I attended a lecture by Aris Spanos to his graduate econometrics class here at Va Tech last week[i]. This course, which Spanos teaches every fall, gives a superb illumination of the disparate pieces involved in statistical inference and modeling, and affords clear foundations for how they are linked together. His slides follow the intro section. Some examples with severity assessments are also included.
Frequentist Hypothesis Testing: A Coherent Approach
1 Inherent difficulties in learning statistical testing
Statistical testing is arguably the most important, but also the most difficult and confusing chapter of statistical inference for several reasons, including the following.
(i) The need to introduce numerous new notions, concepts and procedures before one can paint — even in broad brushes — a coherent picture of hypothesis testing.
(ii) The current textbook discussion of statistical testing is both highly confusing and confused. There are several sources of confusion.
- (a) Testing is conceptually one of the most sophisticated sub-fields of any scientific discipline.
- (b) Inadequate knowledge by textbook writers who often do not have the technical skills to read and understand the original sources, and have to rely on second hand accounts of previous textbook writers that are often misleading or just outright erroneous. In most of these textbooks hypothesis testing is poorly explained as an idiot’s guide to combining off-the-shelf formulae with statistical tables like the Normal, the Student’s t, the chi-square, etc., where the underlying statistical model that gives rise to the testing procedure is hidden in the background.
- (c) The misleading portrayal of Neyman-Pearson testing as essentially decision-theoretic in nature, when in fact the latter has much greater affinity with the Bayesian rather than the frequentist inference.
- (d) A deliberate attempt to distort and cannibalize frequentist testing by certain Bayesian drumbeaters who revel in (unfairly) maligning frequentist inference in their attempts to motivate their preferred view on statistical inference.
(iii) The discussion of frequentist testing is rather incomplete in so far as it has been beleaguered by serious foundational problems since the 1930s. As a result, different applied fields have generated their own secondary literatures attempting to address these problems, but often making things much worse! Indeed, in some fields like psychology it has reached the stage where one has to correct the ‘corrections’ of those chastising the initial correctors!
In an attempt to alleviate problem (i), the discussion that follows uses a sketchy historical development of frequentist testing. To ameliorate problem (ii), the discussion includes ‘red flag’ pointers (¥) designed to highlight important points that shed light on certain erroneous in- terpretations or misleading arguments. The discussion will pay special attention to (iii), addressing some of the key foundational problems.
[i] It is based on Ch. 14 of Spanos (1999) Probability Theory and Statistical Inference. Cambridge[ii].
[ii] You can win a free copy of this 700+ page text by creating a simple palindrome! http://errorstatistics.com/palindrome/march-contest/