CROSSTABS. Notation. Marginal and Cell Statistics

Transcription

1 OSSTABS Notation The notation and statistics efe to bivaiate subtables defined by a ow vaiable X and a column vaiable Y, unless specified othewise. By default, OSSTABS deletes cases with missing values on a table-by-table basis. The following notation is used thoughout this chapte unless othewise stated: X i Y f i c i Distinct values of ow vaiable aanged in ascending ode: X < X < L < X Distinct values of column vaiable aanged in ascending ode: Y < Y < L < Y 6 Sum of cell weights fo cases in cell f i, the th column subtotal i f i, the ith ow subtotal c i, the gand total i Maginal and ell Statistics ount count f i

2 OSSTABS Expected ount Ei c i ow Pecent 3 8 ow pecent fi i olumn Pecent 3 8 column pecent fi c Total Pecent 3 8 total pecent fi esidual i fi Ei Standadized esidual S i i E i

3 OSSTABS 3 Adusted esidual Ai Ei i i c hi-squae Statistics Peason s hi-squae fi E i χ p E i i The degees of feedom ae Likelihood atio χ L f i E i f i ln3 8 i 6 6. The degees of feedom ae Fishe s Exact Test If the table is a table, not esulting fom a lage table with missing cells, with at least one expected cell count less than 5, then the Fishe exact test is calculated. See Appendix 5 fo details.

4 4 OSSTABS Yates ontinuity oected fo x Tables χ c % & ' f f f f 5 f f f f. 7 cc if >.5 The degees of feedom ae. othewise Mantel-Haenszel Test of Linea Association 6 χ MH whee is the Peason coelation coefficient to be defined late. The degees of feedom ae. Othe Measues of Association Phi oefficient Fo a table not ϕ χ p Fo a table only, ϕ is equal to the Peason coelation coefficient so that the sign of ϕ matches that of the coelation coefficients.

5 OSSTABS 5 oefficient of ontingency χ p χ p + amé s V V 6 χ p q whee q min Measues of Popotional eduction in Pedictive Eo Lambda Let f im and f m be the lagest cell count in ow i and column, espectively. Also, let m be the lagest ow subtotal and c m the lagest column subtotal. Define λ YX as the popotion of elative eo in pedicting an individual s Y categoy that can be eliminated by knowledge of the X categoy. λ YX is computed as λ YX i cm im m f c

6 6 OSSTABS The standad eos ae ASE ASE i3 i 8 i i i f δ δ f c f c δ δ + λδ λ c m 3 8 i i m im m YX whee δ i % & ' if is column index fo f othewise im δ % & ' if is index fo c othewise m Lambda fo pedicting X fom Y, λ YX, is obtained by pemuting the indices in the above fomulae. The two asymmetic lambdas ae aveaged to obtain the symmetic lambda. fim + fm cm m i λ m cm

7 OSSTABS 7 The standad eos ae ASE i i i c i c 3 8 im i i! f δ + δ δ δ f + f c c m m m m m " $ # ASE i f i i i c i c i c δ + δ δ δ + λ δ + δ 4λ c m m 3 8 whee δ i % & ' if i is ow index fo f othewise m δ i % & ' if iis index fo othewise m and whee δ i c % & ' if is column index fo f othewise im δ c i % & ' if is index fo c othewise m

8 8 OSSTABS Goodman and uskal s Tau (Goodman & uskal, 954) Similaly defined is Goodman and uskal s tau 6: τ τ YX 4fi i 9 c c with standad eo % & 6 ' 4 ASE fi v fic c f 4 δ i f i i i δ δ, i i in which δ c and v fi i c τ XY and its standad eo can be obtained by intechanging the oles of X and Y. The significance level is based on the chi-squae distibution, since 5 5τ 5 5τ 5 5 YX ~ χ 5 5 XY ~ χ ( ) *

9 OSSTABS 9 Uncetainty oefficient Let U YX be the popotional eduction in the uncetainty (entopy) of Y that can be eliminated by knowledge of X. It is computed as U X U Y U XY UYX + UY whee 5 UX 5 UY i i ln i c c ln and 6 U XY fi fi f ln i >, fo The asymptotic standad eos ae ASE f fi U Y ln U Y 5 i, % & ' 5 i 5 5 i c UX UXY + ln ( ) * ASE UY5 P U X + U Y U XY

10 OSSTABS whee P fi ln c i fi The fomulas fo U XY can be obtained by intechanging the oles of X and Y. A symmetic vesion of the two asymmetic uncetainty coefficients is defined as follows: U UX6+ UY6 UXY6"! UX6+ $ # with asymptotic standad eos % & ' c i fi ASE fi U XY U X U Y ln ln U X + UY o 6 6 ( ) * ASE U X + U Y P U X + U Y U XY ohen s appa ohen s kappa κ fii c i i i i κ c i i i 6, defined only fo squae table 6, is computed as

11 OSSTABS with vaiance va % + & 4 fii94 fii9 4 fii94 fiic i i fiii ci ' 4 c i i9 4 c i i9 "( # $ # ) * " c i i c i ii + ci6 i i $ # fii fi + ci 4 c i i, +! i 4 4 c i i9 va c i i + i c! i i i

12 OSSTABS endall s Tau-b and Tau-c Define D i i D c c f + f i D f + f i P Q hk h< i k< h> i k> hk h< i k> h> i k< i, i, f i f D i i i hk hk Note: the P and Q listed above ae double the usual P (numbe of concodant pais) and Q (numbe of discodant pais). Likewise, D is double the usual P+ Q+ X (the numbe of concodant pais, discodant pais, and pais on which the ow vaiable is tied) and D c is double the usual P+ Q+ Y (the numbe of concodant pais, discodant pais, and pais on which the column vaiable is tied). endall s Tau-b P Q τ b DD c with standad eo

13 OSSTABS 3 ASE c6 3 8 fi D Dc i Di + τbvi τ D + D DD 3 b c

14 4 OSSTABS whee vi i Dc + c D Unde the independence assumption, the standad eo is ASE fi i Di P Q DD c endall s Tau-c 6 6 qp Q τ c q with standad eo ASE q 6 q fi i Di P Q o, unde the independence assumption, ASE q 6 q fi i Di P Q whee q min

15 OSSTABS 5 Gamma Gamma 6 γ is estimated by γ P Q P+ Q with standad eo 4 ASE P+ Q fi Qi PDi o, unde the hypothesis of independence, ASE fi i Di P+ Q P Q 6 Somes d Somes d with ow vaiable X as the independent vaiable is calculated as d YX P Q D with standad eo ASE fi D i Di P Q i D J L o, unde the hypothesis of independence,

16 6 OSSTABS ASE fi i Di D P Q By intechanging the oles of X and Y, the fomulas fo Somes d with X as the dependent vaiable can be obtained. Symmetic vesion of Somes d is d P Q6 Dc + D 6 The standad eo is ASE σ τ b D + Dc 6 DD c whee σ τ b is the vaiance of endall s τ b, ASE 4 fi i Di D D P Q c Peason s The Peason s poduct moment coelation is computed as cov X, Y S S X S Y T

17 OSSTABS 7 whee 6 i i i i i, i cov XY, XYf X Yc 5 i i i i i i SX X X and SY 6 Y c Yc The vaiance of is " %&' $# ()* S va f + 4 i T Xi X Y Y Xi X S Y Y Y S X T T! If the null hypothesis is tue, va whee fi Xi Y fi XiY i X i i cy X i X ii

18 8 OSSTABS and Y Yc Unde the hypothesis that ρ, t is distibuted as a t with degees of feedom. Speaman oelation Eta The Speaman s ank coelation coefficient s is computed by using ank scoes i fo X i and i fo Y. These ank scoes ae defined as follows: + + fo i,,, i k i k < i 6 c + c + fo,,, h h< 3 8 The fomulas fo s and its asymptotic vaiance can be obtained fom the Peason fomulas by substituting i and fo X i and Y, espectively. Asymmetic η with the column vaiable Y as dependent is η Y SY SY 6

19 OSSTABS 9 whee elative isk SY Y fi Y fi i i i, onside a table ( that is, ). In a case-contol study, the elative isk is estimated as f f f f The α6 pecent I fo the elative isk is obtained as exp z α v, exp z α v whee v f f f f The elative isk atios in a cohot study ae computed fo both columns. Fo column, the isk is 6 f f + f f f + f and the coesponding α6 pecent I is exp z α v, exp z α v

20 OSSTABS whee v f + 6 ff + f6 f f f + f McNema s Test The elative isk fo column and the confidence inteval ae computed similaly. Suppose the test sample is ( x, y),( x, y),,( xn, yn). The null hypothesis H is PX ( < Y) PX ( > Y). Let n #{: i xi < y i, n} n #{: i xi > y i, n} and min( n, n ) Notation n Numbe of cases whee xi < y i, n n Numbe of cases whee xi > y i, n min( n, n ) Pobability If thee is no eal diffeence between the two tials, we expect the fequencies n and n to be elated as :. Deviations fom this atio can be tested by using the binomial distibution. The two-tailed pobability level is + n n ( / ) i i n + n Note. This is a genealized vesion of McNema s test. The oiginal vesion is fo a * table.

21 OSSTABS onditional Independence and Homogeneity The ochan s and Mantel-Haenzel statistics test the independence of two dichotomous vaiables, contolling fo one o moe othe categoical vaiables. These othe categoical vaiables define a numbe of stata, acoss which these statistics ae computed. The Beslow-Day statistic is used to test homogeneity of the common odds atio, which is a weake condition than the conditional independence (i.e., homogeneity with the common odds atio of ) tested by ochan s and Mantel-Haenszel statistics. Taone s statistic is the Beslow-Day statistic adusted fo the consistent but inefficient estimato such as the Mantel-Haenszel estimato of the common odds atio. Notation and Definitions The addition of stata equies the following modifications to the notation: f ik c k ik The numbe of stata. Sum of cell weights fo cases in the ith ow of the th column of the kth stata. f ik, the th column of the kth stata subtotal. i f ik, the ith ow of the kth stata subtotal. k ik, the gand total of the kth stata. i n k c E ik ikck E3fik 8, the expected cell count of the ith ow of the th nk column of the kth stata. A statum such that n k is omitted fom the analysis. ( must be modified accodingly.) If n k fo all k, then no computation is done. Peliminaily, define fo each k

22 OSSTABS $p ik fik, ik dk p$ k - p$ k, $p k c k, nk and wk k k. nk ochan s Statistic ochan s (954) statistic is wd k k k k w k wd w p$ ( - p$ ) w w p$ ( - p$ ) k k k k k k k k k k k k k. All statum such that k o k ae excluded, because d k is undefined. If evey statum is such, is undefined. Note that a statum such that k > and k > but that c k o c k is a valid statum, although it contibutes nothing to the denominato o numeato. Howeve, if evey statum is such, is again undefined. So, in ode to compute a non system missing value of, at least one statum must have all non-zeo maginal totals. Altenatively, ochan s statistic can be witten as

23 OSSTABS 3 k k ( f - E ) k k w p$ ( - p$ ) k k k. hen the numbe of stata is fixed as the sample sizes within each statum incease, ochan s statistic is asymptotically standad nomal, and thus its squae is asymptotically distibuted as a chi-squaed distibution with d.f. Mantel and Haeszel s Statistic Mantel and Haenszel s (959) statistic is simply ochan s statistic with smallsample coections fo continuity and vaiance inflation. These coections ae desiable when k and k ae small, but the coections can make a noticeable diffeence even fo elatively lage k and k (Snedeco and ochan, 98, p. 3). The statistic is defined as: M { ( f -E ) -5. } sgn{ ( f -E )} k k k k k kk p$ k( - p$ k) n - k k k, whee sgn is the signum function sgn( x) % & ' if x > if x. - if x < Any statum in which n k is excluded fom the computation. If evey statum is such, then M is undefined. M is also undefined if evey statum is such that k, k, c k, o c k. In ode to compute a non system missing value of M, at least one statum must have all non-zeo maginal totals, ust as fo. hen the numbe of stata is fixed as the sample sizes within each statum incease, o when the sample sizes within each stata ae fixed as the numbe of

24 4 OSSTABS stata inceases, this statistic is asymptotically standad nomal, and thus its squae is asymptotically distibuted as a chi-squaed distibution with d.f. The Beslow-Day Statistic The Beslow-Day statistic fo any estimato $ θ is { ( ; $ fk - E fk ck θ)} ( f k c k; $. V ) k θ E and V ae based on the exact moments, but it is customay to eplace them with the asymptotic expectation and vaiance. Let and Vˆ mean the estimated asymptotic expectation and the estimated asymptotic vaiance, espectively. Given the Mantel-Haenszel common odds atio estimato statistic as the Beslow-Day statistic: θ ˆ MH, we use the following f k - f k c k B { $ ( ; $ E θ MH )}, V$ ( f k c k k ; $ θ ) MH whee E$ ( f ; $ ) $ k ck θ MH fk satisfies the equations f$ ( $ k nk - k - ck + fk) $ θ ( f$ )( c f$ MH, k - k k - k) with constaints such that

25 OSSTABS 5 f$ k, $ k - fk >, c f$ k - k >, n c f$ k - k - k + k ; and $ ( ; $ V fk ck θ MH ) f$ f$ f$ f$ k k k k - with constaints such that f$ k >, f$ k f$ k - k >, f$ c f$ k k - k >, f$ n c f$ k k - k - k + k > ; All statum such that k o c k ae excluded. If evey statum is such, B is undefined. Statum such that f $ k ae also excluded. If evey statum is such, then B is undefined. Beslow-Day s statistic is asymptotically distibuted as a chi-squaed andom vaiable with - degees of feedom unde the null hypothesis of a constant odds atio. Taone s Statistic Taone (985) poposes an adustment to the Beslow-Day statistic when the common odds atio estimato is consistent but inefficient, specifically when we have the Mantel-Haenszel common odds atio estimato. The adusted statistic, Taone s statistic, fo θ ˆ is MH

26 6 OSSTABS f f c { k - $ ( k k ; $ E θ MH )} { f k - $ ( f k c k ; $ k T -! $ # E θ MH )} V$ ( f k c k k ; $ θ ) MH V$ ( f k c k ; $ θ MH ) k " { f k - $ ( f k c k ; $ E θ MH )} k B -! $ #, $ V ( f k c k ; $ θ MH ) k " whee and Vˆ ae as befoe. The equied data conditions ae the same as fo the Beslow-Day statistic computation. T is, of couse, undefined, when B is undefined. T is also asymptotically distibuted as a chi-squaed andom vaiable with - degees of feedom unde the null hypothesis of a constant odds atio. Estimation of the ommon Odds atio Fo stata of tables, wite the tue odds atios as pk( - pk) θ k ( - pk) pk fo k,...,. And, assuming that the tue common odds atio exists, θ θ... θ, Mantel and Haenszel s (959) estimato of this common odds atio is fk fk n k $θ MH k fk fk nk k. If evey statum is such that fk o fk, then $ θ MH is undefined.

27 OSSTABS 7 The (natual) log of the estimated common odds atio is asymptotically nomal. Note, howeve, that if fk o fk in evey statum, then $ θ MH is zeo and log $ θ MH 4 9 is undefined. The Asymptotic onfidence Inteval obins et al. (986) give an estimated asymptotic vaiance fo log 4 $ θ MH 9 that is appopiate in both asymptotic cases: $ σ [ log( $ θ MH )] + + k k k ( f + f ) f f f ( k f k ) n ( f + f ) f f + ( f + f ) f f f ( k fk f )( k fk ) n n ( f + f ) f f f ( k f k ) n k k k k nk k k k k k k k k k nk k k k k k nk k k k k. k k An asymptotic ( - α )% confidence inteval fo log6 θ is log( $ θmh ) z( α / )$[ σ log ( $ θmh )], whee z( α / ) is the uppe α / citical value fo the standad nomal distibution. All these computations ae valid only if $ θ MH is defined and geate than.

28 8 OSSTABS The Asymptotic P-value e compute an asymptotic P-value unde the null hypothesis that θ ( θ k " k) θ o ( > ) against a -sided altenative hypothesis ( θ ž θ o ), using the standad nomal vaiate, as follows P Z > log( $ θmh )- log( θo ) P Z $[ $ > σ log( θ )] MH log( $ θmh )- log( θo ) $[ σ log( $ θ )] MH, given that log 4 $ θ MH 9 is defined. Altenatively, we can conside using $ θ MH and the estimated exact vaiance of $θ MH, which is still consistent in both limiting cases: $ σ [ log ( $ θ )] $ MH θmh. Then, the asymptotic P-value may be appoximated by P Z > $ θmh -θo $[ σ log( $ θ )] θ MH o. efeences The caveat fo this fomula is that $ θ MH may be quite skewed even in modeate sample sizes (obins et al., 986, p. 34). Agest A. (99). ategoical Data Analysis. John iley, New Yok. Agest A. (996). An Intoduction to ategoical Data Analysis. John iley, New Yok. Bishop, Y. M. M., Fienbeg, S. E., and Holland, P Discete multivaiate analysis: Theoy and pactice. ambidge, Mass.: MIT Pess. Beslow, N. E. and Day, N. E. (98). Statistical Methods in ance eseach,, The Analysis of ase-ontol Studies. Intenational Agency fo eseach on ance, Lyon.

29 OSSTABS 9 Bown, M. B The asymptotic standad eos of some estimates of uncetainty in the two-way contingency table. Psychometika, 4(3): 9. Bown, M. B., and Benedett J Sampling behavio of tests fo coelation in two-way contingency tables. Jounal of the Ameican Statistical Association, 7: ochan,. G. (954). Some methods of stengthening the common Biometics,, χ tests. Goodman, L. A., and uskal,. H Measues of association fo cossclassification. Jounal of the Ameican Statistical Association, 49: Goodman, L. A., and uskal,. H. 97. Measues of association fo cossclassification, IV: simplification and asymptotic vaiances, Jounal of the Ameican Statistical Association, 67: Hauck,. (989). Odds atio infeence fom statified samples. ommun. Statist.- Theoy Meth., 8, Somes, G.. and O Bien,. F. (985). Mantel-Haenszel statistic. In Encyclopedia of Statistical Sciences, Vol. 5 (S. otz and N. L. Johnson, eds.) 4-7. John iley, New Yok. Mantel, N. and Haenszel,. (959). Statistical aspects of the analysis of data fom etospective studies of disease. J. Natl. ance Inst.,, obins, J., Beslow, N., and Geenland, S. (986). Estimatos of the Mantel- Haenszel vaiance consistent in both spase data and lage-stata limiting models. Biometics, 4, Snedeco, G.. and ochan,. G. (98). Statistical Methods, 7 th ed. Iowa State Univesity Pess, Ames, Iowa. Taone,. E. (985). On heteogeneity tests based on efficient scoes. Biometika, 7, 9-95.