In August,
2001
we examined the percentile bootstrap, this article demonstrates the
calculation of statistical power using the percentile bootstrap and
robust estimation. The GNU S language, "R" is used to implement
this procedure. R is a statistical programming environment that
is a clone of the S and S-Plus language developed at Lucent
Technologies. In the following document we illustrate the use of a GNU
Web interface to the R engine on the "rss" server, http://rss.acs.unt.edu/cgi-bin/R/Rprog.
This GNU Web interface is a derivative of the "Rcgi" Perl scripts
available for download from the CRAN website, http://www.cran.r-project.org
(the main "R" website). Scripts can be submitted
interactively, edited, and be re-submitted with changed parameters by
selecting the hypertext link buttons that appear below the
figures. For example, clicking the "Run Program" button
below samples 1000 random numbers from a normal distribution, then uses
nonparametric density estimation to fit a density curve to the
data. To view any text output, scroll to the bottom of the
browser window. To view the density curve, select the "Display
Graphic" link. The script can be edited and resubmitted by
changing the script in the form window and then selecting "Run
the R Program". Selecting the browser "back page" button will
return the reader to this document.

The Importance of Power and Effect Size

The techniques of statistical power analysis, effect size
estimation, and sample size estimation are important methods in
statistics and research methodology (Cohen, 1988).Briefly, the power of a statistical test is the probability of
rejecting the null hypothesis given that the alternate hypothesis is
true; the effect size is the degree to which the null hypothesis is
false in relation to the alternate hypothesis; type II error is the
probability of failing to reject the null hypothesis when it needs to
be rejected in favor of the alternate hypothesis; and type I error is
the probability of incorrectly rejecting the null hypothesis. Proper sample size estimation allows one to
achieve an acceptable level of power for a statistical test, thereby
setting the type II error at a pre-specified level. Historically, for the social sciences, neglect of
these topics have led to a long standing controversy surrounding the
interpretation of statistical tests in the research community (Cohen,
1993).Following Jacob Cohen's (1965,
1962) initial work on the power of statistical tests in behavioral
research, many researchers and authors have pointed out the importance
of statistical power analysis. Textbooks
and articles have appeared that provide tables of power and sample
sizes (Cohen, 1988). Additionally,
several computer programs which perform exact power analysis assuming
normal theory have appeared (Bradley, Helmstreet, & Zeigenhagen,
1992; Faul & Erdfelder, 1992). Despite
these recommendations, and availability of resources for power
calculation, Cohen has argued that researchers continue to ignore power
analysis (Cohen, 1994).Sedlmeier and
Gigerenzer, G. (1989) reported a power review of the 1984 volume of the
Journal of Abnormal Psychology showing that there was not any marked
improvement in the power of statistical tests appearing in the
literature. As recent as 1997,a methodological study has found that the power
of statistical tests are not taken into account by researchers and that
they continue to run a high risk of type II error (Clark-Carter, 1997). Cohen (1988) has suggested that the neglect of power analysis
exemplifies the slow movement of methodological advance. Cohen has also suggested a lack of consciousness regarding
effect size, coupled with an overriding concern with the accompanying
"p" value (Cohen, 1992; 1994). Despite
this unawareness on the part of editors and researchers, there has been
a recent shift in the editorial practices of the American Psychological
Association (APA, 1994). The manual notes
that, "Neither of the two types of probability values reflects the
importance or magnitude of an effect because both depend on sample size -- you are encouraged to provide effect-size information (APA,
1994, p.18)." Following these editorial changes,
in 1996 APA established a task force that, among other goals,reexamined the role of statistical hypothesis
testing in the methodological practices of Psychology (http://www.apa.org/science/bsaweb-tfsi.html). The Task Force on
Statistical Inference (TFSI) met twice in two years after which a
preliminary report was circulated that indicated its intention to
examine issues beyond null hypothesis significance testing. After the second meeting, the task force
recommended several possibilities for further action, one of which was
to revise the statistical sections of the American Psychological
Association Publication Manual (APA, 1994). A report was generated following
those meetings (http://www.loyola.edu/library/ref/articles/Wilkinson.pdf). Neglect of power not only decreases the recognition
of interesting effects (type II error), but it also has a negative
effect on the ability of researchers to establish statistical consensus
through replication.
Ottenbacher (1996) points out that, "The apparently paradoxical
conclusion is that the more often we are well guided by theory and
prior observation, but conduct a low power study, the more we decrease
the probability of replication... The responsible investigator must be
concerned with statistical power. A
concern with power, however, cannot end with its calculation.Because the ability to detect treatments
must be optimized, the responsible scientist must also be concerned
with factors that determine effect size". Most treatments of
power analysis deal with the calculation of power for parametric
statistics where normal theory assumptions are required (e.g. t-test,
F-tests). The calculation of power for
robust statistics or nonstandard nonparametric statistics are not
addressed at a practical level. For
example, Cohen's book on power analysis (1988) concentrates mainly on
ANOVA and regression models and some standard nonparametric tests such
as the chi-square test. What is not
addressed is how violations of normality assumptions affect power
estimates. The bootstrap technique can be useful for exploring
how statistical power is affected by non-normality.

Estimating Power with the Bootstrap

Beran (1986) provided
mathematical and simulation results that show that a statistical test
for a null hypothesis can be constructed by bootstrapping the null
distribution for the test statistic.Beran
also proved that the power of the test against an alternative can
itself be estimated by simulation. The
uniform consistency of these simulated power functions is the main
result of Beran's mathematical proof. Additionally,
Beran performed a limited numerical study of the univariate bootstrap
t-test and the associated power function. The
null hypothesis value of the mean difference was zero; the nominal test
level alpha was .05; and the sample size was 20. The
critical value of the bootstrap test was obtained from the simulated
null distribution using 200 bootstrap samples. The
power of the bootstrap t-test was approximated by Monte Carlo
simulation using 1000 standard normal samples. Thus,
the simulation used 200 bootstrap samples for the critical value loop
and 1000 bootstrap samples for the outer loop. Even
at sample size 20, Beran found that the power function of the bootstrap
test is almost indistinguishable from that of the classical t-test. Yuan
(2001) applied Beran's general theory of re-sampling to a covariance
structure analysis framework. Yuan found that, with several data sets,
robust estimators could be combined with the bootstrap to allow
researchers to be in the position of finding an almost optimal
procedure for evaluating covariance structure models (Yuan, 2001). Additionally, based on Beran's results, Yuan
provided an algorithm for determining sample size for a given level of
power. A great advantage of calculating the
critical value from the simulated null sampling distribution is that
the empirical estimate of the critical value is asymptotically
consistent with the true population value, and no assumptions are made
regarding the shape of the null sampling distribution. Consequently, each statistical test (i.e. mean difference
test) that is performed on a simulated bootstrap sample, is compared to
this critical value, and since the critical value was constructed from
the observed data (under the assumption of the null hypothesis), and
according to Beran (1986), is a consistent estimate of the population
critical value, we can expect proper coverage of the mean difference
statistic with the bootstrap confidence intervals, based on this
critical value. This is essential for
calculating power estimates of test statistics whose sampling
distributions are unknown (under the null or the alternate hypothesis),
because of violations of assumptions (i.e. normality) or mathematical
intractability. Re-sampling under the null
hypothesis seems to be the preferred approach in calculating
probability values for an observed test statistic (Hall and Wilson,
1991, p. 757). Hall and Wilson give the
following guidelines for bootstrap testing in univariate situations,
"The first guideline says that care should be taken to ensure that even
if the data might be drawn from a population that fails to satisfy Ho,
re-sampling should be done in
a way that reflects Ho" (Hall and Wilson, 1991). Bootstrapping under the null hypothesis, for a two group
difference test of means, would involve mean centering each group
around their respective group means, and sampling with replacement from
the whole collection of mean centered scores to produce two new groups
of scores (two bootstrap samples) which reflect group differences when
the null hypothesis is true (Westfall and Young, 1993, p. 35-36).
Furthermore, if one is bootstrapping measures of location other than
the mean, one must be sure to create a bootstrap population where the
observations are centered around that alternative measure of location
(Westfall and Young, 1993, p. 143-144). For
example, if one is using a median, or an M-estimate as a measure of
location, then one would center around that measure to insure that the
null hypotheses are true in the bootstrappopulation.

The
General Bootstrap Power Simulation Algorithm

Beran's
(1986) simulation algorithm is presented as a sequence of steps (for a
two-sided difference of location):

Step
1 - Generate the bootstrap null distribution using bootstrap
re-sampling:A) Re-center the data
vector x and the data vector y around their respective measures of
location.B)Stack
the data vectors x and y into a single vector, z. Vector
z is now considered the in-hand, proxy population. C) Re-sample with replacement from vector z to produce a
bootstrap sample for group x1 with length of the original group x. Repeat this re-sampling to produce a group y1.D)Calculate a
measure of location for both groups (e.g. mean, M-estimate, trimmed
mean, or Windsorized mean).E) Subtract
the two location measures.This difference
is one bootstrap sample which comprises the empirical null sampling
distribution. F)Repeat
steps C-E a large number of times to generate the empirical null
distribution (suggestions vary widely, 1000 seems to be a sufficient
number of bootstrap samples; one might resample 10,000 bootstrap
samples to be conservative). The empirical
null distribution will be centered on zero. Step
2. - Calculate the critical scores that correspond to the 2.5^{th}
and 97.5^{th} critical alpha regions under the empirical null
distribution: The critical
scores are the scores that correspond to the 2.5^{th} and 97.5^{th}
percentiles of the empirical null distribution. We
can calculate the percentiles using the following approach:
round((.05/2)x(#bootstrap samples)) for lower percentile; and
round((1-(.05/2))x(#bootstrap samples)). Next,
locate the scores that correspond to those percentiles.
Step 3. - Generate the bootstrap alternative distribution:
A) Re-sample with replacement from vector x with replacement to
generate a bootstrap sample, x1, with length of original vector x.B) Re-sample with replacement from vector y
with replacement to generate a bootstrap sample, y1, with length of
original vector y. C) Calculate measures of location for both bootstrap
samples x1 and y1. D) Subtract the two
measures of location.This is one bootstap
difference, and represents the difference between measures of location
under the empirical alternate distribution. This
empirical distribution is centered on the population difference under
the alternate hypothesis. Step 4. - Calculate the
empirical power of the statistical test: A) Using the
upper and lower critical scores for the empirical null hypothesis
calculated in step 2., Calculate the number of difference scores in the
empirical alternative sampling distribution that are as or more extreme
than the critical scores under the null distribution.B) Take the count tallied in step A) and divide by the total
number of bootstrap samples.This value is
the empirical power for the statistical test that tests differences
between groups using whatever location measure is under consideration.

The Data Set

Doksum & Sievers (1976) report data on a
study designed to assess the effects of ozone on weight gain in rats.The experimental group consisted of 22
seventy-day old rats kept in an ozone environment for 7 days (group y). The control group consisted of 23 rats of the
same age (group x), and were kept in an ozone-free environment. Weight
gain is measured in grams. The following R code produces quantile-quantile plots and non-parametric density plots of the two
groups of data:

Results

The following output is produced. Group x (control group) is
in the upper left panel, and group y (experimental group) is in the
upper right panel. Both groups show substantial deviations away
from normality. Deviations away from the straight line indicate
deviations away from normality. In the lower panel,
non-parametric density estimates of both groups are plotted on the same
graph. The more peaked, narrower density function is the control
group, and the less peaked, more dispersed density function is
experimental group.

Using GNU S ("R") to Calculate
Statistical Power Using the Bootstrap and Robust Estimation

In this section, we use
M-estimation as measures of location for the control and experimental
group. Bootstrap p-values, confidence intervals and
power for the difference between the M-estimates are calculated.
Additionally, a classical t-test is calculated for comparison:

Results

The
following results are produced:

Welch
Independent Two Sample t-test:

data: x and y
t = 2.4585, df = 32.909, p-value
= 0.01938 alternative hypothesis: true difference in means is
not equal to 0 95 percent confidence interval: (1.964178,
20.826336) sample estimates: mean of x mean of y 22.40435 11.00909 Bootstrap statistics based on
difference between M-estimates:

Bootstrap Empirical
P-value >

pvalue.empirical [1] 0

Bootstrap Empirical Power > power.twotail [1] 0.9331104

The M-estimate
confidence intervals are much narrower that the classical confidence
intervals. With 399 bootstrap samples, not one bootstrap sample
exceeded the observed difference, giving a p value less than
1/399=.0025. The non-parametric bootstrap power for the
difference in M-estimates was .933.

Conclusions

The bootstrap and robust estimation
provide a method for improving statistical power whenever the data can
be described as having heavy-tailed distributions. Furthermore,
an estimate of power based on the percentile bootstrap is
non-parametric, and does not depend on normal theory assumptions. Bootstrap power estimation is a general methodology that can be used to
calculate power for many different kinds of statistical estimators
(e.g. mean, median, or
M-estimates).

References

American Psychological Association. (1994).Publication
manual of the American Psychological Association (4^{th} ed.). Washington, DC: Author.

Beran, R
(1986). Simulated Power Functions.The
Annals of Statistics, 14(1), 151-173.

Bradley, D. R., R. L.
Helmstreet, and S. T. Zeigenhagen. 1992. A simulation laboratory for
statistics. Behaviour Research Methods, Instruments, and Computers
24: 190-204.

Clark-Carter, D. (1997). The account taken of statistical power in research published in
the British Journal of Psychology.British
Journal of Psychology, 88, 71-83.

Cohen, J. (1995). The
earth is round (p < .05): Rejoinder. American Psychologist,
49(12), 1103.

Cohen, J. (1994). The earth is round (p <
.05). American Psychologist, 49(12), 997-1003.

Cohen J.
(1992) A power primer. Psychological Bulletin, 112, 155-159.

Cohen, J. (1990). Things I have learned (So Far). American
Psychologist, 45, 1304-1312.

Cohen, J. 1988. Statistical
Power Analysis for the Behavioral Sciences, 2nd Edition. Lawrence
Erlbaum Associates, Inc., Hillsdale, New Jersey.

Cohen, J.
(1965). Some statistical issues in psychological research. In B. B.
Wolman (Ed.), Handbook of clinical psychology (pp. 95-121). New
York: McGraw-Hill.

Cohen, J. (1962). The statistical power of
abnormal-social psychological research: A review. Journal of
Abnormal and Social Psychology, 65, 145-153.

Doksum, K.A. & Sievers, G.L. (1976).
Plotting with confidence: graphical comparisons of two
populations. Biometrika 63, 421-434.

Ottenbacher,
K.J. (1996).The Power of Replications and
Replications of Power. The American
Statistician, 50(3), 271-275.

Hall P, Wilson SR (1991). Two
guidelines for bootstrap hypothesis testing. Biometrics, 47,
757-762.

Sedlmeier, P. & Gigerenzer, G. (1989). Do studies
of statistical power have an effect on the power of studies? Psychological
Bulletin, 105, 309-316.

Yuan, K. (2001). Bootstrap Approach
to inference and power analysis based on three test statistics for
covariance structure models. Under review.

Westfall, P.H.
(1993). Re-sampling based multiple testing: examples & methods
for p-Value adjustment. Wiley, New York.

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