Statistical Methods to Support Difficult Diagnoses
Abstract
:1. Introduction
- Some (especially elderly) people often take a large number of drugs. Often enough, some of these drugs (or combinations of them) can be the reason for further severe problems. Yet many of these interactions are not known well enough. Simply think that for the 1000 most frequent drugs, there are half a million possible interactions, which also differ from patient to patient. Statistics provides an exceptional tool to test many of these interactions at the same time, using sophisticated mathematical methods, such as block designs and matrix calculus. Note that it is useless to find drug interactions for a large number of patients—they have an individual character. For instance, “dizziness” is on almost all package inserts, and hence is useless. We describe this method in more detail in Section 3.1 below. Note that this application does not concern rare diseases, but remarkably frequent cases in treating patients.
- Reactions to the intake of food (components) can provide valuable hints for the diagnosis. However, one cannot expect patients to know that, for instance, they react to an imbalance of magnesium intake. In the section “Statistics Works” below, we describe a case where problems through the intake of too much potassium but too little sodium caused severe and frequents attacks of paralysis for more than 50 years. This led doctors to investigate a gene that was previously not considered as a cause of paralysis (see [3]), as well as to find the defect. The solution of this case and the fact that that this defect seems to be unique worldwide shows the power of the statistical approach. This case was described in detail in [4].
- Combinations of allergens can be tricky. The authors of [5] describe cases in which one allergen is neutral for the patient, another one positive, but the combination is a complete disaster. Our methods can detect cases such as this without problems.
2. Materials and Methods
2.1. Statistical Methods I: Regression Analysis
- Day 1: I tried x1, x3, x4, x7, the result was 23;
- Day 2: I tried x2, x3, x5, x9, the result was 19;
- x1 = exhaust air of the vacuum cleaner (measured in minutes of exposure);
- x2 = intake of certain candies (measured in pieces), …
2.2. Statistical Methods II: Experimental Designs
- (i)
- Each point in P belongs to the same number r of blocks;
- (ii)
- Each Bi has the same number k of elements;
- (iii)
- Each pair pi, pj of points belongs to the same number λ of blocks.
- The points are the factors (e.g., possible triggers for an allergy);
- Every block lists the factors which will be tested simultaneously in a test.
- P = {1, 2, 3, 4, 5, 6, 7} and B consist of the 14 collections
- B1 = {2,4,5}, B2 = {1,3,7}, B3 = {1,2,6}, B4 = {1,5,7}, B5 = {1,3,4}, B6 = {2,3,7}, B7 = {4,5,7},
- B8 = {1,2,4}, B9 = {2,6,7}, B10 = {2,3,5}, B11 = {3,4,6}, B12 = {3,5,6}, B13 = {1,5,6}, B14 = {4,6,7}
- Every test (except #15) involves b = 3 factors (3 dots in every column);
- Each factor is tested in r = 6 tests (6 dots in each row);
- Each pair of factors is tested together in λ = 2 tests.
y = 3 + 51x4 + 19x5 − 41x6 | (Model 1) |
y = 2 + 47x4 − 31x6 + 58x2x5 | (Model 2) |
3. Results
3.1. Statistic Works
3.2. More Examples
- (1)
- Side effects of combinations of drugs: This was also mentioned in the introduction. We had a case of a person (aged 75) who developed a strong and permanent dizziness that did not allow him to drive a car any more. He took 10 types of drugs per day, and we added the consumption of a standardized amount of alcohol as “drug # 11”. Therefore, we used the following (11,22,5,10,4)-block design as in Table 1 and added another “test”, this time”: the usual drug consumption of the patient. Table 3 shows:
- (1)
- Food-dependent, exercise-induced anaphylaxis: The contact with some allergens might be harmless, and physical exercise can help a lot, while the combination can be disastrous. Thus, one factor is neutral for the patient, the other one positive, but the combination is really negative (see Romano et al. [5]).
- (2)
- Phototoxic dermatosis: We usually tolerate sunlight at a usual dose very well since it is essential for our survival. Frequently used medications such as certain antibiotics, non-steroidal anti-inflammatory drugs (NSAIDs), and diuretic and antiarrhythmic drugs have usually no direct side effects at the skin. However, these drugs are known to enhance photosensitivity. In combination with usually well-tolerated sunlight, these drugs can create severe sunburn such as skin reactions (see [9,10]).
- (3)
- Hyperkalemic periodic paralysis: In a mild form, this disease can usually be tolerated, but in combination with a pathogenic gene mutation, it can create severe paralysis.
- (4)
- Stomach problems: A patient complains about stomach pains after some meals. His doctor suspects that seafood might be a reason, but this can hardly explain the pains. Moreover, he can exclude a large number of food components that do not hurt the patient. However, 15 “suspicious” factors remain. The following might be a typical progression of the statistical investigation. A simple regression test as in “Statistical Methods I” quickly excludes eight of them. For the remaining seven components, this test does not give satisfactory results. Therefore, we might use the test in “Statistical Methods II”. Suppose that the remaining seven factors are sugar (= S), apples (= A), lactose (= L), walnut (= W), pepper (= P), crabs (= C), and mustard (= M). So, according to the experimental design in “Statistical Methods II”, the first test would be a meal with S, A, and W. Then, a statistician quickly finds out that C does hurt a bit (as single factor), but the combination A and P is the main reason for the pains, while A and P alone do not really hurt.
4. Discussion
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
- Cleophas, T.J.; Zwinderman, A.H.; Cleophas, T.F.; Cleophas, E.P. Statistics Applied to Clinical Trials; Springer: Dordrecht, The Netherlands, 2009. [Google Scholar]
- Cleophas, T.J.; Zwinderman, S.H. Regression Analysis in Medical Research; Springer International: Berlin/Heidelberg, Germany, 2018. [Google Scholar]
- Kuhn, M.; Jurkatt-Rott, K.; Lehmann-Horn, F. Rare KCNJ18 variants do not explain hypokalaemic periodic paralysis in 263 unrelated patients. J. Neurol. Neurosurg. Psychiatry 2016, 87, 49–52. [Google Scholar] [CrossRef] [PubMed] [Green Version]
- Soufi, M.; Ruppert, V.; Rinné, S.; Mueller, T.; Kurt, B.; Pilz, G.F.; Maieron, A.; Dodel, R.; Decher, N.; Schaefer, J.R. Increased KCNJ18 promoter activity as a mechanism in atypical normokalemic periodic paralysis. Neurol. Genet. 2018, 4, e274. [Google Scholar] [CrossRef] [PubMed] [Green Version]
- Romano, A.; Fonso, M.D.; Giufredda, F.; Pappa, G.; Artesiani, M.C.; Viola, M.; Venuti, A.; Palmieri, V.; Zeppili, P. Food-dependent exercise-induced anaphylaxis: Clinical and laboratory findings in 54 subjects. Int. Arch. Allergy Immunol. 2001, 125, 264–272. [Google Scholar] [CrossRef] [PubMed]
- Morris, M.D. Design of Experiments; CRC Press: Boca Raton, FL, USA, 2011. [Google Scholar]
- Lidl, R.; Pilz, G.F. Applied Abstract Algebra, 2nd ed.; Undergraduate Texts in Mathematics; Springer: New York, NY, USA, 1997. [Google Scholar]
- Ke, W.F.; Pilz, G.F. Abstract algebra in statistics. J. Algebr. Stat. 2010, 1, 6–12. [Google Scholar]
- Gould, J.W.; Mercurio, M.G.; Elmets, C.A. Cutaneous photosensitivity diseases induced by exogenous agents. J. Am. Acad. Dermatol. 1995, 33, 551. [Google Scholar] [CrossRef]
- Hofmann, G.A.; Weber, B. Drug-induced photosensitivity: Culprit drugs, potential mechanisms and clinical consequences. J. Dtsch. Dermatol. Ges. 2021, 19, 19–29, PMCID:PMC7898394. [Google Scholar] [CrossRef] [PubMed]
- Kiefer, J.; Wynn, H.P. Optimum Balanced Block and Latin Square Designs for Correlated Observations. Ann. Statist. 1981, 9, 737–757. [Google Scholar] [CrossRef]
- Holmes, S. Statistical proof? The problem of irreproducibility. Bull. Am. Math. Soc. 2018, 5, 31–55. [Google Scholar] [CrossRef] [Green Version]
- Mueller, W.G. Collecting Spatial Data—Optimum Design of Experiments for Random Fields; Springer: Berlin/Heidelberg, Germany, 2007. [Google Scholar]
- Saunders, M.J.; Wingfield, T.; Datta, S.; Montova, R.; Ramos, E.; Baldwin, M.R.; Tovar, M.A.; Evans, B.E.; Gilman, R.H.; Evans, C.A. A household-level score to predict the risk of tuberculosis among contacts of patients with tuberculosis: A derivation and external validation prospective cohort study. Lancet Infect. Dis. 2019. [Google Scholar] [CrossRef] [Green Version]
Fact.\Test | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
x1 | ● | ● | ● | ● | ● | ● | |||||||||
x2 | ● | ● | ● | ● | ● | ● | |||||||||
x3 | ● | ● | ● | ● | ● | ● | |||||||||
x4 | ● | ● | ● | ● | ● | ● | |||||||||
x5 | ● | ● | ● | ● | ● | ● | |||||||||
x6 | ● | ● | ● | ● | ● | ● | |||||||||
x7 | ● | ● | ● | ● | ● | ● | |||||||||
Results | 49 | −2 | −28 | 3 | 54 | −1 | 51 | 98 | −31 | 69 | 18 | −35 | −25 | 22 | 3 |
Test | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
real | 49 | −2 | −28 | 3 | 54 | −1 | 51 | 98 | −31 | 69 | 18 | −35 | −25 | 22 | 3 |
Mod.1 | 54 | 3 | −38 | 22 | 54 | 3 | 73 | 73 | −38 | 22 | 13 | −19 | −19 | 13 | 3 |
Mod.2 | 49 | 2 | −29 | 2 | 49 | 2 | 49 | 107 | −29 | 60 | 18 | −29 | −29 | 18 | 2 |
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
● | ● | ● | ● | ● | ● | ● | ● | ● | ● | ● | ||||||||||||
● | ● | ● | ● | ● | ● | ● | ● | ● | ● | ● | ||||||||||||
● | ● | ● | ● | ● | ● | ● | ● | ● | ● | ● | ||||||||||||
● | ● | ● | ● | ● | ● | ● | ● | ● | ● | ● | ||||||||||||
● | ● | ● | ● | ● | ● | ● | ● | ● | ● | ● | ||||||||||||
● | ● | ● | ● | ● | ● | ● | ● | ● | ● | ● | ||||||||||||
● | ● | ● | ● | ● | ● | ● | ● | ● | ● | ● | ||||||||||||
● | ● | ● | ● | ● | ● | ● | ● | ● | ● | ● | ||||||||||||
● | ● | ● | ● | ● | ● | ● | ● | ● | ● | ● | ||||||||||||
● | ● | ● | ● | ● | ● | ● | ● | ● | ● | ● | ||||||||||||
● | ● | ● | ● | ● | ● | ● | ● | ● | ● | ● |
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |
© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
Share and Cite
Pilz, G.F.; Weber, F.; Mueller, W.G.; Schaefer, J.R. Statistical Methods to Support Difficult Diagnoses. Diagnostics 2021, 11, 1300. https://doi.org/10.3390/diagnostics11071300
Pilz GF, Weber F, Mueller WG, Schaefer JR. Statistical Methods to Support Difficult Diagnoses. Diagnostics. 2021; 11(7):1300. https://doi.org/10.3390/diagnostics11071300
Chicago/Turabian StylePilz, Guenter F., Frank Weber, Werner G. Mueller, and Juergen R. Schaefer. 2021. "Statistical Methods to Support Difficult Diagnoses" Diagnostics 11, no. 7: 1300. https://doi.org/10.3390/diagnostics11071300