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Recent developments

Granger causality and transfer functions

Prediction versus Causality

An interesting excerpt from Bickel and Li (2006) "Regularization in Statistics":

These methods, often actually have two aims,

(I) To construct a good predictor. The values of coefficients in the regression are then irrelevant.

(II) To give causal interpretations of the factors and determine which variables are “important”.

Regularization is important for both aims. But, as well shall see, the appropriate magnitude of the regularization parameter may be governed by which aim is more important.

Examples of Instrumental Variables

A very interesting example applying instrumental variables to assess the causal effect of watching Fox News on the propensity to vote Republican: It is quite rare that an instrumental variable provides a credible way of extracting an estimate of a causal effect from observational data. This might be one of those rare examples:

In the Toronto Star on January 6, 2015:

Fox News really does make people vote Republican, study says --
An ingenious new study estimates that, if there had been no Fox News on
cable television in 2004 and 2008, the Republican vote share would have
been 4 percentage points lower....

The original article:

Methods for adjustments for confounding factors

Examples of Simpson's Paradox

The between professor relationship between average class size and evaluation is positive (professors with larger average class sizes have higher average evaluations). However, within professors, the relationship is negative: you tend to get lower evaluations from your larger classes than from your smaller classes. The between professor effect is, possibly, a selection effect -- effective professors are recruited for large classes -- and the within professor effect is, possibly, an effect of class size controlling for between-professor confounding effects.
A more thorough analysis of the data might reveal richer possibilities. For example, there are a few classes whose size makes them outliers, potentially very influential in a model that only incorporates a linear effect of class size. A model allowing for a curvilinear relationship with class size suggests that the 'compositional effect' (the between-professor relationshiop with average class size) might be negative in the lower range (10 to 50) and positive in the higher range (100 to 500).
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