Case Study of Faith-Academic Discipline Integration:

Statistical Inference

Andrew M. Hartley, PhDAssociate Statistical Science Director,

PPD

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All of Life Redeemed

• Proverbs 9:10: “The fear of the LORD is the beginning of wisdom, and knowledge of the Holy One is understanding.”

• Kuyper: “There is not a square inch in the whole domain of our human existence over which Christ, who is Sovereign over all, does not cry, ‘Mine!'”

• Christ laid down His life for us; how can we fail to surrender all to Him?

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Medicine to be Redeemed

Kuyper: questions for Medical Doctors that depend on religion1.Tell a dying person they are dying?2.Urge self-control or indulgence for passionate youth?3.Counsel the psychologically distressed, or drug them?

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Statistics, too, to be RedeemedStatistical Qns that depend on religion

1. Can the data alone tell us what to believe about scientific hypotheses? 2. Is every individual free to interpret data the way s/he sees fit? 3. If most people believe in alpha=0.05, does that make it a good rule? 4. Is the human conscience a tabla rasa that can interpret data

completely without bias? Should it be so? 5. Is it true that data are data (concept of the “blind statistician”) so that

we should analyze them the same regardless of their implications?

Connections often missed between religious beliefs & science, because philosophy not recognized as the link.

Religion sets bounds for philosophy…philosophy sets bounds for science.

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Parallels - Sphere Sovereignty & the Philosophy of the Law Idea (PLI)

Note the similarities•Kuyper’s SS: “This perfect Sovereignty of the sinless Messiah at the same time directly denies and challenges all absolute Sovereignty among sinful men on earth, and does so by dividing life into separate spheres, each with its own sovereignty.”•Dooyeweerd’s PLI: Creation reveals itself to us in many distinct yet inter-connected aspects, or kinds of laws & properties. No aspect is more important or real than any other, & to treat it as such is idolatry. More, now, on “kinds of laws & properties…”

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Some of the 14 AspectsAspect = kind of property & law1.Quantitative – how much, how many2.Sensitive3.Biological4.Logical5.Social6.Economic7.Fiducial – trust, reliability, belief

Each science investigates a small number of aspects.Statistical inference – investigates quantitative & fiducial aspects

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(Notice parallels with Sphere Sovereignty) •All aspects equally real & important (“non-reduction”) – Avoid idolizing aspects•Pre-scientific vs Scientific Experience– Pre-scientific (Everyday) experience – multi-

aspectual, coherent, non-abstract– Each science enhances everyday experience, but

does not replace it (“primacy of the pre-scientific”)

Statistical paradigms might respect or violate the above principles

PLI’s Claims about the Aspects

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Driven to WorshipHumans will worship (trust, believe in) either God or something God created, for happiness, fulfillment, security, sure knowledge…•Romans 1 - “For although they knew God, they neither glorified him as God nor gave thanks to him, but their thinking became futile and their foolish hearts were darkened…They exchanged the truth about God for a lie, and worshiped and served created things rather than the Creator…” •Richard Lovelace - “It is the paradox of earthly blessing that because of our own wayward hearts we can worship the gift rather than the giver.”•Bob Dylan: “You’re gonna have to serve somebody. Well, it may be the devil or it may be the Lord.”

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Idolatry Possible on Many LevelsIdolatry: Worshiping something other than God•Personal, e.g., materialism•Cultural , e.g., statism•Scientific, e.g.,– Economism – “Fix the economy & all will be well”– Logicism – “Only logic is trustworthy”– Chemicism – “Chemical reactions explain everything”

•At all levels, recognizing all aspects as equally valid & important helps avoid idolatry

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Aims of Statistical Inference & Select Aspects Involved

• Given– Interesting scientific hypothesis(es)– Sample data

• Conclude– We can (or should) be C% certain that the hypothesis is true

• Example: 75% certain that Vitamin D deficiency increases risk of rickets

• In PLI terms– C% = a quantitative property– Certain = a fiducial property

(However, “statistics” > “statistical inference”)

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Proposed 21st Century Christian Principles of Statistical Inference

1. From the PLI’s Non-reduction Principle: Humans do not set the standards for what to believe; rather, inference constrains permissible beliefs. 2. From the PLI’s Primacy of Pre-scientific Experience: Quantitative data alone are insufficient to indicate what to believe (& what to do); rather, data, and properties and laws from all the aspects are needed.

A statistical inference paradigm that respects these principles…

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1. Applied scientist forms hypothesis (H) to investigate2. Scientist & statistician state initial strength of beliefs

about H quantitatively (recognizing primacy of pre-scientific. Enhance, not replace.)

3. Statistician a) models Data (x), measures evidenceb) combines evidence mathematically with initial beliefs

(identifying lawful constraints on beliefs)c) derives post-analytic certainties about H

(Flow is being discussed among Christian statisticians)

Proposed Christian Process Flow of Statistical Inference

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Proposed Christian Process Flow of Statistical Inference (con’t)

Synthesis of Priors & Evidence

Fiducial

Specific Sci

Quantitative

Modeled Data (x,

Evidence)

Natural & intuitive? Yes & Yes. However, most statistical inference does not follow this pattern.

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Humanism as Analyzed by PLI

• A religion, a faith (=trust) in humans• Central tenets: Humans are – Self-sufficient– Free from external authority– Able to discern (or even decide!) right/wrong

Examples1. “Medical care? Food? Water? Education?

We believe the answer is: All the above.” 2. “I can follow the rules that I set for

myself.”

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Humanism’s Opposing Poles• Personality Pole– Humans are happiest & most fulfilled without external

constraints– Humanity = Self-expression, emotion, freedom

• Control Pole– Humans demonstrate dominance through mastery of the

universe– Humanity = exercise of scientific (esp. logical &

mathematical) reasoning

• Both poles place complete faith in humans’ abilities

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Standard (Frequentist) Statistical Testing

1. Identify a hypothesis (H) to test2. Fix an arbitrary critical region (C) such that

P(x will fall in C|H) is small [P(x will fall in C|H) = α, “alpha”]

3. Decision:If x does not fall into C, “accept” HIf x falls into C, “reject” H at significance level α

Qn: Once we accept or reject H, how strongly should we believe in H?

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Example: Frequentist Testing• Consumer advocate suspects average mass of “1 kg”

packages of candy < 1 kg. Decides to testH: µ ≥1

• C: {sample mean < 0.83}• Pr(C | H) = 0.05• Assess 20 packages• Sample mean = 0.81, H “rejected”• Qns:

– How strongly should we believe µ<1?– Should we act as if µ<1?

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“Frequentist Inference:” An Oxymoron?

• Inference = statements about hypotheses, given data (inductive, outward-looking)

• Frequentist results = statements about data, given hypotheses (deductive, inward-looking). Examples:– Type 1 error rate = probability of rejecting the hypothesis,

if the hypothesis is true– P-value = probability of results as extreme as those

observed, given the tested hypothesis.

• Frequentist results are silent about strengths of belief

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Humanist Control over Interpreting Frequentist “Inferential” Results

• Humanist “Personality” Pole – “Probabilities about x given H constitute

‘evidence’ that people are free to interpret as they wish”

– Leads to “indirect” frequentism

• Humanist “Control” Pole – “Probabilities about x given H convey direct

meanings about H itself”– Leads to “direct” frequentism

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Indirect Frequentism: Examples

• McClean: “Statistics is about judgment.”• RA Fisher: – “We have the duty of…communicating our conclusions…in

recognition of the right of other free minds to utilize them in making their own decisions.”

– “The deviation [might be] in the direction expected for certain influences which seemed to me not improbable.”

• Motulsky: “What conclusion [from p] should you reach? That’s up to you. Statistical calculations provide [p]. You have to interpret it.”

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Direct Frequentism: Statements to ConsiderGiven a p-value of 0.01, which of the following are true (all, none or some)?1.You have absolutely disproved the null hypothesis (i.e., there is no difference between the population means). 2.You have found the probability of the null hypothesis being true. 3.You have absolutely proved your experimental hypothesis (that there is a difference between the population means).4.You can deduce the probability of the experimental hypothesis being true. 5.You know, if you decide to reject the null hypothesis, the probability that you are making the wrong decision.6.You have a reliable experimental finding in the sense that if, hypothetically, the experiment were repeated a great number of times, you would obtain a significant result on 99% of occasions.Ref: Gigerenzer et al., 2004

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Usefulness of Frequentist Results• How can frequentist results aid the applied sciences?• Under some conditions,

– Frequentist p-value approximates P(H|x)– Frequentist 95% confidence interval approximates an interval

with 95% probability of containing the unknown quantity– …

• Correspondences do not hold when, e.g.,– 2-sided Testing – Strong prior information exists about H

Hence, consider discriminating use of results

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The Missing Link, Supplied• Many Christians attempt to integrate faith &

discipline, to show what obedience to God means in their profession

• These attempts often fail; relations between faith & discipline are not revealed

• Missing link = the philosophical framework• Often, efforts fall short because the philosophy

is not examined & made Biblically consistent

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Summary• PLI = a philosophical framework for

– Analyzing relations between religion & science– Identifying biblically consistent science

• Much of science in Western culture coheres with Humanist philosophies– Personality pole: The individual subject is the law-giver– Control pole: Only logical & quantitative reasoning are trustworthy

• When humanity fell, all of creation fell• Christ calls His disciples to re-claim creation for Himself

Andrew M Hartley, PhDAndrew.Hartley@ppdi.com