This document discusses linear regression analysis, a statistical method used to analyze relationships between variables. It can be used to describe, estimate, and predict relationships. The document provides an overview of linear regression, including how it models relationships between dependent and independent variables using equations. It also discusses important considerations for performing and interpreting linear regression analyses correctly. Examples are provided to illustrate key points.
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Similar to Linear regression (1). spss analiisa statistik
Correlation & Regression Analysis using SPSSParag Shah
Concept of Correlation, Simple Linear Regression & Multiple Linear Regression and its analysis using SPSS. How it check the validity of assumptions in Regression
Correlation and regression analysis are statistical methods used to determine if a relationship exists between variables and describe the nature of that relationship. A scatter plot graphs the independent and dependent variables and allows visualization of any trends in the data. The correlation coefficient measures the strength and direction of the linear relationship between variables, ranging from -1 to 1. Regression finds the linear "best fit" line that minimizes the residuals and can be used to predict dependent variable values.
Correlation and regression analysis are statistical methods used to determine if a relationship exists between variables and describe the nature of that relationship. A scatter plot graphs the independent and dependent variables and allows visualization of any trends in the data. The correlation coefficient measures the strength and direction of the linear relationship between variables, ranging from -1 to 1. Regression finds the linear "best fit" line that minimizes the residuals, or differences between observed and predicted dependent variable values. The coefficient of determination measures how much variation in the dependent variable is explained by the regression model.
Regression analysis is a statistical technique used to model relationships between variables. It allows one to predict the average value of a dependent variable based on the value of one or more independent variables. The key ideas are that the dependent variable is influenced by the independent variables in a linear or curvilinear fashion, and regression provides an equation to estimate the dependent variable given values of the independent variables. Common applications of linear regression include forecasting, determining relationships between variables, and estimating how changes in one variable impact another.
Frequency Measures Used in EpidemiologyIntroductionIn e.docxMARRY7
Frequency Measures Used in Epidemiology
Introduction
In epidemiological studies, many qualitative variables have only two possible categories, such as
Alive or dead
Case or control
Exposed and unexposed
The frequency measures for dichotomous variable include:
Ratio
Proportion
Rate
( All the above 3 measure are based on the same formula: )
Ratios, Proportion, and Rates Compared
In a ratio, the values of x and y may be completely independent from each other or x is a part of y
For example , the gender of the children attending a specific program could be compared in one of the following ways:
Proportion is a ratio in which X is included in Y
For example , the gender of the children attending a specific program
Rate is a proportion that measures the occurrence of an event in a population over time
Rate = X
Ratios, Proportion, and Rates Compared
Example 1: The following table was part of an article published by Dr. Mshana and his colleagues. The title of this study is “Outbreak of a novel Enterobacter sp. carrying blaCTX-M-15 in a neonatal unit of a tertiary care hospital in Tanzania. ". Please use this table to answer the following questions.
Source: Mshana SE, Gerwing L, Minde M, Hain T, Domann E, Lyamuya E, et al. Outbreak of a novel Enterobacter sp. carrying blaCTX-M-15 in a neonatal unit of a tertiary care hospital in Tanzania. International journal of antimicrobial agents. 2011;38(3):265-9.
4
Example 1
What is the ratio of males to females? 7 : 10
What proportion of premature babies? 12/17=0.706
What proportion of patients were discharged? 11/17=0.647
What is the ratio of prematurity to birth asphyxia ? 12 : 5
Source: Mshana SE, Gerwing L, Minde M, Hain T, Domann E, Lyamuya E, et al. Outbreak of a novel Enterobacter sp. carrying blaCTX-M-15 in a neonatal unit of a tertiary care hospital in Tanzania. International journal of antimicrobial agents. 2011;38(3):265-9.
5
Example 2:
In 1989, 733,151 new cases of gonorrhea were reported among the United States civilian population. The 1989 mid-year U.S. civilian population was estimated to be 246,552,000. What is the 1989 gonorrhea incidence rate for the U.S. civilian population? (For these data we will use a value of 105 for 10n ).
Answer:
Incidence rate = X
Incidence rate = X = 297.4 per 100,000
6
Measures of association:
They are used to quantify the relationship between exposure and disease among two groups
They are used to compare the disease occurrence among one group with the disease occurrence in the another group
They include the following measures based on the study design:
Risk Ratio (RR):
It also called relative risk
It is used to compare the risk of health related events in two groups
The following formula cis used to calculate the RR:
A risk ratio of 1.0 indicates identical risk in the two groups
A risk ratio greater than 1.0 indicates an increased risk for the numerator group
A risk ratio greater than 1.0 ...
This document provides an introduction to simple logistic regression in R. It discusses that logistic regression is useful for predicting a binary outcome based on predictor variables. The key points are:
1. Logistic regression can be used to predict the presence or absence of an outcome based on predictor variables, when the outcome is binary.
2. It is similar to linear regression but adapted for binary outcome variables.
3. Logistic regression is more robust than discriminant analysis and does not require the same strict assumptions. It can be used when assumptions of discriminant analysis are not met.
Reference/Article
Module 18: Correlational Research
Magnitude, Scatterplots, and Types of Relationships
Magnitude
Scatterplots
Positive Relationships
Negative Relationships
No Relationship
Curvilinear Relationships
Misinterpreting Correlations
The Assumptions of Causality and Directionality
The Third-Variable Problem
Restrictive Range
Curvilinear Relationships
Prediction and Correlation
Review of Key Terms
Module Exercises
Critical Thinking Check Answers
Module 19: Correlation Coefficients
The Pearson Product-Moment Correlation Coefficient: What It Is and What It Does
Calculating the Pearson Product-Moment Correlation
Interpreting the Pearson Product-Moment Correlation
Alternative Correlation Coefficients
Review of Key Terms
Module Exercises
Critical Thinking Check Answers
Module 20: Advanced Correlational Techniques: Regression Analysis
Regression Lines
Calculating the Slope and y-intercept
Prediction and Regression
Multiple Regression Analysis
Review of Key Terms
Module Exercises
Critical Thinking Check Answers
Chapter 9 Summary and Review
Chapter 9 Statistical Software Resources
In this chapter, we discuss correlational research methods and correlational statistics. As a research method, correlational designs allow us to describe the relationship between two measured variables. A correlation coefficient aids us by assigning a numerical value to the observed relationship. We begin with a discussion of how to conduct correlational research, the magnitude and the direction of correlations, and graphical representations of correlations. We then turn to special considerations when interpreting correlations, how to use correlations for predictive purposes, and how to calculate correlation coefficients. Lastly, we will discuss an advanced correlational technique, regression analysis.
MODULE 18
Correlational Research
Learning Objectives
•Describe the difference between strong, moderate, and weak correlation coefficients.
•Draw and interpret scatterplots.
•Explain negative, positive, curvilinear, and no relationship between variables.
•Explain how assuming causality and directionality, the third-variable problem, restrictive ranges, and curvilinear relationships can be problematic when interpreting correlation coefficients.
•Explain how correlations allow us to make predictions.
When conducting correlational studies, researchers determine whether two naturally occurring variables (for example, height and weight, or smoking and cancer) are related to each other. Such studies assess whether the variables are “co-related” in some way—do people who are taller tend to weigh more, or do those who smoke tend to have a higher incidence of cancer? As we saw in Chapter 1, the correlational method is a type of nonexperimental method that describes the relationship between two measured variables. In addition to describing a relationship, correlations also allow us to make predictions from one variable to another. If two variables are correlated, we can pred.
- Regression analysis is a statistical technique for modeling relationships between variables, where one variable is dependent on the others. It allows predicting the average value of the dependent variable based on the independent variables.
- The key assumptions of regression models are that the error terms are normally distributed with zero mean and constant variance, and are independent of each other.
- Linear regression specifies that the dependent variable is a linear combination of the parameters, though the independent variables need not be linearly related. In simple linear regression with one independent variable, the least squares estimates of the intercept and slope are calculated to minimize the sum of squared errors.
correlation or dependence is any statistical relationship, whether causal or not, between two random variables or bivariate data. Although in the broadest sense, "correlation" may indicate any type of association, in statistics it normally refers to the degree to which a pair of variables are linearly related. Familiar examples of dependent phenomena include the correlation between the height of parents and their offspring, and the correlation between the price of a good and the quantity the consumers are willing to purchase, as it is depicted in the so-called demand curve.
Correlations are useful because they can indicate a predictive relationship that can be exploited in practice. For example, an electrical utility may produce less power on a mild day based on the correlation between electricity demand and weather. In this example, there is a causal relationship, because extreme weather causes people to use more electricity for heating or cooling. However, in general, the presence of a correlation is not sufficient to infer the presence of a causal relationship (i.e., correlation does not imply causation).
Formally, random variables are dependent if they do not satisfy a mathematical property of probabilistic independence. In informal parlance, correlation is synonymous with dependence. However, when used in a technical sense, correlation refers to any of several specific types of mathematical operations between the tested variables and their respective expected values. Essentially, correlation is the measure of how two or more variables are related to one another. There are several correlation coefficients, often denoted
ρ
\rho or
r
r, measuring the degree of correlation. The most common of these is the Pearson correlation coefficient, which is sensitive only to a linear relationship between two variables (which may be present even when one variable is a nonlinear function of the other). Other correlation coefficients – such as Spearman's rank correlation – have been developed to be more robust than Pearson's, that is, more sensitive to nonlinear relationships.[1][2][3] Mutual information can also be applied to measure dependence between two variables.
This document defines regression analysis and its key concepts. Regression analysis is used to estimate or predict an unknown dependent variable from known independent variables. There are two types of variables: dependent and independent. Linear regression establishes a linear relationship between a continuous dependent variable and one or more continuous or discrete independent variables using a best-fit straight line. The regression equation represents this relationship as Y=a+bX+e, where a is the intercept, b is the slope, and e is the error term, which can be used to predict the dependent variable based on the independent variables. An example is provided to predict the number of new students joining a school based on the percentage of students returning from the previous year.
Covariance is a measure of the relationship between two random variables, indicating whether they tend to move together or in opposite directions. The covariance formula calculates the average of the products of the deviations from the mean for each variable. A simple linear regression model describes the linear relationship between a dependent variable and a single independent variable, estimating coefficients to find the best fitting straight line through the data points. Linear regression analysis uses observed data to relate a dependent variable to one or more independent variables and build a predictive equation.
- Multinomial logistic regression predicts categorical membership in a dependent variable based on multiple independent variables. It is an extension of binary logistic regression that allows for more than two categories.
- Careful data analysis including checking for outliers and multicollinearity is important. A minimum sample size of 10 cases per independent variable is recommended.
- Multinomial logistic regression does not assume normality, linearity or homoscedasticity like discriminant function analysis does, making it more flexible and commonly used. It does assume independence between dependent variable categories.
Assessment 2 ContextIn many data analyses, it is desirable.docxfestockton
Assessment 2 Context
In many data analyses, it is desirable to compute a coefficient of association. Coefficients of association are quantitative measures of the amount of relationship between two variables. Ultimately, most techniques can be reduced to a coefficient of association and expressed as the amount of relationship between the variables in the analysis. There are many types of coefficients of association. They express the mathematical association in different ways, usually based on assumptions about the data. The most common coefficient of association you will encounter is the Pearson product-moment correlation coefficient (symbolized as the italicized r), and it is the only coefficient of association that can safely be referred to as simply the "correlation coefficient". It is common enough so that if no other information is provided, it is reasonable to assume that is what is meant.
Correlation coefficients are numbers that give information about the strength of relationship between two variables, such as two different test scores from a sample of participants. The coefficient ranges from -1 through +1. Coefficients between 0 and +1 indicate a positive relationship between the two scores, such as high scores on one test tending to come from people with high scores on the second. The other possible relationship, which is every bit as useful, is a negative correlation between -1 and 0. A negative correlation possesses no less predictive power between the two scores. The difference is that high scores on one measure are associated with low scores on the other.
An example of the kinds of measures that might correlate negatively is absences and grades. People with higher absences will be expected to have lower grades. When a correlation is said to be significant, it can be shown that the correlation is significantly different form zero in the population. A correlation of zero means no relationship between variables. A correlation other than zero means the variables are related. As the coefficient gets further from zero (toward +1 or -1), the relationship becomes stronger.Interpreting Correlation: Magnitude and Sign
Interpreting a Pearson's correlation coefficient (rXY) requires an understanding of two concepts:
· Magnitude.
· Sign (+/-).
The magnitude refers to the strength of the linear relationship between Variable X and Variable
The rXY ranges in values from -1.00 to +1.00. To determine magnitude, ignore the sign of the correlation, and the absolute value of rXY indicates the extent to which Variable X and Variable Y are linearly related. For correlations close to 0, there is no linear relationship. As the correlation approaches either -1.00 or +1.00, the magnitude of the correlation increases. Therefore, for example, the magnitude of r = -.65 is greater than the magnitude of r = +.25 (|.65| > |.25|).
In contrast to magnitude, the sign of a non-zero correlation is either negative or positive.
These labels are not interpreted ...
Assessment 2 ContextIn many data analyses, it is desirable.docxgalerussel59292
Assessment 2 Context
In many data analyses, it is desirable to compute a coefficient of association. Coefficients of association are quantitative measures of the amount of relationship between two variables. Ultimately, most techniques can be reduced to a coefficient of association and expressed as the amount of relationship between the variables in the analysis. There are many types of coefficients of association. They express the mathematical association in different ways, usually based on assumptions about the data. The most common coefficient of association you will encounter is the Pearson product-moment correlation coefficient (symbolized as the italicized r), and it is the only coefficient of association that can safely be referred to as simply the "correlation coefficient". It is common enough so that if no other information is provided, it is reasonable to assume that is what is meant.
Correlation coefficients are numbers that give information about the strength of relationship between two variables, such as two different test scores from a sample of participants. The coefficient ranges from -1 through +1. Coefficients between 0 and +1 indicate a positive relationship between the two scores, such as high scores on one test tending to come from people with high scores on the second. The other possible relationship, which is every bit as useful, is a negative correlation between -1 and 0. A negative correlation possesses no less predictive power between the two scores. The difference is that high scores on one measure are associated with low scores on the other.
An example of the kinds of measures that might correlate negatively is absences and grades. People with higher absences will be expected to have lower grades. When a correlation is said to be significant, it can be shown that the correlation is significantly different form zero in the population. A correlation of zero means no relationship between variables. A correlation other than zero means the variables are related. As the coefficient gets further from zero (toward +1 or -1), the relationship becomes stronger.Interpreting Correlation: Magnitude and Sign
Interpreting a Pearson's correlation coefficient (rXY) requires an understanding of two concepts:
· Magnitude.
· Sign (+/-).
The magnitude refers to the strength of the linear relationship between Variable X and Variable
The rXY ranges in values from -1.00 to +1.00. To determine magnitude, ignore the sign of the correlation, and the absolute value of rXY indicates the extent to which Variable X and Variable Y are linearly related. For correlations close to 0, there is no linear relationship. As the correlation approaches either -1.00 or +1.00, the magnitude of the correlation increases. Therefore, for example, the magnitude of r = -.65 is greater than the magnitude of r = +.25 (|.65| > |.25|).
In contrast to magnitude, the sign of a non-zero correlation is either negative or positive.
These labels are not interpreted .
This document discusses correlation and defines it as the statistical relationship between two variables, where a change in one variable results in a corresponding change in the other. It describes different types of correlation including positive, negative, simple, partial and multiple. Methods for studying correlation are also outlined, including scatter diagrams and Karl Pearson's coefficient of correlation (represented by r), which quantifies the strength and direction of the linear relationship between two variables from -1 to 1. The coefficient of determination (r2) is also introduced, which expresses the proportion of variance in one variable that is predictable from the other.
This document discusses correlation analysis and different types of correlation. It defines correlation as a statistical analysis of the relationship between two or more variables. There are three main types of correlation discussed:
1. Positive correlation means that as one variable increases, the other also tends to increase. Negative correlation means that as one variable increases, the other tends to decrease.
2. Simple correlation analyzes the relationship between two variables, while multiple correlation analyzes three or more variables simultaneously. Partial correlation holds the effect of other variables constant.
3. Methods for measuring correlation include scatter diagrams, which graphically show the relationship, and algebraic formulas that calculate a correlation coefficient to quantify the strength and direction of the relationship.
This document discusses correlation and regression analysis. It defines correlation analysis as examining the relationship between two or more variables, and regression analysis as examining how one variable changes when another specific variable changes in volume. It covers positive and negative correlation, linear and non-linear correlation, and how to calculate the coefficient of correlation. Regression analysis and regression equations are introduced for using a known variable to predict an unknown variable. Examples are provided to illustrate key concepts.
Similar to Linear regression (1). spss analiisa statistik (20)
Causes Of Tooth Loss
PERIODONTAL PROBLEMS ( PERIODONTITIS, GINIGIVITIS)
Systemic Causes Of Tooth Loss
1. Diabetes Mellitus
2. Female Sexual Hormones Condition
3. Hyperpituitarism
4. Hyperthyroidism
5. Primary Hyperparathyroidism
6. Osteoporosis
7. Hypophosphatasia
8. Hypophosphatemia
Causes Of Tooth Loss
CARIES/ TOOTH DECAY
Causes Of Tooth Loss
CAUSES OF TOOTH LOSS
Consequence of tooth loss
Anatomic
Loss of ridge volume both height and width
Bone loss :
mandible > maxilla
Posteriorly > anteriorly
Anatomic consequences
Broader mandibular arch with constricting maxilary arch
Attached gingiva is replaced with less keratinised oral mucosa which is more readily traumatized.
Anatomic consequences
Tipping of the adjacent teeth
Supraeruption of the teeth
Traumatic occlusion
Premature occlusal contact
Anatomic Consequences
Anatomic Consequences
Physiologic consequences
Physiologic Consequences
Decreased lip support
Decreased lower facial height
Physiologic Consequences
Physiologic consequences
Education of Patient
Diagnosis, Treatment Planning, Design, Treatment, Sequencing, and Mouth Preparation
Support for Distal Extension Denture Bases
Establishment and Verification of Occlusal Relations and Tooth Arrangements
Initial Placement Procedures
Periodic Recall
Education of Patient
Informing a patient about a health matter to
secure informed consent.
Patient education should begin at the initial
contact with the patient and should continue throughout treatment.
The dentist and the patient share responsibility for the ultimate success of a removable partial denture.
This educational procedure is especially important when the treatment plan and prognosis are discussed with the patient.
Diagnosis, Treatment Planning, Design, Treatment, Sequencing, and Mouth Preparation
Begin with thorough medical and dental histories.
The complete oral examination must include both clinical and radiographic interpretation of:
caries
the condition of existing restorations
periodontal conditions
responses of teeth (especially abutment teeth) and residual ridges to previous stress
The vitality of remaining teeth
Continued…..
Occlusal plan evaluation
Arch form
Evaluation of Occlusal relationship through mounting the diagnostic cast
The dental cast surveyor is an absolute necessity in which patients are being treated with removable partial dentures.
Mouth preparations, in the appropriate sequence, should be oriented toward the goal of
providing adequate support, stability,
retention, and
a harmonious occlusion for the partial denture.
Support for Distal Extension Denture Bases
A base made to fit the anatomic ridge form does not provide adequate support under occlusal loading.
The base may be made to fit the form of the ridge when under function.
Support for Distal Extension Denture Bases
This provides support
TEST BANK For Katzung's Basic and Clinical Pharmacology, 16th Edition By {Tod...rightmanforbloodline
TEST BANK For Katzung's Basic and Clinical Pharmacology, 16th Edition By {Todd W. Vanderah, 2024,} Verified Chapter
TEST BANK For Katzung's Basic and Clinical Pharmacology, 16th Edition By {Todd W. Vanderah, 2024,} Verified Chapter
TEST BANK For Katzung's Basic and Clinical Pharmacology, 16th Edition By {Todd W. Vanderah, 2024,} Verified Chapter
EXPERIMENTAL STUDY DESIGN- RANDOMIZED CONTROLLED TRIALRishank Shahi
Randomized controlled clinical trial is a prospective experimental study.
It essentially involves comparing the outcomes in two groups of patients treated with a test treatment and a control treatment, both groups are followed over the same period of time. Prepare a plan of study or protocol
a. Define clear objectives
b. State the inclusion and exclusion criteria of case
c. Determine the sample size, place and period of study
d. Design of trial (single blind, double blind and triple blind method)
2. Define study population: Most often the patients are chosen from hospital or from the community. For example, for a study for comparison of home and sanatorium treatment, open cases of tuberculosis may be chosen.
3. Selection of participants by defined criteria as per plan:
Selection of participants should be done with precision and should be precisely stated in writing so that it can be replicated by others. For example, out of open cases of tuberculosis those who fulfill criteria for inclusion may be selected (age groups, severity of disease and treatment taken or not, etc.)
Randomization ensures that participants have an equal chance to be assigned to one of two or more groups:
One group gets the most widely accepted treatment (standard treatment/ gold standard)
The other gets the new treatment being tested, which researchers hope and have reason to believe will be better than the standard treatment
Subject variation: First, there may be bias on the part of the participants, who may subjectively feel better or report improvement if they knew they were receiving a new form of treatment.
Observer bias: The investigator measuring the outcome of a therapeutic trial may be influenced if he knows beforehand the particular procedure or therapy to which the patient has been subjected.
Evaluation bias: There may be bias in evaluation - that is, the investigator(Analyzer) may subconsciously give a favorable report of the outcome of the trial.
Co-intervention:
participants use other therapy or change behavior
Study staff, medical providers, family or friends treat participants differently.
Biased outcome ascertainment:
participants may report symptoms or outcomes differently or physicians
Investigators may elicit symptoms or outcomes differently
A technique used to prevent selection bias by concealing the allocation sequence from those assigning participants to intervention groups, until the moment of assignment.
Allocation concealment prevents researchers from influencing which participants are assigned to a given intervention group.
All clinical trials must be approved by Institutional Ethics Committee before initiation
It is mandatory to register clinical trials with Clinical Trials Registry of India
Informed consent from all study participants is mandatory.
A preclinical trial is a stage of research that begins before clinical trials, and during which important feasibility and drug safety data are collected.
Following points high.
JMML is a rare cancer of blood that affects young children. There is a sustained abnormal and excessive production of myeloid progenitors and monocytes.
Why Does Seminal Vesiculitis Causes Jelly-like Sperm.pptxAmandaChou9
Seminal vesiculitis can cause jelly-like sperm. Fortunately, herbal medicine Diuretic and Anti-inflammatory Pill can eliminate symptoms and cure the disease.
High Profile"*Call "*Girls in Kolkata ))86-075-754-83(( "*Call "*Girls in Kol...Nisha Malik Chaudhary
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Kolkata "Call "Girls 74046-34175 "Call "Girl Number in Kolkata | A nutshell review for Hot "Call "Girls in Kolkata . MY experience was superb with them this is the only recommended "Call "Girls service in Kolkata "Call "Girls and again then Russian. so overall my practice was magnificent. The price is also moderate per hour. The plus point is the "Girl comes instantly to your lo"Cation doesn't matter you are in Bur Kolkata or al Nahda or Kolkata or any area she comes undeviatingly to your hotel room. Definitely recommend the "Call "Girls agency. A nutshell review for Hot "Call "Girls in Kolkata . MY experience was superb with them this is the only recommended "Call "Girls service in Kolkata with verified "Call "Girls . I am using their services from past 6 months they never ever disappointed me in any way. Let's just say if i asked them to provide me russian "Call "Girls they fulfilled my request or even beautiful "Call "Girls or indian "Call "Girls in Kolkata . They have their owen drivers who brings the "Call "Girls in less time in any area of Kolkata like bur Kolkata marina or jumeirah or even in jebel ali as well. I'm writing here everything after experience their services in all conditions.
Mainstreaming #CleanLanguage in healthcare.pptxJudy Rees
In healthcare, every day, millions of conversations fail. They fail to cover what’s really important, fail to resolve key issues, miss the point and lead to misunderstandings and disagreements.
Clean Language is one approach that can improve things. It’s a set of precise questions – and a way of asking them – which help us all get clear on what matters, what we’d like to have happen, and what’s needed.
Around 1000 people working in healthcare have trained in Clean Language skills over the past 20+ years. People are using what they’ve learnt, in their own spheres, and share anecdotes of significant successes. But the various local initiatives have not scaled, nor connected with each other, and learning has not been widely shared.
This project, which emerged from work done by the NHS England South-West End-Of-Life Network, with help from the Q Community and especially Hesham Abdalla, aims to fix that.
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Chemical kinetics is the study of the rates at which chemical reactions occur and the factors that influence these rates.
Importance in Pharmaceuticals: Understanding chemical kinetics is essential for predicting the shelf life of drugs, optimizing storage conditions, and ensuring consistent drug performance.
Rate of Reaction: The speed at which reactants are converted to products.
Factors Influencing Reaction Rates:
Concentration of Reactants: Higher concentrations generally increase the rate of reaction.
Temperature: Increasing temperature typically increases reaction rates.
Catalysts: Substances that increase the reaction rate without being consumed in the process.
Physical State of Reactants: The surface area and physical state (solid, liquid, gas) of reactants can affect the reaction rate.
Coronary Circulation and Ischemic Heart Disease_AntiCopy.pdfMedicoseAcademics
In this lecture, we delve into the intricate anatomy and physiology of the coronary blood supply, a crucial aspect of cardiac function. We begin by examining the physiological anatomy of the coronary arteries, which lie on the heart's surface and penetrate the cardiac muscle mass to supply essential nutrients. Notably, only the innermost layer of the endocardial surface receives direct nourishment from the blood within the cardiac chambers.
We then explore the specifics of coronary circulation, including the dynamics of blood flow at rest and during strenuous activity. The impact of cardiac muscle compression on coronary blood flow, particularly during systole and diastole, is discussed, highlighting why this phenomenon is more pronounced in the left ventricle than the right.
Regulation of coronary circulation is a complex process influenced by autonomic and local metabolic factors. We discuss the roles of sympathetic and parasympathetic nerves, emphasizing the dominance of local metabolic factors such as hypoxia and adenosine in coronary vasodilation. Concepts like autoregulation, active hyperemia, and reactive hyperemia are explained to illustrate how the heart adjusts blood flow to meet varying oxygen demands.
Ischemic heart disease is a major focus, with an exploration of acute coronary artery occlusion, myocardial infarction, and subsequent physiological changes. The lecture covers the progression from acute occlusion to infarction, the body's compensatory mechanisms, and the potential complications leading to death, such as cardiac failure, pulmonary edema, fibrillation, and cardiac rupture.
We also examine coronary steal syndrome, a condition where increased cardiac activity diverts blood flow away from ischemic areas, exacerbating the condition. The long-term impact of myocardial infarction on cardiac reserve is discussed, showing how the heart's capacity to handle increased workloads is significantly reduced.
Angina pectoris, a common manifestation of ischemic heart disease, is analyzed in terms of its causes, presentation, and referred pain patterns. We identify factors that exacerbate anginal pain and discuss both medical and surgical treatment options.
Finally, the lecture includes a case study to apply theoretical knowledge to a practical scenario, helping students understand the real-world implications of coronary circulation and ischemic heart disease. The role of biochemical factors in cardiac pain and the interpretation of ECG changes in myocardial infarction are also covered.
1. M E D I C I N E
REVIEW ARTICLE
Linear Regression Analysis
Part 14 of a Series on Evaluation of Scientific Publications
by Astrid Schneider, Gerhard Hommel, and Maria Blettner
SUMMARY
Background: Regression analysis is an important statisti-
cal method for the analysis of medical data. It enables the
identification and characterization of relationships among
multiple factors. It also enables the identification of prog-
nostically relevant risk factors and the calculation of risk
scores for individual prognostication.
Methods: This article is based on selected textbooks of
statistics, a selective review of the literature, and our own
experience.
Results: After a brief introduction of the uni- and multivari-
able regression models, illustrative examples are given to
explain what the important considerations are before a
regression analysis is performed, and how the results
should be interpreted. The reader should then be able to
judge whether the method has been used correctly and
interpret the results appropriately.
Conclusion: The performance and interpretation of linear
regression analysis are subject to a variety of pitfalls,
which are discussed here in detail. The reader is made
aware of common errors of interpretation through practi-
cal examples. Both the opportunities for applying linear
regression analysis and its limitations are presented.
►Cite this as:
Schneider A, Hommel G, Blettner M: Linear regression
analysis—part 14 of a series on evaluation of scientific
publications. Dtsch Arztebl Int 2010; 107(44): 776–82.
DOI: 10.3238/arztebl.2010.0776
The purpose of statistical evaluation of medical
data is often to describe relationships between
two variables or among several variables. For example,
one would like to know not just whether patients have
high blood pressure, but also whether the likelihood of
having high blood pressure is influenced by factors
such as age and weight. The variable to be explained
(blood pressure) is called the dependent variable, or,
alternatively, the response variable; the variables that
explain it (age, weight) are called independent vari-
ables or predictor variables. Measures of association
provide an initial impression of the extent of statistical
dependence between variables. If the dependent and in-
dependent variables are continuous, as is the case for
blood pressure and weight, then a correlation coeffi-
cient can be calculated as a measure of the strength of
the relationship between them (Box 1).
Regression analysis is a type of statistical evaluation
that enables three things:
● Description: Relationships among the dependent
variables and the independent variables can be
statistically described by means of regression
analysis.
● Estimation: The values of the dependent vari-
ables can be estimated from the observed values
of the independent variables.
● Prognostication: Risk factors that influence the
outcome can be identified, and individual prog-
noses can be determined.
Regression analysis employs a model that describes
the relationships between the dependent variables and
the independent variables in a simplified mathematical
form. There may be biological reasons to expect a
priori that a certain type of mathematical function will
best describe such a relationship, or simple assump-
tions have to be made that this is the case (e.g., that
blood pressure rises linearly with age). The best-known
types of regression analysis are the following (Table 1):
● Linear regression,
● Logistic regression, and
● Cox regression.
The goal of this article is to introduce the reader to
linear regression. The theory is briefly explained, and
the interpretation of statistical parameters is illustrated
with examples. The methods of regression analysis are
comprehensively discussed in many standard text-
books (1–3).
Departrment of Medical Biometrics, Epidemiology, and Computer Sciences,
Johannes Gutenberg University, Mainz, Germany: Dipl. Math. Schneider, Prof.
Dr. rer. nat. Hommel, Prof. Dr. rer. nat. Blettner
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2. M E D I C I N E
Cox regression will be discussed in a later article in
this journal.
Methods
Linear regression is used to study the linear relation-
ship between a dependent variable Y (blood pressure)
and one or more independent variables X (age,
weight, sex).
The dependent variable Y must be continuous,
while the independent variables may be either con-
tinuous (age), binary (sex), or categorical (social
status). The initial judgment of a possible relationship
between two continuous variables should always be
made on the basis of a scatter plot (scatter graph).
This type of plot will show whether the relationship is
linear (Figure 1) or nonlinear (Figure 2).
Performing a linear regression makes sense only if
the relationship is linear. Other methods must be used
to study nonlinear relationships. The variable trans-
formations and other, more complex techniques that
can be used for this purpose will not be discussed in
this article.
Univariable linear regression
Univariable linear regression studies the linear rela-
tionship between the dependent variable Y and a
single independent variable X. The linear regression
model describes the dependent variable with a
straight line that is defined by the equation Y = a + b
× X, where a is the y-intersect of the line, and b is its
slope. First, the parameters a and b of the regression
line are estimated from the values of the dependent
variable Y and the independent variable X with the
aid of statistical methods. The regression line enables
one to predict the value of the dependent variable Y
from that of the independent variable X. Thus, for
example, after a linear regression has been perform-
ed, one would be able to estimate a person’s weight
(dependent variable) from his or her height (indepen-
dent variable) (Figure 3).
The slope b of the regression line is called the
regression coefficient. It provides a measure of the
contribution of the independent variable X toward ex-
plaining the dependent variable Y. If the independent
variable is continuous (e.g., body height in cen-
timeters), then the regression coefficient represents
the change in the dependent variable (body weight in
kilograms) per unit of change in the independent vari-
able (body height in centimeters). The proper inter-
pretation of the regression coefficient thus requires
attention to the units of measurement. The following
example should make this relationship clear:
In a fictitious study, data were obtained from 135
women and men aged 18 to 27. Their height ranged
from 1.59 to 1.93 meters. The relationship between
height and weight was studied: weight in kilograms
was the dependent variable that was to be estimated
from the independent variable, height in centimeters.
On the basis of the data, the following regression line
was determined: Y= –133.18 + 1.16 × X, where X is
BOX 1
Interpretation of the correlation coefficient (r)
Spearman’s coefficient:
Describes a monotone relationship
A monotone relationship is one in which the dependent variable either rises or
sinks continuously as the independent variable rises.
Pearson’s correlation coefficient:
Describes a linear relationship
Interpretation/meaning:
Correlation coefficients provide information about the strength and direction of a
relationship between two continuous variables. No distinction between the ex-
plaining variable and the variable to be explained is necessary:
● r = ± 1: perfect linear and monotone relationship. The closer r is to 1 or –1, the
stronger the relationship.
● r = 0: no linear or monotone relationship
● r < 0: negative, inverse relationship (high values of one variable tend to occur
together with low values of the other variable)
● r > 0: positive relationship (high values of one variable tend to occur together
with high values of the other variable)
Graphical representation of a linear relationship:
Scatter plot with regression line
A negative relationship is represented by a falling regression line (regression
coefficient b < 0), a positive one by a rising regression line (b > 0).
TABLE 1
Regression models
Linear regression
Logistic regression
Proportional hazard
regression
(Cox regression)
Poisson regression
Application
Description of a
linear relationship
Prediction of the
probability of
belonging to
groups
(outcome: yes/no)
Modeling of
survival data
Modeling of
counting processes
Dependent
variables
Continuous
(weight,
blood pressure)
Dichotomous
(success of treat-
ment: yes/no)
Survival time
(time from
diagnosis to event)
Counting data:
whole numbers re-
presenting events
in temporal se-
quence (e.g., the
number of times a
woman gave birth
over a certain
period of time)
Independent
variables
Continuous and/or
categorical
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3. M E D I C I N E
height in centimeters and Y is weight in kilograms. The
y-intersect a = –133.18 is the value of the dependent
variable when X = 0, but X cannot possibly take on
the value 0 in this study (one obviously cannot expect
a person of height 0 centimeters to weigh negative
133.18 kilograms). Therefore, interpretation of the con-
stant is often not useful. In general, only values within
the range of observations of the independent vari-
ables should be used in a linear regression model;
prediction of the value of the dependent variable be-
comes increasingly inaccurate the further one goes
outside this range.
The regression coefficient of 1.16 means that, in
this model, a person’s weight increases by 1.16 kg
with each additional centimeter of height. If height
had been measured in meters, rather than in cen-
timeters, the regression coefficient b would have been
115.91 instead. The constant a, in contrast, is inde-
pendent of the unit chosen to express the independent
variables. Proper interpretation thus requires that the
regression coefficient should be considered together
with the units of all of the involved variables. Special
attention to this issue is needed when publications
from different countries use different units to express
the same variables (e.g., feet and inches vs. cen-
timeters, or pounds vs. kilograms).
Figure 3 shows the regression line that represents
the linear relationship between height and weight.
For a person whose height is 1.74 m, the predicted
weight is 68.50 kg (y = –133.18 + 115.91 × 1.74 m).
The data set contains 6 persons whose height is 1.74
m, and their weights vary from 63 to 75 kg.
Linear regression can be used to estimate the
weight of any persons whose height lies within the
observed range (1.59 m to 1.93 m). The data set need
not include any person with this precise height.
Mathematically it is possible to estimate the weight of a
person whose height is outside the range of values ob-
served in the study. However, such an extrapolation is
generally not useful.
If the independent variables are categorical or
binary, then the regression coefficient must be inter-
preted in reference to the numerical encoding of these
variables. Binary variables should generally be en-
coded with two consecutive whole numbers (usually
0/1 or 1/2). In interpreting the regression coefficient,
one should recall which category of the independent
variable is represented by the higher number (e.g., 2,
when the encoding is 1/2). The regression coefficient
reflects the change in the dependent variable that corre-
sponds to a change in the independent variable from 1
to 2.
For example, if one studies the relationship be-
tween sex and weight, one obtains the regression line
Y = 47.64 + 14.93 × X, where X = sex (1 = female, 2
= male). The regression coefficient of 14.93 reflects
the fact that men are an average of 14.93 kg heavier
than women.
When categorical variables are used, the reference
category should be defined first, and all other
categories are to be considered in relation to this cat-
egory.
The coefficient of determination, r2
, is a measure
of how well the regression model describes the ob-
served data (Box 2). In univariable regression analy-
sis, r2
is simply the square of Pearson’s correlation
coefficient. In the particular fictitious case that is de-
scribed above, the coefficient of determination for the
relationship between height and weight is 0.785. This
means that 78.5% of the variance in weight is due to
height. The remaining 21.5% is due to individual
variation and might be explained by other factors that
were not taken into account in the analysis, such as
eating habits, exercise, sex, or age.
FIGURE 1
A scatter plot
showing a linear
relationship
FIGURE 2
A scatter plot show-
ing an exponential
relationship. In this
case, it would not
be appropriate to
compute a coeffi-
cient of determi-
nation or a regres-
sion line
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4. M E D I C I N E
In formal terms, the null hypothesis, which is the
hypothesis that b = 0 (no relationship between vari-
ables, the regression coefficient is therefore 0), can be
tested with a t-test. One can also compute the 95%
confidence interval for the regression coefficient (4).
Multivariable linear regression
In many cases, the contribution of a single independent
variable does not alone suffice to explain the dependent
variable Y. If this is so, one can perform a multivariable
linear regression to study the effect of multiple vari-
ables on the dependent variable.
In the multivariable regression model, the dependent
variable is described as a linear function of the indepen-
dent variables Xi, as follows: Y = a + b1 × X1 + b2 × X2
+…+ bn × Xn . The model permits the computation of a
regression coefficient bi for each independent variable
Xi (Box 3).
Just as in univariable regression, the coefficient of
determination describes the overall relationship
between the independent variables Xi (weight, age,
body-mass index) and the dependent variable Y (blood
pressure). It corresponds to the square of the multiple
correlation coefficient, which is the correlation be-
tween Y and b1 × X1 + ... + bn × Xn.
It is better practice, however, to give the corrected
coefficient of determination, as discussed in Box 2.
Each of the coefficients bi reflects the effect of the
corresponding individual independent variable Xi on
Y, where the potential influences of the remaining
independent variables on Xi have been taken into ac-
count, i.e., eliminated by an additional computation.
Thus, in a multiple regression analysis with age and sex
as independent variables and weight as the dependent
variable, the adjusted regression coefficient for sex
represents the amount of variation in weight that is
due to sex alone, after age has been taken into ac-
count. This is done by a computation that adjusts for
age, so that the effect of sex is not confounded by a
simultaneously operative age effect (Box 4).
In this way, multivariable regression analysis permits
the study of multiple independent variables at the same
time, with adjustment of their regression coefficients for
possible confounding effects between variables.
Multivariable analysis does more than describe a
statistical relationship; it also permits individual prog-
nostication and the evaluation of the state of health of a
given patient. A linear regression model can be used,
for instance, to determine the optimal values for respi-
ratory function tests depending on a person’s age,
body-mass index (BMI), and sex. Comparing a
patient’s measured respiratory function with these com-
puted optimal values yields a measure of his or her state
of health.
Medical questions often involve the effect of a very
large number of factors (independent variables). The
goal of statistical analysis is to find out which of these
factors truly have an effect on the dependent variable.
The art of statistical evaluation lies in finding the vari-
ables that best explain the dependent variable.
One way to carry out a multivariable regression is to
include all potentially relevant independent variables in
the model (complete model). The problem with this
method is that the number of observations that can
practically be made is often less than the model
requires. In general, the number of observations should
be at least 20 times greater than the number of variables
under study.
Moreover, if too many irrelevant variables are in-
cluded in the model, overadjustment is likely to be the re-
sult: that is, some of the irrelevant independent variables
will be found to have an apparent effect, purely by
chance. The inclusion of irrelevant independent variables
in the model will indeed allow a better fit with the data
set under study, but, because of random effects, the find-
ings will not generally be applicable outside of this data
set (1). The inclusion of irrelevant independent variables
also strongly distorts the determination coefficient, so
that it no longer provides a useful index of the quality
of fit between the model and the data (Box 2).
In the following sections, we will discuss how these
problems can be circumvented.
The selection of variables
For the regression model to be robust and to explain Y
as well as possible, it should include only independent
variables that explain a large portion of the variance in
Y. Variable selection can be performed so that only
such independent variables are included (1).
Variable selection should be carried out on the basis
of medical expert knowledge and a good understanding
of biometrics. This is optimally done as a collaborative
FIGURE 3
A scatter plot and the corresponding regression line and regression
equation for the relationship between the dependent variable body
weight (kg) and the independent variable height (m).
r = Pearsons’s correlation coefficient
R-squared linear = coefficient of determination
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5. M E D I C I N E
effort of the physician-researcher and the statistician.
There are various methods of selecting variables:
Forward selection
Forward selection is a stepwise procedure that includes
variables in the model as long as they make an addi-
tional contribution toward explaining Y. This is done
iteratively until there are no variables left that make any
appreciable contribution to Y.
Backward selection
Backward selection, on the other hand, starts with a
model that contains all potentially relevant indepen-
dent variables. The variable whose removal worsens
the prediction of the independent variable of the
overall set of independent variables to the least ex-
tent is then removed from the model. This procedure
is iterated until no dependent variables are left that
can be removed without markedly worsening the
prediction of the independent variable.
BOX 2
Coefficient of determination (R-squared)
Definition:
Let
● n be the number of observations (e.g., subjects in the study)
● ŷi
be the estimated value of the dependent variable for the ith
observation, as computed with the regression equation
● yi
be the observed value of the dependent variable for the i
th
observation
● y be the mean of all n observations of the dependent variable
The coefficient of determination is then defined
as follows:
→ r
2
is the fraction of the overall variance that is explained. The closer the regression model’s estimated values ŷi
lie to the ob-
served values yi
, the nearer the coefficient of determination is to 1 and the more accurate the regression model is.
Meaning: In practice, the coefficient of determination is often taken as a measure of the validity of a regression model or a re-
gression estimate. It reflects the fraction of variation in the Y-values that is explained by the regression line.
Problem: The coefficient of determination can easily be made artificially high by including a large number of independent va-
riables in the model. The more independent variables one includes, the higher the coefficient of determination becomes. This,
however, lowers the precision of the estimate (estimation of the regression coefficients bi
).
Solution: Instead of the raw (uncorrected) coefficient of determination, the corrected coefficient of determination should be gi-
ven: the latter takes the number of explanatory variables in the model into account. Unlike the uncorrected coefficient of deter-
mination, the corrected one is high only if the independent variables have a sufficiently large effect.
BOX 3
Regression line for a multivariable
regression
Y= a + b1
× X1
+ b2
× X2
+ ...+ bn
× Xn
,
where
Y = dependent variable
Xi
= independent variables
a = constant (y-intersect)
b
i
= regression coefficient of the variable Xi
Example: regression line for a multivariable regressi-
on Y = –120.07 + 100.81 × X1
+ 0.38 × X2
+ 3.41 × X3
,
where
X1
= height (meters)
X
2
= age (years)
X3
= sex (1 = female, 2 = male)
Y = the weight to be estimated (kg)
–
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6. M E D I C I N E
Stepwise selection
Stepwise selection combines certain aspects of for-
ward and backward selection. Like forward selec-
tion, it begins with a null model, adds the single in-
dependent variable that makes the greatest contribu-
tion toward explaining the dependent variable, and
then iterates the process. Additionally, a check is
performed after each such step to see whether one of
the variables has now become irrelevant because of
its relationship to the other variables. If so, this vari-
able is removed.
Block inclusion
There are often variables that should be included in
the model in any case—for example, the effect of a
certain form of treatment, or independent variables
that have already been found to be relevant in prior
studies. One way of taking such variables into
account is their block inclusion into the model. In this
way, one can combine the forced inclusion of some
variables with the selective inclusion of further
independent variables that turn out to be relevant
to the explanation of variation in the dependent
variable.
The evaluation of a regression model requires the
performance of both forward and backward selection
of variables. If these two procedures result in the
selection of the same set of variables, then the model
can be considered robust. If not, a statistician should
be consulted for further advice.
Discussion
The study of relationships between variables and the
generation of risk scores are very important elements
of medical research. The proper performance of regres-
sion analysis requires that a number of important fac-
tors should be considered and tested:
1. Causality
Before a regression analysis is performed, the causal
relationships among the variables to be considered
must be examined from the point of view of their con-
tent and/or temporal relationship. The fact that an inde-
pendent variable turns out to be significant says
nothing about causality. This is an especially relevant
point with respect to observational studies (5).
2. Planning of sample size
The number of cases needed for a regression analysis
depends on the number of independent variables and of
their expected effects (strength of relationships). If the
sample is too small, only very strong relationships will
be demonstrable. The sample size can be planned in
the light of the researchers’ expectations regarding the
coefficient of determination (r2
) and the regression
coefficient (b). Furthermore, at least 20 times as many
observations should be made as there are independent
variables to be studied; thus, if one wants to study 2
independent variables, one should make at least 40
observations.
3. Missing values
Missing values are a common problem in medical data.
Whenever the value of either a dependent or an inde-
pendent variable is missing, this particular observation
has to be excluded from the regression analysis. If
many values are missing from the dataset, the effective
sample size will be appreciably diminished, and the
sample may then turn out to be too small to yield
significant findings, despite seemingly adequate
advance planning. If this happens, real relationships
can be overlooked, and the study findings may not be
generally applicable. Moreover, selection effects can
BOX 4
Two important terms
● Confounder (in non-randomized studies): an independent variable that is as-
sociated, not only with the dependent variable, but also with other independent
variables. The presence of confounders can distort the effect of the other inde-
pendent variables. Age and sex are frequent confounders.
● Adjustment: a statistical technique to eliminate the influence of one or more
confounders on the treatment effect. Example: Suppose that age is a con-
founding variable in a study of the effect of treatment on a certain dependent
variable. Adjustment for age involves a computational procedure to mimic a
situation in which the men and women in the data set were of the same age.
This computation eliminates the influence of age on the treatment effect.
BOX 5
What special points require attention in the
interpretation of a regression analysis?
1. How big is the study sample?
2. Is causality demonstrable or plausible, in view of the content or temporal
relationship of the variables?
3. Has there been adjustment for potential confounding effects?
4. Is the inclusion of the independent variables that were used justified, in view of
their content?
5. What is the corrected coefficient of determination (R-squared)?
6. Is the study sample homogeneous?
7. In what units were the potentially relevant independent variables reported?
8. Was a selection of the independent variables (potentially relevant independent
variables) performed, and, if so, what kind of selection?
9. If a selection of variables was performed, was its result confirmed by a second
selection of variables that was performed by a different procedure?
10. Are predictions of the dependent variable made on the basis of extrapolated
data?
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7. M E D I C I N E
be expected in such cases. There are a number of ways
to deal with the problem of missing values (6).
4. The data sample
A further important point to be considered is the com-
position of the study population. If there are subpopu-
lations within it that behave differently with respect to
the independent variables in question, then a real effect
(or the lack of an effect) may be masked from the
analysis and remain undetected. Suppose, for instance,
that one wishes to study the effect of sex on weight, in
a study population consisting half of children under
age 8 and half of adults. Linear regression analysis
over the entire population reveals an effect of sex on
weight. If, however, a subgroup analysis is performed
in which children and adults are considered separately,
an effect of sex on weight is seen only in adults, and
not in children. Subgroup analysis should only be per-
formed if the subgroups have been predefined, and the
questions already formulated, before the data analysis
begins; furthermore, multiple testing should be taken
into account (7, 8).
5. The selection of variables
If multiple independent variables are considered in a
multivariable regression, some of these may turn out to
be interdependent. An independent variable that would
be found to have a strong effect in a univariable regres-
sion model might not turn out to have any appreciable
effect in a multivariable regression with variable selec-
tion. This will happen if this particular variable itself
depends so strongly on the other independent variables
that it makes no additional contribution toward ex-
plaining the dependent variable. For related reasons,
when the independent variables are mutually de-
pendent, different independent variables might end up
being included in the model depending on the particu-
lar technique that is used for variable selection.
Overview
Linear regression is an important tool for statistical
analysis. Its broad spectrum of uses includes relation-
ship description, estimation, and prognostication. The
technique has many applications, but it also has pre-
requisites and limitations that must always be con-
sidered in the interpretation of findings (Box 5).
Conflict of interest statement
The authors declare that they have no conflict of interest as defined by the
guidelines of the International Committee of Medical Journal Editors.
Manuscript submitted on 11 May 2010, revised version accepted on 14 July
2010.
Translated from the original German by Ethan Taub, MD
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Corresponding author
Prof. Dr. rer. nat. Maria Blettner
Department of Medical Biometrics, Epidemiology, and Computer Sciences
Johannes Gutenberg University
Obere Zahlbacher Str. 69
55131 Mainz
Germany
782 Deutsches Ärzteblatt International |Dtsch Arztebl Int 2010; 107(44): 776–82