# STATISTICAL METHODS FOR ASSESSING AGREEMENT BETWEEN TWO METHODS OF CLINICAL MEASUREMENT

Save this PDF as:

Size: px
Start display at page:

Download "STATISTICAL METHODS FOR ASSESSING AGREEMENT BETWEEN TWO METHODS OF CLINICAL MEASUREMENT"

## Transcription

2 PEFR MEASURED WITH WRIGHT PEAK FLOW AND MINI WRIGHT PEAK FLOW METER Wright peak flow meter Mini Wright peak flow meter First PEFR Second PEFR First PEFR Second PEFR Subject (l/min) (l/mi) (l/min) (l/min) Fig 1. PEFR measured with large Wright peak flow meter and mini Wright peak flow meter, with line of equality. INAPPROPRIATE USE OF CORRELATION COEFFICIENT The second step is usually to calculate the correlation coefficient (r) between the two methods. For the data in fig 1, r = 0.94 (p < 0.001). The null hypothesis here is that the measurements by the two methods are not linearly related. The probability is very small and we can safely conclude that PEFR measurements by the mini and large meters are related. However, this high correlation does not mean that the two methods agree: (1) r measures the strength of a relation between two variables, not the agreement between them. We have perfect agreement only if the points in fig 1 lie along the line of equality, but we will have perfect correlation if the points lie along any straight line.

3 () A change in scale of measurement does not affect the correlation, but it certainly affects the agreement. For example, we can measure subcutaneous fat by skinfold calipers. The calipers will measure two thicknesses of fat. If we were to plot calipers measurement against half-calipers measurement, in the style of fig 1, we should get a perfect straight line with slope.0. The correlation would be 1.0, but the two measurements would not agree we could not mix fat thicknesses obtained by the two methods, since one is twice the other. (3) Correlation depends on the range of the true quantity in the sample. If this is wide, the correlation will be greater than if it is narrow. For those subjects whose PEFR (by peak flow meter) is less than 500 l/min, r is 0.88 while for those with greater PEFRs r is Both are less than the overall correlation of 0.94, but it would be absurd to argue that agreement is worse below 500 l/min and worse above 500 l/min than it is for everybody. Since investigators usually try to compare two methods over the whole range of values typically encountered, a high correlation is almost guaranteed. (4) The test of significance may show that the two methods are related, but it would be amazing if two methods designed to measure the same quantity were not related. The test of significance is irrelevant to the question of agreement. (5) Data which seem to be in poor agreement can produce quite high correlations. For example, Serfontein and Jaroszewicz compared two methods of measuring gestational age. Babies with a gestational age of 35 weeks by one method had gestations between 34 and 39.5 weeks by the other, but r was high (0.85). On the other hand, Oldham et al. 3 compared the mini and large Wright peak flow meters and found a correlation of They then connected the meters in series, so that both measured the same flow, and obtained a "material improvement" (0.996). If a correlation coefficient of 0.99 can be materially improved upon, we need to rethink our ideas of what a high correlation is in this context. As we show below, the high correlation of 0.94 for our own data conceals considerable lack of agreement between the two instruments. MEASURING AGREEMENT It is most unlikely that different methods will agree exactly, by giving the identical result for all individuals. We want to know by how much the new method is likely to differ from the old: if this is not enough to cause problems in clinical interpretation we can replace the old method by the new or use the two interchangeably. If the two PEFR meters were unlikely to give readings which differed by more than, say, 10 l/min, we could replace the large meter by the mini meter because so small a difference would not affect decisions on patient management. On the other hand, if the meters could differ by 100 l/min, the mini meter would be unlikely to be satisfactory. How far apart measurements can be without causing difficulties will be a question of judgment. Ideally, it should be defined in advance to help in the interpretation of the method comparison and to choose the sample size. The first step is to examine the data. A simple plot of the results of one method against those of the other (fig 1) though without a regression line is a useful start but usually the data points will be clustered near the line and it will be difficult to assess between-method differences. A plot of the difference between the methods against their mean may be more informative. Fig displays considerable lack of agreement between the large and mini meters, with discrepancies of up to 80 l/min, these differences are not obvious from fig 1. The plot of difference against mean also allows us to investigate any possible relationship between the measurement error and the true value. We do not know the true value, and the mean of the two measurements is the best estimate we have. It would be a mistake to plot the difference against either value separately because the difference will be related to each, a well-known statistical artefact. 4 3

4 Fig. Difference against mean for PEFR data. For the PEFR data, there is no obvious relation between the difference and the mean. Under these circumstances we can summarise the lack of agreement by calculating the bias, estimated by the mean difference d and the standard deviation of the differences ( s ). If there is a consistent bias we can adjust for it by subtracting d from the new method. For the PEFR data the mean difference (large meter minus small meter) is. 1 l/min and s is 38.8 l/min. We would expect most of the differences to lie between d s and d + s (fig ). If the differences are Normally distributed (Gaussian), 95% of differences will lie between these limits (or, more precisely, between d 1. 96s and d s ). Such differences are likely to follow a Normal distribution because we have removed a lot of the variation between subjects and are left with the measurement error. The measurements themselves do not have to follow a Normal distribution, and often they will not. We can check the distribution of the differences by drawing a histogram. If this is skewed or has very long tails the assumption of Normality may not be valid (see below). Provided differences within d ± s would not be clinically important, we could use the two measurement methods interchangeably. We shall refer to these as the "limits of agreement". For the PEFR data we get: d s =.1 ( 38.8) = 79.7 l/min d + s =.1 + ( 38.8) = 75.5 l/min Thus, the mini meter may be 80 l/min below or 76 l/min above the large meter, which would be unacceptable for clinical purposes. This lack of agreement is by no means obvious in fig 1. PRECISION OF ESTIMATED LIMITS OF AGREEMENT The limits of agreement are only estimates of the values which apply to the whole population. A second sample would give different limits. We might sometimes wish to use standard errors and confidence intervals to see how precise our estimates are, provided the differences follow a distribution which is approximately Normal. The standard error of d is s / n where n is the sample size, and the standard error of d s and d + s is about 3s / n. 95% confidence intervals can be calculated by finding the appropriate point of the t distribution with n 1 degrees of freedom, on most tables the columns marked 5% or 0.05,, 4

5 and then the confidence interval will be from the observed value minus t standard errors to the observed value plus t standard errors. For the PEFR data s = The standard error of d is thus 9.4, For the 95% confidence interval we have 16 degrees of freedom and t =.1. Hence the 95% confidence interval for the bias is.1 (.1 9.4) to.1+ (.1 9.4), giving. 0 to 17.8 l/min. The standard error of the limit d s is 16.3 l/min. The 95% confidence interval for the lower limit of agreement is 79.7 ( ) to ( ), giving to 45.1 l/min. For the upper limit of agreement the 95% confidence interval is 40.9 to l/min. These intervals are wide, reflecting the small sample size and the great variation of the differences. They show, however, that even on the most optimistic interpretation there can be considerable discrepancies between the two meters and that the degree of agreement is not acceptable. EXAMPLE SHOWING GOOD AGREEMENT Fig 3 shows a comparison of oxygen saturation measured by an oxygen saturation monitor and pulsed oximeter saturation, a new non-invasive technique. 5 Here the mean difference is 0.4 percentage points with 95% confidence interval 0.13 to Thus pulsed oximeter saturation tends to give a lower reading, by between 0.13 and Despite this, the limits of agreement (. 0 and.8) are small enough for us to be confident that the new method can be used in place of the old for clinical purposes. Fig 3. Oxygen saturation monitor and pulsed saturation oximeter RELATION BETWEEN DIFFERENCE AND MEAN In the preceding analysis it was assumed that the differences did not vary in any systematic way over the range of measurement. This may not be so. Fig 4 compares the measurement of mean velocity of circumferential fibre shortening (VCF) by the long axis and short axis in M- mode echocardiography. 6 The scatter of the differences increases as the VCF increases. We could ignore this, but the limits of agreement would be wider apart than necessary for small VCF and narrower than they should be for large VCF. If the differences are proportional to the mean, a logarithmic transformation should yield a picture more like that of figs and 4, and we can then apply the analysis described above to the transformed data. 5

6 Fig 4. Mean VCF by long and short axis measurements. Fig 5 shows the log-transformed data of fig 4. This still shows a relation between the difference and the mean VCF, but there is some improvement. The mean difference is on the log scale and the limits of agreement are and However, although there is only negligible bias, the limits of agreement have somehow to be related to the original scale of measurement. If we take the antilogs of these limits, we get 0.80 and 1.7. However, the antilog of the difference between two values on a log scale is a dimensionless ratio. The limits tell us that for about 95% of cases the short axis measurement of VCF will be between 0.80 and 1.7 times the long axis VCF. Thus the short axis measurement may differ from the long axis measurement by 0% below to 7% above. (The log transformation is the only transformation giving back-transformed differences which are easy to interpret, and we do not recommend the use of any other in this context.) Fig 5. Data of fig 4 after logarithmic transformation. These numbers were incorrectly printed as 0.008, -0.6, and 0.43 in the Lancet. This mistake arose because when revising the paper we dithered over whether to use natural logs or logs to base 10 and got hopelessly confused. 6

7 Sometimes the relation between difference and mean is more complex than that shown in fig 4 and log transformation does not work. Here a plot in the style of fig is very helpful in comparing the methods. Formal analysis, as described above, will tend to give limits of agreement which are too far apart rather than too close, and so should not lead to the acceptance of poor methods of measurement. REPEATABILITY Repeatability is relevant to the study of method comparison because the repeatabilities of the two methods of measurement limit the amount of agreement which is possible. If one method has poor repeatability i.e. there is considerable variation in repeated measurements on the same subject the agreement between the two methods is bound to be poor too. When the old method is the more variable one, even a new method which is perfect will not agree with it. If both methods have poor repeatability, the problem is even worse. The best way to examine repeatability is to take repeated measurements on a series of subjects. The table shows paired data for PEFR. We can then plot a figure similar to fig, showing differences against mean for each subject. If the differences are related to the mean, we can apply a log transformation. We then calculate the mean and standard deviation of the differences as before. The mean difference should here be zero since the same method was used. (If the mean difference is significantly different from zero, we will not be able to use the data to assess repeatability because either knowledge of the first measurement is affecting the second or the process of measurement is altering the quantity.) We expect 95% of differences to be less than two standard deviations. This is the definition of a repeatability coefficient adopted by the British Standards Institution. 7 If we can assume the main difference to be zero this coefficient is very simple to estimate: we square all the differences, add them up, divide by n, and take the square root, to get the standard deviation of the differences. Fig 6. Repeated measures of PEFR using mini Wright peak flow meter. Fig 6 shows the plot for pairs of measurements made with the mini Wright peak flow meter. There does not appear to be any relation between the difference and the size of the PEFR. There is, however, a clear outlier. We have retained this measurement for the analysis, although we suspect that it was technically unsatisfactory. (In practice, one could omit this subject.) The sum of the differences squared is so the standard deviation of differences between the 17 pairs of repeated measurements is 8. l/min. The coefficient of repeatability 7

8 is twice this, or 56.4 l/min for the mini meter. For the large meter the coefficient is 43. l/min. If we have more than two repeated measurements the calculations are more complex. We plot the standard deviation of the several measurements for that subject against their mean and then use one-way analysis of variance, 8 which is beyond the scope of this article. MEASURING AGREEMENT USING REPEATED MEASUREMENTS If we have repeated measurements by each of the two methods on the same subjects we can calculate the mean for each method on each subject and use these pairs of means to compare the two methods using the analysis for assessing agreement described above. The estimate of bias will be unaffected, but the estimate of the standard deviation of the differences will be too small, because some of the effect of repeated measurement error has been removed. We can correct for this. Suppose we have two measurements obtained by each method, as in the table. We find the standard deviation of differences between repeated measurements for each method separately, s 1 and s, and the standard deviation of the differences between the means for each method, s D. The corrected standard deviation of differences, s c, is 1 1 s D + s1 + s. This is approximately s D, but if there are differences between the 4 4 two methods not explicable by repeatability errors alone (i.e. interaction between subject and measurement method) this approximation may produce an overestimate. For the PEFR, we have s D = 33., s 1 = 1.6, s = 8. l/min. s c is thus or 37.7 l/min. Compare this with the estimate 38.8 l/min which was obtained using a single measurement. On the other hand, the approximation l/min). DISCUSSION s D gives an overestimate (47.0 In the analysis of measurement method comparison data, neither the correlation coefficient (as we show here) nor techniques such as regression analysis 1 are appropriate. We suggest replacing these misleading analyses by a method that is simple both to do and to interpret. Further, the same method may be use to analyse the repeatability of a single measurement method or to compare measurements by two observers. Why has a totally inappropriate method, the correlation coefficient, become almost universally used for this purpose?. Two processes may be at work here --- namely, pattern recognition and imitation. A likely first step in the analysis of such data is to plot a scatter diagram (fig 1). A glance through almost any statistical textbook for a similar picture will lead to the correlation coefficient as a method of analysis of such a plot, together with a test of the null hypothesis of no relationship. Some texts even use pairs of measurements by two different methods to illustrate the calculation of r. Once the correlation approach has been published, others will read of a statistical problem similar to their own being solved in this way and will use the same technique with their own data. Medical statisticians who ask "why did you use this statistical method?" will often be told "because this published paper used it". Journals could help to rectify this error by returning for reanalysis papers which use incorrect statistical techniques. This may be a slow process. Referees, inspecting papers in which two methods of measurement have been compared, sometimes complain if no correlation coefficients are provided, even when the reasons for not doing so are given. This was incorrectly printed as s c in the Lancet and in Biochimica Clinica. 8

9 We thank of our colleagues for their interest and assistance, including Dr David Robson who first brought the problem to us, Dr P. D'Arbela and Dr H. Seeley for the use of their data; and Mrs S Stevens for typing the manuscript. REFERENCES 1. Altman DG, Bland JM. Measurement in medicine: the analysis of method comparison studies. The Statistician 1983; 3, Serfontein GL, Jaroszewicz AM. Estimation of gestational age at birth: comparison of two methods. Arch Dis Child 1978; 53: Oldham HG, Bevan MM, McDermott M. Comparison of the new miniature Wright peak flow meter with the standard Wright peak flow meter. Thorax 1979; 34: Gill JS, Zezulka AV, Beevers DG, Davies P. Relationship between initial blood pressure and its fall with treatment. Lancet 1985; i: Tytler JA, Seeley HF. The Nellcor N-101 pulse oximeter - a clinical-evaluation in anesthesia and intensive-care. Anaesthesia 1986; 41: D'Arbela PG, Silayan ZM, Bland JM. Comparability of M-mode echocardiographic long axis and short axis left ventricular function derivatives. British Heart Journal 1986; 56: British Standards Institution. Precision of test methods 1: Guide for the determination and reproducibility for a standard test method (BS 597, Part 1). London: BSI (1975). 8. Armitage P. Statistical methods in medical research. Oxford: Blackwell Scientific Publications, 1971: chap 7. Reproduced by kind permission of the Lancet. 9

### How do I analyse observer variation studies?

How do I analyse observer variation studies? This is based on work done for a long term project by Doug Altman and Martin Bland. I recommend that you first read our three Statistics Notes on measurement

### Week 4: Standard Error and Confidence Intervals

Health Sciences M.Sc. Programme Applied Biostatistics Week 4: Standard Error and Confidence Intervals Sampling Most research data come from subjects we think of as samples drawn from a larger population.

### AP Physics 1 and 2 Lab Investigations

AP Physics 1 and 2 Lab Investigations Student Guide to Data Analysis New York, NY. College Board, Advanced Placement, Advanced Placement Program, AP, AP Central, and the acorn logo are registered trademarks

### NCSS Statistical Software Principal Components Regression. In ordinary least squares, the regression coefficients are estimated using the formula ( )

Chapter 340 Principal Components Regression Introduction is a technique for analyzing multiple regression data that suffer from multicollinearity. When multicollinearity occurs, least squares estimates

### A practical approach to Bland-Altman plots and variation coefficients for log transformed variables

5 A practical approach to Bland-Altman plots and variation coefficients for log transformed variables A.M. Euser F.W. Dekker S. le Cessie J Clin Epidemiol 2008; 61: 978-982 Chapter 5 Abstract Objective

### Measurement in Medicine: the Analysis of Method Comparison Studies

The Statistician 3 (1983) 307-317 1983 Institute of Statisticians Measurement in Medicine: the Analysis of Method Comparison Studies D. G. ALTMAN and J. M. BLAND Division of Computing and Statistics, MRC

### , then the form of the model is given by: which comprises a deterministic component involving the three regression coefficients (

Multiple regression Introduction Multiple regression is a logical extension of the principles of simple linear regression to situations in which there are several predictor variables. For instance if we

### Simple Linear Regression Chapter 11

Simple Linear Regression Chapter 11 Rationale Frequently decision-making situations require modeling of relationships among business variables. For instance, the amount of sale of a product may be related

### Simple Regression Theory II 2010 Samuel L. Baker

SIMPLE REGRESSION THEORY II 1 Simple Regression Theory II 2010 Samuel L. Baker Assessing how good the regression equation is likely to be Assignment 1A gets into drawing inferences about how close the

### Simple linear regression

Simple linear regression Introduction Simple linear regression is a statistical method for obtaining a formula to predict values of one variable from another where there is a causal relationship between

### The correlation coefficient

The correlation coefficient Clinical Biostatistics The correlation coefficient Martin Bland Correlation coefficients are used to measure the of the relationship or association between two quantitative

### Simple Linear Regression Inference

Simple Linear Regression Inference 1 Inference requirements The Normality assumption of the stochastic term e is needed for inference even if it is not a OLS requirement. Therefore we have: Interpretation

### Hypothesis tests: the t-tests

Hypothesis tests: the t-tests Introduction Invariably investigators wish to ask whether their data answer certain questions that are germane to the purpose of the investigation. It is often the case that

### 4. Introduction to Statistics

Statistics for Engineers 4-1 4. Introduction to Statistics Descriptive Statistics Types of data A variate or random variable is a quantity or attribute whose value may vary from one unit of investigation

### CHAPTER 13 SIMPLE LINEAR REGRESSION. Opening Example. Simple Regression. Linear Regression

Opening Example CHAPTER 13 SIMPLE LINEAR REGREION SIMPLE LINEAR REGREION! Simple Regression! Linear Regression Simple Regression Definition A regression model is a mathematical equation that descries the

T O P I C 1 2 Techniques and tools for data analysis Preview Introduction In chapter 3 of Statistics In A Day different combinations of numbers and types of variables are presented. We go through these

### X X X a) perfect linear correlation b) no correlation c) positive correlation (r = 1) (r = 0) (0 < r < 1)

CORRELATION AND REGRESSION / 47 CHAPTER EIGHT CORRELATION AND REGRESSION Correlation and regression are statistical methods that are commonly used in the medical literature to compare two or more variables.

### 2. Simple Linear Regression

Research methods - II 3 2. Simple Linear Regression Simple linear regression is a technique in parametric statistics that is commonly used for analyzing mean response of a variable Y which changes according

### Paper 208-28. KEYWORDS PROC TRANSPOSE, PROC CORR, PROC MEANS, PROC GPLOT, Macro Language, Mean, Standard Deviation, Vertical Reference.

Paper 208-28 Analysis of Method Comparison Studies Using SAS Mohamed Shoukri, King Faisal Specialist Hospital & Research Center, Riyadh, KSA and Department of Epidemiology and Biostatistics, University

### Inferential Statistics

Inferential Statistics Sampling and the normal distribution Z-scores Confidence levels and intervals Hypothesis testing Commonly used statistical methods Inferential Statistics Descriptive statistics are

BIOSTATISTICS QUIZ ANSWERS 1. When you read scientific literature, do you know whether the statistical tests that were used were appropriate and why they were used? a. Always b. Mostly c. Rarely d. Never

### DATA INTERPRETATION AND STATISTICS

PholC60 September 001 DATA INTERPRETATION AND STATISTICS Books A easy and systematic introductory text is Essentials of Medical Statistics by Betty Kirkwood, published by Blackwell at about 14. DESCRIPTIVE

### Exercise 1.12 (Pg. 22-23)

Individuals: The objects that are described by a set of data. They may be people, animals, things, etc. (Also referred to as Cases or Records) Variables: The characteristics recorded about each individual.

### Introduction to Statistics for Computer Science Projects

Introduction Introduction to Statistics for Computer Science Projects Peter Coxhead Whole modules are devoted to statistics and related topics in many degree programmes, so in this short session all I

### CHAPTER THREE COMMON DESCRIPTIVE STATISTICS COMMON DESCRIPTIVE STATISTICS / 13

COMMON DESCRIPTIVE STATISTICS / 13 CHAPTER THREE COMMON DESCRIPTIVE STATISTICS The analysis of data begins with descriptive statistics such as the mean, median, mode, range, standard deviation, variance,

### Recall this chart that showed how most of our course would be organized:

Chapter 4 One-Way ANOVA Recall this chart that showed how most of our course would be organized: Explanatory Variable(s) Response Variable Methods Categorical Categorical Contingency Tables Categorical

### 1) Write the following as an algebraic expression using x as the variable: Triple a number subtracted from the number

1) Write the following as an algebraic expression using x as the variable: Triple a number subtracted from the number A. 3(x - x) B. x 3 x C. 3x - x D. x - 3x 2) Write the following as an algebraic expression

### business statistics using Excel OXFORD UNIVERSITY PRESS Glyn Davis & Branko Pecar

business statistics using Excel Glyn Davis & Branko Pecar OXFORD UNIVERSITY PRESS Detailed contents Introduction to Microsoft Excel 2003 Overview Learning Objectives 1.1 Introduction to Microsoft Excel

### Business Statistics. Successful completion of Introductory and/or Intermediate Algebra courses is recommended before taking Business Statistics.

Business Course Text Bowerman, Bruce L., Richard T. O'Connell, J. B. Orris, and Dawn C. Porter. Essentials of Business, 2nd edition, McGraw-Hill/Irwin, 2008, ISBN: 978-0-07-331988-9. Required Computing

### Chapter 7 What to do when you have the data

Chapter 7 What to do when you have the data We saw in the previous chapters how to collect data. We will spend the rest of this course looking at how to analyse the data that we have collected. Stem and

### Course Text. Required Computing Software. Course Description. Course Objectives. StraighterLine. Business Statistics

Course Text Business Statistics Lind, Douglas A., Marchal, William A. and Samuel A. Wathen. Basic Statistics for Business and Economics, 7th edition, McGraw-Hill/Irwin, 2010, ISBN: 9780077384470 [This

### American Association for Laboratory Accreditation

Page 1 of 12 The examples provided are intended to demonstrate ways to implement the A2LA policies for the estimation of measurement uncertainty for methods that use counting for determining the number

### 2013 MBA Jump Start Program. Statistics Module Part 3

2013 MBA Jump Start Program Module 1: Statistics Thomas Gilbert Part 3 Statistics Module Part 3 Hypothesis Testing (Inference) Regressions 2 1 Making an Investment Decision A researcher in your firm just

### Basic research methods. Basic research methods. Question: BRM.2. Question: BRM.1

BRM.1 The proportion of individuals with a particular disease who die from that condition is called... BRM.2 This study design examines factors that may contribute to a condition by comparing subjects

### Means, standard deviations and. and standard errors

CHAPTER 4 Means, standard deviations and standard errors 4.1 Introduction Change of units 4.2 Mean, median and mode Coefficient of variation 4.3 Measures of variation 4.4 Calculating the mean and standard

### Appendix E: Graphing Data

You will often make scatter diagrams and line graphs to illustrate the data that you collect. Scatter diagrams are often used to show the relationship between two variables. For example, in an absorbance

### Regression Analysis: A Complete Example

Regression Analysis: A Complete Example This section works out an example that includes all the topics we have discussed so far in this chapter. A complete example of regression analysis. PhotoDisc, Inc./Getty

### E205 Final: Version B

Name: Class: Date: E205 Final: Version B Multiple Choice Identify the choice that best completes the statement or answers the question. 1. The owner of a local nightclub has recently surveyed a random

### Chapter 7: Simple linear regression Learning Objectives

Chapter 7: Simple linear regression Learning Objectives Reading: Section 7.1 of OpenIntro Statistics Video: Correlation vs. causation, YouTube (2:19) Video: Intro to Linear Regression, YouTube (5:18) -

### Using Excel (Microsoft Office 2007 Version) for Graphical Analysis of Data

Using Excel (Microsoft Office 2007 Version) for Graphical Analysis of Data Introduction In several upcoming labs, a primary goal will be to determine the mathematical relationship between two variable

### Unit 24 Hypothesis Tests about Means

Unit 24 Hypothesis Tests about Means Objectives: To recognize the difference between a paired t test and a two-sample t test To perform a paired t test To perform a two-sample t test A measure of the amount

### Case Study in Data Analysis Does a drug prevent cardiomegaly in heart failure?

Case Study in Data Analysis Does a drug prevent cardiomegaly in heart failure? Harvey Motulsky hmotulsky@graphpad.com This is the first case in what I expect will be a series of case studies. While I mention

### Diagrams and Graphs of Statistical Data

Diagrams and Graphs of Statistical Data One of the most effective and interesting alternative way in which a statistical data may be presented is through diagrams and graphs. There are several ways in

### MINITAB ASSISTANT WHITE PAPER

MINITAB ASSISTANT WHITE PAPER This paper explains the research conducted by Minitab statisticians to develop the methods and data checks used in the Assistant in Minitab 17 Statistical Software. One-Way

### STATS8: Introduction to Biostatistics. Data Exploration. Babak Shahbaba Department of Statistics, UCI

STATS8: Introduction to Biostatistics Data Exploration Babak Shahbaba Department of Statistics, UCI Introduction After clearly defining the scientific problem, selecting a set of representative members

### 1. How different is the t distribution from the normal?

Statistics 101 106 Lecture 7 (20 October 98) c David Pollard Page 1 Read M&M 7.1 and 7.2, ignoring starred parts. Reread M&M 3.2. The effects of estimated variances on normal approximations. t-distributions.

### 7. Tests of association and Linear Regression

7. Tests of association and Linear Regression In this chapter we consider 1. Tests of Association for 2 qualitative variables. 2. Measures of the strength of linear association between 2 quantitative variables.

### 1. What is the critical value for this 95% confidence interval? CV = z.025 = invnorm(0.025) = 1.96

1 Final Review 2 Review 2.1 CI 1-propZint Scenario 1 A TV manufacturer claims in its warranty brochure that in the past not more than 10 percent of its TV sets needed any repair during the first two years

### Normality Testing in Excel

Normality Testing in Excel By Mark Harmon Copyright 2011 Mark Harmon No part of this publication may be reproduced or distributed without the express permission of the author. mark@excelmasterseries.com

### A POPULATION MEAN, CONFIDENCE INTERVALS AND HYPOTHESIS TESTING

CHAPTER 5. A POPULATION MEAN, CONFIDENCE INTERVALS AND HYPOTHESIS TESTING 5.1 Concepts When a number of animals or plots are exposed to a certain treatment, we usually estimate the effect of the treatment

### Institute of Actuaries of India Subject CT3 Probability and Mathematical Statistics

Institute of Actuaries of India Subject CT3 Probability and Mathematical Statistics For 2015 Examinations Aim The aim of the Probability and Mathematical Statistics subject is to provide a grounding in

### DESCRIPTIVE STATISTICS. The purpose of statistics is to condense raw data to make it easier to answer specific questions; test hypotheses.

DESCRIPTIVE STATISTICS The purpose of statistics is to condense raw data to make it easier to answer specific questions; test hypotheses. DESCRIPTIVE VS. INFERENTIAL STATISTICS Descriptive To organize,

### Introduction to Quantitative Methods

Introduction to Quantitative Methods October 15, 2009 Contents 1 Definition of Key Terms 2 2 Descriptive Statistics 3 2.1 Frequency Tables......................... 4 2.2 Measures of Central Tendencies.................

### UCLA STAT 13 Statistical Methods - Final Exam Review Solutions Chapter 7 Sampling Distributions of Estimates

UCLA STAT 13 Statistical Methods - Final Exam Review Solutions Chapter 7 Sampling Distributions of Estimates 1. (a) (i) µ µ (ii) σ σ n is exactly Normally distributed. (c) (i) is approximately Normally

### Getting Correct Results from PROC REG

Getting Correct Results from PROC REG Nathaniel Derby, Statis Pro Data Analytics, Seattle, WA ABSTRACT PROC REG, SAS s implementation of linear regression, is often used to fit a line without checking

### Simple Linear Regression in SPSS STAT 314

Simple Linear Regression in SPSS STAT 314 1. Ten Corvettes between 1 and 6 years old were randomly selected from last year s sales records in Virginia Beach, Virginia. The following data were obtained,

### EXPERIMENTAL ERROR AND DATA ANALYSIS

EXPERIMENTAL ERROR AND DATA ANALYSIS 1. INTRODUCTION: Laboratory experiments involve taking measurements of physical quantities. No measurement of any physical quantity is ever perfectly accurate, except

### An Introduction to the Critical Appraisal Section (including example questions)

An Introduction to the Critical Appraisal Section (including example questions) Introduction This section is common to the specialties of Dental Public Health, Oral Medicine, Oral Surgery, Orthodontics,

### Numerical Summarization of Data OPRE 6301

Numerical Summarization of Data OPRE 6301 Motivation... In the previous session, we used graphical techniques to describe data. For example: While this histogram provides useful insight, other interesting

### Problem Solving and Data Analysis

Chapter 20 Problem Solving and Data Analysis The Problem Solving and Data Analysis section of the SAT Math Test assesses your ability to use your math understanding and skills to solve problems set in

### Chapter 3: Central Tendency

Chapter 3: Central Tendency Central Tendency In general terms, central tendency is a statistical measure that determines a single value that accurately describes the center of the distribution and represents

### Good luck! BUSINESS STATISTICS FINAL EXAM INSTRUCTIONS. Name:

Glo bal Leadership M BA BUSINESS STATISTICS FINAL EXAM Name: INSTRUCTIONS 1. Do not open this exam until instructed to do so. 2. Be sure to fill in your name before starting the exam. 3. You have two hours

### STATISTICS 151 SECTION 1 FINAL EXAM MAY

STATISTICS 151 SECTION 1 FINAL EXAM MAY 2 2009 This is an open book exam. Course text, personal notes and calculator are permitted. You have 3 hours to complete the test. Personal computers and cellphones

### Organizing Your Approach to a Data Analysis

Biost/Stat 578 B: Data Analysis Emerson, September 29, 2003 Handout #1 Organizing Your Approach to a Data Analysis The general theme should be to maximize thinking about the data analysis and to minimize

### Standard Deviation Estimator

CSS.com Chapter 905 Standard Deviation Estimator Introduction Even though it is not of primary interest, an estimate of the standard deviation (SD) is needed when calculating the power or sample size of

### International Statistical Institute, 56th Session, 2007: Phil Everson

Teaching Regression using American Football Scores Everson, Phil Swarthmore College Department of Mathematics and Statistics 5 College Avenue Swarthmore, PA198, USA E-mail: peverso1@swarthmore.edu 1. Introduction

### 0.1 Multiple Regression Models

0.1 Multiple Regression Models We will introduce the multiple Regression model as a mean of relating one numerical response variable y to two or more independent (or predictor variables. We will see different

### 11. Analysis of Case-control Studies Logistic Regression

Research methods II 113 11. Analysis of Case-control Studies Logistic Regression This chapter builds upon and further develops the concepts and strategies described in Ch.6 of Mother and Child Health:

### Testing: is my coin fair?

Testing: is my coin fair? Formally: we want to make some inference about P(head) Try it: toss coin several times (say 7 times) Assume that it is fair ( P(head)= ), and see if this assumption is compatible

### Common Tools for Displaying and Communicating Data for Process Improvement

Common Tools for Displaying and Communicating Data for Process Improvement Packet includes: Tool Use Page # Box and Whisker Plot Check Sheet Control Chart Histogram Pareto Diagram Run Chart Scatter Plot

### MAT140: Applied Statistical Methods Summary of Calculating Confidence Intervals and Sample Sizes for Estimating Parameters

MAT140: Applied Statistical Methods Summary of Calculating Confidence Intervals and Sample Sizes for Estimating Parameters Inferences about a population parameter can be made using sample statistics for

### Statistics for Management II-STAT 362-Final Review

Statistics for Management II-STAT 362-Final Review Multiple Choice Identify the letter of the choice that best completes the statement or answers the question. 1. The ability of an interval estimate to

### Chapter 11: Two Variable Regression Analysis

Department of Mathematics Izmir University of Economics Week 14-15 2014-2015 In this chapter, we will focus on linear models and extend our analysis to relationships between variables, the definitions

### Chi Square Tests. Chapter 10. 10.1 Introduction

Contents 10 Chi Square Tests 703 10.1 Introduction............................ 703 10.2 The Chi Square Distribution.................. 704 10.3 Goodness of Fit Test....................... 709 10.4 Chi Square

### Technology Step-by-Step Using StatCrunch

Technology Step-by-Step Using StatCrunch Section 1.3 Simple Random Sampling 1. Select Data, highlight Simulate Data, then highlight Discrete Uniform. 2. Fill in the following window with the appropriate

### Standard Deviation Calculator

CSS.com Chapter 35 Standard Deviation Calculator Introduction The is a tool to calculate the standard deviation from the data, the standard error, the range, percentiles, the COV, confidence limits, or

### Regression Analysis: Basic Concepts

The simple linear model Regression Analysis: Basic Concepts Allin Cottrell Represents the dependent variable, y i, as a linear function of one independent variable, x i, subject to a random disturbance

### AP Statistics 1998 Scoring Guidelines

AP Statistics 1998 Scoring Guidelines These materials are intended for non-commercial use by AP teachers for course and exam preparation; permission for any other use must be sought from the Advanced Placement

### Correlating PSI and CUP Denton Bramwell

Correlating PSI and CUP Denton Bramwell Having inherited the curiosity gene, I just can t resist fiddling with things. And one of the things I can t resist fiddling with is firearms. I think I am the only

### Example G Cost of construction of nuclear power plants

1 Example G Cost of construction of nuclear power plants Description of data Table G.1 gives data, reproduced by permission of the Rand Corporation, from a report (Mooz, 1978) on 32 light water reactor

### Statistics courses often teach the two-sample t-test, linear regression, and analysis of variance

2 Making Connections: The Two-Sample t-test, Regression, and ANOVA In theory, there s no difference between theory and practice. In practice, there is. Yogi Berra 1 Statistics courses often teach the two-sample

### Summarizing Data: Measures of Variation

Summarizing Data: Measures of Variation One aspect of most sets of data is that the values are not all alike; indeed, the extent to which they are unalike, or vary among themselves, is of basic importance

### Univariate Regression

Univariate Regression Correlation and Regression The regression line summarizes the linear relationship between 2 variables Correlation coefficient, r, measures strength of relationship: the closer r is

### Chapter 23. Inferences for Regression

Chapter 23. Inferences for Regression Topics covered in this chapter: Simple Linear Regression Simple Linear Regression Example 23.1: Crying and IQ The Problem: Infants who cry easily may be more easily

### Hypothesis Testing. Chapter Introduction

Contents 9 Hypothesis Testing 553 9.1 Introduction............................ 553 9.2 Hypothesis Test for a Mean................... 557 9.2.1 Steps in Hypothesis Testing............... 557 9.2.2 Diagrammatic

### Name: Date: Use the following to answer questions 3-4:

Name: Date: 1. Determine whether each of the following statements is true or false. A) The margin of error for a 95% confidence interval for the mean increases as the sample size increases. B) The margin

### Lecture Notes Module 1

Lecture Notes Module 1 Study Populations A study population is a clearly defined collection of people, animals, plants, or objects. In psychological research, a study population usually consists of a specific

### 5. Linear regression and correlation

Statistics for Engineers 5-1 5. Linear regression and correlation If we measure a response variable at various values of a controlled variable, linear regression is the process of fitting a straight line

### Uncertainties and Error Propagation

Uncertainties and Error Propagation By Vern Lindberg 1. Systematic and random errors. No measurement made is ever exact. The accuracy (correctness) and precision (number of significant figures) of a measurement

### Descriptive Statistics

Y520 Robert S Michael Goal: Learn to calculate indicators and construct graphs that summarize and describe a large quantity of values. Using the textbook readings and other resources listed on the web

### Chapter 7. One-way ANOVA

Chapter 7 One-way ANOVA One-way ANOVA examines equality of population means for a quantitative outcome and a single categorical explanatory variable with any number of levels. The t-test of Chapter 6 looks

### The Philosophy of Hypothesis Testing, Questions and Answers 2006 Samuel L. Baker

HYPOTHESIS TESTING PHILOSOPHY 1 The Philosophy of Hypothesis Testing, Questions and Answers 2006 Samuel L. Baker Question: So I'm hypothesis testing. What's the hypothesis I'm testing? Answer: When you're

### An Alternative Route to Performance Hypothesis Testing

EDHEC-Risk Institute 393-400 promenade des Anglais 06202 Nice Cedex 3 Tel.: +33 (0)4 93 18 32 53 E-mail: research@edhec-risk.com Web: www.edhec-risk.com An Alternative Route to Performance Hypothesis Testing

### Sample Exam #1 Elementary Statistics

Sample Exam #1 Elementary Statistics Instructions. No books, notes, or calculators are allowed. 1. Some variables that were recorded while studying diets of sharks are given below. Which of the variables

### Experiment #1, Analyze Data using Excel, Calculator and Graphs.

Physics 182 - Fall 2014 - Experiment #1 1 Experiment #1, Analyze Data using Excel, Calculator and Graphs. 1 Purpose (5 Points, Including Title. Points apply to your lab report.) Before we start measuring

### Introduction to Regression and Data Analysis

Statlab Workshop Introduction to Regression and Data Analysis with Dan Campbell and Sherlock Campbell October 28, 2008 I. The basics A. Types of variables Your variables may take several forms, and it

### DEPARTMENT OF PSYCHOLOGY UNIVERSITY OF LANCASTER MSC IN PSYCHOLOGICAL RESEARCH METHODS ANALYSING AND INTERPRETING DATA 2 PART 1 WEEK 9

DEPARTMENT OF PSYCHOLOGY UNIVERSITY OF LANCASTER MSC IN PSYCHOLOGICAL RESEARCH METHODS ANALYSING AND INTERPRETING DATA 2 PART 1 WEEK 9 Analysis of covariance and multiple regression So far in this course,