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Joyful mood is a meritorious deed that cheers up people around you

Joyful mood is a meritorious deed that cheers up people around you like the showering of cool spring breeze. Categorical Data Analysis. Chapter 8: Loglinear Models for Contingency Tables (SAS: Chapter 12). Loglinear Models for Counts. Poisson counts: count ~ Poisson(u)

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Joyful mood is a meritorious deed that cheers up people around you

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  1. Joyful mood is a meritorious deed that cheers up people around you like the showering of cool spring breeze.

  2. Categorical Data Analysis Chapter 8: Loglinear Models for Contingency Tables (SAS: Chapter 12)

  3. Loglinear Models for Counts • Poisson counts: count ~ Poisson(u) • Qualitative factors: X, Y, … • Saturated Model: As usual, the baseline effects are set as 0 for each term

  4. Independence Model • No interaction effect between X and Y on counts; that is, X and Y are independent • As usual, the baseline effects are set as 0 for each term

  5. Interpretation of Parameters • The effect of factor on log(odds) is: • Without XY term: • With XY term:

  6. Associations in 3-way Tables • Let Y be the response, X be the major factor and Z be nuisance factor • The observed marginal association of X on Y might be simply due to the other factor Z • In general we cannot collapse a 3-way table and interpret the 2-way marginal table

  7. Example: Z = Clinic

  8. Example: Z = Sex Partial table Marginal table

  9. Type of Independence of X, Y Conditionally independent with Z Mutually independent with Z Jointly independent with Z Marginally independent

  10. Associations in 3-way Tables Eg. 2x2xK tables • Conditional odds ratio • Marginal odds ratio • Marginal independence of X, Y: • marginal X-Y odds ratios are all 1 • Conditional independence of X, Y given Z: • conditional X-Y odds ratios given Z are all 1

  11. Partial Association (Sec 2.3) • The associations in partial tables are called “partial” associations between X and Y given Z • They are measured by conditional odds ratios

  12. Associations in 3-way Tables • We need to condition on all important variables; but it is not practical. • In randomized experiments this (confounding) problem is less likely to happen. • To study whether an association exists between a primary factor and the response variable AFTER controlling for other possibly confounding variables, such as • Different medical centers • Severity of Condition • Age

  13. Loglinear Models for 3-way Tables • Saturated (also full) model:

  14. Interpreting Model parameters • X: effect of X on (expected) counts • XY: the partial association between X and Y given Z • XYZ: significant  XY depends on Z insignificant  XY does not depend on Z

  15. Interpreting Models

  16. Inference for Loglinear Models • Goodness-of-fit tests • Residuals • Tests for partial associations • Confidence intervals for odds ratios

  17. The Loglinear-Logit Connection • Using logit models to interpret loglinear models • Correspondence between loglinear and logit models

  18. Connection with Logit Models • The loglinear model which corresponds to a logit model is the one with the most general interaction among explanatory variables from the logit model. It has the same association and interaction structure relating the explanatory variables to the response.

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