 00 = r : o correlatio 00 < r < 02: very weak correlatio 02 r < 04: weak correlatio 04 r < 06: moderately strog correlatio 06 r 08: strog correlatio 08 r < 10: very strog correlatio 10 = r : perfect correlatio Exercise 3 Studet recorded the outside temperature X (i Celsius) ad the duratio of his way to uiversity Y (i miutes) x i y i How strog is the correlatio betwee these two features? I additio to Eq (421), it is coveiet to defie a dimesioless symmetric (2 2) correlatio matrix R by ( ) 1 r R := (424) r 1 which is regular ad positive defiite as log as 1 r 2 > 0 The its iverse R 1 is give by R 1 = 1 ( ) 1 r 1 r 2 r 1 (425) Note that for o-correlatig metrically scaled variables X ad Y, ie, whe r = 0, the correlatio matrix degeerates to become a uit matrix, R = 1 Agai, the cocept of a regular ad positive defiite correlatio matrix R, with iverse R 1, geeralises to multivariate joit distributios of metrically scaled m-d(x, Y,, Z), where R R m m is give by 1 r XY r XZ r Y X 1 r Y Z R := (426) r ZX r ZY 1 Note that R is a dimesioless quatity which, hece, is scale-ivariat Exercise 4 Statistical packages are able to calculate correlatios for multiple pairigs of variables, ofte reportig their fidigs i a correlatio matrix Correlatio matrices report correlatio coefficiets for all pairig of quatitative variables We are goig to create a correlatio matrix for the per capita umbers of cigarettes smoked (sold) i 43 states ad the District of Columbia i 1960 ad death rates for various forms of cacer The data, origially from Fraumei et al(1968), ca be dowloaded at

6 24CHAPTER 4 DESCRIPTIVE MEASURES OF ASSOCIATION FOR BIVARIATE DISTRIBUTIONS as a text file Use R to calculate correlatio coefficiets for each variable pairig Iterpret the correlatio coefficiets Which cacers are associated with smokig? Variable Descriptio CIG cigarettes sold per capita BLAD bladder cacer deaths per 100, 000 LUNG lug cacer deaths per 100, 000 KID kidey cacer deaths per 100, 000 LEUK leukemia cacer deaths per 100, Measures of associatio for the ordial scale level At the ordial scale level, raw data {(x i, y i )},, for a 2-D variable (X, Y ) is ot ecessarily quatitative i ature Therefore, i order to be i a positio to defie a sesible quatitative bivariate measure of statistical associatio for ordial variables, oe eeds to itroduce meaigful surrogate data which is umerical This task is realised by meas of defiig so-called raks, which are assiged to the origial ordial data accordig the procedure described i the followig Begi by establishig amogst the observed values {x i },, resp {y i },, their atural hierarchical order, ie, x (1) x (2) x () ad y (1) y (2) y () (427) The, every idividual x i resp y i is assiged a umerical rak which correspods to its positio i the ordered sequeces (427): x i R(x i ), y i R(y i ), for all i = 1,, (428) Should there be ay tied raks due to equality of some x i or y i, oe assigs the arithmetical mea of these raks to all x i resp y i ivolved i the tie Ultimately, by this procedure the etire bivariate raw data udergoes a trasformatio {(x i, y i )},, {[R(x i ), R(y i )]},, (429) yieldig pairs of raks to umerically represet the origial ordial data Give surrogate rak data, the meas of raks always amout to R(x) := 1 R(y) := 1 R(x i ) = R(y i ) = (430) (431) The variaces of raks are defied i accordace with Eqs (??) ad (??), ie, [ ] s 2 R(x) := 1 R 2 (x i ) R 2 (x) [ ] s 2 R(y) := 1 R 2 (y i ) R 2 (y) [ k ] = R 2 (a i )h i+ R 2 (x) (432) = k R 2 (a j )h +j R 2 (y) (433) I additio, to characterise the joit distributio of raks, a covariace of raks is defied i lie with Eqs (416) ad (420) by [ ] s R(x)R(y) := 1 R(x i )R(y i ) R(x)R(y) = k l R(a i )R(b j )h ij R(x)R(y) (434) j=1 j=1

7 44 MEASURES OF ASSOCIATION FOR THE NOMINAL SCALE LEVEL 25 O this fairly elaborate techical backdrop, the Eglish psychologist ad statisticia Charles Edward Spearma FRS ( ) defied a dimesioless rak correlatio coefficiet r S (ord), i aalogy to Eq (423), by (cf Spearma (1904)) r S := s R(x)R(y) s R(x) s R(y) (435) The rage of this rak correlatio coefficiet is 1 r S +1 Agai, while the sig of r S ecodes the directio of a rak correlatio, i iterpretig the stregth of a rak correlatio via the magitude r S oe usually employs the qualitative Rule of thumb: 00 = r S : o rak correlatio 00 < r S < 02: very weak rak correlatio 02 r S < 04: weak rak correlatio 04 r S < 06: moderately strog rak correlatio 06 r S 08: strog rak correlatio 08 r S < 10: very strog rak correlatio 10 r S : perfect rak correlatio Whe o tied raks occur, Eq (435) simplifies to (cf Hartug et al (2005)) r S = 1 6 [R(x i ) R(y i )] 2 ( 2 1) (436) Exercise 5 Two judges at a fete placed the te etries for the best fruit cakes competitio i order as follows (1 deotes first, ) Etry A B C D E F G H I J Judge 1 (x) Judge 2 (y) Is there a liear relatioship betwee the rakigs produced by the two judges? Exercise 6 Fid the value of r S for the followig data Raks x Raks y Measures of associatio for the omial scale level Lastly, let us tur to cosider the case of quatifyig descriptively the degree of statistical associatio i raw data {(x i, y i )},, for a omially scaled 2-D variable (X, Y ) with categories {(a i, b j )},,k;j=1,,l The startig poit are the observed absolute resp relative (cell) frequecies o ij ad h ij of the bivariate joit distributio of (X, Y ), with margial frequecies o i+ resp h i+ for X ad o +j resp h +j for Y The χ 2 - statistic devised by the Eglish mathematical statisticia Karl Pearso FRS ( ) rests o the otio of statistical idepedece of two variables X ad Y i that it takes the correspodig formal coditio provided by Eq (412) as a referece A simple algebraic maipulatio of this coditio obtais h ij = h i+ h +j o ij = o i+ o +j o ij = o i+o +j (437)

8 26CHAPTER 4 DESCRIPTIVE MEASURES OF ASSOCIATION FOR BIVARIATE DISTRIBUTIONS Pearso s descriptive χ 2 -statistic (cf Pearso (1900)) is the defied by χ 2 := k j=1 ( l oij o i+o +j o i+ o +j ) 2 k l (h ij h i+ h +j ) 2 = (438) h i+ h +j j=1 whose rage of values amouts to 0 χ 2 max(χ 2 ), with max(χ 2 ) := [mi(k, l) 1] Remark 441 Provided o i+o +j 5 for all i = 1,, k ad j = 1,, l, Pearso s χ 2 -statistic ca be employed for the aalysis of statistical associatios for 2-D variables (X, Y ) of almost all combiatios of scale levels The problem with Pearso s χ 2 -statistic is that, due to its variable spectrum of values, it is ot clear how to iterpret the stregth of statistical associatios This shortcomig ca, however, be overcome by resortig to the measure of associatio proposed by the Swedish mathematicia, actuary, ad statisticia Carl Harald Cramér ( ), which basically is the result of a special kid of ormalisatio of Pearso s measure Thus, Cramér s V, as it has come to be kow, is defied by (cf Cramér (1946)) χ V := 2 max(χ 2 ), (439) with rage 0 V 1 For the iterpretatio of results, oe may ow employ the qualitative Rule of thumb: 00 V < 02: weak associatio 02 V < 06: moderately strog associatio 06 V 10: strog associatio Exercise 7 The victory of the icumbet, Bill Clito, i the 1996 presidetial electio was attributed to improved ecoomic coditios ad low uemploymet Suppose a survey of 800 adults take soo after the electio resulted i the followig cross-classificatio of fiacial coditio with educatio level: Fiacial Coditio HS Degree or Lower Some College College Degree or Higher Total Worse off ow tha before No differece Better off ow tha before Total At the 05 level of sigificace, is there evidece of a relatioship betwee fiacial coditio ad educatio level? Exercise 8 A radom sample of 50 wome were tested for cholesterol ad classified accordig to age ad cholesterol level The results are give i the followig cotigecy table, where the rows represet age ad the colums represet cholesterol level: < Row total < Colum total Test the followig ull hypothesis at sigificace level α = 005 the hypothesis : age ad cholesterol are idepedet attributes of classificatio

Joint Probability Distributions and Random Samples. Jointly Distributed Random Variables. Chapter { }

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