Analysis of Variance and Analysis of Variance and Design of Experiments of Experiments-I
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Statistical Analysis 8: Two-Way Analysis of Variance (ANOVA)
Statistical Analysis 8: Two-way analysis of variance (ANOVA) Research question type: Explaining a continuous variable with 2 categorical variables What kind of variables? Continuous (scale/interval/ratio) and 2 independent categorical variables (factors) Common Applications: Comparing means of a single variable at different levels of two conditions (factors) in scientific experiments. Example: The effective life (in hours) of batteries is compared by material type (1, 2 or 3) and operating temperature: Low (-10˚C), Medium (20˚C) or High (45˚C). Twelve batteries are randomly selected from each material type and are then randomly allocated to each temperature level. The resulting life of all 36 batteries is shown below: Table 1: Life (in hours) of batteries by material type and temperature Temperature (˚C) Low (-10˚C) Medium (20˚C) High (45˚C) 1 130, 155, 74, 180 34, 40, 80, 75 20, 70, 82, 58 2 150, 188, 159, 126 136, 122, 106, 115 25, 70, 58, 45 type Material 3 138, 110, 168, 160 174, 120, 150, 139 96, 104, 82, 60 Source: Montgomery (2001) Research question: Is there difference in mean life of the batteries for differing material type and operating temperature levels? In analysis of variance we compare the variability between the groups (how far apart are the means?) to the variability within the groups (how much natural variation is there in our measurements?). This is why it is called analysis of variance, abbreviated to ANOVA. This example has two factors (material type and temperature), each with 3 levels. Hypotheses: The 'null hypothesis' might be: H0: There is no difference in mean battery life for different combinations of material type and temperature level And an 'alternative hypothesis' might be: H1: There is a difference in mean battery life for different combinations of material type and temperature level If the alternative hypothesis is accepted, further analysis is performed to explore where the individual differences are. -
Survey Experiments
IU Workshop in Methods – 2019 Survey Experiments Testing Causality in Diverse Samples Trenton D. Mize Department of Sociology & Advanced Methodologies (AMAP) Purdue University Survey Experiments Page 1 Survey Experiments Page 2 Contents INTRODUCTION ............................................................................................................................................................................ 8 Overview .............................................................................................................................................................................. 8 What is a survey experiment? .................................................................................................................................... 9 What is an experiment?.............................................................................................................................................. 10 Independent and dependent variables ................................................................................................................. 11 Experimental Conditions ............................................................................................................................................. 12 WHY CONDUCT A SURVEY EXPERIMENT? ........................................................................................................................... 13 Internal, external, and construct validity .......................................................................................................... -
5. Dummy-Variable Regression and Analysis of Variance
Sociology 740 John Fox Lecture Notes 5. Dummy-Variable Regression and Analysis of Variance Copyright © 2014 by John Fox Dummy-Variable Regression and Analysis of Variance 1 1. Introduction I One of the limitations of multiple-regression analysis is that it accommo- dates only quantitative explanatory variables. I Dummy-variable regressors can be used to incorporate qualitative explanatory variables into a linear model, substantially expanding the range of application of regression analysis. c 2014 by John Fox Sociology 740 ° Dummy-Variable Regression and Analysis of Variance 2 2. Goals: I To show how dummy regessors can be used to represent the categories of a qualitative explanatory variable in a regression model. I To introduce the concept of interaction between explanatory variables, and to show how interactions can be incorporated into a regression model by forming interaction regressors. I To introduce the principle of marginality, which serves as a guide to constructing and testing terms in complex linear models. I To show how incremental I -testsareemployedtotesttermsindummy regression models. I To show how analysis-of-variance models can be handled using dummy variables. c 2014 by John Fox Sociology 740 ° Dummy-Variable Regression and Analysis of Variance 3 3. A Dichotomous Explanatory Variable I The simplest case: one dichotomous and one quantitative explanatory variable. I Assumptions: Relationships are additive — the partial effect of each explanatory • variable is the same regardless of the specific value at which the other explanatory variable is held constant. The other assumptions of the regression model hold. • I The motivation for including a qualitative explanatory variable is the same as for including an additional quantitative explanatory variable: to account more fully for the response variable, by making the errors • smaller; and to avoid a biased assessment of the impact of an explanatory variable, • as a consequence of omitting another explanatory variables that is relatedtoit. -
The Politics of Random Assignment: Implementing Studies and Impacting Policy
The Politics of Random Assignment: Implementing Studies and Impacting Policy Judith M. Gueron Manpower Demonstration Research Corporation (MDRC) As the only nonacademic presenting a paper at this conference, I see it as my charge to focus on the challenge of implementing random assignment in the field. I will not spend time arguing for the methodological strengths of social experiments or advocating for more such field trials. Others have done so eloquently.1 But I will make my biases clear. For 25 years, I and many of my MDRC colleagues have fought to implement random assignment in diverse arenas and to show that this approach is feasible, ethical, uniquely convincing, and superior for answering certain questions. Our organization is widely credited with being one of the pioneers of this approach, and through its use producing results that are trusted across the political spectrum and that have made a powerful difference in social policy and research practice. So, I am a believer, but not, I hope, a blind one. I do not think that random assignment is a panacea or that it can address all the critical policy questions, or substitute for other types of analysis, or is always appropriate. But I do believe that it offers unique power in answering the “Does it make a difference?” question. With random assignment, you can know something with much greater certainty and, as a result, can more confidently separate fact from advocacy. This paper focuses on implementing experiments. In laying out the ingredients of success, I argue that creative and flexible research design skills are essential, but that just as important are operational and political skills, applied both to marketing the experiment in the first place and to helping interpret and promote its findings down the line. -
Key Items to Get Right When Conducting Randomized Controlled Trials of Social Programs
Key Items to Get Right When Conducting Randomized Controlled Trials of Social Programs February 2016 This publication was produced by the Evidence-Based Policy team of the Laura and John Arnold Foundation (now Arnold Ventures). This publication is in the public domain. Authorization to reproduce it in whole or in part for educational purposes is granted. We welcome comments and suggestions on this document ([email protected]). 1 Purpose This is a checklist of key items to get right when conducting a randomized controlled trial (RCT) to evaluate a social program or practice. The checklist is designed to be a practical resource for researchers and sponsors of research. It describes items that are critical to the success of an RCT in producing valid findings about a social program’s effectiveness. This document is limited to key items, and does not address all contingencies that may affect a study’s success.1 Items in this checklist are categorized according to the following phases of an RCT: 1. Planning the study; 2. Carrying out random assignment; 3. Measuring outcomes for the study sample; and 4. Analyzing the study results. 1. Key items to get right in planning the study Choose (i) the program to be evaluated, (ii) target population for the study, and (iii) key outcomes to be measured. These should include, wherever possible, ultimate outcomes of policy importance. As illustrative examples: . An RCT of a pregnancy prevention program preferably should measure outcomes such as pregnancies or births, and not just intermediate outcomes such as condom use. An RCT of a remedial reading program preferably should measure outcomes such as reading comprehension, and not just participants’ ability to sound out words. -
THE ONE-SAMPLE Z TEST
10 THE ONE-SAMPLE z TEST Only the Lonely Difficulty Scale ☺ ☺ ☺ (not too hard—this is the first chapter of this kind, but youdistribute know more than enough to master it) or WHAT YOU WILL LEARN IN THIS CHAPTERpost, • Deciding when the z test for one sample is appropriate to use • Computing the observed z value • Interpreting the z value • Understandingcopy, what the z value means • Understanding what effect size is and how to interpret it not INTRODUCTION TO THE Do ONE-SAMPLE z TEST Lack of sleep can cause all kinds of problems, from grouchiness to fatigue and, in rare cases, even death. So, you can imagine health care professionals’ interest in seeing that their patients get enough sleep. This is especially the case for patients 186 Copyright ©2020 by SAGE Publications, Inc. This work may not be reproduced or distributed in any form or by any means without express written permission of the publisher. Chapter 10 ■ The One-Sample z Test 187 who are ill and have a real need for the healing and rejuvenating qualities that sleep brings. Dr. Joseph Cappelleri and his colleagues looked at the sleep difficul- ties of patients with a particular illness, fibromyalgia, to evaluate the usefulness of the Medical Outcomes Study (MOS) Sleep Scale as a measure of sleep problems. Although other analyses were completed, including one that compared a treat- ment group and a control group with one another, the important analysis (for our discussion) was the comparison of participants’ MOS scores with national MOS norms. Such a comparison between a sample’s mean score (the MOS score for par- ticipants in this study) and a population’s mean score (the norms) necessitates the use of a one-sample z test. -
How to Randomize?
TRANSLATING RESEARCH INTO ACTION How to Randomize? Roland Rathelot J-PAL Course Overview 1. What is Evaluation? 2. Outcomes, Impact, and Indicators 3. Why Randomize and Common Critiques 4. How to Randomize 5. Sampling and Sample Size 6. Threats and Analysis 7. Project from Start to Finish 8. Cost-Effectiveness Analysis and Scaling Up Lecture Overview • Unit and method of randomization • Real-world constraints • Revisiting unit and method • Variations on simple treatment-control Lecture Overview • Unit and method of randomization • Real-world constraints • Revisiting unit and method • Variations on simple treatment-control Unit of Randomization: Options 1. Randomizing at the individual level 2. Randomizing at the group level “Cluster Randomized Trial” • Which level to randomize? Unit of Randomization: Considerations • What unit does the program target for treatment? • What is the unit of analysis? Unit of Randomization: Individual? Unit of Randomization: Individual? Unit of Randomization: Clusters? Unit of Randomization: Class? Unit of Randomization: Class? Unit of Randomization: School? Unit of Randomization: School? How to Choose the Level • Nature of the Treatment – How is the intervention administered? – What is the catchment area of each “unit of intervention” – How wide is the potential impact? • Aggregation level of available data • Power requirements • Generally, best to randomize at the level at which the treatment is administered. Suppose an intervention targets health outcomes of children through info on hand-washing. What is the appropriate level of randomization? A. Child level 30% B. Household level 23% C. Classroom level D. School level 17% E. Village level 13% 10% F. Don’t know 7% A. B. C. D. -
Chapter 5 Experiments, Good And
Chapter 5 Experiments, Good and Bad Point of both observational studies and designed experiments is to identify variable or set of variables, called explanatory variables, which are thought to predict outcome or response variable. Confounding between explanatory variables occurs when two or more explanatory variables are not separated and so it is not clear how much each explanatory variable contributes in prediction of response variable. Lurking variable is explanatory variable not considered in study but confounded with one or more explanatory variables in study. Confounding with lurking variables effectively reduced in randomized comparative experiments where subjects are assigned to treatments at random. Confounding with a (only one at a time) lurking variable reduced in observational studies by controlling for it by comparing matched groups. Consequently, experiments much more effec- tive than observed studies at detecting which explanatory variables cause differences in response. In both cases, statistically significant observed differences in average responses implies differences are \real", did not occur by chance alone. Exercise 5.1 (Experiments, Good and Bad) 1. Randomized comparative experiment: effect of temperature on mice rate of oxy- gen consumption. For example, mice rate of oxygen consumption 10.3 mL/sec when subjected to 10o F. temperature (Fo) 0 10 20 30 ROC (mL/sec) 9.7 10.3 11.2 14.0 (a) Explanatory variable considered in study is (choose one) i. temperature ii. rate of oxygen consumption iii. mice iv. mouse weight 25 26 Chapter 5. Experiments, Good and Bad (ATTENDANCE 3) (b) Response is (choose one) i. temperature ii. rate of oxygen consumption iii. -
ANALYSIS of VARIANCE and MISSING OBSERVATIONS in COMPLETELY RANDOMIZED, RANDOMIZED BLOCKS and LATIN SQUARE DESIGNS a Thesis
ANALYSIS OF VARIANCE AND MISSING OBSERVATIONS IN COMPLETELY RANDOMIZED, RANDOMIZED BLOCKS AND LATIN SQUARE DESIGNS A Thesis Presented to The Department of Mathematics Kansas State Teachers College, Emporia, Kansas In Partial Fulfillment of the Requirements for the Degree Master of Science by Kiritkumar K. Talati May 1972 c; , ACKNOWLEDGEMENTS Sincere thanks and gratitude is expressed to Dr. John Burger for his assistance, patience, and his prompt attention in all directions in preparing this paper. A special note of appreciation goes to Dr. Marion Emerson, Dr. Thomas Davis, Dr. George Poole, Dr. Darrell Wood, who rendered assistance in the research of this paper. TABLE OF CONTENTS CHAPTER I. INTRODUCTION • • • • • • • • • • • • • • • • 1 A. Preliminary Consideration. • • • • • • • 1 B. Purpose and Assumptions of the Analysis of Variance • • • • • • • • •• 1 C. Analysis of Covariance • • • • • • • • • 2 D. Definitions. • • • • • • • • • • • • • • ) E. Organization of Paper. • • • • • • • • • 4 II. COMPLETELY RANDOMIZED DESIGN • • • • • • • • 5 A. Description............... 5 B. Randomization. • • • • • • • • • • • • • 5 C. Problem and Computations •••••••• 6 D. Conclusion and Further Applications. •• 10 III. RANDOMIZED BLOCK DESIGN. • • • • • • • • • • 12 A. Description............... 12 B. Randomization. • • • • • • • • • • • • • 1) C. Problem and Statistical Analysis • • • • 1) D. Efficiency of Randomized Block Design as Compared to Completely Randomized Design. • • • • • • • • • •• 20 E. Missing Observations • • • • • • • • •• 21 F. -
Randomized Experimentsexperiments Randomized Trials
Impact Evaluation RandomizedRandomized ExperimentsExperiments Randomized Trials How do researchers learn about counterfactual states of the world in practice? In many fields, and especially in medical research, evidence about counterfactuals is generated by randomized trials. In principle, randomized trials ensure that outcomes in the control group really do capture the counterfactual for a treatment group. 2 Randomization To answer causal questions, statisticians recommend a formal two-stage statistical model. In the first stage, a random sample of participants is selected from a defined population. In the second stage, this sample of participants is randomly assigned to treatment and comparison (control) conditions. 3 Population Randomization Sample Randomization Treatment Group Control Group 4 External & Internal Validity The purpose of the first-stage is to ensure that the results in the sample will represent the results in the population within a defined level of sampling error (external validity). The purpose of the second-stage is to ensure that the observed effect on the dependent variable is due to some aspect of the treatment rather than other confounding factors (internal validity). 5 Population Non-target group Target group Randomization Treatment group Comparison group 6 Two-Stage Randomized Trials In large samples, two-stage randomized trials ensure that: [Y1 | D =1]= [Y1 | D = 0] and [Y0 | D =1]= [Y0 | D = 0] • Thus, the estimator ˆ ˆ ˆ δ = [Y1 | D =1]-[Y0 | D = 0] • Consistently estimates ATE 7 One-Stage Randomized Trials Instead, if randomization takes place on a selected subpopulation –e.g., list of volunteers-, it only ensures: [Y0 | D =1] = [Y0 | D = 0] • And hence, the estimator ˆ ˆ ˆ δ = [Y1 | D =1]-[Y0 | D = 0] • Only estimates TOT Consistently 8 Randomized Trials Furthermore, even in idealized randomized designs, 1. -
How to Do Random Allocation (Randomization) Jeehyoung Kim, MD, Wonshik Shin, MD
Special Report Clinics in Orthopedic Surgery 2014;6:103-109 • http://dx.doi.org/10.4055/cios.2014.6.1.103 How to Do Random Allocation (Randomization) Jeehyoung Kim, MD, Wonshik Shin, MD Department of Orthopedic Surgery, Seoul Sacred Heart General Hospital, Seoul, Korea Purpose: To explain the concept and procedure of random allocation as used in a randomized controlled study. Methods: We explain the general concept of random allocation and demonstrate how to perform the procedure easily and how to report it in a paper. Keywords: Random allocation, Simple randomization, Block randomization, Stratified randomization Randomized controlled trials (RCT) are known as the best On the other hand, many researchers are still un- method to prove causality in spite of various limitations. familiar with how to do randomization, and it has been Random allocation is a technique that chooses individuals shown that there are problems in many studies with the for treatment groups and control groups entirely by chance accurate performance of the randomization and that some with no regard to the will of researchers or patients’ con- studies are reporting incorrect results. So, we will intro- dition and preference. This allows researchers to control duce the recommended way of using statistical methods all known and unknown factors that may affect results in for a randomized controlled study and show how to report treatment groups and control groups. the results properly. Allocation concealment is a technique used to pre- vent selection bias by concealing the allocation sequence CATEGORIES OF RANDOMIZATION from those assigning participants to intervention groups, until the moment of assignment. -
Random Assignment
7 RANDOM ASSIGNMENT Just as representativeness can be secured by the method of chance . so equivalence may be secured by chance.1distribute —W. A. McCall or LEARNING OBJECTIVES • Understand what random assignment does and how it works. • Produce a valid randomization processpost, for an experiment and describe it. • Critique simple random assignment, blocking, matched pairs, and stratified random assignment. • Explain the importance of counterbalancing. • Describe a Latincopy, square design. ust as the mantra in real estate is “location, location, location,” the motto in experi- Jmental design is “random assignment, random assignment, random assignment.” This notbook has discussed random assignment all throughout. It bears repeating that ran- dom assignment is the single most important thing a researcher can do in an experiment. Everything else pales in comparison to having done this correctly.2 Random assignment is what distinguishes a true experiment from a quasi, natural, or pre-experimental design. In Dochapter 1, experiments were referred to as the gold standard. Without successful random 173 Copyright ©2019 by SAGE Publications, Inc. This work may not be reproduced or distributed in any form or by any means without express written permission of the publisher. 174 Designing Experiments for the Social Sciences assignment, however, they can quickly become “the bronze standard.”3 This chapter will review some of the advantages of random assignment, discuss the details of how to do it, and explore related issues of counterbalancing. THE PURPOSE OF RANDOM ASSIGNMENT People vary. That is, they are different. Were this not so, there would be no reason to study them. Everyone would be the same, reacting the same way to different teaching techniques, advertisements, health interventions, and political messages.