The Poisson trick for matched two-way tables

Transcription

1 The Poisson trick for matched two-way tles a case for putting the fish in the bowl (a case for putting the bird in the cage) Simplice Dossou-Gbété1, Antoine de Falguerolles2,* 1. Université de Pau et des Pays de l Adour 2. Université Paul Satier (Toulouse III) * Antoine -at- Falguerolles.net 31 January 2011

2 Plan Key ideas Matched two-way tles Objectives Poisson trick The suicide data: age, method and gender Data CAs for the two matched tles Plots Bird Fish Bilinear models restricted two-way interaction Case of two matched tles Poisson-Multinomial trick for two matched tles References

3 Key ideas Matched two-way tles Analysis of dissimilarity/similarity between tles Poisson trick

4 Matched two-way tles matched two-way tles The m tles of counts classified by factor A (row) and factor B (column), Yk SAB, their margins Yk SA and Yk SB and total count Yk S y1 SAB y1 SA (y SB 1 ) y S 1... y SAB s y SA s (y SB s ) y S s... y SAB #S y SA #S (y SB #S ) y S #S The marginal two-way tle (and its margins) y AB y A (y B ) y

5 Objectives Objectives Similarity/Dissimilarity between tles row profiles or column profiles May involve some preprocessing of tles by unifying margins by biproportional fitting (RAS, Iterative Proportional Fitting, matrix Raking) row profiles (column profiles) by weighting tles, profiles into tles, common metric

6 Poisson trick Poisson trick Y SAB s independent Poisson E[Ys SAB SAB ] = var(ys ) E[Ys SAB ] = m(βab + restricted(βsab s )) Ys SAB #S s=1 Y s SAB = y AB multinomial with known parameter: y AB probilities: m(β AB + restricted(βsab s )) m(βab m k=1 m(βab + = restricted(βsab s )) + restricted(βsab s )) y AB

7 Poisson trick Poisson trick for two matched tles Particular case: two matched tles (#M = 2) independant Poisson counts E[Ys SAB ] (s = 1, 2) exponential mean function (log link function): m = exp, m 1 = log model: all two-way interactions of A, B and F E[Ys SAB ] = exp(β AB + βsa SA + βsb SB ) Y2 SAB binomial B(y AB, πab 2 ) model: additivity of effects of A and B logit(π2) AB = β2a SA + β2b SB Works also with the inclusion of a reduced rank interaction in the predictor

8 Data Male Method Age c1 c2 c3 c4 c5 c6 c7 c8 c

9 Data Female Method Age c1 c2 c3 c4 c5 c6 c7 c8 c

10 CA Two approaches in CA Peter s trick: ordinary CA of either tle [ M F ] [ M and/or Michael s trick: [ ] M F ordinary CA of tle equivalent to F M F ] ordinary CA of the average tle 1 2 M F adapted CA of tle M (resp. tle F) with respect to 1 2 M F.

11 CA Two approaches in CA (Continued) Implicit in the first stream of approaches are choice of a log-linear model between C + S R and R + S C where R, C, and S denote row, column, matching factors ordinary CA of the tle formed accordingly Implicit in the second stream of approaches are metric choice for the rows (the ages) and the columns (the causes): metrics attached to each tle M, F or (smoothed) metrics attached to the average tle 1 2 M F or...? Metric choice impacts plots and, to a lesser extent, patterns in graphs.

12 CA Peter s plot [ M F ]

13 CA Michael s trick [ M F ] F M

14 CA Peter s trick versus Michael s trick dissimilarity similarity

15 CA Peter s trick versus Michael s trick dissimilarity similarity

16 Bird bird and cage

17 Bird trick

18 Bird bird in cage

19 Fish fish and bowl

20 Fish fish in bowl

21 restricted two-way interaction Notation for a two-way tle Observed #A #B two-way tle y AB of counts cross classified by factor A (row) and factor B (column), and margins y AB y A Profiles: Weights: w AB A-profiles B-profiles = y y A a y y B b y (y B ) y y B A=a b = 1 ya A ya A B=b = 1 yb B y AB y AB can be generalized into γ γ A a γ B a

22 restricted two-way interaction Diet modeling The y AB with are observed values of independant r.v. Y AB expected value: E[Y AB] = µab bi-linear predictor = m(ηab ) reduced rank interaction {}}{ η AB = offset + [β + βa A + βb B +] δ k βk,a A βb k,b k=1,...,r with identification constraints for the βs variance: Var(Y AB) = V (µab ) = allows to replicate most models with rank restricted interaction. = has consequences on the distribution of the profiles.

23 restricted two-way interaction Implementations Current implementations are Benzécri s CA and Goodman s RC. But non-canonical crossovers are possible. CA: µ AB = w AB (1 + η AB ) and V (µab ) = w AB a taste of heteroscedastic Normal distribution with a zest of Poisson RC: µ AB GB: µ AB BG: µ AB = exp (ηab ) and V (µab ) = µab a definite taste of Poisson distribution = exp(ηab ) and V (µab ) = w AB = max{ɛ, w AB (1 + ηab )} and V (µab ) = µab

24 restricted two-way interaction Diet Poisson-Multinomial Y i (i {1,..., n}) are independent r.v. with E[Y i ] = µ i and Var(Y i ) = σ 2 i E[ 1 y [Y 1,..., Y n] Y Y n = y] = 1 y [µ 1,..., µ n] + 1 y i σ2 i Var( 1 y [Y 1,..., Y n] Y Y n = y) = 0 {}}{ (y µ i )[σ1 2,..., σ2 n ] i 1 y 2 σ σ 2 n 1 y 2 i σ2 i σ σ 2 n [ σ 2 1,..., σ2 n ]

25 Poisson-Multinomial trick for two matched tles Poisson-Multinomial trick for two matched tles Poisson counts for the three way tle y SAB = (y1 SAB, y2 SAB ): log(λ SAB 2 ) = β + β2 M + βa A + βb B + βsa 2a + β2b SB + βab δ k ξak A ξb bk k log(λ SAB 1 ) = β + + βa A + βb B βab ( δ k )ξak A ξb bk k + + Binomial model for the two way tle y SAB 2 given the tle y AB (sum of counts of matched cells): logit(π AB ) = log(λsab 2 λ SAB 1 ) = β S 2 + β SA 2a + β SB 2b + 2 k δ k ξ A ak ξb bk

26 Poisson-Multinomial trick for two matched tles What if CA is used? Three way tle y SAB = (y SAB 1, y SAB 2 ) and associated weights w AB = y y a A yb b y y CA of tle y2 SAB with respect to tle 1 2 y AB : offset 1 w AB E[Y2 SAB] = 1 {}}{ 1 w AB ( 2 y AB + k δ kξak A ξb bk ) Interpretation for the reduced rank interaction: 4 w AB yb AB k δ k ξ A ak ξb bk logit(π SAB 2 )

27 Poisson-Multinomial trick for two matched tles Log-odds

28 References Peter van der Heijden and Jan de Leeuw (1985): Correspondence analysis and complementary to loglinear analysis, Psychometrika, 50(4), Michael Greenacre (2003): Singular value decomposition of matched matrices, Journal of Applied Statistics, 30, Simplice Dossou-Gbété (2002): Reduced rank quasi-symmetry and biplots for matched two-way tles, Annales de la Faculté des Sciences, vol. XI (4),

29 Thank you for your atention