# Effect Sizes Based on Means

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 CHAPTER 4 Effect Sizes Based on Means Introduction Raw (unstardized) mean difference D Stardized mean difference, d g Resonse ratios INTRODUCTION When the studies reort means stard deviations, the referred effect size is usually the raw mean difference, the stardized mean difference, or the resonse ratio. These effect sizes are discussed in this chater. RAW (UNSTANDARDIZED) MEAN DIFFERENCE D When the outcome is reorted on a meaningful scale all studies in the analysis use the same scale, the meta-analysis can be erformed directly on the raw difference in means (henceforth, we will use the more common term, raw mean difference). The rimary advantage of the raw mean difference is that it is intuitively meaningful, either inherently (for examle, blood ressure, which is measured on a known scale) or because of widesread use (for examle, a national achievement test for students, where all relevant arties are familiar with the scale). Consider a study that reorts means for two grous (Treated Control) suose we wish to comare the means of these two grous. Let 1 2 be the true (oulation) means of the two grous. The oulation mean difference is defined as D ¼ 1 2 : ð4:1þ In the two sections that follow we show how to comute an estimate D of this arameter its variance from studies that used two indeendent grous from studies that used aired grous or matched designs. Introduction to Meta-Analysis. Michael Borenstein, L. V. Hedges, J. P. T. Higgins H. R. Rothstein 2009 John Wiley & Sons, Ltd. ISBN:

2 22 Effect Size Precision Comuting D from studies that use indeendent grous We can estimate the mean difference D from a study that used two indeendent grous as follows. Let X 1 X 2 be the samle means of the two indeendent grous. The samle estimate of D is just the difference in samle means, namely D ¼ X 1 X 2 : ð4:2þ Note that uercase D is used for the raw mean difference, whereas lowercase d will be used for the stardized mean difference (below). Let S 1 S 2 be the samle stard deviations of the two grous, n 1 n 2 be the samle sizes in the two grous. If we assume that the two oulation stard deviations are the same (as is assumed to be the case in most arametric data analysis techniques), so that , then the variance of D is V D 5 n 1 þ n 2 n 1 n 2 S 2 ooled ; ð4:3þ where sffiffiffiffiffiffiffiffiffiffi S ooled 5 ðn 1 1ÞS 2 1 þðn 2 1ÞS 2 2 : n 1 þ n 2 2 ð4:4þ If we don t assume that the two oulation stard deviations are the same, then the variance of D is V D 5 S2 1 n 1 þ S2 2 n 2 : ð4:5þ In either case, the stard error of D is then the square root of V, ffiffiffiffiffiffi V D: ð4:6þ For examle, suose that a study has samle means X , X , samle stard deviations S , S , samle sizes n 1 5 n The raw mean difference D is D 5 103:00 100:00 5 3:00: If we assume that then the ooled stard deviation within grous is sffiffiffiffi ð50 1Þ5:5 2 þ ð50 1Þ4:5 2 S ooled 5 5 5:0249: 50 þ 50 2 The variance stard error of D are given by 50 þ 50 V D : :0100; 1: :0050:

3 Chater 4: Effect Sizes Based on Means If we do not assume that then the variance stard error of D are given by V D 5 5:52 50 þ 4: :0100 1: :0050: In this examle formulas (4.3) (4.5) yield the same result, but this will be true only if the samle size /or the estimate of the variances is the same in the two grous. Comuting D from studies that use matched grous or re-ost scores The revious formulas are aroriate for studies that use two indeendent grous. Another study design is the use of matched grous, where airs of articiants are matched in some way (for examle, siblings, or atients at the same stage of disease), with the two members of each air then being assigned to different grous. The unit of analysis is the air, the advantage of this design is that each air serves as its own control, reducing the error term increasing the statistical ower. The magnitude of the imact deends on the correlation between (for examle) siblings, with a higher correlation yielding a lower variance ( increased recision). The samle estimate of D is just the samle mean difference, D. If we have the difference score for each air, which gives us the mean difference X diff the stard deviation of these differences (S diff ), then D 5 X diff ; ð4:7þ V D 5 S2 diff n ; ð4:8þ where n is the number of airs, ffiffiffiffiffiffi V D: ð4:9þ For examle, if the mean difference is 5.00 with stard deviation of the difference of n of 50 airs, then D 5 5:0000; V D 5 10: :0000; ð4:10þ ffiffiffiffiffiffiffiffiffi 2:00 5 1:4142: ð4:11þ

4 24 Effect Size Precision Alternatively, if we have the mean stard deviation for each set of scores (for examle, siblings A B), the difference is D ¼ X 1 X 2 : ð4:12þ The variance is again given by V D 5 S2 diff n ; ð4:13þ where n is the number of airs, the stard error is given by ffiffiffiffiffiffi V D: ð4:14þ However, in this case we need to comute the stard deviation of the difference scores from the stard deviation of each sibling s scores. This is given by qffiffiffiffiffiffiffiffiffiffiffiffi S diff 5 S 2 1 þ S2 2 2 r S 1 S 2 ð4:15þ where r is the correlation between siblings in matched airs. If S 1 5 S 2, then (4.15) simlifies to qffiffiffiffiffiffiffiffi S diff 5 2 S 2 ooledð1 rþ: ð4:16þ In either case, as r moves toward 1.0 the stard error of the aired difference will decrease, when r 5 0 the stard error of the difference is the same as it would be for a study with two indeendent grous, each of size n. For examle, suose the means for siblings A B are , with stard deviations 10 10, the correlation between the two sets of scores is 0.50, the number of airs is 50. Then D 5 105:00 100:00 5 5:0000; V D 5 10: :0000; ffiffiffiffiffiffiffiffiffi 2:00 5 1:4142: In the calculation of V D, the S diff is comuted using ffiffiffiffiffiffiffiffiffiffiffi S diff þ : :0000 or qffiffiffiffiffiffiffiffiffi S diff ð1 0:50Þ 5 10:0000: The formulas for matched designs aly to re-ost designs as well. The re ost means corresond to the means in the matched grous, n is the number of subjects, r is the correlation between re-scores ost-scores.

5 Chater 4: Effect Sizes Based on Means Calculation of effect size estimates from information that is reorted When a researcher has access to a full set of summary data such as the mean, stard deviation, samle size for each grou, the comutation of the effect size its variance is relatively straightforward. In ractice, however, the researcher will often be working with only artial data. For examle, a aer may ublish only the -value, means samle sizes from a test of significance, leaving it to the meta-analyst to back-comute the effect size variance. For information on comuting effect sizes from artial information, see Borenstein et al. (2009). Including different study designs in the same analysis Sometimes a systematic review will include studies that used indeendent grous also studies that used matched grous. From a statistical ersective the effect size (D) has the same meaning regardless of the study design. Therefore, we can comute the effect size variance from each study using the aroriate formula, then include all studies in the same analysis. While there is no technical barrier to using different study designs in the same analysis, there may be a concern that studies which used different designs might differ in substantive ways as well (see Chater 40). For all study designs (whether using indeendent or aired grous) the direction of the effect (X 1 X 2 or X 2 X 1 ) is arbitrary, excet that the researcher must decide on a convention then aly this consistently. For examle, if a ositive difference will indicate that the treated grou did better than the control grou, then this convention must aly for studies that used indeendent designs for studies that used re-ost designs. In some cases it might be necessary to reverse the comuted sign of the effect size to ensure that the convention is followed. STANDARDIZED MEAN DIFFERENCE, d AND g As noted, the raw mean difference is a useful index when the measure is meaningful, either inherently or because of widesread use. By contrast, when the measure is less well known (for examle, a rorietary scale with limited distribution), the use of a raw mean difference has less to recommend it. In any event, the raw mean difference is an otion only if all the studies in the meta-analysis use the same scale. If different studies use different instruments (such as different sychological or educational tests) to assess the outcome, then the scale of measurement will differ from study to study it would not be meaningful to combine raw mean differences. In such cases we can divide the mean difference in each study by that study s stard deviation to create an index (the stardized mean difference) that would be comarable across studies. This is the same aroach suggested by Cohen (1969, 1987) in connection with describing the magnitude of effects in statistical ower analysis.

6 Effect Size Precision The stardized mean difference can be considered as being comarable across studies based on either of two arguments (Hedges Olkin, 1985). If the outcome measures in all studies are linear transformations of each other, the stardized mean difference can be seen as the mean difference that would have been obtained if all data were transformed to a scale where the stard deviation within-grous was equal to 1.0. The other argument for comarability of stardized mean differences is the fact that the stardized mean difference is a measure of overla between distributions. In this telling, the stardized mean difference reflects the difference between the distributions in the two grous ( how each reresents a distinct cluster of scores) even if they do not measure exactly the same outcome (see Cohen, 1987, Grissom Kim, 2005). Consider a study that uses two indeendent grous, suose we wish to comare the means of these two grous. Let 1 1 be the true (oulation) mean stard deviation of the first grou let 2 2 be the true (oulation) mean stard deviation of the other grou. If the two oulation stard deviations are the same (as is assumed in most arametric data analysis techniques), so that , then the stardized mean difference arameter or oulation stardized mean difference is defined as : ð4:17þ In the sections that follow, we show how to estimate from studies that used indeendent grous, from studies that used re-ost or matched grou designs. It is also ossible to estimate from studies that used other designs (including clustered designs) but these are not addressed here (see resources at the end of this Part). We make the common assumtion that , which allows us to ool the estimates of the stard deviation, do not address the case where these are assumed to differ from each other. Comuting d g from studies that use indeendent grous We can estimate the stardized mean difference () from studies that used two indeendent grous as d 5 X 1 X 2 S within : ð4:18þ In the numerator, X 1 X 2 are the samle means in the two grous. In the denominator S within is the within-grous stard deviation, ooled across grous, sffiffiffiffiffiffiffiffiffiffi ðn 1 1ÞS 2 1 S within 5 þðn 2 1ÞS 2 2 n 1 þ n 2 2 ð4:19þ where n 1 n 2 are the samle sizes in the two grous, S 1 S 2 are the stard deviations in the two grous. The reason that we ool the two samle

7 Chater 4: Effect Sizes Based on Means estimates of the stard deviation is that even if we assume that the underlying oulation stard deviations are the same (that is ), it is unlikely that the samle estimates S 1 S 2 will be identical. By ooling the two estimates of the stard deviation, we obtain a more accurate estimate of their common value. The samle estimate of the stardized mean difference is often called Cohen s d in research synthesis. Some confusion about the terminology has resulted from the fact that the index, originally roosed by Cohen as a oulation arameter for describing the size of effects for statistical ower analysis is also sometimes called d. In this volume we use the symbol to denote the effect size arameter d for the samle estimate of that arameter. The variance of d is given (to a very good aroximation) by V d 5 n 1 þ n 2 n 1 n 2 d 2 þ 2ðn 1 þ n 2 Þ : ð4:20þ In this equation the first term on the right of the equals sign reflects uncertainty in the estimate of the mean difference (the numerator in (4.18)), the second reflects uncertainty in the estimate of S within (the denominator in (4.18)). The stard error of d is the square root of V d, SE d 5 ffiffiffiffiffi V d: ð4:21þ It turns out that d has a slight bias, tending to overestimate the absolute value of in small samles. This bias can be removed by a simle correction that yields an unbiased estimate of, with the unbiased estimate sometimes called Hedges g (Hedges, 1981). To convert from d to Hedges g we use a correction factor, which is called J. Hedges (1981) gives the exact formula for J, but in common ractice researchers use an aroximation, 3 J 5 1 4df 1 : ð4:22þ In this exression, df is the degrees of freedom used to estimate S within, which for two indeendent grous is n 1 þ n 2 2. This aroximation always has error of less than less than ercent when df 10 (Hedges, 1981). Then, g 5 J d; V g 5 J 2 V d ; ð4:23þ ð4:24þ SE g 5 ffiffiffiffiffi V g: ð4:25þ For examle, suose a study has samle means X , X , samle stard deviations S , S , samle sizes n 1 5 n We would estimate the ooled-within-grous stard deviation as

8 28 Effect Size Precision sffiffiffiffi ð50 1Þ5:5 2 þ ð50 1Þ4:5 2 S within 5 5 5:0249: 50 þ 50 2 Then, d 5 5 0:5970; 5: þ 50 V d þ 0: þ ð 50Þ 5 0:0418; SE d 5 0: :2044: The correction factor (J), Hedges g, its variance stard error are given by 3 J :9923; g 5 0:9923 0: :5924; v g 5 0: : :0411; SE g 5 0: :2028: The correction factor (J) is always less than 1.0, so g will always be less than d in absolute value, the variance of g will always be less than the variance of d. However, J will be very close to 1.0 unless df is very small (say, less than 10) so (as in this examle) the difference is usually trivial (Hedges, 1981). Some slightly different exressions for the variance of d ( g) have been given by different authors even the same authors at different times. For examle, the denominator of the second term of the variance of d is given here as 2(n 1 þ n 2 ). This exression is obtained by one method (assuming the n s become ffiffiffi large with fixed). An alternate derivation (assuming n s become large with n fixed) leads to a denominator in the second term that is slightly different, namely 2(n 1 þ n 2 2). Unless n 1 n 2 are very small, these exressions will be almost identical. Similarly, the exression given here for the variance of g is J 2 times the variance of d, but many authors ignore the J 2 term because it is so close to unity in most cases. Again, while it is referable to include this correction factor, the inclusion of this factor is likely to make little ractical difference. Comuting d g from studies that use re-ost scores or matched grous We can estimate the stardized mean difference () from studies that used matched grous or re-ost scores in one grou. The formula for the samle estimate of d is

9 Chater 4: Effect Sizes Based on Means d 5 Y diff S within 5 Y 1 Y 2 S within : ð4:26þ This is the same formula as for indeendent grous (4.18). However, when we are working with indeendent grous the natural unit of deviation is the stard deviation within grous so this value is tyically reorted (or easily imuted). By contrast, when we are working with matched grous, the natural unit of deviation is the stard deviation of the difference scores, so this is the value that is likely to be reorted. To comute d from the stard deviation of the differences we need to imute the stard deviation within grous, which would then serve as the denominator in (4.26). Concretely, when working with a matched study, the stard deviation within grous can be imuted from the stard deviation of the difference, using S diff S within 5 ffiffiffi ; ð4:27þ 2ð1 rþ where r is the correlation between airs of observations (e.g., the retest-osttest correlation). Then we can aly (4.26) to comute d. The variance of d is given by V d 5 1 n þ d2 21 ð rþ; ð4:28þ 2n where n is the number of airs. The stard error of d is just the square root of V d, SE d 5 ffiffiffiffiffi V d : ð4:29þ Since the correlation between re- ost-scores is required to imute the stard deviation within grous from the stard deviation of the difference, we must assume that this correlation is known or can be estimated with high recision. Otherwise we may estimate the correlation from related studies, ossibly erform a sensitivity analysis using a range of lausible correlations. To comute Hedges g associated statistics we would use formulas (4.22) through (4.25). The degrees of freedom for comuting J is n 1, where n is the number of airs. For examle, suose that a study has re-test ost-test samle means X , X , samle stard deviation of the difference S diff 5 5.5, samle size n 5 50, a correlation between re-test ost-test of r The stard deviation within grous is imuted from the stard deviation of the difference by 5:5 S within 5 ffiffiffiffiffiffiffiffi 5 7:1005: 21 ð 0:7Þ Then d, its variance stard error are comuted as d 5 5 0:4225; 7:1000

10 30 Effect Size Precision v d þ 0:42252 ð21 ð 0:7ÞÞ5 0:0131; 2 50 SE d 5 0: :1143: The correction factor J, Hedges g, its variance stard error are given by 3 J :9846; g 5 0:9846 0: :4160; V g 5 0: : :0127; SE g 5 0: :1126: Including different study designs in the same analysis As we noted earlier, a single systematic review can include studies that used indeendent grous also studies that used matched grous. From a statistical ersective the effect size (d or g) has the same meaning regardless of the study design. Therefore, we can comute the effect size variance from each study using the aroriate formula, then include all studies in the same analysis. While there are no technical barriers to using studies with different designs in the same analysis, there may be a concern that these studies could differ in substantive ways as well (see Chater 40). For all study designs the direction of the effect (X 1 X 2 or X 2 X 1 ) is arbitrary, excet that the researcher must decide on a convention then aly this consistently. For examle, if a ositive difference indicates that the treated grou did better than the control grou, then this convention must aly for studies that used indeendent designs for studies that used re-ost designs. It must also aly for all outcome measures. In some cases (for examle, if some studies defined outcome as the number of correct answers while others defined outcome as the number of mistakes) it will be necessary to reverse the comuted sign of the effect size to ensure that the convention is alied consistently. RESPONSE RATIOS In research domains where the outcome is measured on a hysical scale (such as length, area, or mass) is unlikely to be zero, the ratio of the means in the two grous might serve as the effect size index. In exerimental ecology this effect size index is called the resonse ratio (Hedges, Gurevitch, & Curtis, 1999). It is imortant to recognize that the resonse ratio is only meaningful when the outcome

11 Chater 4: Effect Sizes Based on Means Study A Resonse ratio Log resonse ratio Study B Resonse ratio Log resonse ratio Study C Resonse ratio Log resonse ratio Summary Resonse ratio Summary Log resonse ratio Figure 4.1 Resonse ratios are analyzed in log units. is measured on a true ratio scale. The resonse ratio is not meaningful for studies (such as most social science studies) that measure outcomes such as test scores, attitude measures, or judgments, since these have no natural scale units no natural zero oints. For resonse ratios, comutations are carried out on a log scale (see the discussion under risk ratios, below, for an exlanation). We comute the log resonse ratio the stard error of the log resonse ratio, use these numbers to erform all stes in the meta-analysis. Only then do we convert the results back into the original metric. This is shown schematically in Figure 4.1. The resonse ratio is comuted as R 5 X 1 X 2 ð4:30þ where X 1 is the mean of grou 1 X 2 is the mean of grou 2. The log resonse ratio is comuted as lnr 5 lnðrþ 5 ln X 1 5 ln X 1 ln X2 : ð4:31þ X 2 The variance of the log resonse ratio is aroximately V lnr 5 S 2 ooled þ 2!; ð4:32þ n 1 X 1 n 2 X 2 where S ooled is the ooled stard deviation. The aroximate stard error is SE ln R 5 ffiffiffiffiffiffiffiffi : ð4:33þ Note that we do not comute a variance for the resonse ratio in its original metric. Rather, we use the log resonse ratio its variance in the analysis to yield V lnr

12 32 Effect Size Precision a summary effect, confidence limits, so on, in log units. We then convert each of these values back to resonse ratios using R 5 exðlnrþ; LL R 5 exðll lnr Þ; UL R 5 exðul lnr Þ; ð4:34þ ð4:35þ ð4:36þ where LL UL reresent the lower uer limits, resectively. For examle, suose that a study has two indeendent grous with means X , X , ooled within-grou stard deviation , samle size n 1 5 n Then R, its variance stard error are comuted as R 5 61:515 51: :2058; lnr 5 lnð1:2058þ5 0:1871; V lnr 5 19: ð61:515þ 2 þ 1 10 ð51:015þ 2 SE lnr 5 0: :1581:! 5 0:0246: SUMMARY POINTS The raw mean difference (D) may be used as the effect size when the outcome scale is either inherently meaningful or well known due to widesread use. This effect size can only be used when all studies in the analysis used recisely the same scale. The stardized mean difference (d or g) transforms all effect sizes to a common metric, thus enables us to include different outcome measures in the same synthesis. This effect size is often used in rimary research as well as meta-analysis, therefore will be intuitive to many researchers. The resonse ratio (R) is often used in ecology. This effect size is only meaningful when the outcome has a natural zero oint, but when this condition holds, it rovides a unique ersective on the effect size. It is ossible to comute an effect size variance from studies that used two indeendent grous, from studies that used matched grous (or re-ost designs) from studies that used clustered grous. These effect sizes may then be included in the same meta-analysis.

### The risk of using the Q heterogeneity estimator for software engineering experiments

Dieste, O., Fernández, E., García-Martínez, R., Juristo, N. 11. The risk of using the Q heterogeneity estimator for software engineering exeriments. The risk of using the Q heterogeneity estimator for

### Confidence Intervals for Capture-Recapture Data With Matching

Confidence Intervals for Cature-Recature Data With Matching Executive summary Cature-recature data is often used to estimate oulations The classical alication for animal oulations is to take two samles

### Risk and Return. Sample chapter. e r t u i o p a s d f CHAPTER CONTENTS LEARNING OBJECTIVES. Chapter 7

Chater 7 Risk and Return LEARNING OBJECTIVES After studying this chater you should be able to: e r t u i o a s d f understand how return and risk are defined and measured understand the concet of risk

### Normally Distributed Data. A mean with a normal value Test of Hypothesis Sign Test Paired observations within a single patient group

ANALYSIS OF CONTINUOUS VARIABLES / 31 CHAPTER SIX ANALYSIS OF CONTINUOUS VARIABLES: COMPARING MEANS In the last chater, we addressed the analysis of discrete variables. Much of the statistical analysis

### HOMEWORK (due Fri, Nov 19): Chapter 12: #62, 83, 101

Today: Section 2.2, Lesson 3: What can go wrong with hyothesis testing Section 2.4: Hyothesis tests for difference in two roortions ANNOUNCEMENTS: No discussion today. Check your grades on eee and notify

### Two-resource stochastic capacity planning employing a Bayesian methodology

Journal of the Oerational Research Society (23) 54, 1198 128 r 23 Oerational Research Society Ltd. All rights reserved. 16-5682/3 \$25. www.algrave-journals.com/jors Two-resource stochastic caacity lanning

### Beyond the F Test: Effect Size Confidence Intervals and Tests of Close Fit in the Analysis of Variance and Contrast Analysis

Psychological Methods 004, Vol. 9, No., 164 18 Coyright 004 by the American Psychological Association 108-989X/04/\$1.00 DOI: 10.1037/108-989X.9..164 Beyond the F Test: Effect Size Confidence Intervals

### An Analysis of Reliable Classifiers through ROC Isometrics

An Analysis of Reliable Classifiers through ROC Isometrics Stijn Vanderlooy s.vanderlooy@cs.unimaas.nl Ida G. Srinkhuizen-Kuyer kuyer@cs.unimaas.nl Evgueni N. Smirnov smirnov@cs.unimaas.nl MICC-IKAT, Universiteit

### Loglikelihood and Confidence Intervals

Stat 504, Lecture 3 Stat 504, Lecture 3 2 Review (contd.): Loglikelihood and Confidence Intervals The likelihood of the samle is the joint PDF (or PMF) L(θ) = f(x,.., x n; θ) = ny f(x i; θ) i= Review:

### Effects of Math Tutoring

Requestor: Math Deartment Researcher(s): Steve Blohm Date: 6/30/15 Title: Effects of Math Tutoring Effects of Math Tutoring The urose of this study is to measure the effects of math tutoring at Cabrillo

### Evaluating a Web-Based Information System for Managing Master of Science Summer Projects

Evaluating a Web-Based Information System for Managing Master of Science Summer Projects Till Rebenich University of Southamton tr08r@ecs.soton.ac.uk Andrew M. Gravell University of Southamton amg@ecs.soton.ac.uk

### When Does it Make Sense to Perform a Meta-Analysis?

CHAPTER 40 When Does it Make Sense to Perform a Meta-Analysis? Introduction Are the studies similar enough to combine? Can I combine studies with different designs? How many studies are enough to carry

### THE REVISED CONSUMER PRICE INDEX IN ZAMBIA

THE REVISED CONSUMER PRICE INDEX IN ZAMBIA Submitted by the Central Statistical office of Zambia Abstract This aer discusses the Revised Consumer Price Index (CPI) in Zambia, based on revised weights,

### Investigation of Variance Estimators for Adaptive Cluster Sampling with a Single Primary Unit

Thail Statistician Jul 009; 7( : 0-9 www.statassoc.or.th Contributed aer Investigation of Variance Estimators for Adative Cluster Samling with a Single Primar Unit Urairat Netharn School of Alied Statistics

### Chapter 9, Part B Hypothesis Tests. Learning objectives

Chater 9, Part B Hyothesis Tests Slide 1 Learning objectives 1. Able to do hyothesis test about Poulation Proortion 2. Calculatethe Probability of Tye II Errors 3. Understand ower of the test 4. Determinethe

### An important observation in supply chain management, known as the bullwhip effect,

Quantifying the Bullwhi Effect in a Simle Suly Chain: The Imact of Forecasting, Lead Times, and Information Frank Chen Zvi Drezner Jennifer K. Ryan David Simchi-Levi Decision Sciences Deartment, National

### Price Elasticity of Demand MATH 104 and MATH 184 Mark Mac Lean (with assistance from Patrick Chan) 2011W

Price Elasticity of Demand MATH 104 and MATH 184 Mark Mac Lean (with assistance from Patrick Chan) 2011W The rice elasticity of demand (which is often shortened to demand elasticity) is defined to be the

### where a, b, c, and d are constants with a 0, and x is measured in radians. (π radians =

Introduction to Modeling 3.6-1 3.6 Sine and Cosine Functions The general form of a sine or cosine function is given by: f (x) = asin (bx + c) + d and f(x) = acos(bx + c) + d where a, b, c, and d are constants

### A Multivariate Statistical Analysis of Stock Trends. Abstract

A Multivariate Statistical Analysis of Stock Trends Aril Kerby Alma College Alma, MI James Lawrence Miami University Oxford, OH Abstract Is there a method to redict the stock market? What factors determine

### The impact of metadata implementation on webpage visibility in search engine results (Part II) q

Information Processing and Management 41 (2005) 691 715 www.elsevier.com/locate/inforoman The imact of metadata imlementation on webage visibility in search engine results (Part II) q Jin Zhang *, Alexandra

### Measuring relative phase between two waveforms using an oscilloscope

Measuring relative hase between two waveforms using an oscilloscoe Overview There are a number of ways to measure the hase difference between two voltage waveforms using an oscilloscoe. This document covers

### ChE 120B Lumped Parameter Models for Heat Transfer and the Blot Number

ChE 0B Lumed Parameter Models for Heat Transfer and the Blot Number Imagine a slab that has one dimension, of thickness d, that is much smaller than the other two dimensions; we also assume that the slab

### Large-Scale IP Traceback in High-Speed Internet: Practical Techniques and Theoretical Foundation

Large-Scale IP Traceback in High-Seed Internet: Practical Techniques and Theoretical Foundation Jun Li Minho Sung Jun (Jim) Xu College of Comuting Georgia Institute of Technology {junli,mhsung,jx}@cc.gatech.edu

### Population Genetics I

Poulation Genetics I The material resented here considers a single diloid genetic locus, ith to alleles A and a; their relative frequencies in the oulation ill be denoted as and q (ith q 1 ). Hardy-Weinberg

### Web Application Scalability: A Model-Based Approach

Coyright 24, Software Engineering Research and Performance Engineering Services. All rights reserved. Web Alication Scalability: A Model-Based Aroach Lloyd G. Williams, Ph.D. Software Engineering Research

### HYPOTHESIS TESTING FOR THE PROCESS CAPABILITY RATIO. A thesis presented to. the faculty of

HYPOTHESIS TESTING FOR THE PROESS APABILITY RATIO A thesis resented to the faculty of the Russ ollege of Engineering and Technology of Ohio University In artial fulfillment of the requirement for the degree

### Monitoring Frequency of Change By Li Qin

Monitoring Frequency of Change By Li Qin Abstract Control charts are widely used in rocess monitoring roblems. This aer gives a brief review of control charts for monitoring a roortion and some initial

### Managing specific risk in property portfolios

Managing secific risk in roerty ortfolios Andrew Baum, PhD University of Reading, UK Peter Struemell OPC, London, UK Contact author: Andrew Baum Deartment of Real Estate and Planning University of Reading

### F inding the optimal, or value-maximizing, capital

Estimating Risk-Adjusted Costs of Financial Distress by Heitor Almeida, University of Illinois at Urbana-Chamaign, and Thomas Philion, New York University 1 F inding the otimal, or value-maximizing, caital

### FREQUENCIES OF SUCCESSIVE PAIRS OF PRIME RESIDUES

FREQUENCIES OF SUCCESSIVE PAIRS OF PRIME RESIDUES AVNER ASH, LAURA BELTIS, ROBERT GROSS, AND WARREN SINNOTT Abstract. We consider statistical roerties of the sequence of ordered airs obtained by taking

### Probabilistic models for mechanical properties of prestressing strands

Probabilistic models for mechanical roerties of restressing strands Luciano Jacinto a, Manuel Pia b, Luís Neves c, Luís Oliveira Santos b a Instituto Suerior de Engenharia de Lisboa, Rua Conselheiro Emídio

### lecture 25: Gaussian quadrature: nodes, weights; examples; extensions

38 lecture 25: Gaussian quadrature: nodes, weights; examles; extensions 3.5 Comuting Gaussian quadrature nodes and weights When first aroaching Gaussian quadrature, the comlicated characterization of the

### Compensating Fund Managers for Risk-Adjusted Performance

Comensating Fund Managers for Risk-Adjusted Performance Thomas S. Coleman Æquilibrium Investments, Ltd. Laurence B. Siegel The Ford Foundation Journal of Alternative Investments Winter 1999 In contrast

### Fixed-Effect Versus Random-Effects Models

CHAPTER 13 Fixed-Effect Versus Random-Effects Models Introduction Definition of a summary effect Estimating the summary effect Extreme effect size in a large study or a small study Confidence interval

### A Simple Model of Pricing, Markups and Market. Power Under Demand Fluctuations

A Simle Model of Pricing, Markus and Market Power Under Demand Fluctuations Stanley S. Reynolds Deartment of Economics; University of Arizona; Tucson, AZ 85721 Bart J. Wilson Economic Science Laboratory;

### A MOST PROBABLE POINT-BASED METHOD FOR RELIABILITY ANALYSIS, SENSITIVITY ANALYSIS AND DESIGN OPTIMIZATION

9 th ASCE Secialty Conference on Probabilistic Mechanics and Structural Reliability PMC2004 Abstract A MOST PROBABLE POINT-BASED METHOD FOR RELIABILITY ANALYSIS, SENSITIVITY ANALYSIS AND DESIGN OPTIMIZATION

### Multiperiod Portfolio Optimization with General Transaction Costs

Multieriod Portfolio Otimization with General Transaction Costs Victor DeMiguel Deartment of Management Science and Oerations, London Business School, London NW1 4SA, UK, avmiguel@london.edu Xiaoling Mei

### SOME PROPERTIES OF EXTENSIONS OF SMALL DEGREE OVER Q. 1. Quadratic Extensions

SOME PROPERTIES OF EXTENSIONS OF SMALL DEGREE OVER Q TREVOR ARNOLD Abstract This aer demonstrates a few characteristics of finite extensions of small degree over the rational numbers Q It comrises attemts

### An inventory control system for spare parts at a refinery: An empirical comparison of different reorder point methods

An inventory control system for sare arts at a refinery: An emirical comarison of different reorder oint methods Eric Porras a*, Rommert Dekker b a Instituto Tecnológico y de Estudios Sueriores de Monterrey,

### Pressure Drop in Air Piping Systems Series of Technical White Papers from Ohio Medical Corporation

Pressure Dro in Air Piing Systems Series of Technical White Paers from Ohio Medical Cororation Ohio Medical Cororation Lakeside Drive Gurnee, IL 600 Phone: (800) 448-0770 Fax: (847) 855-604 info@ohiomedical.com

### The fast Fourier transform method for the valuation of European style options in-the-money (ITM), at-the-money (ATM) and out-of-the-money (OTM)

Comutational and Alied Mathematics Journal 15; 1(1: 1-6 Published online January, 15 (htt://www.aascit.org/ournal/cam he fast Fourier transform method for the valuation of Euroean style otions in-the-money

### An Introduction to Risk Parity Hossein Kazemi

An Introduction to Risk Parity Hossein Kazemi In the aftermath of the financial crisis, investors and asset allocators have started the usual ritual of rethinking the way they aroached asset allocation

### Comparing Dissimilarity Measures for Symbolic Data Analysis

Comaring Dissimilarity Measures for Symbolic Data Analysis Donato MALERBA, Floriana ESPOSITO, Vincenzo GIOVIALE and Valentina TAMMA Diartimento di Informatica, University of Bari Via Orabona 4 76 Bari,

### Complex Data Structures

PART 5 Complex Data Structures Introduction to Meta-Analysis. Michael Borenstein, L. V. Hedges, J. P. T. Higgins and H. R. Rothstein 2009 John Wiley & Sons, Ltd. ISBN: 978-0-470-05724-7 CHAPTER 22 Overview

### Meta-Regression CHAPTER 20

CHAPTER 20 Meta-Regression Introduction Fixed-effect model Fixed or random effects for unexplained heterogeneity Random-effects model INTRODUCTION In primary studies we use regression, or multiple regression,

### A 60,000 DIGIT PRIME NUMBER OF THE FORM x 2 + x Introduction Stark-Heegner Theorem. Let d > 0 be a square-free integer then Q( d) has

A 60,000 DIGIT PRIME NUMBER OF THE FORM x + x + 4. Introduction.. Euler s olynomial. Euler observed that f(x) = x + x + 4 takes on rime values for 0 x 39. Even after this oint f(x) takes on a high frequency

### More Properties of Limits: Order of Operations

math 30 day 5: calculating its 6 More Proerties of Limits: Order of Oerations THEOREM 45 (Order of Oerations, Continued) Assume that!a f () L and that m and n are ositive integers Then 5 (Power)!a [ f

### Supplemental material for: Dynamic jump intensities and risk premiums: evidence from S&P500 returns and options

Sulemental material for: Dynamic jum intensities and risk remiums: evidence from S&P5 returns and otions Peter Christo ersen University of Toronto, CBS and CREATES Kris Jacobs University of Houston and

### The Optimal Sequenced Route Query

The Otimal Sequenced Route Query Mehdi Sharifzadeh, Mohammad Kolahdouzan, Cyrus Shahabi Comuter Science Deartment University of Southern California Los Angeles, CA 90089-0781 [sharifza, kolahdoz, shahabi]@usc.edu

### 6.042/18.062J Mathematics for Computer Science December 12, 2006 Tom Leighton and Ronitt Rubinfeld. Random Walks

6.042/8.062J Mathematics for Comuter Science December 2, 2006 Tom Leighton and Ronitt Rubinfeld Lecture Notes Random Walks Gambler s Ruin Today we re going to talk about one-dimensional random walks. In

### Risk in Revenue Management and Dynamic Pricing

OPERATIONS RESEARCH Vol. 56, No. 2, March Aril 2008,. 326 343 issn 0030-364X eissn 1526-5463 08 5602 0326 informs doi 10.1287/ore.1070.0438 2008 INFORMS Risk in Revenue Management and Dynamic Pricing Yuri

### DAY-AHEAD ELECTRICITY PRICE FORECASTING BASED ON TIME SERIES MODELS: A COMPARISON

DAY-AHEAD ELECTRICITY PRICE FORECASTING BASED ON TIME SERIES MODELS: A COMPARISON Rosario Esínola, Javier Contreras, Francisco J. Nogales and Antonio J. Conejo E.T.S. de Ingenieros Industriales, Universidad

### On tests for multivariate normality and associated simulation studies

Journal of Statistical Comutation & Simulation Vol. 00, No. 00, January 2006, 1 14 On tests for multivariate normality and associated simulation studies Patrick J. Farrell Matias Salibian-Barrera Katarzyna

### Frequentist vs. Bayesian Statistics

Bayes Theorem Frequentist vs. Bayesian Statistics Common situation in science: We have some data and we want to know the true hysical law describing it. We want to come u with a model that fits the data.

### Analysis of Effectiveness of Web based E- Learning Through Information Technology

International Journal of Soft Comuting and Engineering (IJSCE) Analysis of Effectiveness of Web based E- Learning Through Information Technology Anand Tamrakar, Kamal K. Mehta Abstract-Advancements of

### Characterizing and Modeling Network Traffic Variability

Characterizing and Modeling etwork Traffic Variability Sarat Pothuri, David W. Petr, Sohel Khan Information and Telecommunication Technology Center Electrical Engineering and Comuter Science Deartment,

### 4 Perceptron Learning Rule

Percetron Learning Rule Objectives Objectives - Theory and Examles - Learning Rules - Percetron Architecture -3 Single-Neuron Percetron -5 Multile-Neuron Percetron -8 Percetron Learning Rule -8 Test Problem

### A Brief Introduction to Design of Experiments

J. K. TELFORD D A Brief Introduction to Design of Exeriments Jacqueline K. Telford esign of exeriments is a series of tests in which uroseful changes are made to the inut variables of a system or rocess

### Joint Production and Financing Decisions: Modeling and Analysis

Joint Production and Financing Decisions: Modeling and Analysis Xiaodong Xu John R. Birge Deartment of Industrial Engineering and Management Sciences, Northwestern University, Evanston, Illinois 60208,

### The Portfolio Characteristics and Investment Performance of Bank Loan Mutual Funds

The Portfolio Characteristics and Investment Performance of Bank Loan Mutual Funds Zakri Bello, Ph.. Central Connecticut State University 1615 Stanley St. New Britain, CT 06050 belloz@ccsu.edu 860-832-3262

### Forensic Science International

Forensic Science International 207 (2011) 135 144 Contents lists available at ScienceDirect Forensic Science International journal homeage: www.elsevier.com/locate/forsciint Analysis and alication of relationshi

### Modeling and Simulation of an Incremental Encoder Used in Electrical Drives

10 th International Symosium of Hungarian Researchers on Comutational Intelligence and Informatics Modeling and Simulation of an Incremental Encoder Used in Electrical Drives János Jób Incze, Csaba Szabó,

### 2.1 Simple & Compound Propositions

2.1 Simle & Comound Proositions 1 2.1 Simle & Comound Proositions Proositional Logic can be used to analyse, simlify and establish the equivalence of statements. A knowledge of logic is essential to the

### High Quality Offset Printing An Evolutionary Approach

High Quality Offset Printing An Evolutionary Aroach Ralf Joost Institute of Alied Microelectronics and omuter Engineering University of Rostock Rostock, 18051, Germany +49 381 498 7272 ralf.joost@uni-rostock.de

### 1 Gambler s Ruin Problem

Coyright c 2009 by Karl Sigman 1 Gambler s Ruin Problem Let N 2 be an integer and let 1 i N 1. Consider a gambler who starts with an initial fortune of \$i and then on each successive gamble either wins

### X How to Schedule a Cascade in an Arbitrary Graph

X How to Schedule a Cascade in an Arbitrary Grah Flavio Chierichetti, Cornell University Jon Kleinberg, Cornell University Alessandro Panconesi, Saienza University When individuals in a social network

### Synopsys RURAL ELECTRICATION PLANNING SOFTWARE (LAPER) Rainer Fronius Marc Gratton Electricité de France Research and Development FRANCE

RURAL ELECTRICATION PLANNING SOFTWARE (LAPER) Rainer Fronius Marc Gratton Electricité de France Research and Develoment FRANCE Synosys There is no doubt left about the benefit of electrication and subsequently

### The Online Freeze-tag Problem

The Online Freeze-tag Problem Mikael Hammar, Bengt J. Nilsson, and Mia Persson Atus Technologies AB, IDEON, SE-3 70 Lund, Sweden mikael.hammar@atus.com School of Technology and Society, Malmö University,

### APPENDIX A Mohr s circle for two-dimensional stress

APPENDIX A ohr s circle for two-dimensional stress Comressive stresses have been taken as ositive because we shall almost exclusively be dealing with them as oosed to tensile stresses and because this

### Learning Human Behavior from Analyzing Activities in Virtual Environments

Learning Human Behavior from Analyzing Activities in Virtual Environments C. BAUCKHAGE 1, B. GORMAN 2, C. THURAU 3 & M. HUMPHRYS 2 1) Deutsche Telekom Laboratories, Berlin, Germany 2) Dublin City University,

### DEPARTMENT OF ECONOMICS DISCUSSION PAPER SERIES

ISSN 1471-0498 DEPARTMENT OF ECONOMICS DISCUSSION PAPER SERIES MARGINAL COST PRICING VERSUS INSURANCE Simon Cowan Number 102 May 2002 Manor Road Building, Oxford OX1 3UQ Marginal cost ricing versus insurance

### COST CALCULATION IN COMPLEX TRANSPORT SYSTEMS

OST ALULATION IN OMLEX TRANSORT SYSTEMS Zoltán BOKOR 1 Introduction Determining the real oeration and service costs is essential if transort systems are to be lanned and controlled effectively. ost information

### On the predictive content of the PPI on CPI inflation: the case of Mexico

On the redictive content of the PPI on inflation: the case of Mexico José Sidaoui, Carlos Caistrán, Daniel Chiquiar and Manuel Ramos-Francia 1 1. Introduction It would be natural to exect that shocks to

### Point Location. Preprocess a planar, polygonal subdivision for point location queries. p = (18, 11)

Point Location Prerocess a lanar, olygonal subdivision for oint location ueries. = (18, 11) Inut is a subdivision S of comlexity n, say, number of edges. uild a data structure on S so that for a uery oint

### Methods for Estimating Kidney Disease Stage Transition Probabilities Using Electronic Medical Records

EDM Forum EDM Forum Community egems (Generating Evidence & Methods to imrove atient outcomes) Publish 12-2013 Methods for Estimating Kidney Disease Stage Transition Probabilities Using Electronic Medical

### ABSOLUTE ZERO

- 36 - ABSOLUE ZEO INODUCION As you may know, for an ideal gas at constant volume, the relation between its ressure and its temerature t, measured in 0 C, is given by: t m b () hus a lot of t versus will

### The Advantage of Timely Intervention

Journal of Exerimental Psychology: Learning, Memory, and Cognition 2004, Vol. 30, No. 4, 856 876 Coyright 2004 by the American Psychological Association 0278-7393/04/\$12.00 DOI: 10.1037/0278-7393.30.4.856

### Forensic Science International

Forensic Science International 214 (2012) 33 43 Contents lists available at ScienceDirect Forensic Science International jou r nal h o me age: w ww.els evier.co m/lo c ate/fo r sc iin t A robust detection

### Methods for Estimating Kidney Disease Stage Transition Probabilities Using Electronic Medical Records

(Generating Evidence & Methods to imrove atient outcomes) Volume 1 Issue 3 Methods for CER, PCOR, and QI Using EHR Data in a Learning Health System Article 6 12-1-2013 Methods for Estimating Kidney Disease

### Optional Strain-Rate Forms for the Johnson Cook Constitutive Model and the Role of the Parameter Epsilon_0 1 AUTHOR

Otional Strain-Rate Forms for the Johnson Cook Constitutive Model and the Role of the Parameter Esilon_ AUTHOR Len Schwer Schwer Engineering & Consulting Services CORRESPONDENCE Len Schwer Schwer Engineering

### Approximate Guarding of Monotone and Rectilinear Polygons

Aroximate Guarding of Monotone and Rectilinear Polygons Erik Krohn Bengt J. Nilsson Abstract We show that vertex guarding a monotone olygon is NP-hard and construct a constant factor aroximation algorithm

### Softmax Model as Generalization upon Logistic Discrimination Suffers from Overfitting

Journal of Data Science 12(2014),563-574 Softmax Model as Generalization uon Logistic Discrimination Suffers from Overfitting F. Mohammadi Basatini 1 and Rahim Chiniardaz 2 1 Deartment of Statistics, Shoushtar

### Working paper No: 23/2011 May 2011 LSE Health. Sotiris Vandoros, Katherine Grace Carman. Demand and Pricing of Preventative Health Care

Working aer o: 3/0 May 0 LSE Health Sotiris Vandoros, Katherine Grace Carman Demand and Pricing of Preventative Health Care Demand and Pricing of Preventative Healthcare Sotiris Vandoros, Katherine Grace

### The Competitiveness Impacts of Climate Change Mitigation Policies

The Cometitiveness Imacts of Climate Change Mitigation Policies Joseh E. Aldy William A. Pizer 2011 RPP-2011-08 Regulatory Policy Program Mossavar-Rahmani Center for Business and Government Harvard Kennedy

DISCUSSION PAPER SERIES IZA DP No. 1549 The Changing Wage Return to an Undergraduate Education Nigel C. O'Leary Peter J. Sloane March 2005 Forschungsinstitut zur Zukunft der Arbeit Institute for the Study

### The Magnus-Derek Game

The Magnus-Derek Game Z. Nedev S. Muthukrishnan Abstract We introduce a new combinatorial game between two layers: Magnus and Derek. Initially, a token is laced at osition 0 on a round table with n ositions.

### A Complete Operational Amplifier Noise Model: Analysis and Measurement of Correlation Coefficient

IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: FUNDAMENTAL THEORY AND APPLICATION, VOL. 47, NO. 3, MARCH 000 40 for ractical alication, oening the ath for widesread adotion of the clock-gating technique

### UNITED STATES OF AMERICA FEDERAL ENERGY REGULATORY COMMISSION

UNITED STATES OF AMERICA FEDERAL ENERGY REGULATORY COMMISSION San Diego Gas & Electric Comany, ) EL00-95-075 Comlainant, ) ) v. ) ) ) Sellers of Energy and Ancillary Services ) Docket Nos. Into Markets

### Implementation of Statistic Process Control in a Painting Sector of a Automotive Manufacturer

4 th International Conference on Industrial Engineering and Industrial Management IV Congreso de Ingeniería de Organización Donostia- an ebastián, etember 8 th - th Imlementation of tatistic Process Control

### ANALYSING THE OVERHEAD IN MOBILE AD-HOC NETWORK WITH A HIERARCHICAL ROUTING STRUCTURE

AALYSIG THE OVERHEAD I MOBILE AD-HOC ETWORK WITH A HIERARCHICAL ROUTIG STRUCTURE Johann Lóez, José M. Barceló, Jorge García-Vidal Technical University of Catalonia (UPC), C/Jordi Girona 1-3, 08034 Barcelona,

### A NONLINEAR SWITCHING CONTROL METHOD FOR MAGNETIC BEARING SYSTEMS MINIMIZING THE POWER CONSUMPTION

Coyright IFAC 5th Triennial World Congress, Barcelona, Sain A NONLINEAR SWITCHING CONTROL METHOD FOR MAGNETIC BEARING SYSTEMS MINIMIZING THE POWER CONSUMPTION Kang-Zhi Liu Λ; Akihiro Ikai Λ Akihiro Ogata

### Pythagorean Triples and Rational Points on the Unit Circle

Pythagorean Triles and Rational Points on the Unit Circle Solutions Below are samle solutions to the roblems osed. You may find that your solutions are different in form and you may have found atterns

### POL 345: Quantitative Analysis and Politics

POL 345: Quantitative Analysis and Politics Precet Handout 8 Week 10 (Verzani Chaters 7 and 8: 7.5, 8.6.1-8.6.2) Remember to comlete the entire handout and submit the recet questions to the Blackboard

### THE RELATIONSHIP BETWEEN EMPLOYEE PERFORMANCE AND THEIR EFFICIENCY EVALUATION SYSTEM IN THE YOTH AND SPORT OFFICES IN NORTH WEST OF IRAN

THE RELATIONSHIP BETWEEN EMPLOYEE PERFORMANCE AND THEIR EFFICIENCY EVALUATION SYSTEM IN THE YOTH AND SPORT OFFICES IN NORTH WEST OF IRAN *Akbar Abdolhosenzadeh 1, Laya Mokhtari 2, Amineh Sahranavard Gargari

### 1.3 Saturation vapor pressure. 1.3.1 Vapor pressure

1.3 Saturation vaor ressure Increasing temerature of liquid (or any substance) enhances its evaoration that results in the increase of vaor ressure over the liquid. y lowering temerature of the vaor we

### c 2009 Je rey A. Miron 3. Examples: Linear Demand Curves and Monopoly

Lecture 0: Monooly. c 009 Je rey A. Miron Outline. Introduction. Maximizing Pro ts. Examles: Linear Demand Curves and Monooly. The Ine ciency of Monooly. The Deadweight Loss of Monooly. Price Discrimination.

### Project Management and. Scheduling CHAPTER CONTENTS

6 Proect Management and Scheduling HAPTER ONTENTS 6.1 Introduction 6.2 Planning the Proect 6.3 Executing the Proect 6.7.1 Monitor 6.7.2 ontrol 6.7.3 losing 6.4 Proect Scheduling 6.5 ritical Path Method

### Working Paper Series

Working Paer Series Measurement Errors in Jaanese Consumer Price Index Shigenori Shiratsuka Working Paers Series Research Deartment (WP-99-2) Federal Reserve Bank of Chicago Measurement Errors in Jaanese