Advanced statistics for master students

Loglinear models IIThe best model selection and models for ordinal variables

3 procedures in SPSS:1)Loglinear (today and next lecture models with ordinal variables)2)Model selection (today) 3)Logit (not included)

Ordinal Loglinear Models

Literature:Agresti (2002), Wiley; Simonof (2003), Springer;Ishii-Kuntz (1994), Sage

Model selection procedure

-try to find „the best“hierarchical model

-logic: based on chi-squre tests which compare LR criteria for 2 nested models

-approach: Start with saturated model and through backward go to the best model (quite opposite strategy can be applied - from model of independence forward method, sometimes not reccomended in literature)

-all variables are treated as nominal

Model selection

-2 tests

1) Test that k-way interaction are all zero2) Test, that k-way and higher interactions are all zero

Model selection procedure in SPSS:1. Saturated model estimates2. 2 tests (see above)3. Proposal for removing nonsignificant the highest order

interactions and computations of new model estimate4. Again step 2 and 3 (finish-the best model)5. Computation of parameters of the best model

Model selection

Limits of procedure

1) Only hierarchical models2) Based on LR tests only, not including principle of

parsimony (see below AIC a BIC etc.)3) Only models for nominal variables

BUT: For most analytical tasks it can be usefull. The procedure is very quick. For the first insight into your data this procedure can be reccomended.

Ordinal Loglinear Models

- One or more variables is treated as ordinal

- We save number of parameters, higher degrees of freedom (e.g. instead of parameters for every row, only parameter for one variable can be used, the same can be applied for interactions)

- There are many models in literature, this lecture only two and three varibles models

-SPSS is limited with the work with ordinal models

Ordinal Loglinear Models

- Row and column effect model – one variable ordinal, one nominal

- Row effect model – row variable is nominal and column variable is ordinal, interactions are created by values of column variable instead of columns (Example table 3x3: for two nominal variables 4 interaction parameters, for row effect model only 2)

- Uniform association- two ordinal variables, interaction is composed by multiplying values of these variables (Example table 3x3: for two nominal variables 4 interaction parameters, for row effect model only 1)

Ordinal Loglinear Models

Formulas, equations - Row and column effect model - Uniform association

Interpretation of parameters and odds in models - Row and column effect model - Uniform association

Model for three variablesModel of independenceModel of constant fluidity, partial asscociationsaturated model

Ordinal Loglinear Models

„The best model!- Tests for LR criterias

Goodman, AIC BIC criteria

Goodman index G = G2/df,

where G2 is LR criteria (overall test of fit)

df-degrees of freedom

Akaike information criteria AIC = G2 + 2p,

Where p is number of parameters in model

Ordinal Loglinear Models

Goodman, AIC, BIC – continue:

Bayes Schwartz information criteria BIC = G2-df (ln n),

Where n is number of respondents

The lower the better –logic of all criterias

Problem – different criteria favour different models

Note: These criteria can be used in many statistical techniques-regression analysis, multilevel models, SEM, etc.

Ordinal Loglinear Models

The best model selection – other methods

- Residuals – tests

- Residuals – charts

- Principle of parsimony

Ordinal Loglinear Models summaryReccomendation for model selection (Ishii-Kuntz 94:53-4)

1) Prefer model with lower number of parameters (parsimony).2) Prefer model with simpler interpretation.3) Prefer model with all parameters statistical significant.4) Higher Sig. for overall test is good but too big Sig. can be sign of

model icludes too much parameters.5) For ordinal variables is reccomended to start with models for

nominal data and then use appropriate model with ordinal varibles.6) The most important rule is to follow the theory and use model proposed

in literature.

Do not apply (or try) all possible models (data driven analysis) but have some hypothesis in advance about model and test this hypothesis (theory driven analysis) – (Petr S. 12.4. slide 12)

HW

1) Try to use general loglinear model procedure and find appropriate model for at lest 3 variables. Interpret results and tests.1) Try to find the best model with Model selection on your data2) Try to use ordinal modelon your data and interpret results. Compare ordinal model and best hierarchical model (degrees of freedom, LR, criterias etc.)