WWW.ABSTRACT.XLIBX.INFO
FREE ELECTRONIC LIBRARY - Abstract, dissertation, book
 
<< HOME
CONTACTS



Pages:   || 2 | 3 | 4 |

«IZA DP No. 3753 Incorporating Cost in Power Analysis for Three-Level Cluster Randomized Designs Spyros Konstantopoulos October 2008 ...»

-- [ Page 1 ] --

DISCUSSION PAPER SERIES

IZA DP No. 3753

Incorporating Cost in Power Analysis for

Three-Level Cluster Randomized Designs

Spyros Konstantopoulos

October 2008

Forschungsinstitut

zur Zukunft der Arbeit

Institute for the Study

of Labor

Incorporating Cost in Power Analysis

for Three-Level Cluster Randomized

Designs

Spyros Konstantopoulos

Boston College

and IZA

Discussion Paper No. 3753

October 2008

IZA

P.O. Box 7240

53072 Bonn

Germany

Phone: +49-228-3894-0 Fax: +49-228-3894-180 E-mail: iza@iza.org Any opinions expressed here are those of the author(s) and not those of IZA. Research published in this series may include views on policy, but the institute itself takes no institutional policy positions.

The Institute for the Study of Labor (IZA) in Bonn is a local and virtual international research center and a place of communication between science, politics and business. IZA is an independent nonprofit organization supported by Deutsche Post World Net. The center is associated with the University of Bonn and offers a stimulating research environment through its international network, workshops and conferences, data service, project support, research visits and doctoral program. IZA engages in (i) original and internationally competitive research in all fields of labor economics, (ii) development of policy concepts, and (iii) dissemination of research results and concepts to the interested public.

IZA Discussion Papers often represent preliminary work and are circulated to encourage discussion.

Citation of such a paper should account for its provisional character. A revised version may be available directly from the author.

IZA Discussion Paper No. 3753 October 2008

ABSTRACT

Incorporating Cost in Power Analysis for Three-Level Cluster Randomized Designs In experimental designs with nested structures entire groups (such as schools) are often assigned to treatment conditions. Key aspects of the design in these cluster randomized experiments include knowledge of the intraclass correlation structure and the sample sizes necessary to achieve adequate power to detect the treatment effect. However, the units at each level of the hierarchy have a cost associated with them and thus researchers need to decide on sample sizes given a certain budget, when designing their studies. This paper provides methods for computing power within an optimal design framework (that incorporates costs of units in all three levels) for three-level cluster randomized balanced designs with two levels of nesting. The optimal sample sizes are a function of the variances at each level and the cost of each unit. Overall, larger effect sizes, smaller intraclass correlations at the second and third level, and lower cost of level-3 and level-2 units result in higher estimates of power.

JEL Classification: I20 Keywords: experimental design, statistical power, optimal sampling

Corresponding author:

Spyros Konstantopoulos Lynch School of Education Boston College Campion Hall, Room 336D 140 Commonwealth Avenue Chestnut Hill, MA 02467 USA E-mail: konstans@bc.edu Many populations of interest in education and the social sciences have multilevel structures. For example, in education students are nested within classrooms, and classrooms are nested within schools. Experiments that involve nested population structures may assign treatment conditions to entire groups. In education, frequently, largescale randomized experiments assign schools to treatment and control conditions and these designs are often called cluster or group randomized designs (see Bloom, 2005; Donner & Klar, 2000; Murray, 1998).

A critical issue in designing experiments is to ensure that the design has sufficient power to detect the intervention effects that are expected if the researchers’ hypotheses were correct. There is an extensive literature on the computation of statistical power (e.g., Cohen, 1988; Lipsey, 1990, Murphy & Myors, 2004). Much of this literature however, involves the computation of power in studies that use simple random samples and thus clustering effects are not included in the power analysis. Software for computing statistical power in single-level designs has also become widely available recently (Borenstein, Rothstein, & Cohen, 2001).

Statistical theory for computing power in two-level designs has also been recently documented and statistical software for two-level balanced designs is currently available (e.g., Hedges & Hedberg, 2007; Murray, 1998; Raudenbush & Liu, 2000, 2001;

Raudenbush, Spybrook, Liu, & Congdon, 2006). However, power analysis in nested designs entails challenges. First, nested factors are usually taken to have random effects, and hence, power computations usually involve the variance components structures (typically expressed via intraclass correlations) of these random effects. Second, there is not one sample size, but several sample sizes at each level of the hierarchy that may affect power differently. For example, in educational studies that assign treatments to schools, the power of the test of the treatment effect depends not only on the number of students within a classroom or a school, but on the number of classrooms or schools as well. Methods for power computations of tests of treatment effects in multi-level designs have also been discussed in the health sciences (e.g., Donner, 1984; Hsieh, 1988; Murray, 1998; Murray, Van Horn, Hawkins, & Arthur, 2006). For example, Murray and colleagues (2006) provided ways for analyzing data with complicated nested structures and discussed posthoc power computations of tests of treatment effects within the ANCOVA framework.





In addition, a more recent study discussed methods for computing power in threelevel balanced cluster randomized designs (Konstantopoulos, 2008). Many factors need to be taken into account when designing randomized experiments with a three-level structure.

For instance, in three-level cluster randomized designs with two levels of clustering (second and third level) researchers need to take into account the clustering effects at both levels and consider trade-offs that involve sampling level-1, level-2, and level-3 units. In such designs maximizing the number of level-3 units in the sample has a larger impact on the power of the test of the treatment effect than maximizing the number of level1 or levelunits (see Konstantopoulos, 2008). Also, clustering effects, often expressed via interclass correlations, affect the power estimates inversely.

In addition, the issue of optimal sampling of units at different levels of the hierarchy to maximize power is critical in designing multilevel experiments. Since larger units such as schools affect power much more than smaller units such as classrooms or students a researcher would be inclined to design large-scale experiments with numerous larger units and fewer smaller units. However, maximizing the number of larger units, such as schools, is more expensive than maximizing smaller units, such as classrooms or students. The researcher then faces the challenge of designing a cost-effective study that will optimize the power of the test of the treatment effect given the budget. This requires incorporating cost-related issues when maximizing power in cluster randomized designs (see Raudenbush, 1997). The present study discusses optimal design considerations that incorporate costs of sample sizes at different levels of the hierarchy when designing threelevel cluster randomized designs with two levels of nesting. Specifically, I follow Cochran (1977) and Raundenbush (1997) and define cost functions that involve the cost ratios among level-1, level-2, and level-3 units, and then I determine the optimal number of level-1, level-2 (and level-3) units to maximize power, given the costs. Following Raudenbush and Liu (2000) I define optimal design, under specific assumptions, a design that results in the highest estimate of power for the treatment effect.

The paper is structured as follows. First, I define the intraclass correlations in threelevel models with two levels of nesting. Second, I present the statistical model and provide an example for computing power in a three-level cluster randomized design. Then, I introduce cost functions that involve level-1, level-2, and level-3 units to maximize power.

Finally, I summarize the usefulness of the methods and draw conclusions.

–  –  –

Suppose that a researcher samples level-3 units at the first stage, samples level-2 units within level-3 units at the second stage, and then samples level-1 units within level-2 units at the third stage. This is a three-stage cluster sample and the variance of the total population is the sum of the within-level-2 unit between-level-1 unit variance, σ e2 ; the within-level-3 unit between-level-2 unit variance, τ 2 ; and the between-level-3 unit variance, ω 2 (see Cochran, 1977; Lohr, 1999). That is, the total variance in the outcome is decomposed into three parts and is defined as σ T = σ e2 + τ 2 + ω 2. In such three-level designs two intraclass correlations are needed to describe the variance component

structure. These are defined as the second level intraclass correlation:

–  –  –

Consider a design where level-3 units are nested within treatment, and level-2 units are nested within level-3 units and treatment (Kirk, 1995), and both level-3 and level-2 units are random effects. A structural model for an outcome Yijkl, the lth level-1 unit in the kth level-2 unit in the jth level-3 unit in the ith treatment can be described in ACOVA notation as

–  –  –

where μ is the grand mean, αi is the (fixed) effect of the ith treatment (i = 1,2), and the last three terms represent level-3, level-2, and level-1 random effects respectively. Specifically, β (i ) j is the random effect of level-3 unit j (j = 1,…, m) within treatment i, γ (ij ) k is the random effect of level-2 unit k (k = 1,…, p) within level-3 unit j within treatment i, and ε (ijk )l is the error term of level-1 unit l (l = 1,…, n) within level-2 unit k, within level-3 unit j, within treatment i. I assume that the level-1, level-2, and level-3 error terms are normally distributed with a mean of zero and residual variances σ e2, τ 2, and ω 2 respectively. For simplicity, I assume that there is one treatment and one control group and that the designs are balanced.

The objective is to examine the statistical significance of the treatment effect, which means to test the hypothesis

–  –  –

The researcher can test this hypothesis by carrying out the usual t-test. Following Konstantopoulos (2008) when the null hypothesis is false, the test statistic has a noncentral t-distribution with 2m-2 degrees of freedom and non-centrality parameter λ (assuming no covariates). The non-centrality parameter is defined as the expected value of the estimate of the treatment effect divided by the square root of the variance of the estimate of the treatment effect, namely

–  –  –

where m is the number of level-3 units in each condition (treatment or control group), p is the number of level-2 units within each level-3 unit, n is the number of level-1 units within each level-2 unit, and δ = α1 - α2 / σ T, where α1 and α2 are the treatment effect parameters from the ANOVA model (defined above) and σT is the population standard deviation.

The power of the two-tailed t-test at level α is

–  –  –

where c(α, ν) is the level α two-tailed critical value of the t-distribution with ν degrees of freedom [ c(0.05,20) = 1.72], and Η(x, ν, λ) is the cumulative distribution function of the non-central t-distribution with ν degrees of freedom and non-centrality parameter λ. The test of the treatment effect and statistical power can also be computed using the F-statistic that has a non-central F-distribution with 1 degree of freedom in the numerator and 2m – 2 degrees of freedom in the denominator and non-centrality parameter λ 2.

–  –  –

3 covariate effects, Xijkl is a column vector of r level-1 covariates, Zijk is a column vector of w level-2 covariates, and Wij is a column vector of q level-3 covariates, and the last three terms represent residuals at the third, second, and first level respectively. The subscript A indicates adjustment due to covariate effects, that is, the level-2 and level-3 random effects are adjusted by level-2 and level-3 covariates respectively and the level-1 error term is adjusted by level-1 covariates. I assume that the covariates at each level are centered at their means to ensure that covariates explain variation in the outcome only at the level at which they are introduced. Note that although in practice covariates could slightly adjust the treatment effect, in principle, due to randomization the treatment effect should be unadjusted. I assume that the adjusted error terms first, second, and third level are normally distributed with a mean of zero and residual variances σ Re, τ R, and ωR, respectively.

The objective in this case is to examine the statistical significance of the treatment effect adjusted by covariates, which means to test the hypothesis

–  –  –

Note that in this case δ and the intraclass correlations are adjusted. Specifically, the numerator of δ remains unchanged (because of orthogonality between the treatment and the covariates), whilst the denominator changes (because the total variance is now residual variance). The intraclass correlations are also adjusted. The second level intraclass correlation is now defined as

–  –  –

where subscript A indicates adjustment and subscript R indicates residual variance. When the null hypothesis is false, the test statistic has the non-central t-distribution with 2m-q-2 degrees of freedom (where q is the number of level-3 covariates) and non-centrality parameter λA. Following Konstantopoulos (2008) the non-centrality parameter is defined now as

–  –  –



Pages:   || 2 | 3 | 4 |


Similar works:

«A Midsummer Night’s Dream By William Shakespeare Family Learning Pack Family Learning Hello! Thank you very much for picking up the Oxford Playhouse Family Learning Pack for A Midsummer Night’s Dream. In this pack you will find a variety of games and activities to give your family a head start on understanding the play. Activities are aimed at children aged 7 to 12 and there’s a story synopsis for the family to read together. FAMILY LEARNING FAMILY LEARNING FAMILY LEARNING FAMILY LEARNING...»

«05-Ritzer-4852.qxd 1/4/2006 3:46 PM Page 46 A Sociology of Rib Joints P. D. Holley and D. E. Wright, Jr. O nly barbeque restaurants that feature ribs may be classified as “rib joints.”... This ideal type includes the following deviant aspects.1. A location that is usually difficult to find.... This joint seldom advertises; patrons learn about it by “word of mouth” and by the occasional “lifestyles” section of some metropolitan newspaper or magazine written by a reporter...»

«SECURITIES AND EXCHANGE COMMISSION [Investment Company Act Release No. 32063; 812-14537] Advisors Asset Management, Inc. and AAM ETF Trust; Notice of Application March 31, 2016 Agency: Securities and Exchange Commission (“Commission”). Action: Notice of an application for an order under section 6(c) of the Investment Company Act of 1940 (the “Act”) for an exemption from sections 2(a)(32), 5(a)(1), 22(d), and 22(e) of the Act and rule 22c-1 under the Act, under sections 6(c) and 17(b) of...»

«Jerzy Bieroński Zbiorniki małej retencji – problemy funkcjonowania Wstęp Do przedsięwzięć technicznych małej retencji zaliczane są w Polsce zbiorniki o pojemności do 5 mln m3. Nale ą do nich obiekty ró nego typu i wielkości – od niewielkich oczek wodnych po podpiętrzenia cieków z wykształconymi zalewami, niekiedy dość rozległymi. Do obiektów technicznych małej retencji zalicza się tak e podpiętrzenia jazami i zastawkami, zazwyczaj o niewielkiej objętości wód...»

«„Das große NEIN zur Schule“ – Schulangst und Schulphobie Erscheinungsformen, Verlauf, Ursachen und Behandlungsmöglichkeiten Schriftliche Hausarbeit im Rahmen der Ersten Staatsprüfung für das Lehramt der Sekundarstufe II dem Staatlichen Prüfungsamt für Erste Staatsprüfungen für Lehrämter an Schulen in Essen (Dienststelle Duisburg) vorgelegt als Gruppenarbeit von: Benjamin Schulz und Pia Anna Weber Duisburg, März, 2006 Themensteller: Dr. Bernd Kern Fachbereich...»

«Class Shelves Listing Amherst College Archives & Special Collections Shepard, Charles U. – Treatise on Mineralogy (Charles W. Clark, owner) Tinker, Rev. Reuben – Sermons of Rev. Reuben Tinker, 1856 Perkins, Justin (trans.) The Four Gospels in the Nestoriac Language (trans.) New and Old Testament in modern and ancient Agriac. (2 New Testament, 1 incl. handwritten note from Perkins.) A Pilgrim’s Progress (modern Agriac) (trans.) Unidentified (modern Agriac?) Riggs, Elias (trans.) The Bible...»

«Geologic Resources Inventory Scoping Summary Geologic Resources Inventory Scoping Summary Glacier Bay National Park, Alaska Chattahoochee River National Recreation Area Georgia Geologic Resources Division PreparedParkKatie KellerLynn National by Service Geologic Resources Division October 31, 2012 the Interior US Department of National Park Service U.S. Department of the Interior The Geologic Resources Inventory (GRI) Program, administered by the Geologic Resources Division (GRD), provides each...»

«Java design patterns 101 Presented by developerWorks, your source for great tutorials ibm.com/developerWorks Table of Contents If you're viewing this document online, you can click any of the topics below to link directly to that section.1. About this tutorial 2. Design patterns overview 3. A brief introduction to UML class diagrams 4. Creational patterns 5. Structural patterns 6. Behavioral patterns 7. Concurrency patterns 8. Wrapup Java design patterns 101 Page 1 of 22 ibm.com/developerWorks...»

«Copyright by Sudeshna Sen The Dissertation Committee for Sudeshna Sen certifies that this is the approved version of the following dissertation: A Joint Multiple Discrete Continuous Extreme Value (MDCEV) Model and Multinomial Logit Model (MNL) for Examining Vehicle Type/Vintage, Make/Model and Usage Decisions of the Household Committee: Chandra R. Bhat, Supervisor Randy B. Machemehl S. Travis Waller Stephen Donald Chandler Stolp A Joint Multiple Discrete Continuous Extreme Value (MDCEV) Model...»

«COURT OF APPEALS OF VIRGINIA Present: Chief Judge Felton, Judges Benton and Petty Argued at Richmond, Virginia PIPER ANN ROUNTREE MEMORANDUM OPINION* BY v. Record No. 0155-06-2 CHIEF JUDGE WALTER S. FELTON, JR. JULY 24, 2007 COMMONWEALTH OF VIRGINIA FROM THE CIRCUIT COURT OF HENRICO COUNTY L.A. Harris, Jr., Judge David B. Hargett (Hargett & Watson, PLC, on brief), for appellant. Susan M. Harris, Assistant Attorney General (Robert F. McDonnell, Attorney General, on brief), for appellee. A jury...»

«Landesgericht für Schriftsatz per webERV eingebracht Zivilrechtssachen Wien Schmerlingplatz 11 1016 Wien Wien, am 12.01.2015 Klagende Partei: Mag. Maximilian Schrems, Doktorand vertreten durch: Proksch & Fritzsche Frank Fletzberger Rechtsanwälte GmbH Tel. 01/877 04 54 Nibelungengasse 11 1010 Wien Code P111395 Beklagte Partei: Facebook Ireland Limited Reg.Nr. 462932 im Unternehmensregister der Republik Irland 4 Grand Canal Square, Grand Canal Harbour, Dublin 2, Irland wegen: Feststellung /...»

«Course Ilt Fireworks 5 0 Basic Anywhere, there love a data that protect all as a meetings during the optimization. This accessible amount with elongating free results is truly apathetic prices are interrupting on their important rates, amounts are tripping to more repayments, training foreclosure tools, pace savings having online, own existence and epub forms depending home-based companies, but direct account applying to events. When it visit analyzed shopping acquire that quantum, hundreds by...»





 
<<  HOME   |    CONTACTS
2016 www.abstract.xlibx.info - Free e-library - Abstract, dissertation, book

Materials of this site are available for review, all rights belong to their respective owners.
If you do not agree with the fact that your material is placed on this site, please, email us, we will within 1-2 business days delete him.