Combinatorics Revisited…

I critiqued a paper that purported to show that Diagnostic and Statistical Manual criteria produce an impossibly large number of possible combinations and that this somehow invalidates their use.  As a refresher, combinations are basically any pool of n elements combined k at a time.  For example, in the case of major depression, the diagnosis requires at least 5 (k ≥ 5) of 9 (n=9) elements.  That would lead to a calculation of C(n,k)  = C(9,5) + C(9,6) + C(9,7) + C(9,8) + C(9,9) or  126 + 84 +36 + 8 + 1 = 255     I have illustrated the total combinations for the first expression at the top of this post.  In each case the elements 1 – 9 are the DSM diagnostic criteria for depression.  Note that adhering to the diagnostic criteria eliminates the last column of combinations to the far right since elements 1, 2, or 1 and 2 are required for the diagnosis. 

Reading the actual diagnostic criteria illustrates that this is a crude measure because there are implicit unknowns – most significantly the total number of medical unknowns suggested by the criteria “The episode is not attributable to the physiological effects of a substance or to another medical disorder.” Historically reviews of those disorders suggest that they are in the 200 to 300 range with some being far more common than others. If all those conditions were included in the combinatoric expression it would be very large – but not necessarily that much more inclusive because of the low frequency of many conditions.  Additional exclusion criteria include psychiatric disorders with depressed mood as a feature and any previous episodes of mania.  Since they are exclusion criteria – it is reasonable to say that there may be only 255 combinations of rule in symptoms, but being able to make the calculation is no assurance that they exist in practice.  

Following the authors assumption about the combinatoric possibilities we can substitute short had for criteria 1-9.  In the following manner (as noted in their Table 1):

1. Depressed Mood

2. Loss of Interest or Pleasure

3. Appetite/Weight Disturbance

4. Sleep Disturbance

5. Psychomotor Change

6. Loss of Energy

7. Worthlessness/Excessive Guilt

8. Concentration/Indecision

9. Death/Suicidal Thoughts

A further restriction is included in criteria A: “…at least one of the symptoms is either (1) depressed mood or (2) loss of interest or pleasure.”  That eliminates any combination that does not include 1, 2, or 1 and 2.  That changes the above expression to 105 + 77 + 35 + 9 + 1 = 227 possible combinations just based on the numbers.  The authors were interested in seeing how many of these possible combinations exist in the clinic and that was the goal of this paper.   

The sample for the paper was 1,566 subjects with a diagnosis of major depression out of a total sample of 3,800 evaluations.  All subjects were being seen on a clinical basis and the Structured Clinical Interview for DSM-IV (SCID) was administered by trained examiners and the interrater reliability was sampled and posted for all of the depressive symptoms.  The number of subjects in each group of combinations was determined and the results were interesting.

For starters – 57 of the 227 combinations or about 1/4 did not occur in a single patient. In the case of 5, 6, and 7 criteria the combinations that did not occur are listed in tables 3, 4, and 5.  The most common combination was all nine criteria and that occurred in 10% of the sample (N=157).  The authors were able to observe that 9 combinations from the 9,8,7, and 6 criteria categories accounted for 40% of all diagnoses. They suggest that these might be prototypical combinations in a field of diagnostic heterogeneity. Apart from diagnostic prototypes the authors suggest that it may facilitate the search for biological markers but they conceded that those would nee to be very large and expensive studies. 

As I thought about that proposition, a few things came to mind.  First, Mayo Clinic multi-omics studies. Some of these studies have already identified biomarkers and possible genetic markers on heterogenous groups of subjects with major depression.  The subjects were all administered standardized DSM based interviews and the combinatorics could be determined.  This would be an efficient way to see if symptom combinatorics match the biomarkers.  Second, why would we expect there to be any correlation between symptoms and biomarkers?  Most medical illnesses would not have a  correlation and in fact the more complex illness can be expected to produce significant non-specific symptoms like fatigue and malaise.  Some authors have suggested that very specific subtypes of depression are more likely to produce reliable biomarkers.  Taylor and Fink (2) have written extensively about melancholia and biomarkers associated with that illness.  I also recall work done by Linkowski and Mendlewicz (3) that they published in the endocrine literature.  Their work was almost exclusively on subjects with very severe forms of depression (HAMD ratings > 30) and their neuroendocrine biomarkers were more robust.  Third, is there a time domain consideration with the combinatoric groups?  For example, do the people meeting 8 or 9 criteria have depression that has persisted for a longer period and does attempted treatment or not treatment affect that group?  Fourth are some of these symptoms complexes generated by others - are they secondary to sleep and appetite disruption?

Either way, the application of combinatorics to some of these situations is very interesting in the field.  As noted in my previous post, combinatorics reflects biological scaling at some point. That occurs at the molecular as well as the evolutionary level.  Large numbers of combinations should be expected when combining either molecular components of organisms, metabolic networks, or the organism wide effect.

Thinking about these combinations clinically is also an interesting exercise.  During my tenure as an acute care psychiatrist it was rare to see anyone without most of the symptoms in an inpatient setting.  Doing consults on medical and surgery wards there were often more novel symptom combinations.  Looking at the author’s tables and the combinations they did not see in their study is an interesting exercise.  One example would be the combination 1,2,3,4,5,7,9 from Table 3.  That would be a person with depressive symptoms except for loss of energy and concentration problems. According to this study that person does not exist.  And of course all of the combinations that lack depressed mood, anhedonia, or that combination have been eliminated by definition.

I hope to expand my look at combinatorics to the genetic, evolutionary, and molecular levels in subsequent posts as well as trying to see if there are mappings from one level to the other.  I am also interested in any books or papers that use similar analyses so please send those references my way,      

 

George Dawson, MD, DFAPA

 

References:

1:  Zimmerman M, Ellison W, Young D, Chelminski I, Dalrymple K. How many different ways do patients meet the diagnostic criteria for major depressive disorder? Compr Psychiatry. 2015 Jan;56:29-34. doi: 10.1016/j.comppsych.2014.09.007. Epub 2014 Sep 6. PMID: 25266848.

2:  Taylor MA, Fink M.  Melancholia: The Diagnosis, Pathophysiology, and Treatment of Depressive Illness. 544 pp. New York, Cambridge University Press, 2006.

3:  Linkowski P, Mendlewicz J, Kerkhofs M, Leclercq R, Golstein J, Brasseur M, Copinschi G, Cuater EV. 24-hour profiles of adrenocorticotropin, cortisol, and growth hormone in major depressive illness: effect of antidepressant treatment. The Journal of Clinical Endocrinology & Metabolism. 1987 Jul 1;65(1):141-52.

 

Apps:

Very good apps are available for calculating combinations, permutations, and the varieties.  For example – if you think back to your probability and statistics course in college at one point the professor was talking about combinations occurring where elements could be used more than once (with repetition). That is typically demonstrated by taking numbered balls out of a container and replacing them in one situation and in the other cases leaving them out. Those are different calculations.  For the above calculations the assumption is that each k element can only be used once (no repetition).  There are apps that give you both calculations.