Foundations of Inferential Statistics

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Foundations of Inferential Statistics
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Two Fundamental Questions

• What Questions Search for General Categories
• E mc2
• Clinical Taxonomy
• Mastery Motivation
• Why Questions Search for Structural Relations
• Weak Relationships Prediction (spurious causes)
• Strong Relationships Causal relationships

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Statistical Theory of Causes

• Naive Causal Theory
• Necessary and Sufficient
• Deterministic
• Yy iff Xx
• Yy iff Xx

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Contemporary Causal Theory

• Probabilistic Causes v. Deterministic
• Shifts in pdf(YyXx)
• Multiple Causes
• INUS Condition Insufficient but Necessary PART
  of an Unnecessary but Sufficient set
• Causal Networks

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A Bio-Psycho-Social Model
Relating Risk and Mechanisms of Alcohol
Teratogenesis
Abel Hannigan (1995)
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Contemporary Causal Theory

• Probabilistic Causes v. Deterministic
• Shifts in pdf(YyXx)
• Multiple Causes
• INUS Condition Insufficient but Necessary PART
  of an Unnecessary but Sufficient set
• Causal Networks
• Implications for knowledge
• Nomothetic v. idiographic

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Inferential Statistics are Logic not Mathematics

• The mathematics of probability are useful to the
  social scientist only to the degree they describe
  reality

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From the World Out There to Inside the Ivory Tower

• A tale of many isms
• Physical v. Mental
• Mono v. Dual (poly just for fun)
• Assuming Physical Monism there is an out there,
  out there
• Stage 1 measurement (indicators of out there)
• Stage 2 representation (refining the indicators
  of out there)
• Stage 3 using a representation (metaphor) to
  describe what is out there

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Logic and Statistics

• Types of Reasoning
• Deductive
• Inductive

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Deductive Reasoning
Revealed Truth

• Mathematics
• High School Geometry Trauma
• Judeo-Christian Theology

prosaic fact
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Inductive Reasoning
Possible Truth - maybe?

• Learning the rules by watching the moves.
• Finding patterns in the blight of randomness and
  meaninglessness

prosaic fact after fact after fact after fact
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Back to Inferential Statistics

• Each observation (observations are more important
  than subjects) provides a piece of evidence.
• Convergence of evidence suggests a pattern.
• Patterns map on to metaphors
• ENLIGHTENMENT ?

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More Philosophy

• Psychological Science and Logic
• Logical Positivism.
• Integrated the positivist philosophy of Auguste
  Comte all knowledge is based on empirical
  (positive) methods of science

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• In the year of our Lord 1432, there arose a
  grievous quarrel among the brethren over the
  number of teeth in the mouth of a horse. For 13
  days the disputation raged without ceasing. All
  the ancient books and chronicles were fetched out
  scholasticism, and wonderful and ponderous
  erudition, such as was never before heard of in
  this region, was made manifest. At the beginning
  of the 14th day, a youthful friar of goodly
  bearing asked his learned superiors for
  permission to add a word, and straightway, to the
  wonderment of the disputants, whose deep wisdom
  he sore vexed, he beseeched them to unbend in a
  manner coarse and unheard-of, and to look in the
  open mouth of a horse and find answer to their
  questionings empiricism. At this, their dignity
  being grievously hurt, they waxed exceedingly
  wroth and joining in a mighty uproar, they flew
  upon him and smote him hip and thigh, and cast
  him out forthwith. For, said they, surely Satan
  hath tempted this bold neophyte to declare unholy
  and unheard-of ways of finding truth contrary to
  all the teachings of the fathers. After many days
  more of grievous strife the dove of peace sat
  upon the assembly, and they as one man, declaring
  the problem to be an everlasting mystery because
  of a grievous dearth of historical and
  theological evidence thereof, so ordered the
  same writ down (Mees, C. E. K. Scientific Thought
  and Social Reconstruction. American Scientist,
  1934, vol. 22. p. 383-384 pages).

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More Philosophy

• Psychological Science and Logic
• Logical Positivism.
• Integrated the positivist philosophy of Auguste
  Comte all knowledge is based on empirical
  (positive) methods of science
• Linked with Bertrand Russells Principia
  Mathematica
• Science was concerned solely with syntax formal
  relations between symbols in accordance with
  precise rules.

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Consequences of Positivism

• Verifiability Criterion of Meaning
• A hypothesis must be verifiable as true to be
  amenable to science
• The syllogism of confirmation
• If T is True, then D ?P
• D ? P
• Therefore, T is True
• Commits the fallacy of affirming the consequent.

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• Operationalization of Variables
• The symbolic operations used in measuring a
  construct equal the construct.
• Depression via the SCL-90
• Depression via the Brief Symptom Checklist
• Two measures ? two constructs

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Karl Popper

• Falsification in Science
• A scientific hypothesis is not valid unless it is
  falsifiable.
• Syllogism of Falsification
• If, T is true, then D ? P
• D do not ? P,
• Therefore T is false
• Science of verisimilitude

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Probabilities and Inferences
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Populations and Estimates

• Parameters are statistical summaries of
  information.

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Populations and Estimates

• Parameters are statistical summaries of
  information.
• We can obtain most of the information we need
  (nomothetic information) from just four
  parameters
• Central Tendency (mean, median, mode)
• Dispersion (Variance, S.D.)
• Symmetry (Skew)
• Density (Kurtosis)

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• Parameters are unknowable so we must use
  estimates of those parameters
• Bootstrapping and the relationship between
  parameters and parameter estimates.

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How then do we know if two estimates which are
not the same value are really different?

• Probability Distributions!
• Conditional Probabilities

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What is a Probability

• Relationship between frequencies, percentages and
  probability.
• Simply a rate of occurrence relative to
  opportunity for occurrence
• Fundamental to statistical inferences

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Rules of probability

• 0 lt P(Xx) lt1
• ?P(Xxn) 1
• P(Xx U Xx) P(Xx) P(Xx)
• P(Xxi) n P(XxJ) P(Xxi) P(XxJ)
• P(XxYy) P(XxYy) CI
• P(XxYy) ? P(XxYy)

Logical OR
Logical AND
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Conditional Probabilities and Popper

• Syllogism of Falsification
• If, T is true, then D ? P
• D do not ? P,
• Therefore T is false
• Conditional Probability
• If, T is true, then P(XxYy) ? P(XxYy)
• P(XxYy) P(XxYy) CI,
• Therefore T is false

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