Causality and causal modeling

General definitional and notational setup

*** Causality is not directly detectable or observable, much like forces or energy aren’t either: it needs to be discovered/uncovered – ‘detected’ through observational consequences.

*** Causality is hardly a statistical matter, and it shows up at the very roots of scientific inquiries: the basic notions of substance and time, e.g. are defined in excessive proximity to  causal relations: e.g. abstract entities (like ‘number 7’) are those that cannot cause anything[i], unlike ‘concrete substances’, and time itself has definitions overlapping with causality [2-6].

*** Causal language[ii] differs drastically by scientific field[iii], and some are more shy than others in using the “C” word. In statistics it has been kept at bay for many decades, and epidemiologists have been to first advocates among statisticians at large (Tyler VanderWeele, e.g., but also Jay Kaufman [8] and [9] in health disparities, and many other nowadays).

*** Even ‘simple’ questions like ‘What causes the universe to expand?’ rely on deeply unobservable constructs, like dark energy.[iv]

*** In medicine, causal questions are broadly aimed at ‘changing the current course of events’ in the future, whereas in astronomy e.g. mere deep descriptive understanding of ‘how things work’ is a goal in itself: finding the laws governing the Universe.[v]

*** The task then is to discover “which variable ‘listens to’ which others“ (BoW p. 7), e.g. whether blood glucose (say measured by hemoglobin A1c) ‘listens to’ one’s body weight (say measured by body mass index, BMI), and responds to it: this metaphor invites deeper delving into how the ‘listening’ works[vi]: is an asteroid ‘listening’ to the Earth’s gravity when heading to it? Does it have a choice not to listen, like biological bodies do? Since biological entities are the first ones to collect information and use it in choosing to respond differently, to benefit themselves, the ‘listening’ appears to make less sense before the first living organisms appeared: all natural bodies and substances before that blindly (and strictly, without choice) followed the tight grip of the natural laws, gravity among them being the central one.[vii]

*** Note that the way BMI relates to A1c is not merely direct and linear, e.g. [47]: FIGURE 1 Dynamic phase of obesity development in the carbohydrate-insulin model[viii].

 

*** Intuitively grasping how causal effects are ‘extracted’ from correlational-kind data, a bit of notational setup will be decisive: for the A1c(BMI) relation of interest, or BMIàA1c, or probability(A1c | BMI), where “|” stands for ‘given’, we want to find out whether lowering BMI would cause a A1c drop, and then by how much. To answer the ‘cause-effect’ research question (RQ), we simply need to compare for each patient i: A1ci If.LowerBMI vs. A1ci If.SameBMI, where superscripts mean ‘it could happen up there’[ix] (vs. what actually happened ‘here on the ground’, in this reality A1ci LoweredBMI and A1ci SameBMI). More details in the footnote, suffice to say that there is a causal effect of BMI on A1c for patient i if: A1ci If.LowerBMI ≠ A1ci If.SameBMI! One of these two quantities however can never be ‘seen’, once reality kicks in: either stays with the same BMI, or lowers their BMI!

***The statistical trickery to still derive a causal effect from such ‘observational’ data (i.e. not from a randomized controlled research design) is to ‘identify’ the average causal effect among those who say lowered their BMI, Average(A1ci If.LowerBMI LoweredBMI – A1ci*If.SameBMI LoweredBMI), in which we added the *, for ‘never visible’: what would have happened to these folks, but… didn’t!

We do this by ‘wiping off’ a bias quantity, the ‘selection bias’[x]. All this math is drastically simplified if one uses Judea Pearl’s notation for ‘setting’ the predictor to a value, so Average(A1c If.LowerBMI) becomes simply Average(A1c | (set (BMI=Lower))[xi], [54, 55], where “set” symbolizes an actual intervention that sets to BMI to a lower value, and E means ‘expected (average) value of’: the challenge of ‘causal inference’ then is ‘simply’ deriving the causal quantity Average(A1c | (set (BMI=Lower)) from the observational one Average(A1c | ‘seeing’ BMI=Lower). Observing BMI=Low is NOT the same as ‘setting’ it to ‘Low’, or ‘intervening’ to ‘do’ that:

Average(A1c | ‘see’ BMI=Lower) ≠ Average(A1c | ‘set’ BMI=Lower)

or Average(A1c | ‘see’ BMI=Lower) ≠ Average(A1c | ‘do’ BMI=Lower)

***Again, statistics can ‘recover’ the causal effects from observational (non-interventional) data when it can turn all ‘set’ (or ‘do’) into ‘see’ in the formulas: then, the math (or software, or AI) can directly yield the actual causal effect: even a simple t-test or a chi-squared test would thus become a ‘causal analysis’.

*** This is a quick overview of several topics that all rely on the simplest possible data setup: a 2by2 binary fields/variables, all using the simple 4-cell setup (0,0), (1,0), (0,1), and (1,1). They make use of the information derived from such a setup completely differently:

***.1. Finding ‘it’ vs. missing ‘it’: the medical sensitivity vs. specificity (& ROC curve – to be added)

The setup here is slightly different than most 2by2 tables(as chi-squared test examples, e.g.), and it has Test(for a disease), on the left, and Disease, on top, and inside:  (1,1) (1,0), first row, then (0,1), (0,0), or the a,b,c, and d, as they are better known (an image is shown). They mean sequentially:

a (1,1) Test is positive, Disease present, so we have (+,+) = Good!

b (1,0),Test is positive, Disease not there, so (+,-) = Bad! first row, then

c (0,1), Test is negative, Disease present, so (-,+) = Bad!

d (0,0), Test is negative, Disease not there, so (-,-) = Good!

Sensitivity reads the 1st column downwards: n(1,1) / [n(1,1) + n(0,1)] or with | meaning ‘given’

P(Test + | Disease +) – this has a more intuitive write-up: percent of all disease+ patients with test+

Specificity reads the 2nd column upwards: n(0,0) / [n(0,0) + n(1,0)] or

P(Test – | Disease -) : percent of all disease- patients with test-

*** Note that it is different to talk about P(Disease + | Test +), than about P(Test + | Disease +), which has a more natural direction; the ‘direction’ differs: we ‘condition on‘ (know) one, or the other.

*** Note that this 2×2 setup is not direction-less[xii], yet here the 1st # talks about the LEFT label (test), second about the TOP label (disease, so (1, 0) means (Test + , Disease -); 2by2 tables read more naturally from left->right and top-down. However, sensitivity and specificity read P(Test | Disease), so the ‘given’ (or ‘knowing’) is the Disease, which ‘happens first. This setup becomes more meaningful when time is incorporated, as in RQ.2. in part 2, which has the before & after of a binary category: here P(Post | Pre) makes sense, but P(Pre | Post) does not.

*** A similar setup is the by-product of a logistic regression analysis, when the predictors of a binary outcome are more/less useful in ‘correctly’ assigning 1’s and 0’s to their category; the ‘classification table’ shows the correct assignments (1,1) & (0,0), and the incorrect ones (1,0), (0,1), and provide an average of percent cases incorrectly classified.

***.2. In formal logic “If Premise – Then Conclusion” or simply “Premise -> Conclusion” follows formal rules, which simply say that the entire statement “Premise -> Conclusion” is false in relation to the 2 ingredients truth value only in 1 circumstance: When “True -> False”: truth cannot lead to falsehood! A  peculiar one however is “False -> True”, this whole statement is ‘True’, (along with the other 2 possibilities “True -> True” and “False -> False”), because, as Claude.ai says “Since the condition is never met, the implication can’t be falsified.”, meaning because we start in an ‘impossible world’, we cannot test its truth value… “If pigs could fly, then anything goes” pretty much…

If we follow the four-cell setup known in the sensitivity/specificity context below (1,1) (1,0), (0,1), (0,0), the ‘first’ is logically the ‘If’ (vertical, from up to down, column), the second the ‘Then’ (horizontal, left to right, row) follow

a (1,1), IF is True, THEN is True too, so we have (+,+) or

b (1,0), IF is True, THEN is False, so (+,-) : this only argument of the 4 is FALSE

c (0,1), IF is False, THEN is True, so (-,+)

d (0,0), IF is False, THEN is False too, so (-,-)

***.3. H0 vs. H1 and what statistics says: rejecting H0 – seeing ‘the effect’ vs. not; unfortunately, the Null Hypothesis language leads to double (and more) negations (fail to reject H0… meaning “we see no effect), so the following emphasizes the ‘Effect seeing’ (or not) phrasing.

If we follow the same four-cell setup known in the sensitivity/specificity context below (1,1) (1,0), (0,1), (0,0), the ‘first’ category is logically ‘there is/not an Effect’ (= ‘H0 (null hypothesis) if true/false’, vertical, from up to down, column), the second the ‘Statistics finds/not an effect’ (= ‘we reject/fail to reject H0 t, horizontal, left to right, row) follow

a: (1,1) or (+ | +): There is an Effect (H0=False), and Statistics finds it (rejects H0 true too); we correctly say ‘There IS an Effect’! This is (1-β) = ‘statistical power’ that we often we need estimated before a study is started: we want some solid/large value here, a good likelihood we will FIND and Effect when it’s there! This is ~= (akin to) the SENSITIVITY of a test, from above!

b: (1,0) or (+ | -): There is NO Effect (H0=True), and Statistics finds it (rejects H0 true); = we make a Type I error: “True null hypothesis is incorrectly rejected”; Type I because it is most serious, we ‘find something that doesn’t exist’ (i.e. ‘make up stuff’)! This is α, the customary .05 (5%) threshold value: we set this upfront, to as small an uncertainty value as we are willing to tolerate.

c: (0,1) or (- | +): There is an Effect (H0=False), BUT Statistics does NOT find it (‘we fail to reject’ H0); we make a make a Type II error: “true null hypothesis is incorrectly rejected” This is β.

d: (0,0) or (- | -): There is NO Effect (H0=True), and Statistics does NOT find it either: we correctly say ‘There if NO effect’! This is (1-α).

***It is advisable to set up expectations based on prior knowledge in the form of a Hypothesis: Overweight -> Diabetes, (a causal statement), or a RQ: “Overweight -> Diabetes?”. The hypothesis testing procedure is a flexibly ‘sharp’ decision process, can be e.g. 2-sigma’, rejecting H0 based on the ‘2 standard errors away’ criterion (‘p < .05’), common in medicine and health research, or more conservatively based on the 5-sigma rule in physics (‘p < .00006’): the  = .05 rule is a ‘rule of thumb’, but thumbs differ widely in size…

 ***I will demonstrate in part 2 several statistical tests and analyses, with an emphasis on the mechanics of each, the ‘how to’, all of them done ‘by hand’ and in Excel alone (no ‘sophisticated’ statistical software needed).

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Footnotes:

[i] “I shall understand by a ‘substance’ a particular thing, capable of causing or being caused, such as Richard Swinburne, or a particular table, or a particular electron, or the planet Venus.” [1] Drive

[ii] “Despite heroic efforts by the geneticist Sewall Wright (1889–1988), causal vocabulary was virtually prohibited for more than half a century. And when you prohibit speech, you prohibit thought and stifle principles, methods, and tools” Pearl’s BoW p. 5.

[iii] What ‘scientific research’ is requires a long detour; a simpler approach is to delineate from ‘non-scientific’ research, which some deem to be the ‘farthest away’, e.g. theology, philosophy or knowledge through art [7] (it is rather paradoxical to consider philosophy non-scientific, as long as it provides all sciences the main argumentation tool, formal logic).

[iv] And this only talks about the ‘observable’ Universe, which is to the entire Universe like an atom is to the observable Universe…

[v] The graphical causal models in medicine are gaining ground: [10] [11]   [12] [12-33]

[11, 27, 34-45]

[vi] “The calculus of causation consists of two languages: causal diagrams, to express what we know, and a symbolic language, resembling algebra, to express what we want to know. The causal diagrams are simply dot-and-arrow pictures that summarize our existing scientific knowledge. The dots represent quantities of interest, called “variables,” and the arrows represent known or suspected causal relationships between those variables—namely, which variable “listens” to which others. “ BoW [46], p. 7

[vii] Listening from a communication (science) standpoint is also an active and deliberate process, not a mere ‘reception of data’, but a meaning-seeking activity. CITE???

[viii] “The relation of energy intake and expenditure to obesity is congruent with the conventional model. However, these components of energy balance are proximate, not root, causes of weight gain. In the compensatory phase (not depicted), insulin resistance increases, and weight gain slows, as circulating fuel concentration rises. (Circulating fuels, as measured in blood, are a proxy for fuel sensing and substrate oxidation in key organs.) Other hormones with effects on adipocytes include sex steroids and cortisol. Fructose may promote hepatic de novo lipogenesis and affect intestinal function, among other actions, through mechanisms independent of, and synergistic with, glucose. Solid red arrows indicate sequential steps in the central causal pathway; associated numbers indicate testable hypotheses as considered in the text. Interrupted red arrows and associated numbers indicate testable hypotheses comprising multiple causal steps. Black arrows indicate other relations. ANS, autonomic nervous  system; GIP, glucose-dependent insulinotropic peptide.” Medical diagnosing has been the focus or early ‘decision science’ works, see the ‘Causal Understanding’ [48] figure e.g.

[ix] This notation was also used by [49] and by [50] p. 244, and by Lok [51]; Reichenbach also used a close-by symbol : “where the events that show a slight variation [from E] are designated by E*:” p.137 [52]

[x] The process is more detailed: we subtract and add one same term to the causal effect, and then ‘take expectations’, i.e. averages the quantities across patients:

E(CausalEffecti) = E (A1ciIf.LowerBMI LoweredBMI – A1ciIf.SameBMI Same.BMI) =

E (A1ciIf.LowerBMI LoweredBMI – A1c*iIf.SameBMI LoweredBMI ) + E (A1c*iIf.SameBMI LoweredBMI – A1ciIf.SameBMI Same.BMI) =

ATT                                      +                            Selection.Bias

Where ATT  = the average treatment effect on the treated: how much the A1c would … not drop among those who did lower their BMI, if… they didn’t! Note, Eric Brunner from UConn stated it using econometric language and labels (𝐷 is Treatment) : “We suspect we won’t be able to learn about the causal effect of college education simply by comparing the average levels of earnings by education status because of selection bias:

𝐸[𝑌𝑖|𝐷𝑖=1]−𝐸[𝑌𝑖|𝐷𝑖=0]= 𝐸[𝑌1𝑖|𝐷𝑖=1]−𝐸[𝑌0𝑖|𝐷𝑖=1] (ATT)

+𝐸[𝑌0𝑖|𝐷𝑖=1]−𝐸[𝑌0𝑖|𝐷𝑖=0] (Selection bias). We suspect that potential outcomes under non-college status are better for those that went to college than for those that did not; i.e. there is positive selection bias”  in his ‘Causal Evaluation’ graduate course handout]

[xi] We use Average for better intuition; mathematically inclined analysts use instead E(A1c), for ‘expectation’, which is the proper way to define a ‘typical patient’s value’, or the mean of a variable. For a variable X with discrete values, the mean is μX = E(X) = Σi [xi·p(xi)], while for one with continuous values μX = E(X) =    ([53], p. 152).

[xii] The confounding of ‘causality’ and ‘time’ is a known definitional problem: e.g. VanFraasen cites Reichenbach defining time in terms of causality ([56], p, 190), and many define time on the basis of the time construct.

*** Reichenbach himself said e.g. “we must now make sure that our definition of “later than”

does not involve circular reasoning. Can we actually recognize what is a cause and what is an effect without knowing their temporal order?

Should we not argue, rather, that of two causally connected events the effect is the later one?

This objection proceeds from the assumption that causality indicates a connection between two events, but does not assign a direction to them. This assumption, however, is erroneous. Causality establishes not a symmetrical but an asymmetrical relation between two events.

If we represent the cause-effect relation by the symbol C, the two cases C(E1, E2) and C(E2, E1) can be distinguished; experience tells us which of the two cases actually occurs. We can state this distinction as follows:

If E1 is the cause of E2. then a small variation (a mark)” in E1 is associated with a small variation in E2. whereas small variations in E2 are not associated with variations in E1.” [52] P. 136