# Interpretation of Somers D under four simple models

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Interpretation of Somers D under four simple models Roger B. Newson 03 September, 04 Introduction Somers D is an ordinal measure of association introduced by Somers (96)[9]. It can be defined in terms of Kendall s τ a (Kendall and Gibbons, 990)[4]. Given a sequence of bivariate random variables (X, Y ) = {(X i, Y i )}, sampled using a sampling scheme for sampling pairs of bivariate pairs from a population of pairs of bivariate pairs, Kendall s τ a is defined as τ(x, Y ) = E [sign(x i X j )sign(y i Y j )] () (where E[ ] denotes expectation), or, equivalently, as the difference between the probability that the two X, Y pairs are concordant and the probability that the two X, Y pairs are discordant. A pair of X, Y pairs is said to be concordant if the larger X value is paired with the larger Y value, and is said to be discordant if the larger X value is paired with the smaller Y value. Somers D of Y with respect to X is defined as D(Y X) = τ(x, Y )/τ(x, X) () or, equivalently, as the difference between the two conditional probabilities of concordance and discordance, assuming that the X values are unequal. Note that Kendall s τ a is symmetric in X and Y, whereas Somers D is asymmetric in X and Y. Somers D plays a central role in rank statistics, and is the parameter behind most of these nonparametric methods, and can be estimated with confidence limits like other parameters. It can also be generalized to sampling probability weighted and/or clustered and/or possibly censored data. (See Newson (00)[7] and Newson (006)[6] for details.) However, many non statisticians appear to have a problem interpreting Somers D, even though a difference between proportions is arguably a simpler concept than an odds ratio, which many of them claim to understand better. Parameters are often easier to understand if they play a specific role in a specific model. Fortunately, in a number of simple standard models, Somers D can be derived from another parameter by a transformation. A confidence interval for Somers D can therefore be transformed, using inverse end point transformation, to give a robust, outlier resistant confidence interval for the other parameter, assuming that the model is true. We will discuss 4 simple models for bivariate X, Y pairs: Binary X, binary Y. Somers D is then the difference between proportions. Binary X, continuous Y, constant hazard ratio. hazard ratio. Somers D is then a transformation of the Binary X, Normal Y, equal variances. Somers D is then a transformation of the mean difference divided by the common standard deviation (SD). (Or, equivalently, a transformation of an interpercentile odds ratio of X with respect to Y.) Bivariate Normal X and Y. Somers D is then a transformation of the Pearson correlation coefficient. Each of these cases has its own Section, and a Figure (or Figures) illustrating the transformation. In each case, the alternative parameter (or its log) is nearly a linear function of Somers D, for values of Somers D between -0 and 0.

2 Interpretation of Somers D under four simple models Binary X, binary Y We assume that there are two subpopulations, Subpopulation A and Subpopulation B, and that X is a binary indicator variable, equal to for observations in Subpopulation A and 0 for observations in Subpopulation B, and that Y is also a binary variable, equal to for successful observations and 0 for failed observations. Define p A = Pr(Y = X = ), p B = Pr(Y = X = 0) (3) to be the probabilities of success in Subpopulations A and B, respectively. Then Somers D is simply D(Y X) = p A p B, (4) or the difference between the two probabilities of success. Figure gives Somers D as the trivial identity function of the difference between proportions. Note that Somers D is expressed on a linear scale of multiples of /, which (as we will see) is arguably a natural scale of reference points for Somers D. 3 Binary X, continuous Y, constant hazard ratio We assume, again, that X indicates membership of Subpopulation A instead of Subpopulation B, and assume, this time, that Y has a continuous distribution in each of the two subpopulations, with cumulative distribution functions F A ( ) and F B ( ), and probability density functions f A ( ) and f B ( ). We imagine Y to be a survival time variable, although we will not consider the possibility of censorship. In the two subpopulations, the survival functions and the hazard functions are given, respectively, by S k (y) = F k (y), h k (y) = f k (y)/s k (y), (5) where y is in the range of Y and k {A, B}. Suppose that the hazard ratio h A (y)/h B (y) is constant in y, and denote its value as R. (This is trivially the case if both subpopulations have an exponential distribution, with h k (y) = /µ k, where µ k is the subpopulation mean. However, it can also be the case if we assume some other distributional families, such as the Gompertz or Weibull families, or even if we do not assume any specific distributional family, but still assume the proportional hazards model of Cox (97)[].) Somers D is then derived as D(Y X) = y h B(y)S A (y)s B (y)dy y R h B(y)S A (y)s B (y)dy y h B(y)S A (y)s B (y)dy + y R h B(y)S A (y)s B (y)dy = ( R)/( + R). (6) Note that Somers D is then the parameter that is zero under the null hypothesis tested using the method of Gehan (965)[]. For finite R, this formula can be inverted to give the hazard ratio R as a function of Somers D by R = [ D(Y X)] / [ + D(Y X)] = [ c(y X)] /c(y X), (7) where c(y X) = [D(Y X) + ]/ is Harrell s c-index, which reparameterizes Somers D to a probability scale from 0 to (Harrell et al., 98)[3]. Note that, for continuous Y and binary X, Harrell s c is the probability of concordance, and that a constant hazard ratio R is the corresponding odds against concordance. Figure gives Somers D as a function of R. Note that R is expressed on a log scale, similarly to the standard practice with logistic regression. Somers D of lifetime with respect to membership of Population A is seen to be a decreasing logistic sigmoid function of the Population A/Population B log hazard ratio, equal to 0 when the log ratio is 0 and the ratio is therefore. A hazard ratio of (as typically found when comparing the lifetimes of cigarette smokers as Population A to lifetimes of nonsmokers as Population B) corresponds to a Somers D of /3, or a Harrell s c of /3. Similarly, a hazard ratio of / (as typically found when comparing lifetimes of nonsmokers as Population A to lifetimes of cigarette smokers as Population B) corresponds to a Somers D of /3, or a Harrell s c of /3. Therefore, although a smoker may possibly survive a non smoker of the same age, the odds are to against this happening. A hazard ratio of 3 corresponds to a Somers D of -0, and a hazard ratio of /3 corresponds to a Somers D of 0. For even more spectacular hazard ratios in either direction, the linearity breaks down, even though the hazard ratio is logged. 4 Binary X, Normal Y, equal variances We assume, again, that X indicates membership of Subpopulation A instead of Subpopulation B, and assume, this time, that Y has a Normal distribution in each of the two subpopulations, with respective means µ A and µ B and standard deviations (SDs) σ A and σ B. Then, the probability of concordance (Harrell s c) is

3 Interpretation of Somers D under four simple models 3 the probability that a random member of Population A has a higher Y value than a random member of Population B, or (equivalently) the probability that the difference between these two Y values is positive. This difference has a Normal distribution, with mean µ A µ B and variance σa + σ B. Somers D is therefore given by the formula ( ) µ A µ B D(Y X) = Φ, (8) σ A + σb where Φ( ) is the cumulative standard Normal distribution function. If the two SDs are both equal (to σ = σ A = σ B ), then this formula simplifies to ( ) ( ) µa µ B δ D(Y X) = Φ σ = Φ, (9) where δ = (µ A µ B )/σ is the difference between the two means, expressed in units of the common SD. The parameter δ can therefore be defined as a function of Somers D by the inverse formula δ = ( ) D(Y X) + Φ, (0) where Φ ( ) is the inverse standard Normal cumulative distribution function. Figure 3 gives Somers D as a function of the mean difference, expressed in SD units. Again, we see a sigmoid curve, but this time Somers D is increasing with the alternative parameter. Note that a mean difference of SD corresponds to a Somers D just above / (approximately ), corresponding to a concordance probability (or Harrell s c) just above 3/4, whereas a mean difference of SDs corresponds to a Somers D of just below / (approximately ), or a Harrell s c just below /4. For mean differences between SD and SD, the corresponding Somers D can be interpolated (approximately) in a linear fashion, with Somers D approximately equal to half the mean difference in SDs. A mean difference of SDs corresponds to a Somers D of approximately (slightly more than 5/6), or a Harrell s c of approximately (slightly over /). A mean difference of 3 SDs corresponds to a Somers D of approximately (slightly more than 9/0), or a Harrell s c of (over 49/50). For mean differences over 3 SDs, the probability of discordance falls to fractions of a percent, and Somers D becomes progressively less useful, as there is very little overlap between subpopulations for Somers D to measure. Within the range of approximate linearity (± SDs), a Somers D of 0, ±/4, ±/3 or ±/ corresponds to a difference of 0, ±.45064, ± or ± SDs, respectively. Note that the above argument applies equally if Y is calculated using a Normalizing and variance stabilizing monotonic transformation on an untransformed variable whose distribution is not Normal. As Somers D is a rank parameter, it is preserved by monotonically increasing transformations. Therefore, if a Normalizing and variance stabilizing increasing transformation exists, then we can estimate δ from the Somers D of the untransformed variable with respect to the binary X by end point transformation of Somers D and its confidence limits, using (0). This can be done without even knowing the Normalizing and variance stabilizing transformation. Therefore, a lot of work can be saved this way, if it was the mean difference in SDs that we wanted to know. 4. Likelihood and odds ratios for diagnostic and case control data The equal variance Normal model is also often used as a toy model for the problem of defining diagnostic tests, based on a continuous marker variable Y, for membership of Subpopulation A instead of Subpopulation B. In this case, Subpopulation A might be diseased individuals, Subpopulation B might be non diseased individuals, and Y might be a quantitative test result. If the true distribution of Y in each subpopulation is Normal, with a common subpopulation variance and different subpopulation means, then the log of the likelihood ratio between Subpopulations A and B is a linear function of the Y -value, given by the formula log LR = µ B µ A σ + ( µa µ B σ ) Y. () Note that the intercept term is the value of the log likelihood ratio if Y = 0, whereas the slope term is equal to δ/σ in our notation. Therefore, δ, the mean difference expressed in standard deviations, is the slope of log LR with respect to Y/σ, the Y value expressed in standard deviations. In Bayesian inference, this log likelihood ratio is added to the log of the prior (pre test) odds of membership of Subpopulation A instead of Subpopulation B, to derive the log of the posterior (post test) odds of membership of Subpopulation A instead of Subpopulation B. The role of Somers D in diagnostic tests is discussed in Newson (00)[7]. Briefly, if we graph true positive rate (sensitivity) against false positive rate ( specificity), and choose

4 Interpretation of Somers D under four simple models 4 points on the graph corresponding to the various possible test thresholds, and join these points in ascending order of candidate threshold value, then the resulting curve is known as the sensitivity specificity curve, or (alternatively) as the receiver operating characteristic (ROC) curve. Harrell s c is the area below the ROC curve, whereas Somers D is the difference between the areas below and above the ROC curve. The likelihood ratio is the slope of the ROC curve (defined as the derivative of true positive rate with respect to false positive rate), and, in the equal variance Normal model, is computed by exponentiating (). The equal variance Normal model, predicting the test result from the disease status, is therefore combined with Bayes theorem to imply a logistic regression model for estimating the disease status from the test result. Figure 3 therefore implies (unsurprisingly) that, other things being equal, the ROC curve becomes higher as the mean difference (in SDs) becomes higher. The equal variance Normal toy model is also useful for interpreting case control studies. In these studies, Subpopulation A is the cases, Subpopulation B is the controls, and Y is a continuous disease predictor. The rare disease assumption implies that the distribution of Y in the controls is a good approximation to the distribution of Y in the population, and that a population relative risk can be estimated using the corresponding odds ratio. The 00qth percentile of Y in the control population is then ξ q = µ B + σφ (q) for q in the interior of the unit interval. It follows from () that, for q < 0, the log of the interpercentile odds ratio between ξ q and ξ q is defined as the difference between the log likelihood ratios at ξ q and ξ q. If we express Y in SD units, then, by (), this difference is given by log OR q = δ [ Φ ( q) Φ (q) ]. () For instance, if q = /4 = 0, then OR q is the interquartile odds ratio, equal to OR 0 = exp { δ [ Φ (0) Φ (0) ] }. (3) The plot of Somers D against the interquartile odds ratio OR 0 (on a log scale) is given as Figure 4. A Somers D of /, /6, /4, /3, 5/ or / is equivalent to an interquartile odds ratio of.0933, , , , or , respectively. Again, this argument is still valid if Y is derived from an original non Normal variable, using a Normalizing and variance stabilizing monotonically increasing transformation. And, again, Somers D can be estimated from the original untransformed variable, without knowing the transformation. This argument about interpercentile odds ratios might possibly be extended to the case where X is a possibly censored survival time variable instead of an uncensored binary variable. In this case, the interpercentile odds ratio would be replaced by an interpercentile hazard ratio, and the equally variable Normal distributions would be assumed to belong to non survivors and survivors, instead of to cases and controls. Note that, unlike the case with a constant hazard ratio for a continuous Y between binary X groups, it would then be X that is the lifetime variable, and the continuous Y that is the predictor variable. 5 Bivariate Normal X and Y We assume, this time, that X and Y have a joint bivariate Normal distribution, with means µ X and µ Y, SDs σ X and σ Y, and a Pearson correlation coefficient ρ. As both X and Y are continuous, Somers D is equal to Kendall s τ a, and is therefore given by the formula D(Y X) = arcsin(ρ), (4) π which can be inverted to give [ π ] ρ = sin D(Y X). (5) This relation is known as Greiner s relation. The curve of (4) is illustrated in Figure 5. Note that Pearson correlations of /, /, 0, / and / correspond to Kendall correlations (and therefore Somers D values) of /, /3, 0, /3 and /, respectively. This implies that audiences accustomed to Pearson correlations may be less impressed when presented with the same correlations on the Kendall Somers scale. A possible remedy for this problem is to use the end point transformation (5) on confidence intervals for Somers D or Kendall s τ a to define outlier resistant confidence intervals for the Pearson correlation. This practice of end point transformation is also useful if we expect variables X and Y not to have a bivariate Normal distribution themselves, but to be transformed to a pair of bivariate Normal variables by a pair of monotonically increasing transformations g X (X) and g Y (Y ). As Somers D and Kendall s τ a are rank parameters, they will not be affected by substituting X for g X (X) and/or substituting Y for g Y (Y ). Therefore, the end point transformation method can be used to estimate the Pearson correlation between

5 Interpretation of Somers D under four simple models 5 g X (X) and g Y (Y ), without even knowing the form of the functions g X ( ) and g Y ( ). This can save a lot of work, if the Pearson correlation between the transformed variables was what we wanted to know. Greiner s relation, or something very similar, is expected to hold for a lot of other bivariate continuous distributions, apart from the bivariate Normal. Kendall (949)[5] showed that Greiner s relation is not affected by odd numbered moments, such as skewness. Newson (987)[8], using a much simpler argument, discussed the case where two variables X and Y are defined as sums or differences of up to 3 latent variables (hidden variables) U, V and W, which were assumed to be sampled independently from an arbitrary common continuous distribution. It was shown that different definitions of X and Y implied the values of Kendall s τ a and Pearson s correlation displayed in Table. These pairs of values all lie along the line of Greiner s relation, as displayed in Figure 5. Table : Kendall and Pearson correlations for X and Y defined in terms of independent continuous latent variables U, V and W. X Y Kendall s τ a Pearson s ρ U ±V 0 0 V + U W ± U ± 3 ± U V ± U ± ± U ±U ± ± 6 Acknowledgement I would like to thank Raymond Boston of Pennsylvania University, PA, USA for raising the issue of interpretations of Somers D, and for prompting me to summarize these multiple interpretations in a single document. References [] Cox DR. Regression models and life tables (with discussion). Journal of the Royal Statistical Society, Series B 97; 34(): [] Gehan EA. A generalized Wilcoxon test for comparing arbitrarily single censored samples. Biometrika 965; 5(/): [3] Harrell FE, Califf RM, Pryor DB, Lee KL, Rosati RA. Evaluating the yield of medical tests. Journal of the American Medical Association 98; 47(8): [4] Kendall MG, Gibbons JD. Rank Correlation Methods. 5th Edition. New York, NY: Oxford University Press; 990. [5] Kendall MG. Rank and product moment correlation. Biometrika 949; 36(/): [6] Newson R. Confidence intervals for rank statistics: Somers D and extensions. The Stata Journal 006; 6(3): [7] Newson R. Parameters behind nonparametric statistics: Kendall s tau, Somers D and median differences. The Stata Journal 00; (): [8] Newson RB. An analysis of cinematographic cell division data using U statistics [DPhil dissertation]. Brighton, UK: Sussex University; 987. [9] Somers RH. A new asymmetric measure of association for ordinal variables. American Sociological Review 96; 7(6):

6 Interpretation of Somers D under four simple models 6 Figure : Somers D and difference between proportions in the two sample binary model Difference between proportions Figure : Somers D and hazard ratio in the two sample constant hazard ratio model Hazard ratio

7 Interpretation of Somers D under four simple models 7 Figure 3: Somers D and mean difference in SDs in the two sample equal variance Normal model Mean difference (SDs) Figure 4: Somers D and interquartile odds ratio in the two sample equal variance Normal model Interquartile odds ratio

8 Interpretation of Somers D under four simple models Figure 5: Somers D and Pearson correlation in the bivariate Normal model Pearson correlation coefficient

### 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:

### Basic Statistics and Data Analysis for Health Researchers from Foreign Countries

Basic Statistics and Data Analysis for Health Researchers from Foreign Countries Volkert Siersma siersma@sund.ku.dk The Research Unit for General Practice in Copenhagen Dias 1 Content Quantifying association

### 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

### Biostatistics: Types of Data Analysis

Biostatistics: Types of Data Analysis Theresa A Scott, MS Vanderbilt University Department of Biostatistics theresa.scott@vanderbilt.edu http://biostat.mc.vanderbilt.edu/theresascott Theresa A Scott, MS

### SUMAN DUVVURU STAT 567 PROJECT REPORT

SUMAN DUVVURU STAT 567 PROJECT REPORT SURVIVAL ANALYSIS OF HEROIN ADDICTS Background and introduction: Current illicit drug use among teens is continuing to increase in many countries around the world.

: Table of Contents... 1 Overview of Model... 1 Dispersion... 2 Parameterization... 3 Sigma-Restricted Model... 3 Overparameterized Model... 4 Reference Coding... 4 Model Summary (Summary Tab)... 5 Summary

### Probability and Statistics Vocabulary List (Definitions for Middle School Teachers)

Probability and Statistics Vocabulary List (Definitions for Middle School Teachers) B Bar graph a diagram representing the frequency distribution for nominal or discrete data. It consists of a sequence

### 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.

### Probability and Statistics Prof. Dr. Somesh Kumar Department of Mathematics Indian Institute of Technology, Kharagpur

Probability and Statistics Prof. Dr. Somesh Kumar Department of Mathematics Indian Institute of Technology, Kharagpur Module No. #01 Lecture No. #15 Special Distributions-VI Today, I am going to introduce

### Survival Analysis Using SPSS. By Hui Bian Office for Faculty Excellence

Survival Analysis Using SPSS By Hui Bian Office for Faculty Excellence Survival analysis What is survival analysis Event history analysis Time series analysis When use survival analysis Research interest

### Dongfeng Li. Autumn 2010

Autumn 2010 Chapter Contents Some statistics background; ; Comparing means and proportions; variance. Students should master the basic concepts, descriptive statistics measures and graphs, basic hypothesis

### Nominal and ordinal logistic regression

Nominal and ordinal logistic regression April 26 Nominal and ordinal logistic regression Our goal for today is to briefly go over ways to extend the logistic regression model to the case where the outcome

### 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

### Nonparametric adaptive age replacement with a one-cycle criterion

Nonparametric adaptive age replacement with a one-cycle criterion P. Coolen-Schrijner, F.P.A. Coolen Department of Mathematical Sciences University of Durham, Durham, DH1 3LE, UK e-mail: Pauline.Schrijner@durham.ac.uk

### 4. Joint Distributions of Two Random Variables

4. Joint Distributions of Two Random Variables 4.1 Joint Distributions of Two Discrete Random Variables Suppose the discrete random variables X and Y have supports S X and S Y, respectively. The joint

### Spearman s correlation

Spearman s correlation Introduction Before learning about Spearman s correllation it is important to understand Pearson s correlation which is a statistical measure of the strength of a linear relationship

### Summary of Formulas and Concepts. Descriptive Statistics (Ch. 1-4)

Summary of Formulas and Concepts Descriptive Statistics (Ch. 1-4) Definitions Population: The complete set of numerical information on a particular quantity in which an investigator is interested. We assume

### Quantitative Data Analysis: Choosing a statistical test Prepared by the Office of Planning, Assessment, Research and Quality

Quantitative Data Analysis: Choosing a statistical test Prepared by the Office of Planning, Assessment, Research and Quality 1 To help choose which type of quantitative data analysis to use either before

### Auxiliary Variables in Mixture Modeling: 3-Step Approaches Using Mplus

Auxiliary Variables in Mixture Modeling: 3-Step Approaches Using Mplus Tihomir Asparouhov and Bengt Muthén Mplus Web Notes: No. 15 Version 8, August 5, 2014 1 Abstract This paper discusses alternatives

### 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

### Yiming Peng, Department of Statistics. February 12, 2013

Regression Analysis Using JMP Yiming Peng, Department of Statistics February 12, 2013 2 Presentation and Data http://www.lisa.stat.vt.edu Short Courses Regression Analysis Using JMP Download Data to Desktop

### UNIVERSITY OF NAIROBI

UNIVERSITY OF NAIROBI MASTERS IN PROJECT PLANNING AND MANAGEMENT NAME: SARU CAROLYNN ELIZABETH REGISTRATION NO: L50/61646/2013 COURSE CODE: LDP 603 COURSE TITLE: RESEARCH METHODS LECTURER: GAKUU CHRISTOPHER

### Semester 1 Statistics Short courses

Semester 1 Statistics Short courses Course: STAA0001 Basic Statistics Blackboard Site: STAA0001 Dates: Sat. March 12 th and Sat. April 30 th (9 am 5 pm) Assumed Knowledge: None Course Description Statistical

### 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

### AMS7: WEEK 8. CLASS 1. Correlation Monday May 18th, 2015

AMS7: WEEK 8. CLASS 1 Correlation Monday May 18th, 2015 Type of Data and objectives of the analysis Paired sample data (Bivariate data) Determine whether there is an association between two variables This

### Analysing Questionnaires using Minitab (for SPSS queries contact -) Graham.Currell@uwe.ac.uk

Analysing Questionnaires using Minitab (for SPSS queries contact -) Graham.Currell@uwe.ac.uk Structure As a starting point it is useful to consider a basic questionnaire as containing three main sections:

### Chris Slaughter, DrPH. GI Research Conference June 19, 2008

Chris Slaughter, DrPH Assistant Professor, Department of Biostatistics Vanderbilt University School of Medicine GI Research Conference June 19, 2008 Outline 1 2 3 Factors that Impact Power 4 5 6 Conclusions

### Probability Calculator

Chapter 95 Introduction Most statisticians have a set of probability tables that they refer to in doing their statistical wor. This procedure provides you with a set of electronic statistical tables that

### II. DISTRIBUTIONS distribution normal distribution. standard scores

Appendix D Basic Measurement And Statistics The following information was developed by Steven Rothke, PhD, Department of Psychology, Rehabilitation Institute of Chicago (RIC) and expanded by Mary F. Schmidt,

### The Big 50 Revision Guidelines for S1

The Big 50 Revision Guidelines for S1 If you can understand all of these you ll do very well 1. Know what is meant by a statistical model and the Modelling cycle of continuous refinement 2. Understand

### Logistic regression: Model selection

Logistic regression: April 14 The WCGS data Measures of predictive power Today we will look at issues of model selection and measuring the predictive power of a model in logistic regression Our data set

### Descriptive Statistics

Descriptive Statistics Suppose following data have been collected (heights of 99 five-year-old boys) 117.9 11.2 112.9 115.9 18. 14.6 17.1 117.9 111.8 16.3 111. 1.4 112.1 19.2 11. 15.4 99.4 11.1 13.3 16.9

### Gamma Distribution Fitting

Chapter 552 Gamma Distribution Fitting Introduction This module fits the gamma probability distributions to a complete or censored set of individual or grouped data values. It outputs various statistics

### Some Essential Statistics The Lure of Statistics

Some Essential Statistics The Lure of Statistics Data Mining Techniques, by M.J.A. Berry and G.S Linoff, 2004 Statistics vs. Data Mining..lie, damn lie, and statistics mining data to support preconceived

### Module 9: Nonparametric Tests. The Applied Research Center

Module 9: Nonparametric Tests The Applied Research Center Module 9 Overview } Nonparametric Tests } Parametric vs. Nonparametric Tests } Restrictions of Nonparametric Tests } One-Sample Chi-Square Test

### CALCULATIONS & STATISTICS

CALCULATIONS & STATISTICS CALCULATION OF SCORES Conversion of 1-5 scale to 0-100 scores When you look at your report, you will notice that the scores are reported on a 0-100 scale, even though respondents

### Models for Count Data With Overdispersion

Models for Count Data With Overdispersion Germán Rodríguez November 6, 2013 Abstract This addendum to the WWS 509 notes covers extra-poisson variation and the negative binomial model, with brief appearances

### Logistic Regression. Jia Li. Department of Statistics The Pennsylvania State University. Logistic Regression

Logistic Regression Department of Statistics The Pennsylvania State University Email: jiali@stat.psu.edu Logistic Regression Preserve linear classification boundaries. By the Bayes rule: Ĝ(x) = arg max

### Variables and Data A variable contains data about anything we measure. For example; age or gender of the participants or their score on a test.

The Analysis of Research Data The design of any project will determine what sort of statistical tests you should perform on your data and how successful the data analysis will be. For example if you decide

### What Does the Correlation Coefficient Really Tell Us About the Individual?

What Does the Correlation Coefficient Really Tell Us About the Individual? R. C. Gardner and R. W. J. Neufeld Department of Psychology University of Western Ontario ABSTRACT The Pearson product moment

### LOGIT AND PROBIT ANALYSIS

LOGIT AND PROBIT ANALYSIS A.K. Vasisht I.A.S.R.I., Library Avenue, New Delhi 110 012 amitvasisht@iasri.res.in In dummy regression variable models, it is assumed implicitly that the dependent variable Y

### Data Analysis. Lecture Empirical Model Building and Methods (Empirische Modellbildung und Methoden) SS Analysis of Experiments - Introduction

Data Analysis Lecture Empirical Model Building and Methods (Empirische Modellbildung und Methoden) Prof. Dr. Dr. h.c. Dieter Rombach Dr. Andreas Jedlitschka SS 2014 Analysis of Experiments - Introduction

### SYSM 6304: Risk and Decision Analysis Lecture 3 Monte Carlo Simulation

SYSM 6304: Risk and Decision Analysis Lecture 3 Monte Carlo Simulation M. Vidyasagar Cecil & Ida Green Chair The University of Texas at Dallas Email: M.Vidyasagar@utdallas.edu September 19, 2015 Outline

### Christfried Webers. Canberra February June 2015

c Statistical Group and College of Engineering and Computer Science Canberra February June (Many figures from C. M. Bishop, "Pattern Recognition and ") 1of 829 c Part VIII Linear Classification 2 Logistic

### Logistic regression modeling the probability of success

Logistic regression modeling the probability of success Regression models are usually thought of as only being appropriate for target variables that are continuous Is there any situation where we might

### 2.0 Lesson Plan. Answer Questions. Summary Statistics. Histograms. The Normal Distribution. Using the Standard Normal Table

2.0 Lesson Plan Answer Questions 1 Summary Statistics Histograms The Normal Distribution Using the Standard Normal Table 2. Summary Statistics Given a collection of data, one needs to find representations

### Non-Inferiority Tests for Two Means using Differences

Chapter 450 on-inferiority Tests for Two Means using Differences Introduction This procedure computes power and sample size for non-inferiority tests in two-sample designs in which the outcome is a continuous

### Statistics in Retail Finance. Chapter 6: Behavioural models

Statistics in Retail Finance 1 Overview > So far we have focussed mainly on application scorecards. In this chapter we shall look at behavioural models. We shall cover the following topics:- Behavioural

### 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,

### Descriptive Statistics

Descriptive Statistics Primer Descriptive statistics Central tendency Variation Relative position Relationships Calculating descriptive statistics Descriptive Statistics Purpose to describe or summarize

### NCSS Statistical Software

Chapter 06 Introduction This procedure provides several reports for the comparison of two distributions, including confidence intervals for the difference in means, two-sample t-tests, the z-test, the

### Overview of Violations of the Basic Assumptions in the Classical Normal Linear Regression Model

Overview of Violations of the Basic Assumptions in the Classical Normal Linear Regression Model 1 September 004 A. Introduction and assumptions The classical normal linear regression model can be written

### 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.

### MTH 140 Statistics Videos

MTH 140 Statistics Videos Chapter 1 Picturing Distributions with Graphs Individuals and Variables Categorical Variables: Pie Charts and Bar Graphs Categorical Variables: Pie Charts and Bar Graphs Quantitative

### Spatial Statistics Chapter 3 Basics of areal data and areal data modeling

Spatial Statistics Chapter 3 Basics of areal data and areal data modeling Recall areal data also known as lattice data are data Y (s), s D where D is a discrete index set. This usually corresponds to data

### Age to Age Factor Selection under Changing Development Chris G. Gross, ACAS, MAAA

Age to Age Factor Selection under Changing Development Chris G. Gross, ACAS, MAAA Introduction A common question faced by many actuaries when selecting loss development factors is whether to base the selected

### SAS Software to Fit the Generalized Linear Model

SAS Software to Fit the Generalized Linear Model Gordon Johnston, SAS Institute Inc., Cary, NC Abstract In recent years, the class of generalized linear models has gained popularity as a statistical modeling

### 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

### 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.

### Data Analysis, Research Study Design and the IRB

Minding the p-values p and Quartiles: Data Analysis, Research Study Design and the IRB Don Allensworth-Davies, MSc Research Manager, Data Coordinating Center Boston University School of Public Health IRB

### 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

### How to choose a statistical test. Francisco J. Candido dos Reis DGO-FMRP University of São Paulo

How to choose a statistical test Francisco J. Candido dos Reis DGO-FMRP University of São Paulo Choosing the right test One of the most common queries in stats support is Which analysis should I use There

### List of Examples. Examples 319

Examples 319 List of Examples DiMaggio and Mantle. 6 Weed seeds. 6, 23, 37, 38 Vole reproduction. 7, 24, 37 Wooly bear caterpillar cocoons. 7 Homophone confusion and Alzheimer s disease. 8 Gear tooth strength.

### 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

### Session 1.6 Measures of Central Tendency

Session 1.6 Measures of Central Tendency Measures of location (Indices of central tendency) These indices locate the center of the frequency distribution curve. The mode, median, and mean are three indices

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

### Statistical Rules of Thumb

Statistical Rules of Thumb Second Edition Gerald van Belle University of Washington Department of Biostatistics and Department of Environmental and Occupational Health Sciences Seattle, WA WILEY AJOHN

### Adequacy of Biomath. Models. Empirical Modeling Tools. Bayesian Modeling. Model Uncertainty / Selection

Directions in Statistical Methodology for Multivariable Predictive Modeling Frank E Harrell Jr University of Virginia Seattle WA 19May98 Overview of Modeling Process Model selection Regression shape Diagnostics

### INTRODUCTORY STATISTICS

INTRODUCTORY STATISTICS FIFTH EDITION Thomas H. Wonnacott University of Western Ontario Ronald J. Wonnacott University of Western Ontario WILEY JOHN WILEY & SONS New York Chichester Brisbane Toronto Singapore

### Class 6: Chapter 12. Key Ideas. Explanatory Design. Correlational Designs

Class 6: Chapter 12 Correlational Designs l 1 Key Ideas Explanatory and predictor designs Characteristics of correlational research Scatterplots and calculating associations Steps in conducting a correlational

### Ordinal Regression. Chapter

Ordinal Regression Chapter 4 Many variables of interest are ordinal. That is, you can rank the values, but the real distance between categories is unknown. Diseases are graded on scales from least severe

### Pearson s correlation

Pearson s correlation Introduction Often several quantitative variables are measured on each member of a sample. If we consider a pair of such variables, it is frequently of interest to establish if there

### PowerTeaching i3: Algebra I Mathematics

PowerTeaching i3: Algebra I Mathematics Alignment to the Common Core State Standards for Mathematics Standards for Mathematical Practice and Standards for Mathematical Content for Algebra I Key Ideas and

### 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

### 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,

### Assumptions. Assumptions of linear models. Boxplot. Data exploration. Apply to response variable. Apply to error terms from linear model

Assumptions Assumptions of linear models Apply to response variable within each group if predictor categorical Apply to error terms from linear model check by analysing residuals Normality Homogeneity

### Permutation Tests for Comparing Two Populations

Permutation Tests for Comparing Two Populations Ferry Butar Butar, Ph.D. Jae-Wan Park Abstract Permutation tests for comparing two populations could be widely used in practice because of flexibility of

### The Kendall Rank Correlation Coefficient

The Kendall Rank Correlation Coefficient Hervé Abdi Overview The Kendall (955) rank correlation coefficient evaluates the degree of similarity between two sets of ranks given to a same set of objects.

### 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

### Lean Six Sigma Analyze Phase Introduction. TECH 50800 QUALITY and PRODUCTIVITY in INDUSTRY and TECHNOLOGY

TECH 50800 QUALITY and PRODUCTIVITY in INDUSTRY and TECHNOLOGY Before we begin: Turn on the sound on your computer. There is audio to accompany this presentation. Audio will accompany most of the online

### Unit 31 A Hypothesis Test about Correlation and Slope in a Simple Linear Regression

Unit 31 A Hypothesis Test about Correlation and Slope in a Simple Linear Regression Objectives: To perform a hypothesis test concerning the slope of a least squares line To recognize that testing for a

### Fairfield Public Schools

Mathematics Fairfield Public Schools AP Statistics AP Statistics BOE Approved 04/08/2014 1 AP STATISTICS Critical Areas of Focus AP Statistics is a rigorous course that offers advanced students an opportunity

### Example: Credit card default, we may be more interested in predicting the probabilty of a default than classifying individuals as default or not.

Statistical Learning: Chapter 4 Classification 4.1 Introduction Supervised learning with a categorical (Qualitative) response Notation: - Feature vector X, - qualitative response Y, taking values in C

### Notes for STA 437/1005 Methods for Multivariate Data

Notes for STA 437/1005 Methods for Multivariate Data Radford M. Neal, 26 November 2010 Random Vectors Notation: Let X be a random vector with p elements, so that X = [X 1,..., X p ], where denotes transpose.

### Chapter 3 RANDOM VARIATE GENERATION

Chapter 3 RANDOM VARIATE GENERATION In order to do a Monte Carlo simulation either by hand or by computer, techniques must be developed for generating values of random variables having known distributions.

### Errata and updates for ASM Exam C/Exam 4 Manual (Sixteenth Edition) sorted by page

Errata for ASM Exam C/4 Study Manual (Sixteenth Edition) Sorted by Page 1 Errata and updates for ASM Exam C/Exam 4 Manual (Sixteenth Edition) sorted by page Practice exam 1:9, 1:22, 1:29, 9:5, and 10:8

### Sampling and Hypothesis Testing

Population and sample Sampling and Hypothesis Testing Allin Cottrell Population : an entire set of objects or units of observation of one sort or another. Sample : subset of a population. Parameter versus

### LAB 4 INSTRUCTIONS CONFIDENCE INTERVALS AND HYPOTHESIS TESTING

LAB 4 INSTRUCTIONS CONFIDENCE INTERVALS AND HYPOTHESIS TESTING In this lab you will explore the concept of a confidence interval and hypothesis testing through a simulation problem in engineering setting.

### 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

### Chapter 14: Analyzing Relationships Between Variables

Chapter Outlines for: Frey, L., Botan, C., & Kreps, G. (1999). Investigating communication: An introduction to research methods. (2nd ed.) Boston: Allyn & Bacon. Chapter 14: Analyzing Relationships Between

### Statistics I for QBIC. Contents and Objectives. Chapters 1 7. Revised: August 2013

Statistics I for QBIC Text Book: Biostatistics, 10 th edition, by Daniel & Cross Contents and Objectives Chapters 1 7 Revised: August 2013 Chapter 1: Nature of Statistics (sections 1.1-1.6) Objectives

### Glossary of Statistical Terms

Department of Biostatistics Vanderbilt University School of Medicine biostat.mc.vanderbilt.edu/clinstat May 26, 2015 Glossary of Statistical Terms adjusting or controlling for a variable: Assessing the

### 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

### Comparing the predictive power of survival models using Harrell s c or Somers D

The Stata Journal (yyyy) vv, Number ii, pp. 1 19 Comparing the predictive power of survival models using Harrell s c or Somers D Roger B. Newson National Heart and Lung Institute, Imperial College London

### Regression Modeling Strategies

Frank E. Harrell, Jr. Regression Modeling Strategies With Applications to Linear Models, Logistic Regression, and Survival Analysis With 141 Figures Springer Contents Preface Typographical Conventions

### Lecture 2 ESTIMATING THE SURVIVAL FUNCTION. One-sample nonparametric methods

Lecture 2 ESTIMATING THE SURVIVAL FUNCTION One-sample nonparametric methods There are commonly three methods for estimating a survivorship function S(t) = P (T > t) without resorting to parametric models:

### 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