Biologist’s Toolbox

Review of UnifyPow.sas

Statistical Power, Sample Sizes, and the Software to Calculate Them Easily UnifyPow.sas is Ralph O'Brien's freeware module that "inspired" SAS Institute to produce PROCs POWER and GLMPOWER. Work on UnifyPow has stopped, but it continues to provide functionality that PROCs POWER and GLMPOWER do not. It can be downloaded at www.Nvestigate.org, along with instructions and examples on use.

KEITH P. LEWIS

The following shortcourse notes provide the primary documentation, which has served thousands for more than a decade. UnifyPow's SAS code provides key technical references.

Many authors have advocated the use of power analysis for designing more sensitive and efficient ecological studies. However, few manuscripts report using power analysis for improving the study design, perhaps because of the wide range of tests routinely used by most ecologists and the limitations of most power analysis software. UnifyPow, an excellent freeware macro that runs within SAS, easily and accurately calculates power or sample sizes for a wide range of statistical tests for a broad range of statistical models. Despite its potential contribution to the field, UnifyPow is virtually unused by ecologists. Perhaps this is because UnifyPow was developed by medical statisticians or because there is no manual, although there is extensive online documentation. I briefly review the flexibility and breadth of UnifyPow; describe how it works; and, using ecological examples, demonstrate how to use it to calculate power for an extremely wide range of statistical tests. Keywords: power analysis, type II error, sample size, software, statistics

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tatistics is an integral tool of modern ecology. Virtually every study reports type I error rates (α) and the statistical significance of tests (p values), but few report the statistical power that the study was designed to provide. Power is defined as 1 – ß, where ß is the probability of committing a type II error (i.e., a false negative). Power is therefore the probability of rejecting the null hypothesis given that some specific alternative hypothesis is true (i.e., the sensitivity of a test to detect an effect; Steidl et al. 1997, Steidl and Thomas 2001). Many authors have thoroughly discussed the concepts of statistical power, advocating the use of power analyses for designing ecological studies (e.g., Steidl and Thomas 2001). Statistical hypothesis testing is based on four interrelated components: power, sample size, α, and effect size, where effect size is the absolute difference between populations in the parameter of interest, scaled by the variance (Steidl and Thomas 2001). Essentially, effect size is the degree of change in the parameter of interest caused by a particular treatment. These components are interrelated, because as effect size, sample size, and α increase, so does the power of the study (Steidl et al. 1997). There are two main types of power analyses: prospective and retrospective. By conducting a power analysis before implementing a study (i.e., performing a prospective power analysis), an investigator can estimate a biologically meaningful effect size, determine the sample size required to produce significant results, determine the precision of the study, and select an efficient study design with adequate statistical power. www.biosciencemag.org

Prospective power analyses force the researcher to explore the relationships among sample size, effect size, and type I error, and how these factors influence the probability of the study to detect an effect. Prospective power analysis can also lead the researcher, often with the help of a statistician, to select a study design that is more sensitive for detecting the effect of interest (O’Brien and Muller 1993). Calculating power after a nonsignificant result is obtained (i.e, retrospective power analysis) has been advocated by many authors (e.g., Peterman 1990) and is often requested by reviewers, but is fundamentally flawed and should not be employed (Hoenig and Heisey 2001, but see Steidl and Thomas 2001 on testing hypotheses with confidence intervals). It is difficult to assess how frequently prospective power analyses are conducted in ecological research. It is possible that many researchers calculate power before designing their studies and simply do not report this exercise, since it is not essential to the replication of the experiment. Alternatively, although ecologists are aware of the need to calculate power, they may believe the computer software to do so is highly limited or unavailable (Thomas and Krebs 1997), and thus may not be using power analyses to design more sensitive and efficient studies. Keith P. Lewis (e-mail: [email protected]) is a postdoctoral fellow in the Department of Biology at Memorial University of Newfoundland, St. John’s, Newfoundland A1C 3X9, Canada. His research interests include interactions among invasive species in the boreal forest, nest predation, and climate change. © 2006 American Institute of Biological Sciences.

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Biologist’s Toolbox Thomas and Krebs (1997) conducted an informative and comprehensive review of software packages that perform power and sample size analyses (hereafter, “power analyses” will refer to both prospective power and sample size analyses). Software packages for conducting power analyses are available as general-purpose statistical software with built-in power analysis capabilities or as stand-alone software that calculates only power. Unfortunately, general-purpose software usually calculates power for just a limited range of tests (Thomas and Krebs 1997). As a whole, the stand-alone software packages cover a wide range of tests, but few packages cover the range of tests that many ecologists regularly use. For example, only 4 of 14 packages reviewed by Thomas and Krebs (1997) calculate power for a three-way ANOVA (analysis of variance), and only 1 of 14 calculates power for one-way ANOVA with contrasts. Packages that can handle more complex ANOVA designs often do not calculate power for many tests of categorical data (e.g., goodness-of-fit tests). Thomas and Krebs (1997) focus their review on the capabilities of several of the best software packages, but apparently none of these computes power for general and generalized linear models (see also O’Brien 1999). In addition, many stand-alone packages often incorrectly calculate power in retrospective power analyses using the actual effect size found in the test (Thomas and Krebs 1997). Because ecologists are using increasingly sophisticated statistical models in their studies (e.g., Schneider 1992, Trexler and Travis 1993) to address conservation and management issues, they need to have an affordable, accessible, accurate, flexible, powerful, user-friendly software package to conduct power analyses. To this end, I suggest UnifyPow, a program that fulfills these criteria (O’Brien 1998a). UnifyPow is available as a freeware macro that runs within SAS (SAS Institute Inc. 1996a) and accurately calculates power or sample size for a wide range of statistical tests for a broad range of statistical models. However, despite its potential contribution, UnifyPow is virtually unused by ecologists. Perhaps ecologists are unaware of this software because it was developed by medical statisticians. (Although UnifyPow falls outside the scope of their article, Thomas and Krebs [1997] do mention it briefly, saying only that it is programmable.) In addition, there is no manual; the online documents that describe UnifyPow are conference-style slide presentations (e.g., O’Brien 1998b). While hundreds of people have learned UnifyPow from these presentations (Ralph G. O’Brien, Department of Biostatistics and Epidemiology, The Cleveland Clinic Foundation, Cleveland, Ohio, personal communication, 22 May 2003), the code may initially appear daunting to those not well versed in power analyses or in command-line driven programs such as SAS (SAS Institute Inc. 1996a). Furthermore, the examples are medical and may be difficult to grasp for novice users with an ecological background. In this article, I briefly review the flexibility and breadth of UnifyPow, describe how it works, and demonstrate, using ecological examples, how to calculate power for an extremely wide range of tests. 608 BioScience • July 2006 / Vol. 56 No. 7

The breadth of UnifyPow The capabilities of UnifyPow are summarized in O’Brien (1998a). Briefly, UnifyPow is a freeware SAS module/macro that can be implemented with relatively simple SAS syntax, and can run in any environment that supports the base SAS System (SAS Institute Inc. 1996a). UnifyPow can conduct analyses of power and sample size on a wide range of nonparametric and parametric tests, account for unbalanced study designs, and report one- and two-tailed tests where appropriate. Power analyses can be performed on many tests, including single proportions, two independent proportions, goodness of fit to multinomial distribution, R x C contingency tables, t test for two samples and matched pairs, WilcoxonMann-Whitney, Wilcoxon signed rank, Pearson correlations, ANOVA, ordinary least-squares regression, and many others (O’Brien 1998a, 1998b). More important, UnifyPow can calculate power for many general (e.g., ANOVA, ANCOVA [analysis of covariance], and linear regression) or generalized linear (e.g., logistic regression and log-linear) models (Shieh and O’Brien 1998). These capabilities give UnifyPow outstanding breadth and flexibility.

UnifyPow: The basics For a complete discussion of the equations used in power analyses for different tests, and of the rationale and approach used in UnifyPow, see O’Brien and Muller (1993) and O’Brien (1998b). To operate UnifyPow, simply download the module from the UnifyPow home page (www.bio.ri.ccf.org/power.html) to a convenient directory on your computer. For many of the simpler and more common tests (e.g., t tests, one-way ANOVA), UnifyPow calculates power in a simple one-step process, but for more complex tests (e.g., twoway ANOVA, logistic regression), a two-step process is required. (For more information on choosing the one- or twostep process, see O’Brien and Muller 1993, O’Brien 1998b.) The methods and rationale for these two different processes are explained below (codes for these and other tests are at www.mun.ca/biology/klewis/appendix1.pdf).

Calculating power with UnifyPow To demonstrate how to use UnifyPow, I present two examples for both the one- and the two-step processes. Data for the t test, R x C contingency table, and ANOVA are from Sokal and Rohlf (1995; see tables 1–3). I use my own data for the logistic regression example (table 4). When conducting a power analysis, the various inputs (parameters) can come from pilot data, from similar studies in the literature, or from a range of possible values using one’s own best scientific judgment. For the purposes of this paper, these data sets (tables 1–4) should be considered as pilot studies, and the power analyses are being used prospectively to design more comprehensive studies. It would be inappropriate to calculate power using the effect size obtained in these studies in a retrospective power analysis (see Hoenig and Heisey 2001, Steidl and Thomas 2001). To simulate different scenarios that a researcher conducting a power www.biosciencemag.org

Biologist’s Toolbox Table 1. Average age in days at the beginning of reproduction in two series of Daphnia longispina. Series I

Mean SD Pooled SD

7.2 7.1 9.1 7.2 7.3 7.2 7.5 7.51 0.71 0.65

Series II

Table 3. Width (in micrometers) of the scutum of larvae of the tick Haemaphysalis leporispalustris in samples of four cottontail rabbits. Host

8.8 7.5 7.7 7.6 7.4 6.7 7.2 7.56 0.64

1

2

380 376 360 368 372 366 374 382

350 356 358 376 338 342 366 350 344 364

372.25 8

354.40 10

SD, standard deviation. Source: Sokal and Rohlf 1995, box 9.5. Mean n

Table 2. Frequencies of color patterns of a species of tiger beetle in different seasons. Season

Bright red

Not bright red

Weight

Early spring Late spring Early summer Late summer

29 (0.72) 273 (0.59) 8 (0.21) 64 (0.5)

11 (0.28) 191 (0.41) 31 (0.79) 64 (0.5)

0.06 0.69 0.06 0.19

Note: The proportion of each cell in a row is given in parentheses. Weight is the proportion of each row in the total sample. Source: Sokal and Rohlf 1995, box 17.8.

3 354 360 362 352 366 372 362 344 342 358 351 348 348 355.31 13

4 376 344 342 372 374 360

361.33 6

Source: Sokal and Rohlf 1995, box 9.1.

Table 4. The proportion of depredated artificial nests surrounded by two different types of exclosures on the ground and in trees. Cage height

Exclosure

Number of depredated nests

Total nests

Ground Ground Tree Tree

Control Treatment Control Treatment

11 3 7 2

21 19 21 20

analysis might explore, and to demonstrate the capabilities of UnifyPow, I have included different effect sizes, sample sizes, levels of type I error, and one- or two-tailed tests when appropriate. After each example, I offer a simple interpretation and suggest possible improvements to the study design. To make the UnifyPow code easier to understand, I provide the names of the various input parameters as they appear in the program in parentheses after their explanations in the text. The results are presented as UnifyPow output in tables 5–8.

The results show that an enormous sample size would be required to achieve adequate statistical power with the observed effect size, but when the standard deviation is reduced (i.e., the effect size is increased), a much smaller sample size would be required. Increasing the type I error rate and choosing a more sensitive study design would decrease the required sample size to some degree (table 5). Although a paired t test is not possible with this example, it is often more sensitive than an independent t test because it controls for more error variance and thus has greater power.

Power analysis: The one-step process

Calculating power for a G-test. Calculating power for an R x C G-test (or contingency table; table 2) is somewhat different from the calculation for the t test. First, calculate the relative proportion of each cell in each row of the contingency table (2WayContTable; these values are presented in table 2 and at www.mun.ca/biology/klewis/appendix1.pdf). Second, determine the proportion of each row to the total sample size (weight). Then determine the appropriate value for the type I error (alpha) and the desired sample size or power (NTotal or power). The results indicate that the sample sizes in future studies would not have to be larger in order to achieve a very high level of power (table 6).

Power analyses for simple tests can be conducted using the one-step process. Below, I demonstrate how to perform these analyses for t tests and R x C G-tests. Calculating power for a t test. To calculate power for a t test (table 1), first calculate or predict the means (mu) of the two groups of interest and the pooled sample standard deviation (sd). Several different standard deviations can be entered simultaneously in this step (e.g., 0.65, 0.1). Second, determine one or more appropriate values for the type I error (alpha, usually 0.05 or 0.1). Third, determine the ratio of the sample size of your treatments (weight). Finally, depending on the parameter of interest, enter one or more desired sample sizes or levels of power (NTotal or power). This format is used for a wide range of tests in UnifyPow and for the second step in more complex tests. www.biosciencemag.org

Power analysis in two steps Power for simple general linear models like one-way ANOVA with contrasts or ordinary least-squares regression can be July 2006 / Vol. 56 No. 7 • BioScience 609

Biologist’s Toolbox Table 5. The influence of alpha and standard deviation on the sample size required (total N) to attain three levels of power for a one- or two-tailed t test. Standard deviation 0.65

0.1

Minimum power Alpha 0.05 0.01

Type 2-tail 1-tail 2-tail 1-tail

t t t t

Minimum power

.700

.800

.900

.700

.800

.900

Total N

Total N

Total N

Total N

Total N

Total N

4176 3184 6502 5498

5314 4184 7906 6792

7112 5798 10070 8810

102 78 158 134

128 102 192 164

172 140 242 212

Note: The formatting of this table corresponds to the presentation of UnifyPow output.

Table 6. The influence of alpha and test type on the sample size required (total N) for an R x C G-test to attain two levels of power.

models, and Shieh and O’Brien 1998 for generalized linear models). This flexibility and breadth is the real strength and value of UnifyPow.

Testing group X category independence Ho [null hypothesis]: All groups have same probability distribution for outcome R x C G-test from Sokal and Rohlf (1995), box 17.2 Scenario: {0.72 0.28} v. {0.59 0.41} v. {0.21 0.79} v. {0.5 0.5}

ANOVA. To create an exemplary data set for an ANOVA, simply calculate the conjectured means of the different treatments and the sample sizes that reflect the cell weights (table 3). Then calculate the sums of squares using the SAS procedure for general Alpha linear models, PROC GLM (SAS Institute 1996b). The 0.05 0.01 Minimum power Minimum power adjusted sums of squares (Type III SS) is the value of 0.900 0.950 0.900 0.950 the SSHe. After calculating the SSHe, the procedure Method Statistic Total N Total N Total N Total N is very similar to the t test, but with several additional inputs (www.mun.ca/biology/klewis/appendix1.pdf). Ordinary Pearson Chi-square 400 500 500 600 First, enter the type of exemplary data set, which is Likelihood ratio Chi-square 400 500 500 600 SSH for general linear models (exemplary SSH). SecNote: The formatting of this table corresponds to the presentation of UnifyPow ond, determine the number of parameters in the output. model (NumParms), and input the sample size of the exemplary data set (Nexemplary). Third, determine calculated in a one-step process similar to that for a t test the standard deviation (sd), the desired level of type I error (O’Brien 1998b). However, for more complex analyses, it is (alpha), and the desired sample size or power (NTotal or necessary to use the two-step process mentioned above to hanpower). Finally, enter the degrees of freedom and the value of dle the noncentrality parameter, an essential but often diffithe SSHe generated in the first step (effects). cult value to calculate in all power analyses. The noncentrality The results indicate that a study with this sample size parameter incorporates both the effect size and the weights would have adequate power for the two lower standard deof the different treatments (i.e., the structure of the study deviations but not for the higher one. Increasing the sample size sign), and becomes more difficult to calculate with more and the type I error rate would produce a modest increase in complex models. Fortunately, this complexity is relatively power (table 7). For simplicity, the power analysis in this exeasily handled by creating an exemplary data set, a data set with ample follows those data in table 3 by using unequal sample sample estimates that are identical to the population estimates sizes, but power is maximized when the study design is bal(i.e., the estimates conform exactly to the assumed model anced (O’Brien and Muller 1993). [Shieh and O’Brien 1998]; see tables 3 and 4 and www.mun.ca/ biology/klewis/appendix1.pdf). The exemplary data set is Logistic regression. Shieh and O’Brien (1998) give several used to calculate the exemplary sums of squares of the hyexamples of how to generate an exemplary data set with pothesis (SSHe), a value that subsumes the complex part of SAS code (see also Shieh 2000). A less elegant, but less computing the noncentrality parameter. The SSHe is calcuprogramming-intensive, method is simply to create an exlated by performing the appropriate test (see “ANOVA” and emplary data set as in the ANOVA example. Then calculate “Logistic regression,” below) on the exemplary data set. The the SSHe, which is the Type III analysis in PROC GENMOD SSHe allows power analyses to be performed relatively easily (SAS Institute Inc. 1996c). The format is similar to the on an extremely broad range of models by unifying the nongeneral linear model, except that exemplary chi-squared centrality parameter across all cases of the general and gen(written as chi**2) substitutes for exemplary SSH, and the lines eralized linear model (see O’Brien and Muller 1993 for a for number of parameters (NUMPARMS) and standard complete explanation of this approach for general linear deviation (sd) are not needed. Finally, enter the degrees of 610 BioScience • July 2006 / Vol. 56 No. 7

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Biologist’s Toolbox Table 7. The influence of alpha and standard deviation on the sample size required (total N) to attain three levels of power for a one-way ANOVA (see table 3). Sokal and Rohlf 1995, box 9.1–1-way ANOVA with unequal sample sizes Scenario: exemplary SSH

35 Power

45 Power

Standard deviation 12.45 Total N 35 45 Power Power

0.999 0.999

0.999 0.999

0.748 0.846

5 Total N

Test Power of 1-way ANOVA

Alpha 0.05 0.1

Type Regular F Regular F

0.867 0.922

20 Total N 35 Power

45 Power

0.340 0.473

0.440 0.576

Note: The formatting of this table corresponds to the presentation of UnifyPow output.

Table 8. The influence of alpha on the power of a two-way logistic regression to detect an effect for three different sample sizes at two levels of alpha (see table 4). Simple 2-way logistic regression Scenario: Exemplary chi**2 Alpha

Effect Ground vs tree

Statistic 2-tail Z

0.05

0.1

Total N

Total N

81 Power

135 Power

225 Power

81 Power

135 Power

225 Power

0.206

0.311

0.473

0.308

0.430

0.598

Note: The formatting of this table corresponds to the presentation of UnifyPow output.

freedom and the value of the SSHe generated in the first step (effects). The results indicate that a study with these sample sizes and type I error rates would not have adequate power (table 8). Improvements in future studies could be made by relaxing type I error rates, increasing sample sizes, or perhaps using a randomized block design to help account for among-site variation (the data in table 4 are pooled among sites).

UnifyPow: The benefits and some cautions Power and sample size analyses are critical for planning efficient, sensitive studies. Many authors have urged a more widespread use of power analyses in ecological studies (e.g., Steidl and Thomas 2001), and Thomas and Krebs (1997) review many programs that calculate power. Still, few authors report conducting power analyses. This may be due to the disparity between the tests covered by many of the available packages and those routinely used by ecologists. To rectify this situation, I recommend UnifyPow. UnifyPow has numerous advantages. It is a freely available macro that runs within the widely used, commercially available SAS system. The output is highly informative, and the tables are easy to read (tables 5–8). UnifyPow has outstanding breadth and flexibility, and can calculate power for an extremely wide range of tests routinely used by ecologists. While extremely powerful, UnifyPow is by no means a panacea and should not be used as a “black box” for several www.biosciencemag.org

reasons. First, a good understanding of statistics, type I and type II errors, effect size, and sample size is required to use this program (see Steidl et al. 1997, Steidl and Thomas 2001). Second, a careful review of the literature is often required to obtain the required parameters when pilot studies are not possible, and many authors do not report their results in sufficient detail for others to use them in power calculations. This leads to guesswork in the power analyses. Third, I urge that all who use this or any other power analysis package carefully examine all input parameters, since errors at this stage will result in false results and spurious conclusions. Even when the rudiments of the program have been mastered, experience (balancing desired power and logistics, comparing potential study designs, and trial and error) is vital to planning efficient, sensitive studies. Consulting a statistician is often appropriate at this stage. Moreover, it is unlikely that one program will satisfy every researcher’s needs, as ecologists use a wide range of statistical tests (O’Brien 1999). If simple statistical tests are routinely used, other programs may be adequate for calculating power. For specialized needs, such as survival analysis or determining changes in species abundance over time, other software may be more appropriate (e.g., Monitor [Gibbs 1995], TRENDS [Gerrodette 1993]). Furthermore, traditionally SAS, and therefore UnifyPow, must be programmed and does not run in a graphical user interface environment; thus, some will find it more difficult than menu-driven programs. July 2006 / Vol. 56 No. 7 • BioScience 611

Biologist’s Toolbox However, newer versions of SAS have a more user-friendly interface. Finally, UnifyPow was developed by medical researchers, and there is no manual, although there is substantial online documentation. I hope that the ecological examples, along with the SAS code, will provide ecologists with the knowledge of UnifyPow and the ability to employ it in their own studies. SAS currently provides some tools for computing power and sample size in the SAS 9.1 release. PROC POWER calculates power for simpler analyses such as t tests, one-way ANOVA, and multiple regression (SAS Institute Inc. 2003a). PROC GLMPOWER calculates power for models in PROC GLM but cannot handle continuous explanatory variables (SAS Institute Inc. 2003b). PROC GLMPOWER is very similar to UnifyPow, using the concept of exemplary data sets. However, while these PROCs have many advantages over UnifyPow, including the ability to calculate power for confidence intervals and two-sided equivalency testing, they lack the breadth of UnifyPow in terms of the number of statistical tests for which they can calculate power.

Conclusion Statistics are integral to modern ecology, and we should consider type II errors along with type I errors (Steidl et al. 1997). I believe that most ecologists would calculate power if there were a relatively straightforward way to do so; many researchers may have failed to analyze the power of their study designs because they lacked a flexible software package that covers the wide range of statistical tests routinely used by ecologists. This software is readily available in UnifyPow, and knowledge of its existence, and how to use it, should facilitate a greater use of power analysis in ecology.

Acknowledgments Greg Robertson and Francis Wiese were invaluable to my understanding of the programming aspects of UnifyPow. Ellen Jedry, Stephen Lewis, Bill Montevecchi, Ralph O’Brien, Mark Renkawitz, Dave Schneider, Len Thomas, and Susan Walling reviewed drafts of this manuscript. I thank Ralph O’Brien for his help and for continuing to make UnifyPow free and accessible.

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