Glossary of Testing, Measurement, and Statistical Terms
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Analysis of Variance; *********43***1****T******T
DOCUMENT RESUME 11 571 TM 007 817 111, AUTHOR Corder-Bolz, Charles R. TITLE A Monte Carlo Study of Six Models of Change. INSTITUTION Southwest Eduoational Development Lab., Austin, Tex. PUB DiTE (7BA NOTE"". 32p. BDRS-PRICE MF-$0.83 Plug P4istage. HC Not Availablefrom EDRS. DESCRIPTORS *Analysis Of COariance; *Analysis of Variance; Comparative Statistics; Hypothesis Testing;. *Mathematicai'MOdels; Scores; Statistical Analysis; *Tests'of Significance ,IDENTIFIERS *Change Scores; *Monte Carlo Method ABSTRACT A Monte Carlo Study was conducted to evaluate ,six models commonly used to evaluate change. The results.revealed specific problems with each. Analysis of covariance and analysis of variance of residualized gain scores appeared to, substantially and consistently overestimate the change effects. Multiple factor analysis of variance models utilizing pretest and post-test scores yielded invalidly toF ratios. The analysis of variance of difference scores an the multiple 'factor analysis of variance using repeated measures were the only models which adequately controlled for pre-treatment differences; ,however, they appeared to be robust only when the error level is 50% or more. This places serious doubt regarding published findings, and theories based upon change score analyses. When collecting data which. have an error level less than 50% (which is true in most situations)r a change score analysis is entirely inadvisable until an alternative procedure is developed. (Author/CTM) 41t i- **************************************4***********************i,******* 4! Reproductions suppliedby ERRS are theibest that cap be made * .... * '* ,0 . from the original document. *********43***1****t******t*******************************************J . S DEPARTMENT OF HEALTH, EDUCATIONWELFARE NATIONAL INST'ITUTE OF r EOUCATIOp PHISDOCUMENT DUCED EXACTLYHAS /3EENREPRC AS RECEIVED THE PERSONOR ORGANIZATION J RoP AT INC. -
A Review of Standardized Testing Practices and Perceptions in Maine
A Review of Standardized Testing Practices and Perceptions in Maine Janet Fairman, Ph.D. Amy Johnson, Ph.D. Ian Mette, Ph.D Garry Wickerd, Ph.D. Sharon LaBrie, M.S. April 2018 Maine Education Policy Research Institute Maine Education Policy Research Institute McLellan House, 140 School Street 5766 Shibles Hall, Room 314 Gorham, ME 04038 Orono, ME 04469-5766 207.780.5044 or 1.888.800.5044 207.581.2475 Published by the Maine Education Policy Research Institute in the Center for Education Policy, Applied Research, and Evaluation (CEPARE) in the School of Education and Human Development, University of Southern Maine, and the Maine Education Policy Research Institute in the School of Education and Human Development at the University of Maine at Orono. The Maine Education Policy Research Institute (MEPRI), is jointly funded by the Maine State Legislature and the University of Maine System. This institute was established to conduct studies on Maine education policy and the Maine public education system for the Maine Legislature. Statements and opinions by the authors do not necessarily reflect a position or policy of the Maine Education Policy Research Institute, nor any of its members, and no official endorsement by them should be inferred. The University of Maine System does not discriminate on the basis of race, color, religion, sex, sexual orientation, national origin or citizenship status, age, disability, or veteran's status and shall comply with Section 504, Title IX, and the A.D.A in employment, education, and in all other areas of the University. The University provides reasonable accommodations to qualified individuals with disabilities upon request. -
6 Modeling Survival Data with Cox Regression Models
CHAPTER 6 ST 745, Daowen Zhang 6 Modeling Survival Data with Cox Regression Models 6.1 The Proportional Hazards Model A proportional hazards model proposed by D.R. Cox (1972) assumes that T z1β1+···+zpβp z β λ(t|z)=λ0(t)e = λ0(t)e , (6.1) where z is a p × 1 vector of covariates such as treatment indicators, prognositc factors, etc., and β is a p × 1 vector of regression coefficients. Note that there is no intercept β0 in model (6.1). Obviously, λ(t|z =0)=λ0(t). So λ0(t) is often called the baseline hazard function. It can be interpreted as the hazard function for the population of subjects with z =0. The baseline hazard function λ0(t) in model (6.1) can take any shape as a function of t.The T only requirement is that λ0(t) > 0. This is the nonparametric part of the model and z β is the parametric part of the model. So Cox’s proportional hazards model is a semiparametric model. Interpretation of a proportional hazards model 1. It is easy to show that under model (6.1) exp(zT β) S(t|z)=[S0(t)] , where S(t|z) is the survival function of the subpopulation with covariate z and S0(t)isthe survival function of baseline population (z = 0). That is R t − λ0(u)du S0(t)=e 0 . PAGE 120 CHAPTER 6 ST 745, Daowen Zhang 2. For any two sets of covariates z0 and z1, zT β λ(t|z1) λ0(t)e 1 T (z1−z0) β, t ≥ , = zT β =e for all 0 λ(t|z0) λ0(t)e 0 which is a constant over time (so the name of proportional hazards model). -
18.650 (F16) Lecture 10: Generalized Linear Models (Glms)
Statistics for Applications Chapter 10: Generalized Linear Models (GLMs) 1/52 Linear model A linear model assumes Y X (µ(X), σ2I), | ∼ N And ⊤ IE(Y X) = µ(X) = X β, | 2/52 Components of a linear model The two components (that we are going to relax) are 1. Random component: the response variable Y X is continuous | and normally distributed with mean µ = µ(X) = IE(Y X). | 2. Link: between the random and covariates X = (X(1),X(2), ,X(p))⊤: µ(X) = X⊤β. · · · 3/52 Generalization A generalized linear model (GLM) generalizes normal linear regression models in the following directions. 1. Random component: Y some exponential family distribution ∼ 2. Link: between the random and covariates: ⊤ g µ(X) = X β where g called link function� and� µ = IE(Y X). | 4/52 Example 1: Disease Occuring Rate In the early stages of a disease epidemic, the rate at which new cases occur can often increase exponentially through time. Hence, if µi is the expected number of new cases on day ti, a model of the form µi = γ exp(δti) seems appropriate. ◮ Such a model can be turned into GLM form, by using a log link so that log(µi) = log(γ) + δti = β0 + β1ti. ◮ Since this is a count, the Poisson distribution (with expected value µi) is probably a reasonable distribution to try. 5/52 Example 2: Prey Capture Rate(1) The rate of capture of preys, yi, by a hunting animal, tends to increase with increasing density of prey, xi, but to eventually level off, when the predator is catching as much as it can cope with. -
Nine Facts About the SAT That Might Surprise You
Nine Facts About the SAT That Might Surprise You By Lynn Letukas COLLEGE BOARD RESEARCH Statistical Report RESEARCH Executive Summary The purpose of this document is to identify and dispel rumors that are frequently cited about the SAT. The following is a compilation of nine popular rumors organized into three areas: Student Demographics, Test Preparation/Test Prediction, and Test Utilization. Student Demographics Student demographics claims are those that primarily center around a specific demographic group that is said to over-/underperform on the SAT. Presumably, it is reasoned, if the SAT were a fair test, no student demographic characteristics would matter, as average scores would be similar across groups. Therefore, some people assume that any difference in SAT performance by demographics must indicate test bias for/against a demographic group. Rumor 1: The SAT Is a Wealth Test. According to Peter Sacks, author of Standardized Minds, “one can make a good guess about a child’s standardized test scores simply by looking at how many degrees her parents have and what kind of car they drive.”1 This comment is illustrative of frequent criticism that the SAT is merely a “wealth test” because there is a correlation between student scores and socioeconomic status (i.e., parental income and educational attainment). Proponents of this claim often support their arguments by stating that the SAT only covers content that is learned by wealthy students, rather than material covered in high school.2 Fact: Rigorous Course Taking in High School Better Explains Why Students Do Well (or Not) on the SAT. While SAT scores may be correlated with socioeconomic status (parental income and education), correlation does not mean two phenomena are causally related (e.g., parental income causes students to do well on the SAT). -
Optimal Variance Control of the Score Function Gradient Estimator for Importance Weighted Bounds
Optimal Variance Control of the Score Function Gradient Estimator for Importance Weighted Bounds Valentin Liévin 1 Andrea Dittadi1 Anders Christensen1 Ole Winther1, 2, 3 1 Section for Cognitive Systems, Technical University of Denmark 2 Bioinformatics Centre, Department of Biology, University of Copenhagen 3 Centre for Genomic Medicine, Rigshospitalet, Copenhagen University Hospital {valv,adit}@dtu.dk, [email protected], [email protected] Abstract This paper introduces novel results for the score function gradient estimator of the importance weighted variational bound (IWAE). We prove that in the limit of large K (number of importance samples) one can choose the control variate such that the Signal-to-Noise ratio (SNR) of the estimator grows as pK. This is in contrast to the standard pathwise gradient estimator where the SNR decreases as 1/pK. Based on our theoretical findings we develop a novel control variate that extends on VIMCO. Empirically, for the training of both continuous and discrete generative models, the proposed method yields superior variance reduction, resulting in an SNR for IWAE that increases with K without relying on the reparameterization trick. The novel estimator is competitive with state-of-the-art reparameterization-free gradient estimators such as Reweighted Wake-Sleep (RWS) and the thermodynamic variational objective (TVO) when training generative models. 1 Introduction Gradient-based learning is now widespread in the field of machine learning, in which recent advances have mostly relied on the backpropagation algorithm, the workhorse of modern deep learning. In many instances, for example in the context of unsupervised learning, it is desirable to make models more expressive by introducing stochastic latent variables. -
Bayesian Methods: Review of Generalized Linear Models
Bayesian Methods: Review of Generalized Linear Models RYAN BAKKER University of Georgia ICPSR Day 2 Bayesian Methods: GLM [1] Likelihood and Maximum Likelihood Principles Likelihood theory is an important part of Bayesian inference: it is how the data enter the model. • The basis is Fisher’s principle: what value of the unknown parameter is “most likely” to have • generated the observed data. Example: flip a coin 10 times, get 5 heads. MLE for p is 0.5. • This is easily the most common and well-understood general estimation process. • Bayesian Methods: GLM [2] Starting details: • – Y is a n k design or observation matrix, θ is a k 1 unknown coefficient vector to be esti- × × mated, we want p(θ Y) (joint sampling distribution or posterior) from p(Y θ) (joint probabil- | | ity function). – Define the likelihood function: n L(θ Y) = p(Y θ) | i| i=1 Y which is no longer on the probability metric. – Our goal is the maximum likelihood value of θ: θˆ : L(θˆ Y) L(θ Y) θ Θ | ≥ | ∀ ∈ where Θ is the class of admissable values for θ. Bayesian Methods: GLM [3] Likelihood and Maximum Likelihood Principles (cont.) Its actually easier to work with the natural log of the likelihood function: • `(θ Y) = log L(θ Y) | | We also find it useful to work with the score function, the first derivative of the log likelihood func- • tion with respect to the parameters of interest: ∂ `˙(θ Y) = `(θ Y) | ∂θ | Setting `˙(θ Y) equal to zero and solving gives the MLE: θˆ, the “most likely” value of θ from the • | parameter space Θ treating the observed data as given. -
Public School Admissions and the Myth of Meritocracy: How and Why Screened Public School Admissions Promote Segregation
BUERY_FIN.DOCX(DO NOT DELETE) 4/14/20 8:46 PM PUBLIC SCHOOL ADMISSIONS AND THE MYTH OF MERITOCRACY: HOW AND WHY SCREENED PUBLIC SCHOOL ADMISSIONS PROMOTE SEGREGATION RICHARD R. BUERY, JR.* Public schools in America remain deeply segregated by race, with devastating effects for Black and Latinx students. While residential segregation is a critical driver of school segregation, the prevalence of screened admissions practices can also play a devastating role in driving racial segregation in public schools. New York City, one of the most segregated school systems in America, is unique in its extensive reliance on screened admissions practices, including the use of standardized tests, to assign students to sought-after public schools. These screens persist despite their segregative impact in part because they appeal to America’s embrace of the idea of meritocracy. This Article argues that Americans embrace three conceptions of merit which shield these screens from proper scrutiny. The first is individual merit—the idea that students with greater ability or achievement deserve access to better schools. The second is systems merit—the idea that poor student performance on an assessment is a failure of the system that prepared the student for the assessment. The third is group merit—the idea that members of some groups simply possess less ability. Each of these ideas has a pernicious impact on perpetuating racial inequality in public education. INTRODUCTION ................................................................................. 102 I. THE APPEAL OF THREE DIMENSIONS OF MERIT .................... 104 A. Merit of the Individual .................................................... 104 B. Merit of the System ......................................................... 107 C. Merit of the Group .......................................................... 109 II. THE MYTH OF MERITOCRACY IN K12 ADMISSIONS ............. -
Simple, Lower-Variance Gradient Estimators for Variational Inference
Sticking the Landing: Simple, Lower-Variance Gradient Estimators for Variational Inference Geoffrey Roeder Yuhuai Wu University of Toronto University of Toronto [email protected] [email protected] David Duvenaud University of Toronto [email protected] Abstract We propose a simple and general variant of the standard reparameterized gradient estimator for the variational evidence lower bound. Specifically, we remove a part of the total derivative with respect to the variational parameters that corresponds to the score function. Removing this term produces an unbiased gradient estimator whose variance approaches zero as the approximate posterior approaches the exact posterior. We analyze the behavior of this gradient estimator theoretically and empirically, and generalize it to more complex variational distributions such as mixtures and importance-weighted posteriors. 1 Introduction Recent advances in variational inference have begun to Optimization using: make approximate inference practical in large-scale latent Path Derivative ) variable models. One of the main recent advances has Total Derivative been the development of variational autoencoders along true φ with the reparameterization trick [Kingma and Welling, k 2013, Rezende et al., 2014]. The reparameterization init trick is applicable to most continuous latent-variable mod- φ els, and usually provides lower-variance gradient esti- ( mates than the more general REINFORCE gradient es- KL timator [Williams, 1992]. Intuitively, the reparameterization trick provides more in- 400 600 800 1000 1200 formative gradients by exposing the dependence of sam- Iterations pled latent variables z on variational parameters φ. In contrast, the REINFORCE gradient estimate only de- Figure 1: Fitting a 100-dimensional varia- pends on the relationship between the density function tional posterior to another Gaussian, using log qφ(z x; φ) and its parameters. -
Rethinking School Accountability: Opportunities for Massachusetts
RETHINKING SCHOOL ACCOUNTABILITY OPPORTUNITIES FOR MASSACHUSETTS UNDER THE EVERY STUDENT SUCCEEDS ACT Senator Patricia D. Jehlen, Chair Senate Sub-committee to the Joint Committee on Education May 2018 The special Senate Subcommittee to the Joint Committee on Education was established per order of the Massachusetts State Senate in February, 2017: Ms. Chang-Díaz presented the following order, to wit: Ordered, That there shall be a special Senate sub-committee to the Joint Committee on Education, to consist of five members from the current Senate membership on the Joint Committee on Education, chaired by the Senate vice chair of the Joint Committee on Education, for the purpose of making an investigation and study of the Commonwealth’s alignment with and opportunities presented by the Every Student Succeeds (ESSA) Act, Public Law 114-95. The subcommittee shall submit its report and related legislation, if any, to the joint committee on Education once its report is completed. Senate Sub-committee chair and report author Senator Patricia D. Jehlen, with gratitude to those who contributed ideas, data, and comments Senate Sub-committee Members: Senator Michael J. Barrett Senator Jason M. Lewis Senator Barbara A. L'Italien Senator Patrick M. O'Connor Staff: Victoria Halal, Matthew Hartman, Emily Wilson, Kat Cline, Dennis Burke, Erin Riley, Sam Anderson, Daria Afshar Sponsored Events: (6/13/17) Panel Discussion: Life & Learning in MA Turnaround Schools https://www.youtube.com/watch?v=KbErK6rLQAY&t=2s (12/12/17) Mildred Avenue School Site Visit -
How and Why Standardized Tests Systematically Underestimate
St. John's Law Review Volume 80 Number 1 Volume 80, Winter 2006, Number 1 Article 7 How and Why Standardized Tests Systematically Underestimate African-Americans' True Verbal Ability and What to Do About It: Towards the Promotion of Two New Theories with Practical Applications Dr. Roy Freedle Follow this and additional works at: https://scholarship.law.stjohns.edu/lawreview This Symposium is brought to you for free and open access by the Journals at St. John's Law Scholarship Repository. It has been accepted for inclusion in St. John's Law Review by an authorized editor of St. John's Law Scholarship Repository. For more information, please contact [email protected]. THE RONALD H. BROWN CENTER FOR CIVIL RIGHTS AND ECONOMIC DEVELOPMENT SYMPOSIUM HOW AND WHY STANDARDIZED TESTS SYSTEMATICALLY UNDERESTIMATE AFRICAN-AMERICANS' TRUE VERBAL ABILITY AND WHAT TO DO ABOUT IT: TOWARDS THE PROMOTION OF TWO NEW THEORIES WITH PRACTICAL APPLICATIONS DR. ROY FREEDLEt INTRODUCTION In this Article, I want to raise a number of issues, both theoretical and practical, concerning the need for a total reassessment of especially the verbal intelligence of minority individuals. The issues to be raised amount to a critical reappraisal of standardized multiple-choice tests of verbal intelligence, such as the Law School Admissions Test ("LSAT"). I want to probe very deeply into why such standardized tests systematically underestimate verbal intelligence. This leads me first to review the prospects for a new standardized test of verbal intelligence associated with the studies of Joseph Fagan and Cynthia Holland.1 These studies show us that the races are equal; this result leads us to question the construct validity of many current standardized tests of verbal aptitude. -
STAT 830 Likelihood Methods
STAT 830 Likelihood Methods Richard Lockhart Simon Fraser University STAT 830 — Fall 2011 Richard Lockhart (Simon Fraser University) STAT 830 Likelihood Methods STAT 830 — Fall 2011 1 / 32 Purposes of These Notes Define the likelihood, log-likelihood and score functions. Summarize likelihood methods Describe maximum likelihood estimation Give sequence of examples. Richard Lockhart (Simon Fraser University) STAT 830 Likelihood Methods STAT 830 — Fall 2011 2 / 32 Likelihood Methods of Inference Toss a thumb tack 6 times and imagine it lands point up twice. Let p be probability of landing points up. Probability of getting exactly 2 point up is 15p2(1 − p)4 This function of p, is the likelihood function. Def’n: The likelihood function is map L: domain Θ, values given by L(θ)= fθ(X ) Key Point: think about how the density depends on θ not about how it depends on X . Notice: X , observed value of the data, has been plugged into the formula for density. Notice: coin tossing example uses the discrete density for f . We use likelihood for most inference problems: Richard Lockhart (Simon Fraser University) STAT 830 Likelihood Methods STAT 830 — Fall 2011 3 / 32 List of likelihood techniques Point estimation: we must compute an estimate θˆ = θˆ(X ) which lies in Θ. The maximum likelihood estimate (MLE) of θ is the value θˆ which maximizes L(θ) over θ ∈ Θ if such a θˆ exists. Point estimation of a function of θ: we must compute an estimate φˆ = φˆ(X ) of φ = g(θ). We use φˆ = g(θˆ) where θˆ is the MLE of θ.