Statistics and Linear Algebra

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Presentation transcript:

Statistics and Linear Algebra

Cohen (1968): "If you should say to a mathematical statistician that you have discovered that linear multiple regression analysis and the analysis of variance (and covariance) are identical systems, he would mutter something like 'Of course general linear model,' and you might have trouble maintaining his attention. If you should say this to a typical psychologist, you would be met with incredulity, or worse."

General Linear Model

Science (adapted from Coombs, 1983)

Empirical Systems T Segment of the real world Empirical generalization Language development Attitude measurement Memory Illusions T Empirical generalization Segment of the real world Testing by observation or experiment To draw conclusion from the data about the existence of empirical regularities

Empirical Systems Can make inferences Bring novelty Descriptions of observations T Empirical generalization Segment of the real world Extrapolation beyond the bounds of the observation is unjustified Have no explanatory power

Mathematical Systems M Axiom system Logical consequences Set of abstract objects Unproven assertions to avoid circularity of reasoning (e.g. element, belong to, set, point, and or) M Logical consequences Axiom system Mathematical reasoning (proof) Consequences that are derived from the axiom system (theorems)

Mathematical Systems (Model) Totally deductive Brings nothing new Seeks logical consistency Can explain M Logical consequences Axiom system

Empirical and Mathematical Systems Empirical generalization Segment of the real world A I (interpretation) (abstraction) M Logical consequence Axiom system

Model A I T M Segment of the real world Axiom system Empirical generalization Logical consequence

Theory A I T M Segment of the real world Axiom system Empirical generalization Logical consequence

The empirical world is rich, the mathematical world is powerful The empirical world is rich, the mathematical world is powerful. The match is fruitful. If AMI = T, then scientific knowledge has increased A I M Logical consequence Axiom system T Empirical generalization Segment of the real world

Psychology (linear, independent)