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Plasma Electrolyte Distributions in Humans—Normal or Skewed?
CLINICAL INVESTIGATION
Plasma Electrolyte Distributions in Humans—Normal or Skewed? Mark Feldman, MD and Beverly Dickson, MD ABSTRACT Background: It is widely believed that plasma electrolyte levels are normally distributed. Statistical tests and calculations using plasma electrolyte data are often reported based on this assumption of normality. Examples include t tests, analysis of variance, correlations and confidence intervals. The purpose of our study was to determine whether plasma sodium (Naþ), potassium (Kþ), chloride (Cl−) and bicarbonate ðHCO3 − Þ distributions are indeed normally distributed. Methods: We analyzed plasma electrolyte data from 237 consecutive adults (137 women and 100 men) who had normal results on a standard basic metabolic panel which included plasma electrolyte measurements. The skewness of each distribution (as a measure of its asymmetry) was compared to the zero skewness of a normal (Gaussian) distribution. Results: The plasma Naþ distribution was skewed slightly to the right, but the skew was not significantly different from zero skew. The plasma Cl− distribution was skewed slightly to the left, but again the skew was not significantly different from zero skew. On the contrary, both the plasma Kþ and HCO3 − distributions were significantly skewed to the right (P o 0.01 zero skew). There was also a suggestion from examining frequency distribution curves that Kþ and HCO3 − distributions were bimodal. Conclusions: In adults with a normal basic metabolic panel, plasma potassium and bicarbonate levels are not normally distributed and may be bimodal. Thus, statistical methods to evaluate these 2 plasma electrolytes should be nonparametric tests and not parametric ones that require a normal distribution. Key Indexing Terms: Electrolyte distributions; Skewness; Normal (Gaussian). [Am J Med Sci 2017;](]):]]]–]]].]
INTRODUCTION odium (Naþ), potassium (Kþ), chloride (Cl−), bicarbonate ðHCO3 − Þ are the predominant electrolytes in the plasma and extracellular space. Physiologic levels of these monovalent cations and anions are maintained within relatively narrow ranges by numerous regulatory mechanisms. It is often tacitly assumed that the levels of these 4 plasma electrolytes are normally distributed (closely resembling a bell-shaped Gaussian distribution), although normal distributions are considered rare in medicine.1 A normal distribution has a single peak at its center, is symmetric with zero skew, and has mean, median and mode levels that are identical.2,3 Investigators often analyze plasma electrolyte data using tests, which rely on an assumption of normality, including t tests, correlation, regression, analysis of variance and confidence intervals; 4 recent examples are cited.4-7 To our knowledge, this normality assumption has not been examined rigorously in a sample of individuals with normal plasma electrolyte levels. Therefore, we analyzed the distributions of plasma electrolyte levels in a sample of individuals who had normal findings on a basic metabolic panel (BMP), and we calculated their degree of asymmetry (skewness).2,8-12 Our null hypothesis was that the skewness of plasma electrolyte distributions in our sample do not differ significantly from the zero skewness of a normal (Gaussian) distribution. The alternate hypothesis was that 1 or more of these plasma electrolytes had significant skewness and
S
therefore had not likely been sampled from a normally distributed population.
METHODS WITH STATISTICAL CONSIDERATIONS Laboratory Measurements BMPs were performed on an Abbott Architect C 16000 chemistry analyzer (Abbott Laboratories, Abbott Park, IL) using plasma samples collected in BD vacutainer lithium heparin tubes. Samples were run on an automated robotic line which centrifuges the specimen, decaps the vacutainer tube and runs the samples immediately. Sodium, potassium and chloride were measured by ion selective electrode (indirect method), and the total carbon dioxide was measured by the phenylpyruvate carboxylase method and expressed as bicarbonate. The results included in the statistical sample were chosen by using an a posteriori indirect sampling technique as described in the CLSI EP28A3C document.13 Laboratory results were imported from the clinical chemistry laboratory into a spreadsheet for further computations (Microsoft Excel 2016, Microsoft Corporation, Redmond, WA). All patient identifiers were removed by the laboratory personnel before sending the spreadsheets to the investigators, and the study was therefore exempt from institutional review board approval. The only information provided to the investigator was the sex of the patient, as reference ranges for plasma creatinine and urea nitrogen are
Copyright © 2017 Southern Society for Clinical Investigation. Published by Elsevier Inc. All rights reserved. www.amjmedsci.com www.ssciweb.org
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Feldman and Dickson
TABLE 1. Laboratory reference rangesa. Plasma analyte þ
Na Kþ Cl− CO2 content (HCO3 − ) Creatinine Urea nitrogen Glucose
Reference range 135-145 mmol/L 3.5-5.0 mmol/L 98-110 mmol/L 22-34 mmol/L 0.55-1.11 mg/dL (women) 0.72-1.25 mg/dL (men) 10-20 mg/dL (women) 9-25 mg/dL (men) 65-139 mg/dL
a Reference ranges are for subjects ages 19 years and above. The upper boundary of the glucose reference range is for the postprandial state, as some blood samples could have been collected after a recent meal. Reference ranges were used to select our sample and as the x-axes in the frequency distribution curves for the 4 serum electrolytes listed (Figures 1-4).
sex-specific. For each of the 7 analytes in the BMP, the analyzer manufacturer's suggested reference ranges were verified by performing analysis of at least 40 venous plasma samples of apparently healthy individuals (Table 1). Population Sample Our sample included 237 consecutive women (n ¼ 137) and men (n ¼ 100) ages 19 and above who had completely normal BMP results. They were selected from 994 consecutive BMP results. The other 757 individuals were excluded because they were younger than age 19, had an abnormal Naþ, Kþ, Cl− and HCO3 − or any of these levels (i.e., outside the reference range), or had a plasma creatinine, urea nitrogen or glucose level above the reference range. Statistical and Graphical Methods Skewness was calculated using Excel and 2 online calculators,8,9 with only slight differences between results. The Z test we employed to evaluate our null hypothesis was originally proposed by Joanes and Gill10 and advocated by other authors.2,11 Specifically, we determined whether the skewness of the electrolyte distribution was ≠ 0 at a 2-sided probability level of 1%.2,10-12 This P of 0.01 (rather than 0.05) was chosen based on our sample size and its small standard errors.2 Skewness was divided by the standard error of skewness (0.158 for a sample size of 237).11 The quotient, Z, has a critical value of 2.58 at the 1% significance level.2,11 Thus, a calculated skewness ≥ þ0.408 or ≤ −0.408 would provide statistical support for rejecting our null hypothesis, as 0.408 / 0.158 exceeds 2.58. We also graphed frequency distribution curves using Excel's histogram analysis tool kit and compared the curves to those of a Gaussian distribution using the mean level for each electrolyte distribution and its standard deviation to construct the normal curves.
RESULTS Table 2 summarizes the mean (± standard deviation), median and modal plasma electrolyte levels for women
2
(n ¼ 137), men (n ¼ 100) and the combined sample (n ¼ 237). Women and men had almost identical results for each of these parameters. Naþ Mean, median and modal plasma Naþ levels in the sample were virtually identical in the women and men (Table 2). The calculated Naþ skewness for the combined group ranged from þ0.280 to þ0.282, which was not significantly different from zero skewness (Table 3). The Naþ frequency distribution curve was fairly close to the theoretical Gaussian curve (Figure 1). Kþ Mean, median and modal plasma Kþ levels in the study sample were not identical in either the women or the men (Table 2). The calculated skewness for the combined group ranged from þ0.550 to þ0.555, which was significantly different from zero skewness (Table 3). The Kþ frequency distribution curve differed considerably from the theoretical normal curve, with a possible bimodal distribution (Figure 2). Cl− Mean, median and modal plasma Cl− levels in the sample were identical in the women and the men (Table 2). Calculated Cl− skewness for the combined group ranged from −0.203 to −0.205, which was not significant (Table 3). The Cl− frequency distribution curve was moderately close the theoretical Gaussian curve (Figure 3). HCO3 − Mean, median and modal plasma HCO3 − levels in the study sample were not identical in either the women TABLE 2. Mean, median and modal plasma electrolyte levels (mmol/ L) in the women (n = 137), men (n = 100) and the combined sample (n = 237). Electrolyte
Mean (SD)
Median
Mode
139 (2) 139 (2) 139 (2)
139 139 139
139 138 139
4.1 (0.4) 4.1 (0.4) 4.1 (0.4)
4.0 4.0 4.0
3.8 3.7 3.8
105 (2.5) 105 (2.4) 105 (2.5)
105 105 105
105 105 105
25 (2.2) 25 (2.1) 25 (2.1)
25 25 25
23 23 23
þ
Na Women Men Combined Kþ Women Men Combined Cl− Women Men Combined HCO3 − Women Men Combined
SD, standard deviation of the mean; to calculate the standard error of the mean, divided by the square root of n.
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TABLE 3. Skewness of plasma electrolyte levels in the study sample (n = 237). Electrolyte Naþ Kþ Cl− HCO3 −
Skewness, excel þ0.282 þ0.555 −0.205 þ0.529
(ns) (o0.01) (ns) (o0.01)
Skewness8 þ0.280 þ0.550 −0.203 þ0.524
(ns) (o0.01) (ns) (o0.01)
Skewness9 þ0.280 þ0.551 −0.204 þ0.525
(ns) (o0.01) (ns) (o0.01)
Numbers in parentheses refer to P value. A P o 0.01 indicates that the overall population is very likely skewed and not symmetric (Gaussian), justifying rejection of the null hypothesis. A P value that is not significant (ns) indicates that one cannot reach any conclusion about the skewness of the overall population and about whether to accept or reject the null hypothesis.1
or the men (Table 2). The calculated skewness for the combined group ranged from þ0.524 to þ0.529, which was significantly different from zero skewness (Table 3). The HCO3 − frequency distribution curve differed considerably from the theoretical normal curve, with a possibly bimodal distribution. Creatinine, Urea Nitrogen and Glucose Although not the primary focus of the study, skewness of plasma creatinine and urea nitrogen levels was not significantly different from zero skewness. The plasma glucose skewness was significant (P o 0.01 versus zero skewness), possibly because the samples were not uniformly collected after an overnight fast.
DISCUSSION When sampling from a normal (Gaussian) distribution, it is unlikely that the distribution of the sample will recapitulate a normal distribution due to random sampling variability. Thus, some asymmetry is expected in random samples from a normal distribution, even with sample sizes in the 100-500 range.1 As sample size further increases, the less likely a sizable sampling error will occur and the more likely the sample's distribution will begin to resemble the overall population's true (but never known) distribution. Thus, we can only sample from the overall population and calculate or judge
whether the sample distribution is essentially the same as (or different from) the theoretical normal distribution. The primary statistic used in this study was the skewness of the sample compared to the zero skewness of a theoretical normal distribution.10-12 Accordingly, we rejected the null hypothesis (and accepted the alternate hypothesis) in the cases of plasma Kþ and HCO3 − . We did not reject the null hypothesis in the cases of plasma Naþ and Cl−, although failure to reject the null hypothesis does not prove that the overall population distribution of serum Naþ and Cl− is normal (Gaussian). We considered the possibility that a preanalytical error such as hemolysis or a prolonged tourniquet time might have contributed to Kþ skewness. Hemolysis raises plasma Kþ levels proportionate to the amount of Kþ-containing hemolysate entering the plasma from red blood cells.14 Thus, if some specimens were hemolyzed and others were not, the resultant Kþ distribution may be bimodal (or even multimodal). A mixture of 2 normal distributions with similar but unequal means may appear bimodal. Such bimodal distributions typically have negative kurtosis, since the 2 modes on either side of the center of the distribution effectively lowers the central peak and flattens out the distribution, making it platykurtic.11,15 The kurtosis of our Kþ distribution was −0.356 (mildly platykurtic). Another possible explanation for K skewness is that some of the individuals in the sample tested may have been taking diuretics (e.g., for hypertension or edema), which may have shifted their
FIGURE 1. Frequency distribution curve for serum Naþ (black solid line). In this and subsequent figures, the x axis represents possible test results within the laboratory reference range (Table 1). The y-axis represents the frequency (f) of each test result among the 237 individuals in the sample. A theoretical normal distribution using the mean Naþ level and its standard deviation is shown for comparison as a blue dotted line.
Copyright © 2017 Southern Society for Clinical Investigation. Published by Elsevier Inc. All rights reserved. www.amjmedsci.com www.ssciweb.org
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Feldman and Dickson
FIGURE 2. Frequency distribution curve for serum Kþ (black solid line). A theoretical normal distribution using the mean Kþ level and it standard deviation is shown for comparison as a blue dotted line.
plasma K levels toward the lower end of the reference range. The explanation for the significant skew of the plasma HCO3 − distribution is unclear. Red blood cell hemolysis does not affect plasma HCO3 − measurements.14 A bimodal HCO3 − distribution remains possible. As with the Kþ distribution, the HCO3 − distribution was mildly platykurtic, with a kurtosis of −0.354. One possible explanation is that differences in dietary acid intake among the subjects in our sample may have led to a skewed and possibly bimodal plasma HCO3 − distribution.16The acid-base status of an individual can influence plasma HCO3 − and Cl− levels in a reciprocal manner.17 Thus, if an individual tends toward alveolar hypoventilation and respiratory acidosis (e.g., a person who is morbidly obese), their plasma HCO3 − level will rise with a reciprocal and equal fall in the plasma Cl− level. Conversely, if an individual tends toward alveolar hyperventilation with respiratory alkalosis (e.g., a pregnant woman), their plasma HCO3 − level will fall with a reciprocal and equal rise in the plasma Cl− level. Thus, intersubject variations in alveolar ventilation rates could have contributed to some variability and asymmetry of the HCO3 − distribution. Although it is true that the HCO3 − distribution was skewed right, and the Cl− distribution was skewed left, the former skew was considerably more prominent than the latter, suggesting that another unidentified factor
contributed to significant right skew for plasma bicarbonate levels and its possible bimodality. Plasma sodium concentrations are maintained in a narrow physiological range (to within 3-4% of the mean level), primarily by the hypothalamus and the neurohypophysis. When there is dehydration and a rise in plasma osmolality, thirst is stimulated and antidiuretic hormone (ADH, vasopressin) is released. Thirst leads to water intake and ADH leads to renal water retention by the collecting ducts and distal convoluted tubules, concentrating the urine. Conversely, when there is overhydration and a fall in plasma osmolality, thirst and ADH release are suppressed, with reduced water intake and excretion of a dilute urine. Our study results support a normal distribution of plasma Naþ, the major cation contributing to plasma osmolality. This was also the case for plasma Cl−, the major anion contributing to plasma osmolality. Thus, parametric statistical tests are probably acceptable when analyzing changes or differences in plasma Naþ and Cl− levels. In summary, it is highly unlikely that the distributions of plasma Kþ and HCO3 − levels observed in our sample of people with normal BMPs were derived from a normally distributed population. Although the mechanism(s) for these asymmetric Kþ and HCO3 − distributions remain(s) to be
FIGURE 3. Frequency distribution curve for serum CI−(black solid line). A theoretical normal distribution using the mean CI− level and its standard deviation is shown for comparison as a blue dotted line.
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Are Plasma Electrolytes Normal or Skewed?
FIGURE 4. Frequency distribution curve for serum HCO3 − (black solid line). A theoretical normal distribution using the mean HCO3 − level and its standard deviation is shown for comparison as a blue dotted line.
elucidated, statistical tests involving plasma Kþ and HCO3 − levels should be nonparametric ones that use order and rank of the electrolytes rather than their actual, measured concentration (e.g., Wilcoxon signed rank test, Mann-Whitney U test and Spearman rank order correlation). The more popular parametric tests, particularly t tests frequently performed on plasma Kþ and HCO3 − data5-7,17,18 should be avoided to reduce statistical errors, which are common in scientific journals.19
ACKNOWLEDGMENTS The authors thank Andrea Sablica-Phillips and Margaret Preston for their valuable assistance.
REFERENCES 1. Altman DG, Bland JM. The normal distribution. Br Med J 1995;310:298. 2. Ghasemi A, Zahedias S. Normality tests for statistical analysis: a guide for non-statisticians. Int J Endocrinol Metab 2012;10:486–9. 3. National Institute of Standards and Technology. Measure of location. http://www.itl.nist.gov/div898/handbook/eda/section3/eda351.htm. Accessed May 10, 2017. 4. Patel S, Nguyen D-S, Rastogi A, et al. Treatment of cirrhosisassociated hyponatremia with midodrine and octreotide. Front Med (Lausanne) 2017;4:17. 5. Demirtunc R, Ustun E, Karatoprak C, et al. Effect of transfusion of washed red blood cells on serum potassium level in hemodialysis patients. Turk J Med Sci 2017;47:407–11. 6. Rehman AM, Woodd SL, Heimburger DC, et al. Changes in serum phosphate and potassium and their effects on mortality in malnourished African HIV-infected adults starting antiretroviral therapy and given vitamins and minerals in lipid-based nutritional supplements: secondary analysis from the Nutritional Support for African Adults Starting Antiretroviral Therapy (NUSTART) trial. Br J Nutr 2017;117:814–21. 7. Ferreira JP, Girerd N, Duarte K, et al. Serum chloride and sodium interplay in patients with acute myocardial infarction and heart failure with reduced ejection fraction. An analysis from the high-risk myocardial infarction database initiative. Cir Heart Fail 2017;10:e003500. 8. http://www.endmemo.com/statistics/skewness.php. Accessed May 20, 2017.
9. http://www.numberempire.com/statisticalcalculator.php. Accessed May 20, 2017. 10. Joanes DM, Gill CA. Comparing measures of sample skewness and kurtosis. Statistician 1998;47:183–9. 11. Brown S. Measures of shape: skewness and kurtosis. https://www. brownmath.com/stat/shape.htm. Accessed May 10, 2017. 12. National Institute of Standards and Technology. 1.3.5.11. Measures of skewness and kurtosis. http://www.itl.nist.gov/div898/handbook/eda/sec tion3/eda356.htm. Accessed May 10, 2017. 13. CLSI. Defining, Establishing, and Verifying Reference Intervals in the Clinical Laboratory; Approved Guideline-Third Edition. Wayne, PA: Clinical and Laboratory Standards Institute; 2008 [CLSI document EP28A3c]. 14. Frank JJ, Bermes EW, Bickel MJ, et al. Effect of in vitro hemolysis on chemical values for serum. Clin Chem 1978;24:1966–70. 15. Multimodal distribution. Mixture of two normal distributions. https://en. wikipedia.org/wiki/Multimodaldistribution. Accessed May 18, 2017. 16. Amodu A, Abramowitz MK. Dietary acid, age, and serum bicarbonate levels among adults in the United States. Clin J Am Soc Nephrol 2013;8: 2034–42. 17. Feldman M, Soni N, Dickson B. Use of sodium concentration and anion gap to improve correlation between serum chloride and bicarbonate concentrations. J Clin Lab Anal 2006;20:154–9. 18. Zawada ET, Williams L, McClung DE, et al. Renal-metabolic consequences of antihypertensive therapy with diltiazem versus hydrochlorothiazide. Mineral Electrolyte Metab 1987;13:72–7. 19. Curran-Everett D, Benos DJ. Guidelines for reporting statistics in journals by the American Physiological Society. Am J Physiol Endocrinol Metab 2004;287:E189–91.
From the Department of Internal Medicine (MF) and Department of Clinical Pathology (BD), Texas Health Presbyterian Hospital Dallas, Dallas, Texas. Submitted May 24, 2017; accepted July 25, 2017. The authors have no financial or other conflicts of interest to disclose. This work was supported by the William O. Tschumy, Jr. M.D. Chair of Internal Medicine (M.F.). Correspondence: Mark Feldman, MD, Department of Internal Medicine, Texas Health Presbyterian Hospital Dallas, 8200 Walnut Hill Lane, Dallas, TX 75232 (E-mail: [email protected]).
Copyright © 2017 Southern Society for Clinical Investigation. Published by Elsevier Inc. All rights reserved. www.amjmedsci.com www.ssciweb.org
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Plasma Electrolyte Distributions in Humans—Normal or Skewed? Mark Feldman, MD and Beverly Dickson, MD ABSTRACT Background: It is widely believed that plasma electrolyte levels are normally distributed. Statistical tests and calculations using plasma electrolyte data are often reported based on this assumption of normality. Examples include t tests, analysis of variance, correlations and confidence intervals. The purpose of our study was to determine whether plasma sodium (Naþ), potassium (Kþ), chloride (Cl−) and bicarbonate ðHCO3 − Þ distributions are indeed normally distributed. Methods: We analyzed plasma electrolyte data from 237 consecutive adults (137 women and 100 men) who had normal results on a standard basic metabolic panel which included plasma electrolyte measurements. The skewness of each distribution (as a measure of its asymmetry) was compared to the zero skewness of a normal (Gaussian) distribution. Results: The plasma Naþ distribution was skewed slightly to the right, but the skew was not significantly different from zero skew. The plasma Cl− distribution was skewed slightly to the left, but again the skew was not significantly different from zero skew. On the contrary, both the plasma Kþ and HCO3 − distributions were significantly skewed to the right (P o 0.01 zero skew). There was also a suggestion from examining frequency distribution curves that Kþ and HCO3 − distributions were bimodal. Conclusions: In adults with a normal basic metabolic panel, plasma potassium and bicarbonate levels are not normally distributed and may be bimodal. Thus, statistical methods to evaluate these 2 plasma electrolytes should be nonparametric tests and not parametric ones that require a normal distribution. Key Indexing Terms: Electrolyte distributions; Skewness; Normal (Gaussian). [Am J Med Sci 2017;](]):]]]–]]].]
INTRODUCTION odium (Naþ), potassium (Kþ), chloride (Cl−), bicarbonate ðHCO3 − Þ are the predominant electrolytes in the plasma and extracellular space. Physiologic levels of these monovalent cations and anions are maintained within relatively narrow ranges by numerous regulatory mechanisms. It is often tacitly assumed that the levels of these 4 plasma electrolytes are normally distributed (closely resembling a bell-shaped Gaussian distribution), although normal distributions are considered rare in medicine.1 A normal distribution has a single peak at its center, is symmetric with zero skew, and has mean, median and mode levels that are identical.2,3 Investigators often analyze plasma electrolyte data using tests, which rely on an assumption of normality, including t tests, correlation, regression, analysis of variance and confidence intervals; 4 recent examples are cited.4-7 To our knowledge, this normality assumption has not been examined rigorously in a sample of individuals with normal plasma electrolyte levels. Therefore, we analyzed the distributions of plasma electrolyte levels in a sample of individuals who had normal findings on a basic metabolic panel (BMP), and we calculated their degree of asymmetry (skewness).2,8-12 Our null hypothesis was that the skewness of plasma electrolyte distributions in our sample do not differ significantly from the zero skewness of a normal (Gaussian) distribution. The alternate hypothesis was that 1 or more of these plasma electrolytes had significant skewness and
S
therefore had not likely been sampled from a normally distributed population.
METHODS WITH STATISTICAL CONSIDERATIONS Laboratory Measurements BMPs were performed on an Abbott Architect C 16000 chemistry analyzer (Abbott Laboratories, Abbott Park, IL) using plasma samples collected in BD vacutainer lithium heparin tubes. Samples were run on an automated robotic line which centrifuges the specimen, decaps the vacutainer tube and runs the samples immediately. Sodium, potassium and chloride were measured by ion selective electrode (indirect method), and the total carbon dioxide was measured by the phenylpyruvate carboxylase method and expressed as bicarbonate. The results included in the statistical sample were chosen by using an a posteriori indirect sampling technique as described in the CLSI EP28A3C document.13 Laboratory results were imported from the clinical chemistry laboratory into a spreadsheet for further computations (Microsoft Excel 2016, Microsoft Corporation, Redmond, WA). All patient identifiers were removed by the laboratory personnel before sending the spreadsheets to the investigators, and the study was therefore exempt from institutional review board approval. The only information provided to the investigator was the sex of the patient, as reference ranges for plasma creatinine and urea nitrogen are
Copyright © 2017 Southern Society for Clinical Investigation. Published by Elsevier Inc. All rights reserved. www.amjmedsci.com www.ssciweb.org
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Feldman and Dickson
TABLE 1. Laboratory reference rangesa. Plasma analyte þ
Na Kþ Cl− CO2 content (HCO3 − ) Creatinine Urea nitrogen Glucose
Reference range 135-145 mmol/L 3.5-5.0 mmol/L 98-110 mmol/L 22-34 mmol/L 0.55-1.11 mg/dL (women) 0.72-1.25 mg/dL (men) 10-20 mg/dL (women) 9-25 mg/dL (men) 65-139 mg/dL
a Reference ranges are for subjects ages 19 years and above. The upper boundary of the glucose reference range is for the postprandial state, as some blood samples could have been collected after a recent meal. Reference ranges were used to select our sample and as the x-axes in the frequency distribution curves for the 4 serum electrolytes listed (Figures 1-4).
sex-specific. For each of the 7 analytes in the BMP, the analyzer manufacturer's suggested reference ranges were verified by performing analysis of at least 40 venous plasma samples of apparently healthy individuals (Table 1). Population Sample Our sample included 237 consecutive women (n ¼ 137) and men (n ¼ 100) ages 19 and above who had completely normal BMP results. They were selected from 994 consecutive BMP results. The other 757 individuals were excluded because they were younger than age 19, had an abnormal Naþ, Kþ, Cl− and HCO3 − or any of these levels (i.e., outside the reference range), or had a plasma creatinine, urea nitrogen or glucose level above the reference range. Statistical and Graphical Methods Skewness was calculated using Excel and 2 online calculators,8,9 with only slight differences between results. The Z test we employed to evaluate our null hypothesis was originally proposed by Joanes and Gill10 and advocated by other authors.2,11 Specifically, we determined whether the skewness of the electrolyte distribution was ≠ 0 at a 2-sided probability level of 1%.2,10-12 This P of 0.01 (rather than 0.05) was chosen based on our sample size and its small standard errors.2 Skewness was divided by the standard error of skewness (0.158 for a sample size of 237).11 The quotient, Z, has a critical value of 2.58 at the 1% significance level.2,11 Thus, a calculated skewness ≥ þ0.408 or ≤ −0.408 would provide statistical support for rejecting our null hypothesis, as 0.408 / 0.158 exceeds 2.58. We also graphed frequency distribution curves using Excel's histogram analysis tool kit and compared the curves to those of a Gaussian distribution using the mean level for each electrolyte distribution and its standard deviation to construct the normal curves.
RESULTS Table 2 summarizes the mean (± standard deviation), median and modal plasma electrolyte levels for women
2
(n ¼ 137), men (n ¼ 100) and the combined sample (n ¼ 237). Women and men had almost identical results for each of these parameters. Naþ Mean, median and modal plasma Naþ levels in the sample were virtually identical in the women and men (Table 2). The calculated Naþ skewness for the combined group ranged from þ0.280 to þ0.282, which was not significantly different from zero skewness (Table 3). The Naþ frequency distribution curve was fairly close to the theoretical Gaussian curve (Figure 1). Kþ Mean, median and modal plasma Kþ levels in the study sample were not identical in either the women or the men (Table 2). The calculated skewness for the combined group ranged from þ0.550 to þ0.555, which was significantly different from zero skewness (Table 3). The Kþ frequency distribution curve differed considerably from the theoretical normal curve, with a possible bimodal distribution (Figure 2). Cl− Mean, median and modal plasma Cl− levels in the sample were identical in the women and the men (Table 2). Calculated Cl− skewness for the combined group ranged from −0.203 to −0.205, which was not significant (Table 3). The Cl− frequency distribution curve was moderately close the theoretical Gaussian curve (Figure 3). HCO3 − Mean, median and modal plasma HCO3 − levels in the study sample were not identical in either the women TABLE 2. Mean, median and modal plasma electrolyte levels (mmol/ L) in the women (n = 137), men (n = 100) and the combined sample (n = 237). Electrolyte
Mean (SD)
Median
Mode
139 (2) 139 (2) 139 (2)
139 139 139
139 138 139
4.1 (0.4) 4.1 (0.4) 4.1 (0.4)
4.0 4.0 4.0
3.8 3.7 3.8
105 (2.5) 105 (2.4) 105 (2.5)
105 105 105
105 105 105
25 (2.2) 25 (2.1) 25 (2.1)
25 25 25
23 23 23
þ
Na Women Men Combined Kþ Women Men Combined Cl− Women Men Combined HCO3 − Women Men Combined
SD, standard deviation of the mean; to calculate the standard error of the mean, divided by the square root of n.
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TABLE 3. Skewness of plasma electrolyte levels in the study sample (n = 237). Electrolyte Naþ Kþ Cl− HCO3 −
Skewness, excel þ0.282 þ0.555 −0.205 þ0.529
(ns) (o0.01) (ns) (o0.01)
Skewness8 þ0.280 þ0.550 −0.203 þ0.524
(ns) (o0.01) (ns) (o0.01)
Skewness9 þ0.280 þ0.551 −0.204 þ0.525
(ns) (o0.01) (ns) (o0.01)
Numbers in parentheses refer to P value. A P o 0.01 indicates that the overall population is very likely skewed and not symmetric (Gaussian), justifying rejection of the null hypothesis. A P value that is not significant (ns) indicates that one cannot reach any conclusion about the skewness of the overall population and about whether to accept or reject the null hypothesis.1
or the men (Table 2). The calculated skewness for the combined group ranged from þ0.524 to þ0.529, which was significantly different from zero skewness (Table 3). The HCO3 − frequency distribution curve differed considerably from the theoretical normal curve, with a possibly bimodal distribution. Creatinine, Urea Nitrogen and Glucose Although not the primary focus of the study, skewness of plasma creatinine and urea nitrogen levels was not significantly different from zero skewness. The plasma glucose skewness was significant (P o 0.01 versus zero skewness), possibly because the samples were not uniformly collected after an overnight fast.
DISCUSSION When sampling from a normal (Gaussian) distribution, it is unlikely that the distribution of the sample will recapitulate a normal distribution due to random sampling variability. Thus, some asymmetry is expected in random samples from a normal distribution, even with sample sizes in the 100-500 range.1 As sample size further increases, the less likely a sizable sampling error will occur and the more likely the sample's distribution will begin to resemble the overall population's true (but never known) distribution. Thus, we can only sample from the overall population and calculate or judge
whether the sample distribution is essentially the same as (or different from) the theoretical normal distribution. The primary statistic used in this study was the skewness of the sample compared to the zero skewness of a theoretical normal distribution.10-12 Accordingly, we rejected the null hypothesis (and accepted the alternate hypothesis) in the cases of plasma Kþ and HCO3 − . We did not reject the null hypothesis in the cases of plasma Naþ and Cl−, although failure to reject the null hypothesis does not prove that the overall population distribution of serum Naþ and Cl− is normal (Gaussian). We considered the possibility that a preanalytical error such as hemolysis or a prolonged tourniquet time might have contributed to Kþ skewness. Hemolysis raises plasma Kþ levels proportionate to the amount of Kþ-containing hemolysate entering the plasma from red blood cells.14 Thus, if some specimens were hemolyzed and others were not, the resultant Kþ distribution may be bimodal (or even multimodal). A mixture of 2 normal distributions with similar but unequal means may appear bimodal. Such bimodal distributions typically have negative kurtosis, since the 2 modes on either side of the center of the distribution effectively lowers the central peak and flattens out the distribution, making it platykurtic.11,15 The kurtosis of our Kþ distribution was −0.356 (mildly platykurtic). Another possible explanation for K skewness is that some of the individuals in the sample tested may have been taking diuretics (e.g., for hypertension or edema), which may have shifted their
FIGURE 1. Frequency distribution curve for serum Naþ (black solid line). In this and subsequent figures, the x axis represents possible test results within the laboratory reference range (Table 1). The y-axis represents the frequency (f) of each test result among the 237 individuals in the sample. A theoretical normal distribution using the mean Naþ level and its standard deviation is shown for comparison as a blue dotted line.
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Feldman and Dickson
FIGURE 2. Frequency distribution curve for serum Kþ (black solid line). A theoretical normal distribution using the mean Kþ level and it standard deviation is shown for comparison as a blue dotted line.
plasma K levels toward the lower end of the reference range. The explanation for the significant skew of the plasma HCO3 − distribution is unclear. Red blood cell hemolysis does not affect plasma HCO3 − measurements.14 A bimodal HCO3 − distribution remains possible. As with the Kþ distribution, the HCO3 − distribution was mildly platykurtic, with a kurtosis of −0.354. One possible explanation is that differences in dietary acid intake among the subjects in our sample may have led to a skewed and possibly bimodal plasma HCO3 − distribution.16The acid-base status of an individual can influence plasma HCO3 − and Cl− levels in a reciprocal manner.17 Thus, if an individual tends toward alveolar hypoventilation and respiratory acidosis (e.g., a person who is morbidly obese), their plasma HCO3 − level will rise with a reciprocal and equal fall in the plasma Cl− level. Conversely, if an individual tends toward alveolar hyperventilation with respiratory alkalosis (e.g., a pregnant woman), their plasma HCO3 − level will fall with a reciprocal and equal rise in the plasma Cl− level. Thus, intersubject variations in alveolar ventilation rates could have contributed to some variability and asymmetry of the HCO3 − distribution. Although it is true that the HCO3 − distribution was skewed right, and the Cl− distribution was skewed left, the former skew was considerably more prominent than the latter, suggesting that another unidentified factor
contributed to significant right skew for plasma bicarbonate levels and its possible bimodality. Plasma sodium concentrations are maintained in a narrow physiological range (to within 3-4% of the mean level), primarily by the hypothalamus and the neurohypophysis. When there is dehydration and a rise in plasma osmolality, thirst is stimulated and antidiuretic hormone (ADH, vasopressin) is released. Thirst leads to water intake and ADH leads to renal water retention by the collecting ducts and distal convoluted tubules, concentrating the urine. Conversely, when there is overhydration and a fall in plasma osmolality, thirst and ADH release are suppressed, with reduced water intake and excretion of a dilute urine. Our study results support a normal distribution of plasma Naþ, the major cation contributing to plasma osmolality. This was also the case for plasma Cl−, the major anion contributing to plasma osmolality. Thus, parametric statistical tests are probably acceptable when analyzing changes or differences in plasma Naþ and Cl− levels. In summary, it is highly unlikely that the distributions of plasma Kþ and HCO3 − levels observed in our sample of people with normal BMPs were derived from a normally distributed population. Although the mechanism(s) for these asymmetric Kþ and HCO3 − distributions remain(s) to be
FIGURE 3. Frequency distribution curve for serum CI−(black solid line). A theoretical normal distribution using the mean CI− level and its standard deviation is shown for comparison as a blue dotted line.
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Are Plasma Electrolytes Normal or Skewed?
FIGURE 4. Frequency distribution curve for serum HCO3 − (black solid line). A theoretical normal distribution using the mean HCO3 − level and its standard deviation is shown for comparison as a blue dotted line.
elucidated, statistical tests involving plasma Kþ and HCO3 − levels should be nonparametric ones that use order and rank of the electrolytes rather than their actual, measured concentration (e.g., Wilcoxon signed rank test, Mann-Whitney U test and Spearman rank order correlation). The more popular parametric tests, particularly t tests frequently performed on plasma Kþ and HCO3 − data5-7,17,18 should be avoided to reduce statistical errors, which are common in scientific journals.19
ACKNOWLEDGMENTS The authors thank Andrea Sablica-Phillips and Margaret Preston for their valuable assistance.
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9. http://www.numberempire.com/statisticalcalculator.php. Accessed May 20, 2017. 10. Joanes DM, Gill CA. Comparing measures of sample skewness and kurtosis. Statistician 1998;47:183–9. 11. Brown S. Measures of shape: skewness and kurtosis. https://www. brownmath.com/stat/shape.htm. Accessed May 10, 2017. 12. National Institute of Standards and Technology. 1.3.5.11. Measures of skewness and kurtosis. http://www.itl.nist.gov/div898/handbook/eda/sec tion3/eda356.htm. Accessed May 10, 2017. 13. CLSI. Defining, Establishing, and Verifying Reference Intervals in the Clinical Laboratory; Approved Guideline-Third Edition. Wayne, PA: Clinical and Laboratory Standards Institute; 2008 [CLSI document EP28A3c]. 14. Frank JJ, Bermes EW, Bickel MJ, et al. Effect of in vitro hemolysis on chemical values for serum. Clin Chem 1978;24:1966–70. 15. Multimodal distribution. Mixture of two normal distributions. https://en. wikipedia.org/wiki/Multimodaldistribution. Accessed May 18, 2017. 16. Amodu A, Abramowitz MK. Dietary acid, age, and serum bicarbonate levels among adults in the United States. Clin J Am Soc Nephrol 2013;8: 2034–42. 17. Feldman M, Soni N, Dickson B. Use of sodium concentration and anion gap to improve correlation between serum chloride and bicarbonate concentrations. J Clin Lab Anal 2006;20:154–9. 18. Zawada ET, Williams L, McClung DE, et al. Renal-metabolic consequences of antihypertensive therapy with diltiazem versus hydrochlorothiazide. Mineral Electrolyte Metab 1987;13:72–7. 19. Curran-Everett D, Benos DJ. Guidelines for reporting statistics in journals by the American Physiological Society. Am J Physiol Endocrinol Metab 2004;287:E189–91.
From the Department of Internal Medicine (MF) and Department of Clinical Pathology (BD), Texas Health Presbyterian Hospital Dallas, Dallas, Texas. Submitted May 24, 2017; accepted July 25, 2017. The authors have no financial or other conflicts of interest to disclose. This work was supported by the William O. Tschumy, Jr. M.D. Chair of Internal Medicine (M.F.). Correspondence: Mark Feldman, MD, Department of Internal Medicine, Texas Health Presbyterian Hospital Dallas, 8200 Walnut Hill Lane, Dallas, TX 75232 (E-mail: [email protected]).
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