Respect your elders: Lykken's (1968) correlated ambient noise: Do fractal scaling and violations of the ergodic condition evidence the crud factor?

Lykken (1968) estimated that the “unrelated” molar variables involved in most studies in psychology share 4-5% common variance, meaning, with 0 measurement error a correlation of about .20 can be expected between any one of them. This really depends on the field of inquiry, but it has been suggested that estimates between .15 and .35 are by no means an exaggeration.

The origins of such correlations are debated (and of course disputed), but I consider them as an example of the violation of the ergodic theorems for studying human behaviour and development (Molenaar & Campbell, 2009; Molenaar, 2008). The ergodic condition applies to systems whose current state in a state/phase space (that describes all the theoretically possible states a system could be in), is very weakly, or not at all influenced by its history, or its initial conditions. Hidden Markov models are an example of such systems. These systems have no "memory" for their initial state and formally this means their time averaged trajectories through phase space are about equal to their space averaged trajectories. Given enough time, they will visit all the regions of the phase space (formally there's a difference between phase and state space, which I will ignore here).

For Psychological Science the ergodic assumptions related to probability theory are important: In an ergodic system it does not matter if you measure a property of the system 100 times as a repeated measurement (time average), or, you measure the property of 100 ergodic systems at the same point in time (space average). The latter is of course the sample of participants from which inferences are drawn social science. The former would be repeated measurements within a single subject. In ergodic system, the averages of these different types of measurement would be the same. It does not matter for the expected averaged result whether you roll 1 die 100 times in a row, or 100 dice in 1 throw.

Trick or treat?

Now, the trick question is, do you think such is the case for psychological variables? Would I get the same developmental trajectory if I measured IQ in a single human being each year from 1 to 80 (assuming I have a 0-error, unbiased, IQ measuring instrument and a very long lifespan) as when I would draw a sample of 80 participants aged 1 through 80 and measured their IQ on one occasion. Very few scientists would predict I would obtain the same results, in both situations, but in social science we do act as if such would be the case. To me, any evidence of a system's future state being influenced by a state at a previous point in time (memory), is a violation of the ergodic condition and basically should indicate to a scientist to stop using central tendency measures and sampling theory to infer knowledge about the properties of this system. If you do not want to go that far, but still feel uncomfortable about my IQ example, you should probably accept that there may be some truth to Lykken's suggestion about a default ambient correlation between variables in social science. Simply put, if you walk like a duck, there is a small base expectancy that you will also talk like a duck. 

Another line of evidence revealing that everything is correlated (over time), or "has memory", is of course ubiquitous fractal scaling in repeated measurements of human physiology and performance (e.g., Kello et al., 2010). If measurements are interdependent rather than independent it does not necessarily point to a violation of the ergodic condition, but combined, the two frameworks do predict very different measurement outcomes in certain contexts (e.g., Diniz, et al., 2011). My money is still on the "long memory" interpretation. 

Based on the lower estimates of Lykken's correlation, the expected difference between any sample-based averages would be about 0.5 standard deviations. The test against a null hypothesis of “no association” is often a test against a “straw man” null hypothesis because it can be known in advance that an assumption of no association at all is false. Therefore, a researcher can maximize his chances to corroborate any weak prediction of association between variables, by making sure a large enough number of data points are collected. You know, those statistical power recommendations you have been hearing about for a while now. A genuine “crud factor” (cf. Meehl, 1990) implies a researcher has a chance of 1 in 4 to evidence an association using a sample size of 100 data points, without even needing a truth-like theory to predict an association or its sign.

Figure 1. A simulation of the effect of sampling from different regions of a population distribution (Npop = 500000) in the presence of a crud factor, a population-level correlation between any two random variables. Each dot represents the number of significant results (p < .05) observed in 100 t-tests for independent groups of the size represented on the x-axis (10 – 100). Two random variables were generated for each population correlation: .1, .2, .3 (columns). One random variable was used to sample data points in the 10th (top row) or 25th (bottom row), and between the 25th and 75th percentile (comparison group). The means concern the aggregated values of the second random variable for each sampled case. The directional hypothesis tested against the null was (M[.25,.75] – M[0,.10]) > 0  or (M[.25,.75] – M[0,.25]) > 0 .

Psychologists need to change the way they theorise about reality

The crud factor, or the violation of the ergodic condition, are not statistical errors that one can resolve by changing the way psychologists analyse their data. It requires adopting a different formalism about measurement of properties of non-ergodic systems, it requires theories that make different kinds of predictions. No worries, such theories already exist and there are social scientists who use them. To encourage others, here are some examples of what can happen if one continues to assume the ergodic condition is valid and use the prediction of signs of associations between variables (or group differences) as the ultimate epistemic tool for inferring scientific knowledge.

Suppose two variables x (e.g., a standardised reading ability test) and y (amount of music training received in childhood) were measured in samples drawn from a population that was cut into regions in order to compare dyslexic readers (the 10^th percentile and lower, and the 25^th and lower on variable x) and average readers (between the 25^th and 75^th percentile on variable x) on variable y. The sample size for each group was varied from 10 to 100 data points and 100 tests were performed for each group size. For each test a new random group sample was drawn.

Figure 1 represents the number of significant (p < .05) t tests found in the series of 100 tests conducted for each group size. If the crud factor were .1, then comparing to the samples from the 10th and 25th percentile would yield 25% significant results at group sizes of 44 and 58 data points respectively. The total study sample size would be 88 and 116. At this crud factor level the chances do not get much better than 1 in 4 corroborative events without there being any theory to pat on the back and grant some verisimilitude. When the correlation is .2, 25% significant tests can be expected at group sizes of 12 (10th) and 23 (25th) and at a correlation of .3 it’s 10 (10th) and 12 (25th) participants in each group to find 25% significant differences. The crud factor of .3 even implies that 100% of the conducted tests could give a significant result if the group size is larger than 87 and the dyslexic group is drawn from the 10th percentile of the population distribution of reading ability.

So, what's the use of a priori sample size calculations again? To get a sample size that will allow you to evidence just about anything you can(not) think of, as long as you limit your predictions to signs of associations (Figure 2). A real treat.

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Figure 2. Same simulations as described in Figure 1, but for a range of crud factors between 0 and 0.4.

References

Diniz, A., Wijnants, M.L., Torre, K., Barreiros, J., Crato, N., Bosman, A.M.T., Hasselman, F., Cox, R.F.A., Van Orden, G.C., & Delignières, D. (2011). Contemporary theories of 1/f noise in motor control. Human movement science, 30(5), 889–905. doi:10.1016/j.humov.2010.07.006

Kello, C. T., Brown, G. D. A., Ferrer-i-Cancho, R., Holden, J. G., Linkenkaer-Hansen, K., Rhodes, T., & Van Orden, G. C. (2010). Scaling laws in cognitive sciences. Trends in Cognitive Sciences, 14(5), 223–232. 

Lykken, D. T. (1968). Statistical significance in psychological research. Psychological bulletin, 70(3), 151–9.

Meehl, P. E. (1990). Why Summaries of Research on Psychological Theories Are Often Uninterpretable. Psychological Reports, 66(1), 195. doi:10.2466/PR0.66.1.195-244

Molenaar, P. C. M. (2008). On the Implications of the Classical Ergodic Theorems : Analysis of Developmental Processes has to Focus on Intra-Individual Variation. Developmental Psychobiology, 50(1), 60–69. doi:10.1002/dev

Molenaar, P. C. M., & Campbell, C. G. (2009). The New Person-Specific Paradigm in Psychology. Current Directions in Psychological Science, 18(2), 112–117. doi:10.1111/j.1467-8721.2009.01619.x

Respect your elders: First, you watch Meehl's videotaped philosophy of psychology lectures - then we'll discuss your "pseudo-intellectual bunch of nothing"

I've never understood how it was possible a reviewer or editor of a scientific journal could write something like: This subject matter is too difficult and complex for our reader audience (to be interested in). I even heard a colleague exclaim once that a mathematical ps. journal found his mathematics too complex. 

That can only mean the audience does not want to get educated on things they do not know about yet, which is strange behaviour for scientists. An editor should invite an author to write a primer possibly as supplementary material, I've seen some examples of that recently, the p-curve article to appear in JEP: General is one of them.

More often than not, psychological theories and their predictions are evaluated for their descriptive value, which means: can the reviewer relate what it is about to his own preferred theories. This should not matter in science, theories should (as long as they do not claim a re-write of well-established theories based on some statistical oddities: Bem, 2011), be evaluated for the precision of the predictions they make, their empirical accuracy and their logical structure.

Problem is, we do not get educated on these matters in psychology. Whether you do or not seems to trivially depend on whether there's a professor at your university who knows about theses things.

(How lucky they were in Minnesota!)

It's plain and simple: If we really want psychology to be taken seriously as a scientific endeavour, we need to discuss it at the level of metatheory, how do we evaluate theories, what is their verisimilitude, their similarities so we can hope to unify them. 

We need to discuss it at the level Paul Meehl discussed it.

Now, his list of publications is long, so are the publications themselves and my list of quotes I would like to paste here is endless and going by the popular journals, our generation of scientists is likely to doze off  by anything longer than 5000 words anyway.

How about some video then? 

Meehl's Philosophical Psychology Seminar is available online as video!

12 lectures of about 1.5 hours and you'll know all you need to know to have a proper discussion about the credibility of the theory you use to study the phenomena you are interested in.

(You do know TED last only 20 minutes or so?),

Ok, get through the first 7 at least (this will not be a difficult task, I even enjoyed to hear him speak about the practicalities of the course)

The videos are here: http://www.psych.umn.edu/meehlvideos.php 

Recommendations of Meehl's work by others:

"After reading Meehl (1967) [and other psychologists] one wonders whether the function of statistical techniques in the social sciences is not primarily to provide a machinery for producing phony corroborations and thereby a semblance of ‘scientific progress’ where, in fact, there is nothing but an increase in pseudo-intellectual garbage." (Lakatos, 1978, pp. 88–9)

Just one quote sums it up for me

Whenever I try to evaluate what someone is claiming about the world based on their data, or "theory" from the perspective of theory-evaluation, they look at me like a dog who has just been shown a card trick. It's so unreal that I cannot use a word like ontology or epistemology or ask about the measurement theory or rules of inference someone used to make a claim about the way the universe works that I considered leaving academia, but I guess leaving without trying to change the world is not how I was raised, or genetically determined,  The quote below summarises how I feel almost exactly:

"I am prepared to argue that a tremendous amount of taxpayer money goes down the drain in research that pseudotests theories in soft psychology and that it would be a material social advance as well as a reduction in what Lakatos has called “intellectual pollution” (Lakatos, 1970, fn. 1 on p. 176) if we would quit engaging in this feckless enterprise. 

I think that if psychologists would face up to the full impact of the above criticisms, something worthwhile would have been achieved in convincing them of it. Besides, before one can motivate many competent people to improve an unsatisfactory cognitive situation by some judicious mixture of more powerful testing strategies and criteria for setting aside complex substantive theory as “not presently testable,” it is necessary to face the fact that the present state of affairs is unsatisfactory. 

My experience has been that most graduate students, and many professors, engage in a mix of defense mechanisms (most predominantly, denial), so that they can proceed as they have in the past with a good scientific conscience. The usual response is to say, in effect, “Well, that Meehl is a clever fellow and he likes to philosophize, fine for him, it’s a free country. But since we are doing all right with the good old tried and true methods of Fisherian statistics and null hypothesis testing, and since journal editors do not seem to have panicked over such thoughts, I will stick to the accepted practices of my trade union and leave Meehl’s worries to the statisticians and philosophers.” 

I cannot strongly fault a 45-year-old professor for adopting this mode of defense, even though I believe it to be intellectually dishonest, because I think that for most faculty in soft psychology the full acceptance of my line of thought would involve a painful realization that one has achieved some notoriety, tenure, economic security and the like by engaging, to speak bluntly, in a bunch of nothing." (Meehl, 1990, emphasis and markup added)

References

Meehl, P. E. (1990). Why Summaries of Research on Psychological Theories Are Often Uninterpretable. Psychological Reports, 66(1), 195. doi:10.2466/PR0.66.1.195-244

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