The Multigraph for Loglinear Models Harry Khamis Statistical Consulting Center Wright State University Dayton, Ohio, USA

OUTLINE 1.LOGLINEAR MODEL (LLM) - two-way table - three-way table - examples 2.MULTIGRAPH - construction - maximum spanning tree - conditional independencies - collapsibility 3.EXAMPLES 2

Loglinear Model Goal Identify the structure of associations among a set of categorical variables. 3

LLM: two variables Y 123…JTotal n 11 n 12 n 13 …n 1J n 1+ 2n 21 n 22 n 23 …n 2J n X In I1 n I2 n I3 … n IJ n I+ Total n +1 n +2 n +3 … n +J n 4

LLM: two variables Example Survey of High School Seniors in Dayton, Ohio Collaboration: WSU Boonshoft School of Medicine and United Health Services of Dayton Marijuana Use? YesNoTotal Yes Cigarette Use? No Total

LLM: two variables 6 Two discrete variables, X and Y Model of independence: generating class is [X][Y]

LLM: two variables LLM of independence: 7

LLM: two variables Saturated LLM: generating class is [XY]: 8

LLM: two variables GeneratingProbabilistic InterpretationClassModel X and Y independent[X][Y]p ij = p i+ p +j X and Y dependent[XY]p ij 9

LLM: three variables Example: Dayton High School Data AlcoholCigarette Marijuana Use UseUseYesNo YesYes No NoYes 3 43 No

1111 LLM: three variables Saturated LLM, [XYZ]:

LLM: three variables Generating Probabilistic InterpretationClassModel mutual independence[X][Y][Z]p ijk = p i++ p +j+ p ++k joint independence[XZ][Y]p ijk = p i+k p +j+ conditional independence[XY][XZ]p ijk = p ij+ p i+k /p i++ homogeneous association * [XY][XZ][YZ] * saturated model[XYZ]p ijk * nondecomposable model 12

Decomposable LLMs  closed-form expression for MLEs  closed-form expression for asymptotic variances (Lee, 1977) asymptotic variances (Lee, 1977)  conditional G 2 statistic simplifies  allow for causal interpretations  easier to interpret the LLM 13

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3 Categorical Variables: X, Y, and Z If [X ⊗ Y] and [Y ⊗ Z] then [X ⊗ Z] FALSE! 15

LLM: three variables Generating Probabilistic InterpretationClassModel mutual independence[X][Y][Z]p ijk = p i++ p +j+ p ++k joint independence[XZ][Y]p ijk = p i+k p +j+ conditional independence[XY][XZ]p ijk = p ij+ p i+k /p i++ homogeneous association[XY][XZ][YZ]p ijk = ψ ij φ ik ω jk saturated model[XYZ]p ijk 16

3 Categorical Variables: X, Y, and Z If [Y ⊗ Z] for all X = 1, 2, …. then [Y ⊗ Z] FALSE! 17

LLM: three variables Generating Probabilistic InterpretationClassModel mutual independence[X][Y][Z]p ijk = p i++ p +j+ p ++k joint independence[XZ][Y]p ijk = p i+k p +j+ conditional independence[XY][XZ]p ijk = p ij+ p i+k /p i++ homogeneous association[XY][XZ][YZ]p ijk = ψ ij φ ik ω jk saturated model[XYZ]p ijk 18

3 Categorical Variables: X, Y, and Z If [Y ⊗ Z] then [Y ⊗ Z] for all X = 1, 2, 3, … FALSE! 19

Which Treatment is Better? TRIAL 1 TRIAL 2 CURED? CURED? YesNoTotalYesNoTotal A40 (.20) (.85) TREATMENT B30 (.15) (.75) Combine TRIALS 1 and 2: CURED? Yes NoTotal A125 (.42) TREATMENT B330 (.55) “Ask Marilyn”, PARADE section, DDN, pages 6-7, April 28,

Florida Homicide Convictions Resulting in Death Penalty ML Radelet and GL Pierce, Florida Law Review 43: 1-34, 1991 Death Penalty Yes No White53 (0.11) 430 Defendant’s Race Black15 (0.08) 176 White VictimBlack VictimDeath Penalty YesNoYesNo White53 (0.11)414White 0 (0.00) 16 Defendant’s Race Black11 (0.23) 37Black 4 (0.03)139 21

Multigraph Representation of LLMs   Vertices = generators of the LLM   Multiedges = edges that are equal in number to the number of indices shared by the two vertices being joined 22

Multigraph: three variables [XY][XZ]XY XZ 23

Examples of Multigraphs 24 [AS][ACR][MCS][MAC] ASACR MACMCS

Examples of Multigraphs 25 [ABCD][ACE][BCG][CDF] ABCD CDF ACEBCG

Maximum Spanning Tree The maximum spanning tree of a multigraph M: tree (connected graph with no circuits) includes each vertex sum of the edges is maximum 26

Examples of maximum spanning trees 27 [XY][XZ]XYXZ

Examples of maximum spanning trees 28 [AS][ACR][MCS][MAC] ASACR MACMCS

Examples of maximum spanning trees 29 [ABCD][ACE][BCG][CDF] ABCD CDF ACEBCG

Fundamental Conditional Independencies for a Decomposable LLM 1.Let S be the set of indices in a branch of the maximum spanning tree 2.Remove each factor of S from the multigraph, M; the resulting multigraph is M/S 3.An FCI is determined as: where C 1, C 2, …, C k are the sets of factors in the components of M/S 30

31 FCIs [XY][XZ]XYXZ X S = {X} M/S: Y Z [Y ⊗ Z|X]

Collapsibility Conditions Consider a conditional independence relationship of the form [C 1 ⊗ C 2 |S]. If the levels of all factors in C 1 are collapsed, then all relationships among the remaining factors are undistorted EXCEPT for relationships among factors in S. 32

33 FCIs [XY][XZ]XYXZ X S = {X} M/S: Y Z [Y ⊗ Z|X]

Example: Ob-Gyn Study Example: Ob-Gyn Study (Darrocca, et al., 1996) n = 201 pregnant mothers Variables: E: EGA (Early, Late) B: Bishop score (High, Low) T: Treatment (Prostin, Placebo) 34

Example: Ob-Gyn Study BISHOP SCORE (B) HighLow EGA (E) EGA (E) TREATMENT (T) Early Late Early Late Prostin Placebo Best-fitting model: [E][TB] 35

Example: Ob-Gyn Study Generating Class: [E][TB] Multigraph: ETB FCI: [E ⊗ T,B] 36

Example: Ob-Gyn Study Collapsed Table (collapse over EGA): BISHOP SCORE (B) HighLowTotal Prostin58 (0.55) TREATMENT (T) Placebo38 (0.40) P =

Example: WSU-United Way Study M: Marijuana (No, Yes) A: Alcohol (No, Yes) C: Cigarettes (No, Yes) R: Race (Other, White) S: Sex (Female, Male) Observed cell frequencies (n = 2,276):

Example: WSU-United Way Study Generating class: [ACE][MAC][MCG] Multigraph, M: ACE MCGMAC 39

Example: WSU-United Way Study M: S = {A,C} ACE M/S: E A C MGM MCG MAC [E ⊗ M,G|A,C] A = AlcoholC = CigaretteE = Ethnic G = GenderM = Marijuana 40

Example: WSU PASS Program “Preparing for Academic Success” GPA below 2.0 at the end of first quarter 41

Example: WSU PASS Program Variables (n = 972): FACTORLABELLEVELS RetentionR1=No, 2=Yes CohortC1, 2, 3, 4 PASS ParticipationP1=No, 2=Yes Ethnic GroupE1=Caucasian, 2=African-American, 3=Other GenderG1=Male, 2=Female 42

Example: WSU PASS Program The best-fitting LLM has generating class [EG][CP][RC][PG] Multigraph, M: G EGPG P RC C CP 43

Example: WSU PASS Program M: S = {C} EG PG RC CP R P C M M/S [E,G,P ⊗ R|C] C = CohortE = EthnicG = Gender P = PASS ParticipationR = Retention 44

Example: Affinal Relations in Bosnia-Herzegovina Example: Affinal Relations in Bosnia-Herzegovina Data courtesy of Dr. Keith Doubt, Department of Sociology, Wittenberg University, Springfield, Ohio N = 861 couples from Bosnia-Herzegovina are surveyed concerning affinal relations. M: Marriage Type (traditional, elopement) L: Location of Man and Wife (same, different) E: Ethnicity (Bosniak, Serb, Croat) S: Settlement (rural, urban) Best-fitting model: [MLES] Consider structural associations among M, L, and S for each ethnic group (E) separately. 45

Example: Affinal Relations in Bosnia-Herzegovina Bosniaks:[ML][LS] Serbs:[MS][SL] Croats:[M][L][S] M: Marriage TypeL: Location of Man and WifeS: Settlement 46

Conclusions   The generator multigraph uses mathematical graph theory to analyze and interpret LLMs in a facile manner   Properties of the multigraph allow one to: – –Find all conditional independencies – –Determine all collapsibility conditions REFERENCE Khamis, H.J. (2011). The Association Graph and the Multigraph for Loglinear Models, SAGE series Quantitative Applications in the Social Sciences, No

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