SPECIAL CONTRIBUTION biostatistics
Introduction to Biostatistics: Part 1, Basic Concepts Statistical methods commonly used to analyze data presented in journal articles should be understood by both medical scientists and practicing clinicians. Inappropriate data analysis methods have been reported in 42% to 78% of original publications in critical reviews of selected medical journals. The only way to halt researchers' misuse of statistics and improve the clinician's knowledge of statistics is through education. This is the first of a six-part series of articles intended to provide the reader with a basic, yet fundamental knowledge of common biomedical statistical methods. The series will cover basic concepts of statistical analysis, descriptive statistics, statistical inference theory, comparison of means, X2, and correlational and regression techniques. A conceptual explanation will accompany discussion of the appropriate use of these techniques. [Gaddis ML, Gaddis GM: Introduction to biostatistics: Part I, basic concepts. Ann Emerg Med January 1990;19:86-89.] INTRODUCTION The practicing physician must remain abreast of new information and techniques and meet standards for continuing medical education. A major source of new information is the medical journal. Given the competition involved in achieving publication of research, as well as the review and editing process, the reader might justifiably assume that published studies are correct in their methodology, regardless of the conclusions made. Although this assumption seems reasonable, it is not correct. Between 1979 and 1984, 42% to 78% of original publications from selected medical journals used inappropriate statistical analysis methods. 1-5 Statistical analysis involves organization and mathematical manipulation of data. This process can describe characteristics studied and help to infer conclusions from the data, thus guiding the acceptance or rejection of a given treatment or theory. Incorrect data analysis is a grave error in the research process, often leading to inappropriate conclusions, continued study of erroneous hypotheses, and curtailed study of viable therapies and therapeutic adjuncts. Additional dangers ensue when a physician uses nonefficacious treatment on a patient. From this, it becomes apparent that a basic knowledge of statistics can be an important tool for any clinician, whether in performing research or simply reading about it. However, statistical analysis of data is a task commonly delegated to statistical consultants. This is often justified by citing that there are those more qualified to perform this function than the principal investigator of a study. Though this seems logical, the principal investigator remains the individual ultimately responsible for the content and conclusions of a research project. The principal investigator cannot effectively meet this obligation fully if he does not have an adequate working knowledge of biomedical statistics. The investigator cannot plead innocence through ignorance when serious errors are made. Thus, there is an obvious need for statistical education among clinicians, not only to provide for a better understanding when reading the biomedical literature but also to aid medical researchers in communication with consulting statisticians and for selection of appropriate data analysis techniques.
19:1 January 1990
Annals of Emergency Medicine
Monica L Gaddis, PhD Gary M Gaddis, MD, PhD Kansas City, Missouri From the Departments of Surgery and Emergency Health Services, Truman Medical Center, University of Missouri, Kansas City. Received for publication September 1, 1989. Accepted for publication October 4, 1989. Address for reprints: Monica L Gaddis, PhD, Department of Surgery, Truman Medical Center, 2301 Holmes, Kansas City, Missouri 64108.
86/143
BIOSTATISTICS Gaddis & Gaddis
FIGURE I. A bell-shaped or normal
distribution curve. FIGURE 2. Frequency distributions: A, bimodal; B, rectangular; C, positively skewed; D, negatively skewed. (Hopkins KD, Glass GV: Basic Statistics for the Behavioral Sciences. Englewood Cliffs, N e w Jersey, PrenticeHall Inc, 1978, p 36.) S T A T E M E N T OF PURPOSE It is our purpose to present a sixpart series discussing the basics of biomedical statistics, with the intent to familiarize the reader with the terminology and appropriate use of data analysis techniques commonly used in original papers published in Annals of Emergency M e d i c i n e and o t h e r c l i n i c a l m e d i c a l journals. Learning will be directed toward a conceptual understanding of statistical analysis m e t h o d s rather than computational exercises. Part 1, in this issue of Annals, presents some basic concepts of statistical analysis. Knowledge of these building blocks is necessary for a complete understanding of biomedical statistics and the topics of future articles in this series. Part 2 will address descriptive statistics. These include measures of central tendency (mean, median, and mode), measures of variability (standard deviation and standard error of the mean), and confidence intervals. Appropriate uses of these will be discussed. Part 3 will introduce statistical inference theory. An explanation of the concept of hypothesis testing, definition of the probability value P, a discussion of the terms alpha, beta, and power, and a discussion of clinical versus statistical significance will be included. Sensitivity, specificity, and p r e d i c t i v e value will also be addressed. Part 4 will present parametric and nonparametric methods used for the comparison of means. Included will be analysis of variance (ANOVA), the Student's t test, the Mann-Whitney U test, and m e t h o d s of multiple comparisons. Part 5 will present a discussion of X2 and Fisher's exact tests. Both statistical methods should be understood by readers of biomedical research involving a study of the effic a c y of e x p e r i m e n t a l m e d i c a l treatments. 144/87
1
B,
I
I
o
2 Part 6 will give a basic discussion of correlation and regression, along with the series' concluding remarks. SAMPLE V E R S U S POPULATION It is clearly impossible for most scientific studies to survey all individuals of a group about which conclusions are to be drawn. Cost and time considerations force the researcher to settle for studying a subset of a group in order to form conclusions about the entire group. For example, if one wished to know the systolic, diastolic, and mean blood pressures of the entire population of the United States, this testing could be done as part of the next census, in which all members of the population would be surveyed. However, the cost of training census takers for this time-consuming task and the need to hire additional census takers because of the increased time required to check each individual would make sampling the entire population an impractical task. Similarly, in biomedical research, time and financial costs preclude study of every member of the population. However, if a representative sample of an appropriate population can be obtained and studied, conclusions Annals of Emergency Medicine
about the sample can be properly extrapolated to the defined population. The key to obtaining a representative sample of that population lies in random selection of a study sample from the applicable population. Randomization can be accomplished by sample subject selection using a random number table, drawing numbers or names out of a hat, or the like. Random sampling implies that all individuals in a population have an equal chance to be included in the sample. It is when random allocation of study subjects to treatment groups is violated that bias is introduced. Bias can easily lead to erroneous conclusions because control and treatment groups may inherently differ in relevant characteristics before the study is initiated. Therefore, post-treatment differences between groups can erroneously be ascribed to an effect of the experimental treatment! Improper allocation of subjects to c o n trol and treatment groups remains a significant problem confounding current biomedical research. DATA SCALES Before an analysis method can be selected, the type of data that will be generated by the research process must be defined. Data will fit a nom19:1 January 1990
F I G U R E 3. Summary of biostatisti-
Statistical The organization and mathematical manipulation of analysis data, used to describe characteristics studied and/or Population
to help infer conclusions from the data A large group possessing a given characteristic or set of characteristics. A population may be finite (the states of the United States) or infinite (blood pressure measurements of all infants born in New York)6
The studied subset of members of a defined population 6 The process of selection of a sample from a Random sample population whereby each member of the population has an equal and independent chance of being chosen 6 Nominal scale Numbers are arbitrarily assigned to characteristics for data classification Ordinal scale Numbers are used to denote rank-order, without defining a magnitude of difference between numbers Interval scale Numbers denote units of equal magnitude as well as rank order on a scale without an absolute zero Ratio scale Numbers denote units of equal magnitude as well as rank order on a scale with an absolute zero Distribution The systematic organization of a collection of data Normal A grouping of data that is graphically symmetrical and distribution bell shaped. Many human anatomic and physiologic characteristics are normally distributed Parametric Used when the data studied are from a sample or methods population that is normally distributed. Data must be of an interval or ratio scale
Sample
Nonparametric methods
Used when the data studied are from a sample or population that deviates from a normal distribution. Ordinal data are analyzed using nonparametric methods of analysis
inal, ordinal, interval, or ratio scale.
Nominal Scale N o m i n a l is the m o s t p r i m i t i v e of t h e data scales. I n f o r m a t i o n classified b y an a s s i g n e d n u m b e r or code to m a k e the data n u m e r i c fits a n o m i n a l scale. For instance, gender m a y be defined as 1 for " f e m a l e " and 2 for " m a l e . " Clinical diagnoses m a y also be assigned r e p r e s e n t a t i v e n u m b e r s , s u c h as 1 for " r e n a l failure," 2 for "congestive heart failure," 3 for "diabetes," and so forth. It m u s t be emphasized that the numbers selected are purely arbitrary and are chosen at the discretion of the researcher witho u t regard to any order of ranking of severity.6, 7
Ordinal Scale Ordinal scale data can be ranked in a specific order, be it low to high or h i g h to low. A n e x a m p l e w o u l d be data from a questionnaire in w h i c h a 19:1 January 1990
response of "strongly agree" is scored as 5, "agree" is scored 4, "no opini o n " is scored 3, "disagree" is scored 2, and "strongly disagree" is scored 1. In t h i s e x a m p l e , t h e r e s p o n s e s are s c o r e d on a c o n t i n u u m , w i t h o u t a c o n s i s t e n t level of m a g n i t u d e of differences between ranks. However, u n l i k e the n o m i n a l scale, n u m b e r s of an ordinal scale are not arbitrary because the order of n u m b e r s is meaningful. 6 For instance, in the above example, p r o g r e s s i v e l y larger n u m b e r s i n d i c a t e p r o g r e s s i v e l y greater agreement with the question presented. T h e G l a s g o w C o m a Score s allots a progressively decreased classification number that implies progressively worsened obtundation. Other examp l e s of o r d i n a l s c a l e s f a m i l i a r to many emergency physicians would i n c l u d e t h e T r a u m a Score 9 and the Injury Severity Score. 1° A v e r a g e v a l u e s of o r d i n a l s c a l e data are often calculated but are usu-
Annals of Emergency Medicine
cal terms. ally misleading because of the lack of any c o n s i s t e n t m a g n i t u d e of differe n c e b e t w e e n u n i t s of t h e scale, u Also, it is n o t u n c o m m o n in t r a u m a o u t c o m e studies to see such data reported with standard deviation values. C a l c u l a t i o n of such statistics f r o m n o n p a r a m e t r i c , and t h u s nonn o r m a l l y d i s t r i b u t e d data, is h i g h l y questionable because use of t h e standard deviation assumes that the data are n o r m a l l y distributed. H
Interval Scale Interval scale data are a step m o r e s o p h i s t i c a t e d t h a n ordinal scale data. N o t only is there a p r e d e t e r m i n e d order to the n u m b e r i n g of the scale, but also t h e r e is a c o n s i s t e n t l e v e l of m a g n i t u d e of difference described bet w e e n the observed data units. 6 Interval data also have a clearly defined u n i t of measure. However, " t h e zero p o i n t on t h e scale is arbitrary, and does n o t c o r r e s p o n d to a t o t a l absence of t h e characteristic m e a s u r e d . . . . ,,6 T h e Farenheit scale for temperature is interval in nature, as the n u m b e r i n g of the scale is consistent, y e t the zero v a l u e is arbitrary. Because of t h e consistent n u m b e r i n g of t h e scale a n d e q u a l m a g n i t u d e bet w e e n m e a s u r e m e n t u n i t s , average values and measures of variability of i n t e r v a l scale d a t a are m e a n i n g f u l . A n interval scale can be converted to an ordinal scale to show ranking, but an ordinal scale ordinarily cannot be converted to an interval scale. 6
Ratio Scale T h e ratio scale is s i m p l y an interval scale w i t h an a b s o l u t e zero. 6 A p r e d e t e r m i n e d order to the numbering of the scale is present, as is a cons i s t e n t level of m a g n i t u d e b e t w e e n each u n i t of measure. Ratio data can be c o n v e r t e d to o r d i n a l data. H e a r t rate, blood pressure, distance, time, and degrees Kelvin r e p r e s e n t exampies of ratio scale data.
DISTRIBUTIONS O n c e d a t a are collected, t h e y can be organized into a d i s t r i b u t i o n , or g r a p h of f r e q u e n c y of o c c u r r e n c e . T h i s is a v i s u a l l y d e s c r i p t i v e t o o l that allows the researcher to begin to define and analyze data. Figure 1 is a theoretical frequency d i s t r i b u t i o n of resting h e a r t rate of 88/145
BIOSTATISTICS Gaddis & Gaddis
e m e r g e n c y p h y s i c i a n s . Its shape is s y m m e t r i c a l and bell-shaped, and is defined as the " n o r m a l distribution." M a n y h u m a n a n a t o m i c and physiologic c h a r a c t e r i s t i c s approach the normal distribution.6 Other terminology used synonymously with "normal distribution" includes G a u s s i a n curve, curve of error, and n o r m a l probability curve. A n understanding of the characteristics of the n o r m a l d i s t r i b u t i o n is f u n d a m e n t a l i n the d e v e l o p m e n t of even a basic l~nowledge of biostatistics. This topic will be discussed in greater detail in Part 2 of this series. Other distributions are depicted in Figure 2. 6 Bimodal distributions have two peaks of cluster, or areas w i t h a high frequency level. For example, if the weights of A m e r i c a n adults were plotted, there w o u l d be two definitive points of cluster, one for female weight and one for male weight. Data that are rectangularly distributed show equal frequency of occurrence for all levels of a characteristic. The date of birth (month and day) of a sample of all p a t i e n t s seen i n an e m e r g e n c y d e p a r t m e n t approaches this distribution pattern. Skewed data are those that tail off to either the high or low end of meas u r e m e n t units. The a n n u a l i n c o m e of patients seen i n the ED of an inner-city c o m m u n i t y hospital will be defined by a distribution that is posi t i v e l y skewed, s h o w i n g a high frequency of lower a n n u a l incomes. A negatively skewed distribution has a cluster of data on the high end of the u n i t scale and tails off toward the low end. 6 G i v e n the nature of data collected in medical science research, the norm a l d i s t r i b u t i o n is t h e o n e w i t h which the clinician will be most familiar. Most statistical methods ap-
146/89
plied to interval or ratio data assume t h a t the data are n o r m a l l y distributed. It is w h e n this a s s u m p t i o n is violated that significant controversy exists i n the statistical c o m m u n i t y regarding t h e proper a p p l i c a t i o n of statistical tests.
PARAMETRIC VERSUS NONPARAMETRIC METHODS Statistical methods are defined as being p a r a m e t r i c or n o n p a r a m e t r i c . T h e type of a n a l y s i s m e t h o d t h a t should be selected is d e p e n d e n t on the nature of the data to be analyzed. Parametric methods use data extrapolated from a sample of the population studied to n u m e r i c a l l y describe some characteristic of a population. P a r a m e t r i c m e t h o d s are v a l i d o n l y w h e n t h a t characteristic follows or nearly follows the n o r m a l distribut i o n i n t h e p o p u l a t i o n s t u d i e d . 12 Thus, parametric methods can be applied properly to m o s t i n t e r v a l and r a t i o scale data w h e n t h o s e data come from a sample of a n o r m a l l y distributed population. Parametric methods can be used to derive measures of c e n t r a l t e n d e n c y and variability, w h i c h will be discussed i n Part 2 of the series. N o n p a r a m e t r i c m e t h o d s are app l i e d to n o n - n o r m a l l y d i s t r i b u t e d data and/or data that do not meet the c r i t e r i a for t h e u s e of p a r a m e t r i c m e t h o d s . D a t a t h a t fit the o r d i n a l scale d e f i n i t i o n should be analyzed by n o n p a r a m e t r i c m e t h o d s . Examples i n c l u d e Glasgow C o m a Scale, 8 T r a u m a Score, 9 the Injury Severity Score, lo a n d o t h e r s i m i l a r o r d i n a l scales data.
SUMMARY This has been the introductory ins t a l l m e n t of a series of articles outlining the proper use of biostatistics.
Annals of Emergency Medicine
We have introduced some basic concepts regarding data and its classification, w h i c h will be needed for understanding of topics to be presented s u b s e q u e n t l y (Figure 3). Measures of central tendency, m e a s u r e s of variability, confidence intervals, and the a p p r o p r i a t e use of t h e s e s t a t i s t i c a l concepts will be discussed i n part 2 of this series.
REFERENCES 1. Glantz SA: Biostatistics: How to detect, correct, and prevent errors in the medical literature. Circulation 1980;61:1-7. 2. FelsonDT, Cupples LA, Meenan RF: Misuse of statistical methods in arthritis and rheumatism: 1982 versus 1967-68. Arthritis R h e u m 1984;27:1018-1022. 3. Thorn MD, Pulliam CC, Symons MJ, et al: Statistical and research quality of the medical and pharmacy literature. A m J Hosp Pharm 1985;42:1077-1082. 4. Avram MJ, Shanks CA, Dykes MHM, et ah Statistical methods in anesthesia articles: An evaluationof two Americanjournalsduringtwo six-month periods. A n e s t h Analg 1985;64: 607-611. 5. MacArthur RD, Jackson GG: An evaluation of the use of statistical methodologyin the Journal of Infectious Diseases. J Infect Dis 1984; 149:349~354. 6. Hopkins KD, Glass GV: Basic Statistics for the Behavioral Sciences. EnglewoodCliffs, New Jersey, Prentice-HallInc, 1978, 7. Elenbaas RM, ElenbaasJK, Cuddy PG: Evaluating the medical literature. Part II: Statistical analysis. Ann Emerg Med 1983;12:610-620. 8. Teasdale G, Jennett B: Assessment of coma and impaired consciousness:A practical scale. Lancet 1974;2:81-84. 9. Champion HR, 8acco WJ, Carnazzo AJ, et al: Trauma score. Crit Care Med 1981;9:672-676. 10. Baker SP, O'Neill B, HaddonW Jr, et al: The Injury Severity Score: A method for describing patients with multiple injuries and evaluating emergency care. I Trauma 1974;14:187q96. ll. Sokal RR, Rohlf FJ: Biometry, ed 2. New York, WH Freeman and Co, 198!, pp 39-52. 12. Glantz SA: Primer of Biostatistics, ed 2. New York, McGraw-HillBook Co, 1987.
19:1 January 1990
Introduction to Biostatistics: Part 1, Basic Concepts Statistical methods commonly used to analyze data presented in journal articles should be understood by both medical scientists and practicing clinicians. Inappropriate data analysis methods have been reported in 42% to 78% of original publications in critical reviews of selected medical journals. The only way to halt researchers' misuse of statistics and improve the clinician's knowledge of statistics is through education. This is the first of a six-part series of articles intended to provide the reader with a basic, yet fundamental knowledge of common biomedical statistical methods. The series will cover basic concepts of statistical analysis, descriptive statistics, statistical inference theory, comparison of means, X2, and correlational and regression techniques. A conceptual explanation will accompany discussion of the appropriate use of these techniques. [Gaddis ML, Gaddis GM: Introduction to biostatistics: Part I, basic concepts. Ann Emerg Med January 1990;19:86-89.] INTRODUCTION The practicing physician must remain abreast of new information and techniques and meet standards for continuing medical education. A major source of new information is the medical journal. Given the competition involved in achieving publication of research, as well as the review and editing process, the reader might justifiably assume that published studies are correct in their methodology, regardless of the conclusions made. Although this assumption seems reasonable, it is not correct. Between 1979 and 1984, 42% to 78% of original publications from selected medical journals used inappropriate statistical analysis methods. 1-5 Statistical analysis involves organization and mathematical manipulation of data. This process can describe characteristics studied and help to infer conclusions from the data, thus guiding the acceptance or rejection of a given treatment or theory. Incorrect data analysis is a grave error in the research process, often leading to inappropriate conclusions, continued study of erroneous hypotheses, and curtailed study of viable therapies and therapeutic adjuncts. Additional dangers ensue when a physician uses nonefficacious treatment on a patient. From this, it becomes apparent that a basic knowledge of statistics can be an important tool for any clinician, whether in performing research or simply reading about it. However, statistical analysis of data is a task commonly delegated to statistical consultants. This is often justified by citing that there are those more qualified to perform this function than the principal investigator of a study. Though this seems logical, the principal investigator remains the individual ultimately responsible for the content and conclusions of a research project. The principal investigator cannot effectively meet this obligation fully if he does not have an adequate working knowledge of biomedical statistics. The investigator cannot plead innocence through ignorance when serious errors are made. Thus, there is an obvious need for statistical education among clinicians, not only to provide for a better understanding when reading the biomedical literature but also to aid medical researchers in communication with consulting statisticians and for selection of appropriate data analysis techniques.
19:1 January 1990
Annals of Emergency Medicine
Monica L Gaddis, PhD Gary M Gaddis, MD, PhD Kansas City, Missouri From the Departments of Surgery and Emergency Health Services, Truman Medical Center, University of Missouri, Kansas City. Received for publication September 1, 1989. Accepted for publication October 4, 1989. Address for reprints: Monica L Gaddis, PhD, Department of Surgery, Truman Medical Center, 2301 Holmes, Kansas City, Missouri 64108.
86/143
BIOSTATISTICS Gaddis & Gaddis
FIGURE I. A bell-shaped or normal
distribution curve. FIGURE 2. Frequency distributions: A, bimodal; B, rectangular; C, positively skewed; D, negatively skewed. (Hopkins KD, Glass GV: Basic Statistics for the Behavioral Sciences. Englewood Cliffs, N e w Jersey, PrenticeHall Inc, 1978, p 36.) S T A T E M E N T OF PURPOSE It is our purpose to present a sixpart series discussing the basics of biomedical statistics, with the intent to familiarize the reader with the terminology and appropriate use of data analysis techniques commonly used in original papers published in Annals of Emergency M e d i c i n e and o t h e r c l i n i c a l m e d i c a l journals. Learning will be directed toward a conceptual understanding of statistical analysis m e t h o d s rather than computational exercises. Part 1, in this issue of Annals, presents some basic concepts of statistical analysis. Knowledge of these building blocks is necessary for a complete understanding of biomedical statistics and the topics of future articles in this series. Part 2 will address descriptive statistics. These include measures of central tendency (mean, median, and mode), measures of variability (standard deviation and standard error of the mean), and confidence intervals. Appropriate uses of these will be discussed. Part 3 will introduce statistical inference theory. An explanation of the concept of hypothesis testing, definition of the probability value P, a discussion of the terms alpha, beta, and power, and a discussion of clinical versus statistical significance will be included. Sensitivity, specificity, and p r e d i c t i v e value will also be addressed. Part 4 will present parametric and nonparametric methods used for the comparison of means. Included will be analysis of variance (ANOVA), the Student's t test, the Mann-Whitney U test, and m e t h o d s of multiple comparisons. Part 5 will present a discussion of X2 and Fisher's exact tests. Both statistical methods should be understood by readers of biomedical research involving a study of the effic a c y of e x p e r i m e n t a l m e d i c a l treatments. 144/87
1
B,
I
I
o
2 Part 6 will give a basic discussion of correlation and regression, along with the series' concluding remarks. SAMPLE V E R S U S POPULATION It is clearly impossible for most scientific studies to survey all individuals of a group about which conclusions are to be drawn. Cost and time considerations force the researcher to settle for studying a subset of a group in order to form conclusions about the entire group. For example, if one wished to know the systolic, diastolic, and mean blood pressures of the entire population of the United States, this testing could be done as part of the next census, in which all members of the population would be surveyed. However, the cost of training census takers for this time-consuming task and the need to hire additional census takers because of the increased time required to check each individual would make sampling the entire population an impractical task. Similarly, in biomedical research, time and financial costs preclude study of every member of the population. However, if a representative sample of an appropriate population can be obtained and studied, conclusions Annals of Emergency Medicine
about the sample can be properly extrapolated to the defined population. The key to obtaining a representative sample of that population lies in random selection of a study sample from the applicable population. Randomization can be accomplished by sample subject selection using a random number table, drawing numbers or names out of a hat, or the like. Random sampling implies that all individuals in a population have an equal chance to be included in the sample. It is when random allocation of study subjects to treatment groups is violated that bias is introduced. Bias can easily lead to erroneous conclusions because control and treatment groups may inherently differ in relevant characteristics before the study is initiated. Therefore, post-treatment differences between groups can erroneously be ascribed to an effect of the experimental treatment! Improper allocation of subjects to c o n trol and treatment groups remains a significant problem confounding current biomedical research. DATA SCALES Before an analysis method can be selected, the type of data that will be generated by the research process must be defined. Data will fit a nom19:1 January 1990
F I G U R E 3. Summary of biostatisti-
Statistical The organization and mathematical manipulation of analysis data, used to describe characteristics studied and/or Population
to help infer conclusions from the data A large group possessing a given characteristic or set of characteristics. A population may be finite (the states of the United States) or infinite (blood pressure measurements of all infants born in New York)6
The studied subset of members of a defined population 6 The process of selection of a sample from a Random sample population whereby each member of the population has an equal and independent chance of being chosen 6 Nominal scale Numbers are arbitrarily assigned to characteristics for data classification Ordinal scale Numbers are used to denote rank-order, without defining a magnitude of difference between numbers Interval scale Numbers denote units of equal magnitude as well as rank order on a scale without an absolute zero Ratio scale Numbers denote units of equal magnitude as well as rank order on a scale with an absolute zero Distribution The systematic organization of a collection of data Normal A grouping of data that is graphically symmetrical and distribution bell shaped. Many human anatomic and physiologic characteristics are normally distributed Parametric Used when the data studied are from a sample or methods population that is normally distributed. Data must be of an interval or ratio scale
Sample
Nonparametric methods
Used when the data studied are from a sample or population that deviates from a normal distribution. Ordinal data are analyzed using nonparametric methods of analysis
inal, ordinal, interval, or ratio scale.
Nominal Scale N o m i n a l is the m o s t p r i m i t i v e of t h e data scales. I n f o r m a t i o n classified b y an a s s i g n e d n u m b e r or code to m a k e the data n u m e r i c fits a n o m i n a l scale. For instance, gender m a y be defined as 1 for " f e m a l e " and 2 for " m a l e . " Clinical diagnoses m a y also be assigned r e p r e s e n t a t i v e n u m b e r s , s u c h as 1 for " r e n a l failure," 2 for "congestive heart failure," 3 for "diabetes," and so forth. It m u s t be emphasized that the numbers selected are purely arbitrary and are chosen at the discretion of the researcher witho u t regard to any order of ranking of severity.6, 7
Ordinal Scale Ordinal scale data can be ranked in a specific order, be it low to high or h i g h to low. A n e x a m p l e w o u l d be data from a questionnaire in w h i c h a 19:1 January 1990
response of "strongly agree" is scored as 5, "agree" is scored 4, "no opini o n " is scored 3, "disagree" is scored 2, and "strongly disagree" is scored 1. In t h i s e x a m p l e , t h e r e s p o n s e s are s c o r e d on a c o n t i n u u m , w i t h o u t a c o n s i s t e n t level of m a g n i t u d e of differences between ranks. However, u n l i k e the n o m i n a l scale, n u m b e r s of an ordinal scale are not arbitrary because the order of n u m b e r s is meaningful. 6 For instance, in the above example, p r o g r e s s i v e l y larger n u m b e r s i n d i c a t e p r o g r e s s i v e l y greater agreement with the question presented. T h e G l a s g o w C o m a Score s allots a progressively decreased classification number that implies progressively worsened obtundation. Other examp l e s of o r d i n a l s c a l e s f a m i l i a r to many emergency physicians would i n c l u d e t h e T r a u m a Score 9 and the Injury Severity Score. 1° A v e r a g e v a l u e s of o r d i n a l s c a l e data are often calculated but are usu-
Annals of Emergency Medicine
cal terms. ally misleading because of the lack of any c o n s i s t e n t m a g n i t u d e of differe n c e b e t w e e n u n i t s of t h e scale, u Also, it is n o t u n c o m m o n in t r a u m a o u t c o m e studies to see such data reported with standard deviation values. C a l c u l a t i o n of such statistics f r o m n o n p a r a m e t r i c , and t h u s nonn o r m a l l y d i s t r i b u t e d data, is h i g h l y questionable because use of t h e standard deviation assumes that the data are n o r m a l l y distributed. H
Interval Scale Interval scale data are a step m o r e s o p h i s t i c a t e d t h a n ordinal scale data. N o t only is there a p r e d e t e r m i n e d order to the n u m b e r i n g of the scale, but also t h e r e is a c o n s i s t e n t l e v e l of m a g n i t u d e of difference described bet w e e n the observed data units. 6 Interval data also have a clearly defined u n i t of measure. However, " t h e zero p o i n t on t h e scale is arbitrary, and does n o t c o r r e s p o n d to a t o t a l absence of t h e characteristic m e a s u r e d . . . . ,,6 T h e Farenheit scale for temperature is interval in nature, as the n u m b e r i n g of the scale is consistent, y e t the zero v a l u e is arbitrary. Because of t h e consistent n u m b e r i n g of t h e scale a n d e q u a l m a g n i t u d e bet w e e n m e a s u r e m e n t u n i t s , average values and measures of variability of i n t e r v a l scale d a t a are m e a n i n g f u l . A n interval scale can be converted to an ordinal scale to show ranking, but an ordinal scale ordinarily cannot be converted to an interval scale. 6
Ratio Scale T h e ratio scale is s i m p l y an interval scale w i t h an a b s o l u t e zero. 6 A p r e d e t e r m i n e d order to the numbering of the scale is present, as is a cons i s t e n t level of m a g n i t u d e b e t w e e n each u n i t of measure. Ratio data can be c o n v e r t e d to o r d i n a l data. H e a r t rate, blood pressure, distance, time, and degrees Kelvin r e p r e s e n t exampies of ratio scale data.
DISTRIBUTIONS O n c e d a t a are collected, t h e y can be organized into a d i s t r i b u t i o n , or g r a p h of f r e q u e n c y of o c c u r r e n c e . T h i s is a v i s u a l l y d e s c r i p t i v e t o o l that allows the researcher to begin to define and analyze data. Figure 1 is a theoretical frequency d i s t r i b u t i o n of resting h e a r t rate of 88/145
BIOSTATISTICS Gaddis & Gaddis
e m e r g e n c y p h y s i c i a n s . Its shape is s y m m e t r i c a l and bell-shaped, and is defined as the " n o r m a l distribution." M a n y h u m a n a n a t o m i c and physiologic c h a r a c t e r i s t i c s approach the normal distribution.6 Other terminology used synonymously with "normal distribution" includes G a u s s i a n curve, curve of error, and n o r m a l probability curve. A n understanding of the characteristics of the n o r m a l d i s t r i b u t i o n is f u n d a m e n t a l i n the d e v e l o p m e n t of even a basic l~nowledge of biostatistics. This topic will be discussed in greater detail in Part 2 of this series. Other distributions are depicted in Figure 2. 6 Bimodal distributions have two peaks of cluster, or areas w i t h a high frequency level. For example, if the weights of A m e r i c a n adults were plotted, there w o u l d be two definitive points of cluster, one for female weight and one for male weight. Data that are rectangularly distributed show equal frequency of occurrence for all levels of a characteristic. The date of birth (month and day) of a sample of all p a t i e n t s seen i n an e m e r g e n c y d e p a r t m e n t approaches this distribution pattern. Skewed data are those that tail off to either the high or low end of meas u r e m e n t units. The a n n u a l i n c o m e of patients seen i n the ED of an inner-city c o m m u n i t y hospital will be defined by a distribution that is posi t i v e l y skewed, s h o w i n g a high frequency of lower a n n u a l incomes. A negatively skewed distribution has a cluster of data on the high end of the u n i t scale and tails off toward the low end. 6 G i v e n the nature of data collected in medical science research, the norm a l d i s t r i b u t i o n is t h e o n e w i t h which the clinician will be most familiar. Most statistical methods ap-
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plied to interval or ratio data assume t h a t the data are n o r m a l l y distributed. It is w h e n this a s s u m p t i o n is violated that significant controversy exists i n the statistical c o m m u n i t y regarding t h e proper a p p l i c a t i o n of statistical tests.
PARAMETRIC VERSUS NONPARAMETRIC METHODS Statistical methods are defined as being p a r a m e t r i c or n o n p a r a m e t r i c . T h e type of a n a l y s i s m e t h o d t h a t should be selected is d e p e n d e n t on the nature of the data to be analyzed. Parametric methods use data extrapolated from a sample of the population studied to n u m e r i c a l l y describe some characteristic of a population. P a r a m e t r i c m e t h o d s are v a l i d o n l y w h e n t h a t characteristic follows or nearly follows the n o r m a l distribut i o n i n t h e p o p u l a t i o n s t u d i e d . 12 Thus, parametric methods can be applied properly to m o s t i n t e r v a l and r a t i o scale data w h e n t h o s e data come from a sample of a n o r m a l l y distributed population. Parametric methods can be used to derive measures of c e n t r a l t e n d e n c y and variability, w h i c h will be discussed i n Part 2 of the series. N o n p a r a m e t r i c m e t h o d s are app l i e d to n o n - n o r m a l l y d i s t r i b u t e d data and/or data that do not meet the c r i t e r i a for t h e u s e of p a r a m e t r i c m e t h o d s . D a t a t h a t fit the o r d i n a l scale d e f i n i t i o n should be analyzed by n o n p a r a m e t r i c m e t h o d s . Examples i n c l u d e Glasgow C o m a Scale, 8 T r a u m a Score, 9 the Injury Severity Score, lo a n d o t h e r s i m i l a r o r d i n a l scales data.
SUMMARY This has been the introductory ins t a l l m e n t of a series of articles outlining the proper use of biostatistics.
Annals of Emergency Medicine
We have introduced some basic concepts regarding data and its classification, w h i c h will be needed for understanding of topics to be presented s u b s e q u e n t l y (Figure 3). Measures of central tendency, m e a s u r e s of variability, confidence intervals, and the a p p r o p r i a t e use of t h e s e s t a t i s t i c a l concepts will be discussed i n part 2 of this series.
REFERENCES 1. Glantz SA: Biostatistics: How to detect, correct, and prevent errors in the medical literature. Circulation 1980;61:1-7. 2. FelsonDT, Cupples LA, Meenan RF: Misuse of statistical methods in arthritis and rheumatism: 1982 versus 1967-68. Arthritis R h e u m 1984;27:1018-1022. 3. Thorn MD, Pulliam CC, Symons MJ, et al: Statistical and research quality of the medical and pharmacy literature. A m J Hosp Pharm 1985;42:1077-1082. 4. Avram MJ, Shanks CA, Dykes MHM, et ah Statistical methods in anesthesia articles: An evaluationof two Americanjournalsduringtwo six-month periods. A n e s t h Analg 1985;64: 607-611. 5. MacArthur RD, Jackson GG: An evaluation of the use of statistical methodologyin the Journal of Infectious Diseases. J Infect Dis 1984; 149:349~354. 6. Hopkins KD, Glass GV: Basic Statistics for the Behavioral Sciences. EnglewoodCliffs, New Jersey, Prentice-HallInc, 1978, 7. Elenbaas RM, ElenbaasJK, Cuddy PG: Evaluating the medical literature. Part II: Statistical analysis. Ann Emerg Med 1983;12:610-620. 8. Teasdale G, Jennett B: Assessment of coma and impaired consciousness:A practical scale. Lancet 1974;2:81-84. 9. Champion HR, 8acco WJ, Carnazzo AJ, et al: Trauma score. Crit Care Med 1981;9:672-676. 10. Baker SP, O'Neill B, HaddonW Jr, et al: The Injury Severity Score: A method for describing patients with multiple injuries and evaluating emergency care. I Trauma 1974;14:187q96. ll. Sokal RR, Rohlf FJ: Biometry, ed 2. New York, WH Freeman and Co, 198!, pp 39-52. 12. Glantz SA: Primer of Biostatistics, ed 2. New York, McGraw-HillBook Co, 1987.
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