Adv Categorical Data Analysis
Adv Categorical Data Analysis STA 7853
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Date Created: 10/29/15
Chapter 8 Loglinear Models for Contingency Tables Lecture 1 Twoway tables Consider an I x J two way table based on a sample of size n with 1 cell probabilities TIM 2 cell counts 3 expected cell count uzj mTZj The loglinear models use the expected cell counts so they equally apply to the Poisson sampling for the N I J independent cell counts Yij with MHAEY Independence Model Under independence model Hij Mai j For example for multinomial sampling uZj furl u a multiplicative form In logarithmic form the independence model is additive loam A A A 1 Where X denotes the row variable and Y denotes the column variable 1 AgX are the row effects and are the column effects For identi ability A 0 and A 0 The ML tted values for the ex pected frequencies are One can use the X2 and G2 goodness of t tests for loglinear models as well Interpretations of Parameters 1 Loglinear models are GLM s that treat the N cell counts as independent observations of a Poisson variable rather than classi cation of 71 subjects 2 The expected cell counts relate to the explanatory variables through the log link 3 The model does not distinguish between the response and explanatory vari ables treats both jointly as responses and models the expected cell count in terms of their level combinations For interpretation however we treat them as response and explanatory vari able For an I x 2 table with independence model we have P Y l X Z39 logl l M2 109 Mn 109 M2 AH which does not depend on 239 the level of X Thus the independence model implies logithO 1X oz hence in each row the odds of response in column 1 is ea BAT7A5 Similarly for J gt 2 differences for two parameters for a given variable relate to the log odds of making one response relative to the other on that variable Saturated Model A dependence model between X and Y satis es 109017 A AgX A Aggy 2 The terms Aggy are the association terms reflecting deviation from the inde pendence model Equation 2 resembles the ANOVA model with interaction and AgY represent interaction between X and Y whereby the effect of one variable on uZj depends on the levels of the other variable The independence model 1 is a special case with Aij 0 How Many Parameters Do We Have 1 With A 0 there are I 1 effects of X 3 2With A 0 there are J 1 effects of Y 3 With AU 0139 l2I and A1 0 j l2J there are I lJ l nonredundant interaction parameters 4 Tests of independence is H0 Ag 0 and is based on I lJ 1 df 5 The number of parameters in model 2 is l lJ l lJ l I J same as the number of cells hence the model is saturated 6 For this model association parameters can be written in terms Ag For example in a 2 x 2 table the log odds ratio is 1090 109 109 11 109 22 109 M12 109 M21 7 XY XY XY XY 11 A22 A12 A21 As usual unsaturated models are preferable as they are easier to interpret and their t smooths the data Hierarchical Moel A model is hierarchical if it includes all lower order terms composed from variables contained in a higher order model term For example model 2 is hierarchical whereas the model 1090127 A AgX A33 is not hierarchical Such models force unnatural behavior of expected fre quencies with the constrained used for parameters For example with the parameters being zero at the last levels log 1 A for all j Such models are not used much It is like using an ANOVA model with interaction without the corresponding main effects Note that with the interaction effects present we focus on interpreting these effects since the main effects are not meaningful Alternative Parameter Constraints One could use sum to zero constraints 2 AgX 0 Z 0 Z AfjY 0 j 12 J and Z AfjY 0 139 12 1 Parameter estimates change however contrasts remain the same Multinomial Model for Cell Probabilities Conditional on n being xed the Poisson loglinear model the mean becomes a multinomial model with cell probabilities 7TH For example for 5 model 2 6AA A Af y 7Tz j W Ea Eb 6AAa b Aab Loglinear Models for Three Way Tables Consider an I x J x K table with response variables X Y and Z Assume multinomial sampling with cell probabilities Wijk Z Z Zk Wijk 1 Or one can assume independent Poisson sampling with cell means pHk Types of Independence 1 Mutual Independence X Y and Z are mutually independent if 72 Wi7Tj7Tk7 for all 239 j k The loglinear model for mutual independence is zogmzjk A A A Ag 3 2 One Variable Jointly Independent of the Other Two Variable Y is independent jointly of X and Z if Mk 7Tj7Tik for all 239 j k This is equivalent to two way independence of Y with another variable with K levels The loglinear model is lamm A A A A Ag 4 One can similarly de ne the independence of X jointly with Y and Z 7 and that of Z jointly with X and Y Note that the mutual independence implies the independence of any one variable jointly with the others Conditional Independence X and Y are conditionally independent given Z if X and Y are independent within each partial table xed levels of Z Let then the conditional independence of X and Y given Z implies Tri k Wilk7rjlk for all 239 j k In terms of the joint probabilities of the table thuis implies Wijk 7Tik7Tjk 7Tk for all 239 j k The loglinear model is lamm A A A A A Ag 5 Summary of the Three Types of Independence Probabilistic Association Terms Model Form of EM in Loglinear Model Interpretation 3 7Ti7Tj7Tk None Variables mutually independent 4 7Tj7Tik Viz Y indep of X and Z 5 Ag A X and Y indep given Z Thus Mutual lndep gt Jointly lnd of Z gt Conditional lndep given Z and Marginal Independence The term with double subscript re ects conditional dependence of the vari ables with those subscripts given the third The model permitting all the three pairs to be conditionally dependent is longjk A A A A A Ag A Exponentiating both sides gives Wm ij jkwik This is a homogeneous association model the OR between any pair of vari ables are the same at every level of the third or a model with no three factor interaction The general three factor loglinear model saturated is longjk A A A A A Ag A Aggy The total number of non redundant parameters is IJK lI lJ lK lI lJ lI lK l J lK l I lJ lK 1 Some Possible Models for Threeway Tables Loglinear Modle Symbol lamM MA A A XIX 1090112776 AA AfA Af Y XY Z longjk A A A A AggY A XY YZ zogmzjk A A A A A3 A Ang XY YZ XZ lamm A A A A A2 V A A XYZ Interpretation of Model Parameters We use the highest order terms in a model for interpretation Consider model X Y X Z 7 YZ We use the two factor terms to describe the conditional OR s For example at a xed level k of Z 7 the conditional association between X and Y uses the I lJ 1 local OR s 6 i WijWiLjLk W i 7WLk7TiLM 9 l S 139 S I l l S j S J 1 Similarly there are I lK l OR s 02 to describe the XZ conditional association for xed j and J lK l OR s 0iJk to describe the conditional YZ association for xed 239 Conditional lndependence Conditional independence of X and Y is equivalent to 0k11gz39gI 113ng 1k12K Also the two factor parameters relate to the conditional OR s Hijk iJrLjJrLk Hi7j17k i17j7k 7 XY XY z39j Ai1j1 10992 jk 1le AXY XY A i17j 239j1 which is independent of k Hence absence of a three factor interaction implies homogeneous association m mm for all 239 j Similarly the model X Y YZ X Z also implies 94m 94m m 0mm for all 239 k and OWM0 M39 6 M
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