Theory of Thyroid Statistics - DRAFT 02/14/2025

The distribution of many measured values subject to random independent variations tends to follow a "Gaussian," "normal," or "bell-shaped" distribution that approximates a binomial distribution. In particular, if the lab test results of many people are affected by random individual variation, then a graph of the values will be approximately normally distributed, and standard statistical techniques are applicable. In this view, the null hypothesis is most likely to be satisfied at the center of the "hump" of the bell curve and is progressively less likely to be satisfied as the patient's lab value moves left or right toward the "tails" of the bell curve.

This bell-shaped distribution assumption is not a perfect reflection of reality. However, it is a better approximation of reality than the uniform rectangular distribution implied by the conventional approach used by allopathic MDs as described above. As will be discussed below, fuzzy medical decision-making based on the assumption of a normal distribution of test results (e.g., statements like the patient's lab has a Z-score of -1, which means we are about 68% certain that the patient's lab value is abnormal) is expected to be more precise than Boolean medical decision-making using the "in the box/out of the box" approach of assuming a uniform rectangular distribution (which for that same patient we would say "the patient is normal)."

Any Gaussian distribution curve can be characterized by the population mean (μ) and the standard deviation (σ). The population mean is the value that represents the top of the hump of the bell curve; the standard deviation is a measure of "how wide" the bell curve is. Without delving into all the mathematics, it can be shown that there is a simple approximate relationship between the reference range described in the conventional medical view and the population mean and standard deviation:

Let L and U represent the lower and upper bounds of the 95% reference range obtained as above (when applied to data that follows an approximately normal distribution).

Then the population mean = μ = (L+U)/2, and the standard deviation = σ = (U-L)/2 .

It is convenient to convert (transform) a patient's lab values into Z-scores.

The following formula converts the measured lab value (denoted V) to its corresponding Z-score (denoted Z):

Z = 2 * (V - μ) / σ

These transformed values have the following convenient properties according to the 68-95 rule:

• Z = 0 means that the patient's lab value is in the middle of the reference range and is most likely normal (the patient is normal from an allopathic perspective);
• Z < -2 means that the patient's lab value is lower than the 95% reference range - reject the null hypothesis at a level of p=0.05 (we are more than 95% certain the patient has a diagnosable disorder from an allopathic perspective);
• Any Z value between -2 and +2 lies within the 95% reference range (within 2 standard deviations of the middle) - accept the null hypothesis at a level of p=0.05 (we are less than 95% certain the patient has a diagnosable disorder, so the patient is considered normal from an allopathic perspective);
• Z > +2 means that the patient's lab value is greater than the 95% reference range - reject the null hypothesis at a level of p=0.05 (we are more than 95% certain the patient has a diagnosable disorder from an allopathic perspective);
• Z-scores are real numbers with a continuum of values representing shades of gray (naturopathic and functional medical perspective) - not just true or false (allopathic medical perspective).

An advantage of Z-scores is that the reference range is always from -2 to +2, so it is easy to tell where a lab value lies relative to the reference range (low, normal, high). Even more powerful, since Z-scores are continuous, degrees of belief in the null hypothesis can be expressed by intermediate values. If we assume that the test data is approximately normally distributed (which follows from the Central Limit Theorem of statistics), then given the 95% upper and lower bounds of the test data (L and U) and the patient's test value (V), then we can calculate a Z value as follows. For example, consider the case of TSH (reference range = 0.45 to 4.5) and a measured lab value = 3.5, as follows:

Example:

L = 0.5, U = 4.5, and V = 3.5; then

Z = 2 * (V - ?) / ? = 2 * (3.5 - 2.5) / 2 = +1.0

I.e., the patient's lab value is 1 standard deviation higher than the mid-range.

Based on the 68-95 rule, we are 68% certain the null hypothesis can be rejected, which means we are 68% certain the patient has a problem that deserves intervention. Do we wait until we are more than 95% certain the patient has a problem, or do we begin mild interventions sooner rather than wait for the patient to cross the line into 95% certainty of abnormality? Where do we draw the line between intervention and watchful waiting?

Standard parametric statistical methods depend on the assumption that the probability density function (PDF) is normal (Gaussian).

In the case of TSH [Fontes2013  🕮 ] reports that the Kolmogorov-Smirnov Test shows the PDF of TSH to be non-Gaussian (lognormal) (exhibits kurtosis [Blanca2013]) but can be transformed into an approximately Gaussian distribution using a logarithmic transformation. The Kolmogorov-Smirnov Test is a non-parametric test that can determine whether two sample data sets came from the same probability distribution. The logarithmic transformation has been criticized by [Feng2014  🕮 ], [Feng2019  🕮 ], and ResearchGate.

In the case of freeT4, [Fontes2013  🕮 ] reports that freeT4 exhibits a Gaussian distribution.

Further literature research (or access to appropriate raw data) is necessary to clarify the PDF of the remaining thyroid analytes. The biggest problem is that raw data is generally unavailable to determine the shape of the PDFs. Presumably, the labs providing the tests have the required data, which they used to establish their reference ranges. However, can we find labs willing to share this information?

In the absence of any specific information to the contrary, we assume that the optimal value for a lab value is the center of the reference range, where Z = 0. In other words, the center of "normal" = "optimal." For example, in the thyroid system, a patient is in optimum balance when Z(TSH) = Z(T4) = Z(fT4) = Z(T3) = Z(fT3) = Z(rT3) = 0.

• It is assumed that at each step of the pathway TSH → T4 → T3 + reverseT3, there is a causative correlation between the titer of the precursor and product. This assumption has been confirmed by [Fontes2013  🕮 ] in a healthy population for the case of TSH → freeT4, where the authors report a "high level of significance" inverse correlation between log10(TSH) and fT4 using the Pearson correlation coefficient test, which showed an r-value between -0.3862 and -0.4946.
  
  However, [Johansen1978  🕮 ] reports that in a hypothyroid population the serum TSH or log10(TSH) vs freeT4 and freeT3 showed a nonlinear inverse relationship.
  
  [Ryder1980  🕮 ] reports that TSH correlates better with T4 than T3.
  
  Similarly, [Kahn1978  🕮 ] reports a better reciprocal correlation between TSH and T4 than between TSH and T3.
  
  Dr. Weyrich is not surprised since he expects that the TSH → T4 and T4 → T3 correlations are imperfect due to different confounding factors, so the TSH → T3 correlation should be doubly imperfect.
• [Kahn1978  🕮 ] reports that various patterns of T4 and T3 are associated with elevated TSH:
  1. normal T4 and normal T3 (19.5%);
  2. low T4 and elevated T3 (8%);
  3. low T4 and normal T3 (37%);
  4. low T4 and low T3 (35.5%).
  These authors note that many patients in category 3 had normal T3 despite striking clinical evidence of hypothyroidism and conclude that "a normal serum T3 in the absence of a normal serum T4 is not generally sufficient for maintenance of a euthyroid state."
  
  Likewise, [Kumar1977  🕮 ] concludes that "T4 and T3 may function together to maintain euthyroidism, and that in addition to serum TSH, T4 ... has more diagnostic value than ... T3."
• [Ferrari1987  🕮 ] reports that TSH has a significant inverse correlation with freeT4, T4/TBG ratio, T4, and freeT3 in hypothyroid patients, but freeT4 is the variable that discriminates best between control subjects and hypothyroid patients.
  
  Similarly, [Kahn1978  🕮 ] reports a better reciprocal correlation between TSH and T4 than between TSH and T3.
• [Erfurth1986  🕮 ] observes that the mean ratio between T3 and T4 in euthyroid subjects was less than in hypothyroid subjects, independent of TSH levels.
  
  This observation suggests some mechanism preserving T4 better than T3 in primary thyroid failure. Dr. Weyrich suggests that this reinforces the finding of [Erfurth1986  🕮 ] that freeT4 is a stronger indicator of hypothyroid status than TSH or freeT3.
• [Pekary1980  🕮 ] reports that an inverse linear relationship between TSH and T4 or T3 is found for individuals when TSH levels are manipulated. When a collection of individual trend lines is averaged, the result is the "familiar hyperbolic relationship between thyrotrophin and thyroid hormone levels."
  
  Dr. Weyrich notes that this is an important finding, which suggests that when confounding factors are constant within an individual, the transfer functions between TSH → T4 and T4 → T3 are linear.
• [Aizawa1978  🕮 ] notes a correlation between the size of the sella turcica and the severity of a patient's hypothyroid condition. Dr. Weyrich presumes that this correlation is mediated by increased hypothalamic release of TRH, which drives the pituitary to produce more TSH.
• [Bemben1994  🕮 ] reviewed 283 elderly patients with no previous history of thyroid disease and found that 15% of these patients exhibited subclinical hypothyroidism, which they defined as elevated TSH levels of 5.0 to 14.9 mU/L and normal free T4 levels of 0.7 to 2.0 ng/dL. By these criteria, they found no significant differences in the frequencies of any of the clinical signs and symptoms of hypothyroidism between euthyroid and hypothyroid patients. They concluded that "thyroid status could not be predicted from clinical signs and symptoms in this sample of elderly community-dwelling patients."
  
  Dr. Weyrich counters that an alternative conclusion is that the TSH and T4 criteria used fail to explain the clinical signs and symptoms of hypothyroidism and the criteria should be reconsidered.

Dr. Weyrich notes that a perfect inverse correlation would have r = -1, which suggests that there are also significant confounding factors affecting the correlation between log10(TSH) and freeT4, which may include:

• [Fontes2013  🕮 ] reports that TSH increases very significantly with age, while fT4 slightly decreases with age.
  
  [Carle2007  🕮 ] reports similar findings.
  
  Dr. Weyrich notes that this suggests that the negative feedback loop between the thyroid and the pituitary is mostly successful in maintaining homeostasis (compensation) of fT4 despite the apparent age-related loss of thyroid function. A competing hypothesis is that with increasing age, the hypothalamus/pituitary becomes more active, which causes the thyroid gland to down-regulate to maintain homeostasis, which appears less likely to Dr. Weyrich. In other words, in most cases, thyroid function appears to be the independent variable, rather than pituitary function, which reacts to negative feedback.
• [Alevizaki2005  🕮 ] notes an inverse correlation between TSH and SHBG and also reports the "slopes of the regression lines of T3 to TSH were significantly different in the control group and the hypothyroid group: thus, for the same TSH levels, T3 levels were lower in the hypothyroid group.
  
  This observation argues that using TSH levels to monitor T4 replacement therapy may be misleading as a measure of adequate levels of thyroid hormone in tissues such as the liver because intracellular T3 status in the pituitary [due to the action of DIO2] may not reflect the T3 status in the periphery [due to the action of DIO1].
• Iodine nutritional status [Shahid2025].
  
  [Philippou1992  🕮 ] reports that the daily administration of 150 mg of potassium iodide to hyperthyroid patients caused T4, T3, and reverseT3 to decrease and thyroxine-binding globulin to increase. Trends were smaller and less predictable for euthyroid and hypothyroid patients.
  
  [Waldhausl1976  🕮 ] reports that the administration of 25 mg iodide (daily for 2 weeks) to euthyroid patients reduced T4 and T3 levels.
• Increased estrogen levels can increase binding proteins and thereby increase the ratio of T4 to freeT4 (assuming compensatory negative feedback increases TSH) [Shahid2025].
• TSH receptor antibodies (Graves disease) [Shahid2025].
• Antibodies to thyroid peroxidase or thyroglobulin (Hashimoto thyroiditis) [Shahid2025].
• Ectopic production of thyroid hormone in some conditions, leading to increased thyroid hormones and compensatory TSH decrease [Shahid2025].
• High doses of biotin supplementation interfere with the immunoassay tests for TSH, resulting in suppressed lab value (but not affecting the physiology of the thyroid system itself) [Ardabilygazir2018  🕮 ].
• [Kahana1983  🕮 ] has observed a decrease in TSH in a group of high cortisol patients versus low cortisol patients under the stress of an acute myocardial infarction.
  
  However, [Sinha2023  🕮 ], [Walter2012  🕮 ], [Abdulateef2019  🕮 ] report a positive correlation between TSH and cortisol.
• [Borst1983  🕮 ] reports that fasting or poor caloric intake in the critically ill can lower TSH and mask hypothyroid states.
  
  However, [Shulkin1985  🕮 ] reports that "caloric restriction in both untreated and T4-treated hypothyroid patients is accompanied by ... reduced serum T3 concentrations, as it is euthyroid subjects, and ... no alterations in ... TSH secretion."
• [Yamauchi1984  🕮 ] suggests, "Systolic time intervals (ET/PEP) can discriminate between euthyroid and hyperthyroid states. ... T4 doses [should be] adjusted to maintain normal ET/PEP rather than normal serum TSH levels, especially in older patients in whom T4 may aggravate angina pectoris or provoke myocardial infarction." Systolic time intervals can be measured using pulsed Doppler echocardiography [Boudoulas1990  🕮 ]. See also [Nuutila1992  🕮 ].
• [Chaudhary2023  🕮 ] reports an inverse correlation between thyroid density evaluated by Computed Tomography and TSH.

Factors that can affect conversion of T4 → T3 and T4 → reverseT3 include:

• The enzymes DIO1 (in the liver and kidney) and DIO2 (in the central nervous system, pituitary, brown adipose tissue, and muscle) convert T4 → T3 [Peeters2017].
• Genetic deficiency of DIO2 causes central resistance to exogenous T4, which is characterized by elevated TSH that is not responsive to administration of exogenous T4 [Lacamara2020  🕮 ].
• The enzyme DIO3 (in the brain) converts T4 → rT3 and T3 → 3,3'-T2 [Peeters2017].
• The enzyme DIO1 also degrades rT3 → 3,3'-T2 [Peeters2017].
• T4 is also metabolized by (phase II detoxification) sulfation and glucuronidation [Peeters2017].
• The iodothyronine sulfates and glucuronides are excreted in the bile. They can be hydrolyzed back to iodothyronines by bacterial β-glucuronidases and bacterial sulfatases in the intestines and then reabsorbed (enterohepatic recirculation) [Peeters2017]. This process may be modulated by gut dysbiosis.
• The iodothyronine deiodinases DIO1, DIO2, and DIO3 all contain selenoproteins at their active site, so selenium nutritional deficiency impairs the production of T4, T3, etc. [Peeters2017].
• In non-thyroidal illness (NTI) and malnutrition/extreme dieting, plasma T3 often decreases, and plasma rT3 increases [Peeters2017].
  
  Similarly, [Wadwekar2004  🕮 ] reports that during acute illness, "Serum T4, T3 declined to a nadir and serum rT3 rose to its peak by day 3 of hospitalization before returning to pre-admission euthyroid levels. Serum TSH declined initially but rose to supernormal levels on day 7 before normalization."
  
  [Tibaldi1985  🕮 ] suggests that "because the changes in thyroid hormone metabolism that occur in non-thyroidal disease probably represent adaptive changes to the illness, treatment with L-thyroxine to restore serum thyroid concentrations to the normal range is not indicated."
  
  [Mahashabde2024  🕮 ] refers to this condition as Sick Euthyroid Syndrome (SES), which commonly occurs in critically ill patients, such as heart failure, chronic kidney disease, and severe sepsis. Typically, these patients present with low T3 and normal or low levels of thyroxine T4 and TSH. As discussed by [Tibaldi1985  🕮 ], SES appears to be an adaptive response to reduce the body's metabolic rate.
• Drugs, including propylthiouracil, dexamethasone, propranolol, iodinated radiographic contrast, and amiodarone, inhibit the various iodothyronine deiodinases by several mechanisms [Peeters2017].
• Genetic variations in deiodinase activity have been reported, as well as variations in the SECIS-binding protein SBP2, which is required for the synthesis of the selenoproteins found in iodothyronine deiodinases [Bianco2019  🕮 ].
• Zinc or copper deficiencies are associated with reduced production of T4. Paradoxically, thyroid hormone is required for the absorption of zinc. Furthermore, zinc is required for the T3 receptor to function [Betsy2013  🕮 ], [Maxwell2007  🕮 ].
• Iron deficiency anemia has been associated with decreased thyroperoxidase activity, reduced T4 and T3 production, and increased reverseT3 production. The mechanism is unclear [Soliman2017  🕮 ].
• High doses of biotin supplementation interfere with the immunoassay tests for T4 and T3, resulting in elevated lab values (but not affecting the physiology of the thyroid system itself) [Ardabilygazir2018  🕮 ]
• In a rat model, [Helmreich2011  🕮 ] reports that stress was associated with decreases in peripheral TSH, T4, and T3 but not reverseT3. Dr. Weyrich notes that this implies that the reverseT3 / T3 ratio was increased.
• [Kahana1983  🕮 ] has observed a decrease in T3 and an increase in rT3 in a group of high cortisol patients versus low cortisol patients under the stress of acute myocardial infarction.
  
  [Sinha2023  🕮 ] reports a negative correlation between serum T4 and T3 levels and serum cortisol in hypothyroidism.
  
  [Khandelwal2012  🕮 ] states that hypocortisolism must be addressed prior to initiating T4/T3 replacement therapy.
• A study of COVID-19-infected outpatients had significantly lower serum T3 but higher T4 than non-infected participants [Naghashpour2022  🕮 ].
  
  Another study reported that adrenal insufficiency, low T3, low TSH, and hyperprolactinemia were common in COVID-19 hospitalized patients. "hsCRP showed a rising trend with disease severity while IL-6 did not" [Kumar2021  🕮 ].
• [Inada1980  🕮 ] showed that administering exogenous T3 can reduce reverseT3.
• [Douyon2002  🕮 ] States that less than 10% of obese individuals are hypothyroid, but overfeeding may increase T3 and decrease reverseT3. On the other hand, hypocaloric diets decrease T4, T3, freeT3 and increase reverseT3 (resembling sick euthyroid syndrome).
• High cortisol levels can raise reverseT3 levels and block the conversion of T4 into T3 [need cite]. They are also associated with decreased TSH and free T4 [Cai2020  🕮 ].
• Stress can also cause the body to convert more T4 into reverseT3 instead of T3. This conversion is often a protective mechanism to prevent the body from going into overdrive. Inflammation may also be associated with elevated reverseT3 [need cite].
• Liver conditions like non-alcoholic fatty liver disease can sometimes increase T4 to T3 conversion as a compensatory mechanism, but generally, liver problems can decrease this conversion [Leo; need cite].

In the absence of any specific information to the contrary, we assume that for a process in which one precursor is converted through one or more steps to a product, if the conversion pathway proceeds at a normal rate, then the Z value of the precursor should be proportional to the Z value of the product. Note that if Z(precursor) > Z(product), it may imply either an abnormally active subsequent step is siphoning off the product or the conversion is impaired. For example, in the thyroid pathway, Z(T4) should equal Z(T3) and also equal Z(rT3) in the case of "normal rate of conversion."

In the absence of any specific information to the contrary, we assume that if a product exerts a negative feedback loop that suppresses the production of a control substance, the Z(control) should be inversely proportional to the Z(product). This is in the case of Z(TSH) and Z(T4).