Combinatorics Summary....

 

 

I realized that I have a combinatorics thread running through my blog across several subjects.  I have been interested in combinatorics since I sent an email to Robert Spitzer on the various combinations of diagnostic criteria.  His only comment was “Interesting”.  Since then, I have commented on a post that purported to discredit psychiatric diagnoses based on combination of diagnostic criteria (too many), a study of the real combinations of major depression diagnoses, and character and word phrase combinations for encryption and password protection.  I went as far as getting dice and using them to construct passphrases of varying length using the Electronic Frontier Foundation (EFF) word list for that purpose.

If you have no experience with combinations or it has been a long time since your college statistics course – dice are a good place to start.  Each die has 6 sides with corresponding numbers. The total combinations possible are 6ⁿ, where n = the number of dice rolled at once.  The EFF world list is 6,667 word long and that happens to be 6^5.  So, to generate passphrases – 5 dice are rolled and the corresponding number is looked up on the word list and recorded.  The process is repeated until the desired phrase length is generated.  The only downside to this method is that some sites still insist on additional numbers and special characters.  They can still be inserted in the passphrase, but other systems like hexadecimal may be more convenient.  The advantage to passphrases is that they are theoretically easier to memorize and type without error.  That breaks down with very long phrases.

In biology and medicine, combinatorics can be applied at several levels. Some have more meaning than others.  On this blog, I responded to a paper suggesting that the possible combinations of diagnostic criteria meant that psychiatric diagnoses were meaningless and unscientific.  The lesson from this post is to have an idea of what you are counting and what it means. The total combinations of verbal criteria depend a lot on the phrasing and the total number of criteria whether large or small is not necessarily disqualifying as illustrated in this post.  The combinatorial upper limit can be unrealistically large based on how it is defined and just running the numbers does not mean that all possible combinations will be found.  There also seems to be some magical thinking involved – just because you count something does not say anything about what that means.  It is quite literally an exercise in the map is not the territory. 

I looked at a second paper where the authors looked at a lower number of combinations based on the DSM diagnostic criteria for major depression.  In that case the total number of diagnoses was much lower at 227 combinations.  The authors of that second paper did standardized interviews on 3,800 people and of the 1,566 with major depression – just 10 of those combinations accounted for 50% of the cases.  About ¼ of the possible combinations (57/227) did not occur in any group.  This paper is a stark reminder that just counting things in biology or medicine doesn’t necessarily mean anything.

That brings me to the concept of how we make sense out of the most valid combinatorial explosions in medicine. For me validity is baked into the biology and not a verbal description of things.  The backing for that comes from biological taxonomy and the fact that molecular biology and genomics is solving problems that could not be solved by the verbal description of direct observations in the Linnean tradition.  To that end I am reproducing a table below that is all about the polygenic risk for bipolar disorder. 

Note that in this table the authors are estimating the total possible combinations of 803 polygenes. The theoretical number of possible combinations can be calculated using the formula n! / r!((nr)!,where n represents the number of genetic variants analyzed in a study, and r represents the number of genetic variants per combination. In the case of  SNP genotypes,3^r.the formula is n! / r!(n-r)! ×3^r.  The authors point out that the lowest value for r is 2 but the upper limit is unknown.  They also show how the number of combinations can be limited experimentally.  Of the 57,911,211 combinations found only in patients and not controls they could all be random but there were a significant number of SNPs associated with different groupings in bipolar disorder.    

Using the equations from above in a more readable graphic form:

 

 

Substitution yields the following:

- from the top equation, for 100 variants the theoretical 10-variant combinations would be 1.73 x 10¹³

- from the bottom equation, for 500,000 SNPs analyzed there would be 2.3 × 10¹² two-variant combinations and 3.4 × 10¹⁸ three variant combinations.

The application of practical measure includes scanning SNPs for varying combination lengths in the population of interest relative to controls. At lower numbers those combinations can be taken out scanning for longer combinations. A further simplification is to scan only for combinations found in patient populations.  An example of that study is included in the tables below for 803 SNPs in 607 bipolar disorder patients and  1,354 controls. 

Cluster and subgroup analysis is required in very heterogeneous conditions to analyze clusters containing a specific SNP, the distribution of SNP genotypes relative to controls, and cluster selection that contains an SNP for a specific biological function.  Using this kind of analysis 73/609 bipolar disorder patients had these clusters compared to none in the control population. 

While the SNP and variant analysis in 2017 is a good example of combinatoric applications – it did not address the problem of missing heritability.  Missing heritability is the difference between what is observed in familial heritability studies and what is predicted with genetic analysis.  Looking at the predictions from SNP based analysis only a low percentage of familial inheritance was predicted.  That improved with more sensitive analytical techniques that considered additional genetic mechanisms.  The additional mechanisms included SNV (single nucleotide variation), insertions or deletions (indels), SVs (structural variations), CNV (copy number variations), and STR (short tandem repeat (3-5).  Applications that identify all these variations are much more likely to predict the heritability of the pedigree than earlier techniques.  I hope to revisit some of these genetic innovations in an upcoming post about the DSM-6 proposals.

 

George Dawson, MD, DFAPA 

 

References:

1:  Mellerup E, Møller GL. Combinations of Genetic Variants Occurring Exclusively in Patients. Comput Struct Biotechnol J. 2017 Mar 10;15:286-289. doi: 10.1016/j.csbj.2017.03.001. PMID: 28377798; PMCID: PMC5367802.

2:  Koefoed P, Andreassen OA, Bennike B, Dam H, Djurovic S, Hansen T, Jorgensen MB, Kessing LV, Melle I, Møller GL, Mors O, Werge T, Mellerup E. Combinations of SNPs related to signal transduction in bipolar disorder. PLoS One. 2011;6(8):e23812. doi: 10.1371/journal.pone.0023812. Epub 2011 Aug 29. PMID: 21897858; PMCID: PMC3163586.

3:  Behera S, Catreux S, Rossi M, Truong S, Huang Z, Ruehle M, Visvanath A, Parnaby G, Roddey C, Onuchic V, Finocchio A, Cameron DL, English A, Mehtalia S, Han J, Mehio R, Sedlazeck FJ. Comprehensive genome analysis and variant detection at scale using DRAGEN. Nat Biotechnol. 2025 Jul;43(7):1177-1191. doi: 10.1038/s41587-024-02382-1. Epub 2024 Oct 25. PMID: 39455800; PMCID: PMC12022141.

4:  Wainschtein P, Zhang Y, Schwartzentruber J, Kassam I, Sidorenko J, Fiziev PP, Wang H, McRae J, Border R, Zaitlen N, Sankararaman S, Goddard ME, Zeng J, Visscher PM, Farh KK, Yengo L. Estimation and mapping of the missing heritability of human phenotypes. Nature. 2026 Jan;649(8099):1219-1227. doi: 10.1038/s41586-025-09720-6. Epub 2025 Nov 12. PMID: 41225014; PMCID: PMC12851931.

5:  Grotzinger AD, Werme J, Peyrot WJ, Frei O, de Leeuw C, Bicks LK, Guo Q, Margolis MP, Coombes BJ, Batzler A, Pazdernik V, Biernacka JM, Andreassen OA, Anttila V, Børglum AD, Breen G, Cai N, Demontis D, Edenberg HJ, Faraone SV, Franke B, Gandal MJ, Gelernter J, Hatoum AS, Hettema JM, Johnson EC, Jonas KG, Knowles JA, Koenen KC, Maihofer AX, Mallard TT, Mattheisen M, Mitchell KS, Neale BM, Nievergelt CM, Nurnberger JI, O'Connell KS, Peterson RE, Robinson EB, Sanchez-Roige SS, Santangelo SL, Scharf JM, Stefansson H, Stefansson K, Stein MB, Strom NI, Thornton LM, Tucker-Drob EM, Verhulst B, Waldman ID, Walters GB, Wray NR, Yu D; Anxiety Disorders Working Group of the Psychiatric Genomics Consortium; Attention-Deficit/Hyperactivity Disorder (ADHD) Working Group of the Psychiatric Genomics Consortium; Autism Spectrum Disorders Working Group of the Psychiatric Genomics Consortium; Bipolar Disorder Working Group of the Psychiatric Genomics Consortium; Eating Disorders Working Group of the Psychiatric Genomics Consortium; Major Depressive Disorder Working Group of the Psychiatric Genomics Consortium; Nicotine Dependence GenOmics (iNDiGO) Consortium; Obsessive-Compulsive Disorder and Tourette Syndrome Working Group of the Psychiatric Genomics Consortium; Post-Traumatic Stress Disorder Working Group of the Psychiatric Genomics Consortium; Schizophrenia Working Group of the Psychiatric Genomics Consortium; Substance Use Disorders Working Group of the Psychiatric Genomics Consortium; Lee PH, Kendler KS, Smoller JW. Mapping the genetic landscape across 14 psychiatric disorders. Nature. 2026 Jan;649(8096):406-415. doi: 10.1038/s41586-025-09820-3. Epub 2025 Dec 10. PMID: 41372416; PMCID: PMC12779569.

 

Graphics Credit:

Table 1 is reused from open access reference 1 above per  CC BY license (http://creativecommons.org/licenses/by/4.0/). 

 

Recent Opinion About Diagnostic Heterogeneity – Gets It Wrong

There was an opinion piece about categorical diagnosis in psychiatry and diagnostic heterogeneity that was published in Psychiatric Research weeks ago (1), that generated a lot of controversy.  The controversy started when an online publication characterized the article as showing that Psychiatric Diagnoses Found to Be "Scientifically Meaningless".  The author of that article subsequently posted that the article was written by science undergraduates re-purposed as science writers.  If this was supposed to be investigative journalism it failed at several levels not the least of which is the apparent conflict of interest by the authors. Instead the internet article basically quotes the authors as factual and scientific rather than a rhetorical opinion piece.  What follows is my take on the Psychiatric Research Article.

The first sign of bias that a reader may encounter in the original article is right in the abstract. The concluding sentence reads:

“A pragmatic approach to psychiatric assessment, allowing for recognition of individual experience, may therefore be a more effective way of understanding distress than maintaining commitment to a disingenuous categorical system.” (my emphasis added).

When I read this sentence, it was difficult for me to believe that peer reviewers for a psychiatric journal could allow it to pass. In one sentence the authors are allowed to distort and discredit psychiatric clinical methods and diagnostic methods that have been carefully developed for over a century.  I won’t belabor the definition of “disingenuous” but it is safe to say that the expenditures in terms of brainpower and money as well as the transparency of the process make the production of the DSM 5 one of the more rigorous approaches to a diagnostic system in medicine. The people sitting on the DSM 5 committees for each section were acknowledged experts in their fields with decades of experience and published research.  Production of the DSM-5 was also a multiyear process that took 14 years to develop prior to its publication in 2015 (2).  During that time there was a multiyear grant that sponsored 13 international conferences on specific diagnostic issues.  Guiding principles and conceptual issues were examined.  Public input was solicited. Hundreds of clinicians and researchers were involved.  There was transparency about potential conflicts of interest. It was not just an intense effort – it was a unique diagnostic effort in terms of overall vigor and resource utilization.   Describing the output of all of this work as “disingenuous” and getting that in print lead me to question the peer review and editorial process.  Are the editors and reviewers ignorant of the effort that went into the diagnostic categories or don’t they care? It is clear that the authors of this article don’t.

The second red flag in the paper to anyone familiar with typical antipsychiatry arguments is the mention of Foucault and the suggestion that psychiatric classification occurs within wider sociocultural developments and that these roots have resulted in diagnostic heterogeneity.  In fact, Foucault’s observations of psychiatry were inaccurate at the time and have not held up at all over the course of time. The authors seem to ignore the actual reasons for categorical diagnosis in the first place and list none of those references.  Practically all modern DSM work can be traced back to the reference generally referred to as the Feighner criteria (3).  Reading those papers, clearly describes categorical diagnosis as a work in progress and the importance of diagnosis. The authors also describe five phases for the validation of psychiatric diagnoses.  They have this comment on diagnostic heterogeneity:

“In the absence of known etiology or pathogenesis, which is true of the more common psychiatric disorders, marked differences in outcome, such as between complete recovery and chronic illness, suggests that the group is not homogeneous. This latter point is not as compelling in suggesting diagnostic heterogeneity as is the finding of a change in diagnosis. The same illness may have variable prognosis, but until we know more about the fundamental nature of common psychiatric illnesses, marked differences in outcome should be regarded as a challenge to the validity of the original diagnosis.” p 57.

These authors suggested 5 phases to establish the diagnostic validity of psychiatric illness including the clinical description, laboratory studies, delimitation from other disorders, follow-up studies, and family studies.  There are entire texts dedicated to some of these markers on epidemiology and family studies.  One of the mandates of the DSM-5 committees was to review all of this data and compile it into the most clinically useful form.  In the interim they happened to pare the total number of diagnoses from a maximum of 297 in DSM-IV to 157 in DSM-5 (see reference 2, p xxiii).  This is the basis of categorical diagnosis – not the narrative of a philosopher.

Contrary to the idea that the current authors and the like-minded authors they have referenced have discovered diagnostic heterogeneity it has been widely acknowledged from the outset and by all current psychiatrists. There are no surprises here especially for people trained as physicians. Practically every complex biological illness is heterogeneous with heterogeneous outcomes as well as polygenic etiologies.  Their Foucauldian criticism also ignores the fact that the Washington University group was based on empirical research as opposed to the psychoanalytic process of the day.
 

The example of the empirical approach is illustrated by tracing the development of Major Depression criteria from 1950 to 1980. In fact, many in that group were highly skeptical of psychoanalysis as a possible diagnostic process at all. As they started to publish research article, one of their original articles was highly edited by a psychoanalyst/editor to remove any reference to the term diagnosis. 

The second acknowledged aspect of psychiatric diagnosis and treatment that is given short shrift by the authors is the issue the value of both diagnosis and formulation or as Kendler, et al discuss:

“However, neither we nor, we think, the developers of the criteria would claim that assessing operationalized diagnostic criteria is all there is to a good psychiatric evaluation. While critical, a diagnosis does not reflect everything we want to know about a patient. Our diagnostic criteria, however detailed, never contain all the important features of psychiatric illness that we should care about.” (see reference 4 p. 141.)

The authors’ research method is an exercise in subjectivity.  They basically read five chapters in the DSM 5 (schizophrenia spectrum and other psychotic disorders, anxiety disorders, bipolar and related disorders, trauma and stressor related disorders, and anxiety disorders) and use a technique called “thematic analysis” “to code themes or patterns of meaning across diagnostic categories being analyzed, with a particular focus on the heterogeneity or differences across types of diagnostic criteria.”  You don’t need an advanced research seminar to figure out what is wrong with that picture. Here is a group of psychologists several of whom make a career out of criticizing psychiatry and who are building a case that psychiatric diagnoses are inferior to their own vague diagnostic system using a qualitative technique that even their reference (5) refers to as having “no particular kudos as an analytic method – this, we argue, stems from the very fact that it is poorly demarcated and claimed, yet widely used”.  What outcome would any objective observer expect?

The combinatorics argument:

The authors make it seem like large combinations of diagnostic features mean categorical diagnoses are problematic.  Although they don’t say it explicitly - referring to more diagnoses greater than the number of stars in the solar system - suggests improbability.  But do large combinations of number preclude reasonable human use?  A chess board for example has an 8 x 8 square configuration and by some estimates - 10¹³⁷ moves are possible.  And yet players at all levels seem to be able to negotiate a chess board and determine win, lose or draw.  Master players can develop strategies that make them more likely to win.  Is there similar evidence that diagnoses with large combinations can be managed the same way?  What follows is a mixed table of a psychiatric diagnosis (PTSD) that yields a large number of combinations of diagnostic criteria on the left, a dimensional scale for depression (DEP) from a standard psychological test (MMPI), two different criteria for systemic lupus erythematosus (SLE), and criteria for asthma. Qualifiers for each column are listed at the bottom.

Disorder	PTSD (1)	MMPI-DEP (2)	SLE (ACR) (3)	SLE (SLICC) (4)	Asthma (5)
Criteria	Presence of 1 (or more) of the following symptoms: 1. 2. 3. 4. 5. One or both of the following symptoms: 1. 2. Two (or more) of the following: 1. 2. 3. 4. 5. 6. 7. Two (or more) of the following: 1. 2. 3. 4. 5. 6.	15/26 items	4 of 11 criteria: 1. 2. 3. 4. 5. 6. 7.   A or B 8.   A or B 9.   A or B 10. A or B or C or D 11. A or B or C or D or E	4 of 17 criteria including at least 1 clinical criterion and 1 immunologic criterion; or biopsy proven lupus nephritis: 1.   A or B 2.   A or B 3.   4.   A or B 5.   A or B 6.   A or B 7.   A or B 8.   A or B 9.   10. A or B 11. 12. 13. 14. 15. 16. A or B 17.	1. A or B or C or D 2. A1 or A2 or A3  or B or C
Minimal Combinations	3,150	7.726160e6	330	2,380	36
Total Possible Combinations	636,120	7.726160e6 + 5.311735e6 + 3.124550e6 + 1.562275e6 + 657800 + 230230 + 65780 + 14950 + 2600 + 325	12,555	321,489	46

Footnotes:

1.  This column is from the reference: Galatzer-Levy, I.R., Bryant, R.A., 2013. 636,120 Ways to have posttraumatic stress disorder. Perspect. Psychol. Sci. 8, 651–662.

2.  I have several opinions from different psychologists on the current use of this MMPI scale and the raw cut-off scores. I understand that there are different raw scores for men and women. I can recalculate this scale based on any numbers that may be deemed more reliable. Just email them to me along with the evidence.

3.  American College of Rheumatology (ACR) classification criteria for Systemic Lupus Erythematosus

4.  Systemic Lupus International Collaborating Clinics (SLICC) proposed revised classification criteria for Systemic Lupus Erythematosus

5.  There are numerous endophenotyping classifications for asthma.  It is clear at this point there is no comprehensive system of clinical classification.

What can be observed from this table?   

Apart from waxing poetically they seem to not recognize that common psychological approaches scale to an even larger extent – much greater than 10¹⁸. I have also demonstrated that the way diagnostic criteria are worded makes a big difference in counting word combinations.  Just using the DSM phrasing “or more” greatly increases the number of combinations.  Criteria designed like the SLE criteria as a series of “A or B” choices that greatly reduce the number of possible combinations.  On the other hand dimensional criteria like a single scale from a popular psychological test – greatly increases the number of possible combinations because that scale is a many n and many k.  Using a 15/26 item scale results in 10⁷ combinations.  Using that as a ball park estimate for the other clinical scales results in numbers far larger than used by the authors to criticize categorical diagnosis.  The other aspect of this table is that less combinations is not necessarily better. With asthma for example, these numbers are based on very basic diagnostic criteria.  There are at least 2 other 6 item endophenotype systems and an additional cough variant asthma, but currently experts in the field have not developed a way to incorporate that level of clinical complexity into diagnostic criteria that would be useful to clinicians. Low number of combinations of diagnoses criteria are not necessarily better than higher numbers – especially when the disease complexity is not captured.  

The second issue with combinatorics is that they are not predictive of anything. Great strides in treating post-traumatic stress disorder have occurred in the past 30 years using criteria with a high number of combinations.  That obviously does not preclude patient selection or monitoring in clinical trials of either psychotherapy or pharmacotherapy. It does not prevent the successful diagnosis and treatment of patients in clinical settings in many cases where severe and potentially fatal psychiatric illness exists.  As an example, delirious mania had a fatality rate of 75% in 1849 in the United States (7). That number has fallen to zero with psychiatric treatment based on categorial diagnosis and the clinical training of psychiatrists to recognize severe illness. Many of those improvements have occurred in the past 30-50 years. 

  

In the authors selection strategy, large sections of the DSM 5 that clearly disprove the author’s contentions are omitted. The elimination of Neurocognitive Disorders, Sleep-Wake Disorders, and Substance Related and Addictive Disorders for example also eliminates biological markers and autopsy validation of criteria of diagnoses.  Table 1 (p. 482 of DSM-5) contains 127 discrete categorical diagnoses across 10 categories of substances. 

But the larger misunderstanding here is that what the authors disparage as heterogeneity is an expected part of medicine. Every physician knows that no two patients with asthma, benign prostatic hypertrophy, or gout are the same. There are a collection of illness features with some overlap but no truly homogeneous categories – even in clinical trials that attempt to minimize it. Biological systems especially the brain are designed to scale in various ways including based on combinatorics of various biological elements.  The author’s use of the term quadrillion, happens to be the estimated number of synapses in the brain but that is just a starting point of how systems in the human brain can scale.  The endothelial system in the human body has more cells than the brain and massive heterogeneity that allows for regulation of the vascular beds the human body. The hematopoietic and immune systems have similar levels of scaling that could also result in very large number of combinations. In none of these cases do the number of combinations of cell types, connections, tissue behavior, or descriptions preclude diagnoses, research or treatment.  A very small sample of this heterogeneity is suggested by the table below.  

Heterogeneity In Normal Functioning And Disease States In Human Biology (very partial list)

Endothelial cells	Diabetic nephropathy
Hematopoietic Stem Cells	Hepatitis C virus
Neuroendocrine Neoplasms	Ischemic Stroke
Leukemia - Clonal and Intraclonal cell types	Prostate Cancer
Aphasia syndromes	Mitochondrial Myopathies
Atrial Fibrillation Syndromes	Asthma
Immunodeficiency syndromes	Coriticobasal Degeneration
Diabetes Mellitus Type I and II	Viral Syndromes
Congestive Heart Failure	Cryptospridium genus and species

The authors ignore clinical heterogeneity that physicians have to address in their patients every day.  Very few physicians see clinical trials subjects as patients requesting assistance. That means comorbid physical illnesses, variations in patient tolerance of medical and psychological interventions, pharmacokinetic and pharmacodynamic factors, heart disease, liver disease, renal disease, substance use disorders, traumatic brain injuries, old age, pediatric age, suicide risk, aggression risk, impaired functional capacity, and even pregnancy have to be addressed in patients being seen every day by psychiatrists and adjustments have to be made. Only physicians schooled in heterogeneity would be able to treat those people.  Only physicians schooled in heterogeneity would realize that the people in clinical trials are rarely the people being seen in the office.  

In conclusion, the authors have a poor understanding of diagnostic heterogeneity and why it is a central part of medicine.  Some of their arguments are similar to arguments offered up by the critics of Kraepelin in the early 20^th century.  Other arguments - like the combinatorial ones reflect a poor understanding of biological systems and how they scale as well as a lack of understanding of medicine. Physicians know for example that diagnostic models are not completely explanatory, that over time - the explanations change, but that science exists at some level of that explanation or treatment. That is the nature of biological as opposed to physical systems. Anyone interested in these issues can find a rich literature out there that describes these problems and even the involved philosophy. Unfortunately, only one of the authors referenced (out of 28) is written by anyone authoritative in that area.

The only disappointment greater than an article like this being published is the fact that it was published in the journal Psychiatric Research.  It has little to do with psychiatry or research and it is shocking that the obvious problems with article were overlooked. On the other hand, this journal was never at the top of my reading list and this may be why.

George Dawson, MD, DFAPA

References:

1: Allsopp K, Read J, Corcoran R, Kinderman P. Heterogeneity in psychiatric diagnostic classification. Psychiatry Res. 2019 Sep;279:15-22. doi: 10.1016/j.psychres.2019.07.005. Epub 2019 Jul 2. PubMed PMID: 31279246.

2:  Black DW, Grant JE.  DSM-5 Guidebook. American Psychiatric Publishing, Arlington, VA: pp 543.

3: Feighner JP, Robins E, Guze SB, Woodruff RA Jr, Winokur G, Munoz R. Diagnostic criteria for use in psychiatric research. Arch Gen Psychiatry. 1972 Jan;26(1):57-63. PubMed PMID: 5009428.

4: Kendler KS, Muñoz RA, Murphy G. The development of the Feighner criteria: a historical perspective. Am J Psychiatry. 2010 Feb;167(2):134-42. doi: 10.1176/appi.ajp.2009.09081155. Epub 2009 Dec 15. PubMed PMID: 20008944.

5: Braun, V., Clarke, V., 2006. Using thematic analysis in psychology. Qual. Res. Psychol. 3, 77–101. https://doi.org/10.1191/1478088706qp063oa.

6: Kendler KS, Engstrom EJ. Criticisms of Kraepelin's Psychiatric Nosology: 1896-1927. Am J Psychiatry. 2018 Apr 1;175(4):316-326. doi: 10.1176/appi.ajp.2017.17070730. Epub 2017 Dec 15. PubMed PMID: 29241358.

7: Bell, L., 1849. On a form of disease resembling some advanced stage of mania and fever. Am. J. Insanity 6, 97–127.