The Poisson trick for matched two-way tables a case for putting the - PowerPoint PPT Presentation

Plan Key ideas Example Plots Bilinear models Case of two matched tables References The Poisson trick for matched two-way tables a case for putting the fish in the bowl (a case for putting the bird in the cage) e1, Antoine de Falguerolles2,* Simplice Dossou-Gb´ et´ 1. Universit´ e de Pau et des Pays de l’Adour 2. Universit´ e Paul Sabatier (Toulouse III) * Antoine -at- Falguerolles.net 31 January 2011 1. Universit´ e de Pau et des Pays de l’Adour 2. Universit´ e Paul Sabatier (Toulouse III) * Antoine -at- Falguerolles.net

Plan Key ideas Example Plots Bilinear models Case of two matched tables References Plan Key ideas Matched two-way tables Objectives Poisson trick The suicide data: age, method and gender Data CAs for the two matched tables Plots Bird Fish Bilinear models restricted two-way interaction Case of two matched tables Poisson-Multinomial trick for two matched tables References 1. Universit´ e de Pau et des Pays de l’Adour 2. Universit´ e Paul Sabatier (Toulouse III) * Antoine -at- Falguerolles.net

Plan Key ideas Example Plots Bilinear models Case of two matched tables References Key ideas ◮ Matched two-way tables ◮ Analysis of dissimilarity/similarity between tables ◮ Poisson trick 1. Universit´ e de Pau et des Pays de l’Adour 2. Universit´ e Paul Sabatier (Toulouse III) * Antoine -at- Falguerolles.net

Plan Key ideas Example Plots Bilinear models Case of two matched tables References Matched two-way tables matched two-way tables The m tables of counts classified by factor A (row) and factor B (column), Y SAB , their margins Y SA and Y SB and total count Y S k k k k y SAB y SA y SAB y SA y SAB y SA 1 1 s s # S # S . . . . . . 1 ) ′ ) ′ # S ) ′ ( y SB y S ( y SB y S ( y SB y S 1 s s # S The marginal two-way table (and its margins) y AB y A ( y B ) ′ y ∅ 1. Universit´ e de Pau et des Pays de l’Adour 2. Universit´ e Paul Sabatier (Toulouse III) * Antoine -at- Falguerolles.net

Plan Key ideas Example Plots Bilinear models Case of two matched tables References Objectives Objectives Similarity/ Dissimilarity between tables row profiles or column profiles May involve some preprocessing of ◮ tables by unifying margins by biproportional fitting (RAS, Iterative Proportional Fitting, matrix Raking) ◮ row profiles (column profiles) by weighting tables, profiles into tables, common metric 1. Universit´ e de Pau et des Pays de l’Adour 2. Universit´ e Paul Sabatier (Toulouse III) * Antoine -at- Falguerolles.net

Plan Key ideas Example Plots Bilinear models Case of two matched tables References Poisson trick Poisson trick ◮ Y SAB independent Poisson sab E [ Y SAB var ( Y SAB sab ] = sab ) E [ Y SAB m ( β AB ab + restricted( β SAB sab ] = sab )) sab | � # S ◮ Y SAB s =1 Y SAB = y AB multinomial with sab ab ◮ known parameter: y AB ab ◮ probabilities: m ( β AB ab + restricted( β SAB sab )) = m ( β AB ab + restricted( β SAB sab )) sab )) � m k =1 m ( β AB ab + restricted( β SAB y AB ab 1. Universit´ e de Pau et des Pays de l’Adour 2. Universit´ e Paul Sabatier (Toulouse III) * Antoine -at- Falguerolles.net

Plan Key ideas Example Plots Bilinear models Case of two matched tables References Poisson trick Poisson trick for two matched tables Particular case: two matched tables (# M = 2) ◮ independant Poisson counts E [ Y SAB 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 [ Y SAB exp( β AB ab + β SA sa + β SB sab ] = sb ) ◮ Y SAB binomial B ( y AB ab , π AB 2 ab ) 2 ab ◮ model: additivity of effects of A and B logit( π AB β SA 2 a + β SB 2 ab ) = 2 b Works also with the inclusion of a reduced rank interaction in the predictor 1. Universit´ e de Pau et des Pays de l’Adour 2. Universit´ e Paul Sabatier (Toulouse III) * Antoine -at- Falguerolles.net

Plan Key ideas Example Plots Bilinear models Case of two matched tables References Data Male Method Age c1 c2 c3 c4 c5 c6 c7 c8 c9 10-15 4 0 0 247 1 17 1 6 9 15-20 348 7 67 578 22 179 11 74 175 20-25 808 32 229 699 44 316 35 109 289 25-30 789 26 243 648 52 268 38 109 226 30-35 916 17 257 825 74 291 52 123 281 35-40 1118 27 313 1278 87 293 49 134 268 40-45 926 13 250 1273 89 299 53 78 198 45-50 855 9 203 1381 71 347 68 103 190 50-55 684 14 136 1282 87 229 62 63 146 55-60 502 6 77 972 49 151 46 66 77 60-65 516 5 74 1249 83 162 52 92 122 65-70 513 8 31 1360 75 164 56 115 95 70-75 425 5 21 1268 90 121 44 119 82 75-80 266 4 9 866 63 78 30 79 34 80-85 159 2 2 479 39 18 18 46 19 85-90 70 1 0 259 16 10 9 18 10 90+ 18 0 1 76 4 2 4 6 2 1. Universit´ e de Pau et des Pays de l’Adour 2. Universit´ e Paul Sabatier (Toulouse III) * Antoine -at- Falguerolles.net

Plan Key ideas Example Plots Bilinear models Case of two matched tables References Data Female Method Age c1 c2 c3 c4 c5 c6 c7 c8 c9 10-15 28 0 3 20 0 1 0 10 6 15-20 353 2 11 81 6 15 2 43 47 20-25 540 4 20 111 24 9 9 78 67 25-30 454 6 27 125 33 26 7 86 75 30-35 530 2 29 178 42 14 20 92 78 35-40 688 5 44 272 64 24 14 98 110 40-45 566 4 24 343 76 18 22 103 86 45-50 716 6 24 447 94 13 21 95 88 50-55 942 7 26 691 184 21 37 129 131 55-60 723 3 14 527 163 14 30 92 92 60-65 820 8 8 702 245 11 35 140 114 65-70 740 8 4 785 271 4 38 156 90 70-75 624 6 4 610 244 1 27 129 46 75-80 495 8 1 420 161 1 29 129 35 80-85 292 3 2 223 78 0 10 84 23 85-90 113 4 0 83 14 0 6 34 2 90+ 24 1 0 19 4 0 2 7 0 1. Universit´ e de Pau et des Pays de l’Adour 2. Universit´ e Paul Sabatier (Toulouse III) * Antoine -at- Falguerolles.net

Plan Key ideas Example Plots Bilinear models Case of two matched tables References CA Two approaches in CA ◮ Peter’s trick: � M � M ′ � � ordinary CA of either table and/or F ′ F ◮ Michael’s trick: � M � F ordinary CA of table equivalent to F M ◮ ordinary CA of the ‘average’ table 1 2 M + 1 2 F ◮ adapted CA of table M (resp. table F) with respect to 1 2 M + 1 2 F . 1. Universit´ e de Pau et des Pays de l’Adour 2. Universit´ e Paul Sabatier (Toulouse III) * Antoine -at- Falguerolles.net

Plan Key ideas Example Plots Bilinear models Case of two matched tables References 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 table 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 table M, F or (smoothed) metrics attached to the ‘average’ table 1 2 M + 1 2 F or . . . ? Metric choice impacts plots and, to a lesser extent, patterns in graphs. 1. Universit´ e de Pau et des Pays de l’Adour 2. Universit´ e Paul Sabatier (Toulouse III) * Antoine -at- Falguerolles.net

Plan Key ideas Example Plots Bilinear models Case of two matched tables References CA Peter’s plot � M � F 1. Universit´ e de Pau et des Pays de l’Adour 2. Universit´ e Paul Sabatier (Toulouse III) * Antoine -at- Falguerolles.net

Plan Key ideas Example Plots Bilinear models Case of two matched tables References CA Michael’s trick � M � F F M 1. Universit´ e de Pau et des Pays de l’Adour 2. Universit´ e Paul Sabatier (Toulouse III) * Antoine -at- Falguerolles.net

Plan Key ideas Example Plots Bilinear models Case of two matched tables References CA Peter’s trick versus Michael’s trick dissimilarity similarity 1. Universit´ e de Pau et des Pays de l’Adour 2. Universit´ e Paul Sabatier (Toulouse III) * Antoine -at- Falguerolles.net

Plan Key ideas Example Plots Bilinear models Case of two matched tables References CA Peter’s trick versus Michael’s trick dissimilarity similarity 1. Universit´ e de Pau et des Pays de l’Adour 2. Universit´ e Paul Sabatier (Toulouse III) * Antoine -at- Falguerolles.net

Plan Key ideas Example Plots Bilinear models Case of two matched tables References Bird bird and cage 1. Universit´ e de Pau et des Pays de l’Adour 2. Universit´ e Paul Sabatier (Toulouse III) * Antoine -at- Falguerolles.net

Plan Key ideas Example Plots Bilinear models Case of two matched tables References Bird trick 1. Universit´ e de Pau et des Pays de l’Adour 2. Universit´ e Paul Sabatier (Toulouse III) * Antoine -at- Falguerolles.net

Plan Key ideas Example Plots Bilinear models Case of two matched tables References Bird bird in cage 1. Universit´ e de Pau et des Pays de l’Adour 2. Universit´ e Paul Sabatier (Toulouse III) * Antoine -at- Falguerolles.net

Plan Key ideas Example Plots Bilinear models Case of two matched tables References Fish fish and bowl 1. Universit´ e de Pau et des Pays de l’Adour 2. Universit´ e Paul Sabatier (Toulouse III) * Antoine -at- Falguerolles.net