Testing Research Hypotheses using Linear Discriminant Function

1. Testing Research Hypotheses using Linear Discriminant Function • Nested & non-nested model comparison • the 3 Model fit indices • Comparing Nested Models • Comparing Non-nested Models

2. As for multiple regression, nearly any research hypotheses can be assessed by the proper comparison of two ldf models • There are two types of model comparisons… • comparing nested models -- the smaller model involves discriminator variables that are a proper subset of those in the larger model • comparing non-nested models -- those models may be of the same or different numbers of discriminator variables • Unlike with multiple regression, there are three different indices of how well the data fit the model that might be used… •  value (remember - smaller means model “works better”) • Rc -- basically the same index as used in multiple regression • % correct re-classification

3. Comparing Nested Models Starting with the simple case of 2-groups • comparing  • since the models are nested, the X² values used to test the  of the models are nested • nested X² values are tested by comparing the difference between the X² from the models to a X²-critical based on the difference between the df of the two models • e.g., the larger 8-var model has  = .78, X² (8) = 12.2 the smaller 5-var model has  = .89, X² (5) = 4.2 • To test the models X² = 12.2 - 4.2 = 8.0 • The X²-critical based on df = 8 - 5 = 3 is 7.815 • Since X² > X²-critical, we would conclude the larger model “does better”

4. comparing Rc • since the models are nested, the R² F-test can be used • ( RL² - RS² ) / ( kL - kS ) RL² - R² from larger mode F = ------------------------------- RS² - R² from smaller model ( 1 - RL² ) / ( N - kL-1) kL - # vars in larger model kS - # vars in smaller model N - number of cases • F is compared to F-critical based on df (kL - kS &N - kL -1)

5. comparing % correct re-classification • we expect that the larger model will have a higher %, but need to test if the difference is larger than would be expected by chance • However, often those folks who are mis-classified by the smaller model are often not a proper subset of those mis- classified by the larger model. • So, statistically the question becomes, “Are there fewer folks who are uniquely mis-classified by the larger model than there are those who are uniquely mis-classified by the smaller model?” • This conforms to the H0: tested by McNemar’s X² test. • There is a worked example on the ldf handout...

6. Comparing nested models from k-group designs… • this can get ugly quickly… • when there is a concentrated structure, you can use the same techniques as for a 2-group design (shown above) • when there is a diffuse structure (in one or both models), comparing  values is of limited utility, since the difference between the models might be in the second ldf (while the 1st  values from the two models are nested, the second  values are not, and so the X² test won’t work) • when there is a diffuse structure, comparing the RC values has the same problem • however, the models can be directly and properly compared using the test of different % correct classification and McNemar’s X²

7. Comparing Non-nested ldf Models • For 2-group models and concentrated k-group models • The % correct classifications can be compared using McNemar’s X² • The RC from the two model can be compared using Hotelling’s t-test or the Meng and Rosenthal Z-test • For diffuse k-group models (one or both models are diffuse) • limited to the comparison of % reclassification and McNemar’s X² test