20168392 Regression, ANOVA, And Estimates of Effect Size 2000

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Regression, ANOVA, and Estimates of Effect Size

Author(s): Stephen F. MatterSource: Bulletin of the Ecological Society of America, Vol. 81, No. 1 (Jan., 2000), pp. 74-76Published by: Ecological Society of AmericaStable URL: http://www.jstor.org/stable/20168392 .

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One feature that I have used a lotrecently is the Regression Wizard(not new to this version-it was alsoin 4.0). If you already have your datain a worksheet, in as few as five

mouse clicks you can select an equation to fit to the data, choose the datacolumns to use, generate a report withregression statistics and ANOVA,and create a graph with the data andregression line. This is a quick andeasy way to look at relationshipsamong data variables.

I have used SigmaPlot to creategraphs that were then printed on avariety of laser printers, printed to aslide maker, cut and pasted to a

Word document to e-mail them to aco-author, and photographed fromthe computer monitor for developingthe next day to take to a meeting. Ialso use it to generate graphs forPowerPoint slide shows for talks at

meetings or departmental seminars.All parts of a graph can be assignedcolors or gray scales, which makes iteasy to create versions appropriatefor different kinds of presentations.

The user-definable templates can facilitate creating these different versions of the same graph.

The program does still have somefeatures I don't like. I found that occasionally there were problems withthe axis legends when I cut andpasted a graph into Word. (The spacing between letters was changed fromthe SigmaPlot page.) Another inconvenience I found was that if you insert a column into a data file, graphsalready created using columns in thefile then appear to call for the wrongcolumns and have to be corrected.The fact that the "undo" feature onlygoes back one step is an inconvenience. Problems with the programcrashing appear to have been solvedby a patch downloaded from theSPSS web site (upgrading from version 5.0 to 5.05).System requirements for the program include Windows 95, Windows98, or NT 4.0; 32 Mb of RAM, 20 Mbof hard disk space; and SVGA or better graphics. SigmaPlot comes with a287-page programming guide that de

scribes the program's math, data manipulation, regression, and curve-fitting features, itsMacro Recorder, anda 448-page user' smanual. I found theuser' s manual well organized andeasy to use. Technical support isavailable via telephone (not toll-free),e-mail, fax, or mail. I found that responses to e-mail were prompt when Ihad questions, although the technicians weren't always able to solve myproblems (e.g., the incorrect spacingin legends when a graph was cut andpasted). Although this version ofSigmaPlot may be overkill if all youneed is a simple graph, the learningcurve isn't very steep and you'll sooncome to appreciate its range of capabilities if you use it for very long. Afree demo version is available at.

Reviewed by David W. InouyeDepartment of BiologyUniversity of Maryland

College Park, MD [email protected]

Ecology 1Note: Dr. Harold Ornes is the edi

tor of Ecology 101. Anyone wishingto contribute articles or reviews tothis section should contact him at the

Office of the Dean, College of Science, SB 310A, Southern Utah University, Cedar City, UT 84720;(435)586-7921; fax (435) 865-8550;e-mail: [email protected].

Among us are ecologists whospend a lifetime looking for effectsof experimental treatments on organisms. (I am of that ilk). The goodnews is that we often use relativelysimple statistical designs that can beanalyzed using linear regressionand/or analysis of variance (ANOVA)techniques. The bad news is that lifeain't that simple, and some precautions should be taken before apply

ing even simple statistical techniques.Please consider Professor StephenMatter's (University of Alberta)

thoughts on regression, ANOVA, andestimates of effect size.

In our second article, ProfessorsHilary Callahan of Barnard Collegeand Susan Will-Wolf and Timothy

Allen of the University of WisconsinMadison offer an exercise designed to

teach undergraduates about Landscape Ecology. This exercise offersstudents an opportunity to integrateseveral levels of ecological organization, work in groups and later as a

whole class, and build their mappingskills. In addition to this paper and

pencil exercise, the authors offer aweb site address with additional information and interesting links related to both Botany and Ecology.

REGRESSION, ANOVA,AND ESTIMATES OFEFFECTSIZEFor ecologists, linear regression

and analysis of variance (ANOVA)are among our most common statistical tools. ANOVA designs are usedto test for differences among meansof categorical independentvariables,such as sex or species. ANOVA is often also used in situations where thefactor "levels" are continuous variables transformed into categoricalvariables, such as the amount of fertilizer applied at "high," "medium,"and "low" levels. In such situations,either ANOVA or regression could beused, provided the amount of fertilizer applied is known. The application of ANOVA using these transformed continuous variables is quite

74 Bulletin f theEcologicalocietyfAmerica

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common. In a randomly selected, recent Ecology issue (Volume 78[1]), Ifound 19 ANOVAs, 9 of which usedANOVA with continuous ndependentvariables that had been made categorical.

Generally, the choice of ANOVAor linear regression in these situations has little bearing on the conclusions drawn from the analysis.

The main difference is that regression specifies a functional form (e.g.,linear), whereas ANOVA does not.

However, the tests differ in estimatesof variance explained or "effect size."Estimates of effect size help to enhance the interpretability of resultsby providing ameasure of themagnitude of experimental effects, while

guarding against significance attained simply by large sample size(Dwyer 1974). Use and comparisonof effect sizes has become commonplace in other disciplines such aspsychology. Effect size metrics canalso be used in the calculation ofstatistical power and meta-analyses(Cohen1988).

The difference in estimates of effectsize between regression and ANOVAdesigns can be illustrated simply. I constructed a hypothetical linear relationship where Y = (2 x X) + 100. For fourlevels of X, (X = 0, 4, 24, and 60), Idrew four random samples of Y froma normal distribution with a mean set

by the linear equation and a standarddeviation of 40.0. I analyzed the datausing both a one-way ANOVA and asimple linear regression (Table 1). Ithen made comparisons of effect sizesestimated from each analysis. ForANOVA designs, the three most common estimates of effect size are l,

2, and 0)2.Eta squared is calculatedas 11 = SSbetweenssto and is equivalentto R2 from regression (Camp andMaxwell 1983, Keppel 1991). Epsilonsquared = ssbetween((k-1) mswihn)/S

where k is the number of groups.Epsilon squared is equivalent toregression's adjusted or shrunken R2(Camp and Maxwell 1983, Keppel1991). Omega squared = ssbetween-(k-1)MS . .)/ssto + MSw. . .Omega squaredhas no analogue in regression analysis, but can be calculated as 1- (NIN -k -RI + 1)(1- R2), where Nis the totalnumber of observations (Camp and

Maxwell 1983). Iwill use R 2to referto this term.

That estimates of effect size derived from ANOVA and regressiondiffer is quite clear (Table 2). In general, ANOVA will produce larger effect sizes (more variance explained)thanwill regression for the same data.

The difference between analyses inthe total variance explained (R2 vs.7 2) is due to differences in the errorvariance. For regression, the totalerror is partitioned into pure error,

Table 1. A comparison of a one-way ANOVA and a regression using the samehypotheticaldata.

Source df Ss MS F PANOVA

Between groups 3 39,785.03 13,261.68 9.14 0.00Within groups 12 17,401.70 1,450.14Total 15 57,186.73

RegressionRegression 1 36,215.94 36,215.94 24.18 0.00Residual

Lack of fit 2 3,569.09 292.46 0.20 0.98Pure error 12 17,401.70 1,450.14

Table 2. Effect size estimates derived from the hypothetical ANOVAand regression analyses.

ANOVA Regressiona2= 0.70 R2 = 0.63E2= 0.62 Radj2=0.61c2-= 0.60 R 2= 0.53(

which is equivalent to the error termin theANOVA, plus error due to lack

of fit. This added error term reducesthe total "explained variance" for theregression, and thus the effect sizein comparison to ANOVA (Table 1).

Differences between the other effectsize metrics (F2vs. Radj2and w02vs. R.J2)are due both to difference in the overall amount of variation explained andto differences in the calculation of the

mean square variance (Ms). It is important to point out that no effect sizerepresents the true population effect.

As with the means calculated fromthe experiment, the estimate of an effect size is also a sample statistic ofthe "true" population value (Maxwellet al. 1981).

As an example of the problemsthat could be encountered from comparing effect sizes between regression and ANOVA, suppose that wewanted to know if the effect of fertilizer application on plant growth varied with latitude for a particular species. For the sake of this argument, weassume that there is an effect of fertilizer but it does not vary with latitude.

We find 10 studies, 5 ANOVAs and 5regressions, using the same experi

mental design. If, by chance, the fiveANOVAs were conducted at higherlatitudes and there is not too much between-experiment variation, we mighterroneously conclude that there is astronger relationship between fertilizer and growth at high latitudes.This report ismeant to be moreprophylactic than proscriptive. It is,of course, the experimenter's rerogative to analyze data as he or she seesfit.Both ANOVA and regressiondesigns will be used to analyze similartypes of data depending on the par

January 2000 75



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ticular objective of the research.When working with effect sizes, cau

tion must be exercised concerning theexperimental designs employed. Asthis study shows, it is inappropriate todirectly combine or compare effectsizes that rely on variance explainedfrom regression and ANOVA modelsunless researchers provide estimatesof lack of fit for regression models.Other effect size metrics that do notrely on variance explained are available (Cohen 1988, Hedges et al.1999). These metrics may be ofgreater use when comparing or combining effects between ANOVA andregressionmodels.Literature citedCamp, C. J., and S. E. Maxwell.

1983. A comparison of variousstrength of association measurescommonly used in gerontologicalresearch. Journal of Gerontology38:3-7.

Cohen, J. 1988. Statistical poweranalysis for the behavioral sciences. Second edition. LawrenceErlbaum Associates, Hillsdale,New Jersey, USA.

Dwyer, J. H. 1974. Analysis of variance and themagnitude of effects:a general approach. PsychologicalBulletin 81:731-737.Hedges, L. V., J.Gurevitch, and P. S.Curtis. 1999. The meta-analysis ofresponse ratios in experimentalecology. Ecology 80:1150-1156.

Keppel, G. 1991. Design and analysis: a researcher's handbook.Third edition. Prentice-Hall,

Englewood Cliffs, New Jersey,USA.Maxwell, S. E., C. J.Camp, and R. D.

Arvey. 1981. Measures of strengthof association: a comparative examination. Journal of AppliedPsychology 66:525-534.

Stephen F. MatterDepartment of Biological Sciences

University of AlbertaEdmonton, Alberta, Canada

T6G 2E9(780)492-0084

Fax: (780) 492-9234E-mail: [email protected]

TEACHINGABOUT LANDSCAPE ECOLOGY:DEVELOPINGAND APPLYINGNEUTRALMODELS

We offer a paper and pencil laboratory exercise to introduce landscapeecology to undergraduate natural science majors. This exercise has beenused since 1994 as part of a laboratory class for natural science majorsat the University of Wisconsin-Madison. It has been used in a similar classat the University of Tennessee.

Why teach landscape ecologyin a general ecology class?Curricula and popular textbooks for

general ecology courses tend to ignorelandscape ecology, instead dividingecology into three basic areas: populations, communities, and ecosystems(Begon et al. 1994, Krebs 1994, Stiling1996). Landscape ecology, rather thanunifying these areas, is mentioned onlyin introductory sections (Krohne 1998),or perhaps as part of population ecology (e.g., dispersion in heterogeneouslandscapes; icklefs 1997:307-312) orcommunity ecology (Brewer 1994:402-403). Beyond lack of coverage intextbooks, some instructors feel uncomfortable teaching a subject deemedtoo new, too computer-intensive, andalternately either too abstract or too descriptive.

We think landscape ecology belongs in general ecology courses because it is new, fresh, and exciting.One new general ecology textbook(Ecology by Dodson et al. 1998; seereviews; Ribbens 1999), also from the

University of Wisconsin-Madison, includes a major chapter on landscapeecology. To assist those who wouldlike to cover landscape ecology, wehave developed an appealing laboratory or group homework exercise thatrequires no computers, yet forces students to think quantitatively and interms of null hypotheses and neutralmodels. We believe that any topicthat bridges abstract and descriptivemodes of inquiryprovides an excellent opportunity to show studentshow ecology reallyworks.

Landscape ecology: a quickoverviewLandscape ecology is a discipline

defined by two characteristics: it typically studies ecological processes overlarge areas (e.g., Southern Appalachia,or all of Yellowstone National Park);and it explicitly studies the causes andeffects of spatial patterning (Turner1989). The region studied by a landscape ecologist may include variouscommunities or habitats, and she focuses on the totality of this variety.

Landscape ecologists often focuson intuitive attributes of landscapessuch as total area of a given habitattype, the number of patches, size distributions of patches, or edge-to-arearatios for a patch or set of patches.

After choosing an attribute, it is alsoessential to be able to measure andstudy it quantitatively across differentlandscapes and through time. Sincesome sort of meaningful "yardstick"is clearly required, some contemporarylandscape ecologists emphasize thedevelopment of neutral models. Such

models often involve simulation studies that generate spatial pattern randomly, or include only minimal directional effects due to spatial patterning,such as the north-south tendencies of

migrating birds. Neutral models otherwise lack effects due to ecologicalor biologically relevant processes suchas disturbance history, behaviors thataffect the speed of dispersal or migration, etc. (Gardner et al. 1989)Percolation theory as a neutralmodel

Percolation theory studies the properties of clusters, or patches, across atwo- or three-dimensional space. It isparticularly concerned with connectivity; other applications include materials science, where engineers mighttry to decide how much metal must beplated randomly across a surface sothat electricity can flow across it. Theengineer would want to have justenough gold to maintain conductivity, but perhaps not extra amounts,because of the cost.Ecologists are also interested inquestions thatdeal with connectivity.

76 Bulletinof theEcologicalSociety ofAmerica


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20168392 Regression, ANOVA, And Estimates of Effect Size 2000