The Multigraph for Loglinear Models

Harry KhamisStatistical Consulting Center

Wright State UniversityDayton, Ohio, USA

OUTLINE1. LOGLINEAR MODEL (LLM)

- two-way table- three-way table- examples

2. MULTIGRAPH- construction- maximum spanning tree- conditional independencies- collapsibility

3. EXAMPLES

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Loglinear ModelLoglinear Model

Goal

Identify the structure of associations among a set of categorical variables.

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LLM: two variables Y1 2 3 … J Total

------------------------------------------------------------------------------1 n11 n12 n13 … n1J n1+

2 n21 n22 n23 … n2J n2+

. . . . . .

X . . . . . .. . . . . .I nI1 nI2 nI3 … nIJ nI+

Total n+1 n+2 n+3 … n+J n

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LLM: two variablesExample

Survey of High School Seniors in Dayton, OhioCollaboration: WSU Boonshoft School of Medicine and

United Health Services of Dayton

Marijuana Use?Yes No Total---------------------------------------------------------------------Yes 914 581 1495Cigarette Use?No 46 735 781Total 960 1316 2276

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LLM: two variables

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Two discrete variables, X and Y

Model of independence: generating class is [X][Y]

LLM: two variables

LLM of independence:

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0

log

j

Yj

i

Xi

Yj

Xiij

where

LLM: two variablesSaturated LLM: generating class is [XY]:

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RatioOddsNote

where

XYij

j

XYij

i

XYij

j

Yj

i

Xi

XYij

Yj

Xiij

:

0

log

LLM: two variables

Generating ProbabilisticInterpretation Class Model-------------------------------------------------------------------------------------X and Y independent [X][Y] pij = pi+p+j

X and Y dependent [XY] pij

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LLM: three variablesExample: Dayton High School Data

Alcohol Cigarette Marijuana UseUse Use Yes No----------------------------------------------------------------------------------Yes Yes 911 538

No 44 456

No Yes 3 43No 2 279

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11111111

LLM: three variables

Saturated LLM, [XYZ]:

0...

log

k

XYZijk

j

XYij

i

XYij

j

Yj

i

Xi

XYZijk

YZjk

XZik

XYij

Zk

Yj

Xiijk

where

LLM: three variablesGenerating Probabilistic

Interpretation Class Model------------------------------------------------------------------------------------mutual independence [X][Y][Z] pijk = pi++p+j+p++k

joint independence [XZ][Y] pijk = pi+kp+j+

conditional independence [XY][XZ] pijk = pij+pi+k/pi++

homogeneous association* [XY][XZ][YZ] *

saturated model [XYZ] pijk

*nondecomposable model1212

Decomposable LLMs closed-form expression for MLEsclosed-form expression for MLEs

closed-form expression for closed-form expression for asymptotic variances (Lee, 1977)asymptotic variances (Lee, 1977)

conditional Gconditional G22 statistic simplifies statistic simplifies

allow for causal interpretationsallow for causal interpretations

easier to interpret the LLM easier to interpret the LLM

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3 Categorical Variables: X, Y, and Z3 Categorical Variables: X, Y, and Z

If [X Y] and [Y Z] ⊗ ⊗then [X Z]⊗

FALSE!

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LLM: three variables

Generating ProbabilisticInterpretation Class Model------------------------------------------------------------------------------------mutual independence [X][Y][Z] pijk = pi++p+j+p++k

joint independence [XZ][Y] pijk = pi+kp+j+

conditional independence [XY][XZ] pijk = pij+pi+k/pi++

homogeneous association [XY][XZ][YZ] pijk = ψijφikωjk

saturated model [XYZ] pijk

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3 Categorical Variables: X, Y, and Z3 Categorical Variables: X, Y, and Z

If [Y Z] for all X = 1, 2, ….⊗then [Y Z]⊗

FALSE!

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LLM: three variables

Generating ProbabilisticInterpretation Class Model------------------------------------------------------------------------------------mutual independence [X][Y][Z] pijk = pi++p+j+p++k

joint independence [XZ][Y] pijk = pi+kp+j+

conditional independence [XY][XZ] pijk = pij+pi+k/pi++

homogeneous association [XY][XZ][YZ] pijk = ψijφikωjk

saturated model [XYZ] pijk

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3 Categorical Variables: X, Y, and Z3 Categorical Variables: X, Y, and Z

If [Y Z] ⊗then

[Y Z] for all X = 1, 2, 3, …⊗FALSE!

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Which Treatment is Better?Which Treatment is Better? TRIAL 1 TRIAL 2 CURED? CURED?Yes No Total Yes No Total---------------------------------------------- ----------------------------------------A 40 (.20) 160 200 85 (.85) 15 100

TREATMENTB 30 (.15) 170 200 300 (.75) 100 400

Combine TRIALS 1 and 2: CURED?Yes No Total-----------------------------------------------A 125 (.42) 175 300TREATMENTB 330 (.55) 270 600

“Ask Marilyn”, PARADE section, DDN, pages 6-7, April 28, 1996

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Florida Homicide Convictions Resulting in Death PenaltyML Radelet and GL Pierce, Florida Law Review 43: 1-34, 1991

Death PenaltyYes No

----------------------------------------White 53 (0.11) 430

Defendant’s RaceBlack 15 (0.08) 176

White Victim Black Victim

Death Penalty Death PenaltyYes No Yes No

------------------------------------- --------------------------------------White 53 (0.11) 414 White 0 (0.00) 16

Defendant’s RaceBlack 11 (0.23) 37 Black 4 (0.03) 139

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Multigraph Representation of LLMsMultigraph 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

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Multigraph: three variablesMultigraph: three variables

[XY][XZ] XY XZ

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Examples of MultigraphsExamples of Multigraphs

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[AS][ACR][MCS][MAC]

AS ACR

MAC MCS

Examples of MultigraphsExamples of Multigraphs

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[ABCD][ACE][BCG][CDF]

ABCD

CDF

ACE BCG

Maximum Spanning TreeMaximum 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

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Examples of maximum spanning trees Examples of maximum spanning trees

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[XY][XZ] XY XZ

Examples of maximum spanning trees Examples of maximum spanning trees

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[AS][ACR][MCS][MAC]

AS ACR

MAC MCS

Examples of maximum spanning trees Examples of maximum spanning trees

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[ABCD][ACE][BCG][CDF]

ABCD

CDF

ACE BCG

Fundamental Conditional IndependenciesFundamental Conditional Independenciesfor a Decomposable LLMfor 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 C1, C2, …, Ck are the sets of factors in the components of M/S

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FCIs FCIs

[XY][XZ] XY XZX

S = {X}

M/S:Y Z

[Y⊗Z|X]

Collapsibility ConditionsCollapsibility Conditions

Consider a conditional independence relationship of the form

[C1 C⊗ 2|S].

If the levels of all factors in C1 are collapsed, then all relationships among the remaining factors are

undistorted EXCEPT for relationships among factors in S.

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FCIs FCIs

[XY][XZ] XY XZX

S = {X}

M/S:Y Z

[Y⊗Z|X]

Example: Ob-Gyn StudyExample: 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)

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Example: Ob-Gyn StudyExample: Ob-Gyn Study

BISHOP SCORE (B)High Low

EGA (E) EGA (E)TREATMENT (T) Early Late Early Late

------------------------------------------------------------------------------------------------------Prostin 34 24 27 21

Placebo 22 16 35 22

Best-fitting model: [E][TB]

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Example: Ob-Gyn StudyExample: Ob-Gyn Study

Generating Class: [E][TB]

Multigraph:

E TB

FCI: [E T,B]⊗

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Example: Ob-Gyn StudyExample: Ob-Gyn StudyCollapsed Table (collapse over EGA):

BISHOP SCORE (B) High Low Total

-------------------------------------------------Prostin 58 (0.55) 48 106

TREATMENT (T)Placebo 38 (0.40) 57 95

P = 0.037

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Example: WSU-United Way StudyExample: 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):

12 0 19 2 1 0 23 23117 1 218 13 17 1 268 40517 0 18 1 8 1 19 30133 1 201 28 17 1 228 453

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Example: WSU-United Way StudyExample: WSU-United Way Study

Generating class: [ACE][MAC][MCG]

Multigraph, M:

ACE MCG MAC

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Example: WSU-United Way StudyExample: WSU-United Way StudyM: S = {A,C}

ACE M/S: E A C MG M

MCG MAC [E M,G|⊗ A,C]

A = Alcohol C = Cigarette E = EthnicG = Gender M = Marijuana

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Example: WSU PASS ProgramExample: WSU PASS Program

“Preparing for Academic Success”

GPA below 2.0 at the end of first quarter

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Example: WSU PASS ProgramExample: WSU PASS Program

Variables (n = 972):

FACTOR LABEL LEVELS--------------------------------------------------------------------------------------------------------------Retention R 1=No, 2=YesCohort C 1, 2, 3, 4PASS Participation P 1=No, 2=YesEthnic Group E 1=Caucasian, 2=African-American, 3=OtherGender G 1=Male, 2=Female

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Example: WSU PASS ProgramExample: WSU PASS Program

The best-fitting LLM has generating class [EG][CP][RC][PG]

Multigraph, M: G

EG PG P

RC C CP 4343

Example: WSU PASS ProgramExample: WSU PASS ProgramM: S = {C}

EG PG EG PG

RC CP R PC M M/S

[E,G,P⊗R|C]

C = Cohort E = Ethnic G = GenderP = PASS Participation R = Retention

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Example: Affinal Relations in Bosnia-HerzegovinaExample: Affinal Relations in Bosnia-HerzegovinaData 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.

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Example: Affinal Relations in Bosnia-Herzegovina Example: Affinal Relations in Bosnia-Herzegovina

Bosniaks: [ML][LS]

Serbs: [MS][SL]

Croats: [M][L][S]

M: Marriage Type L: Location of Man and Wife S: Settlement

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ConclusionsConclusions 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

REFERENCEKhamis, H.J. (2011). The Association Graph and the Multigraph for Loglinear Models,

SAGE series Quantitative Applications in the Social Sciences, No. 167.

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Without data, you’re just one more person with an

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