“I’ve Got the Power”: How Anyone Can Do a Power Analysis...

“I’ve Got the Power”: How Anyone Can Do a Power Analysis of Any Type of Study Using Simulation

Sean P. Lane Erin P. Hennes Tessa V. West

University of Missouri Purdue University New York University

San Diego, CA SPSP January 29, 2016

Power Analysis Using Simulation 1

https://youtu.be/_BRv9wGf5pk

https://youtu.be/_BRv9wGf5pk

Part IGeneral Approaches to Power Analysis


What is Power?

• Power: The probability of rejecting a false null hypothesis• The inverse of the probability of a Type II Error1 – β

• 1 – β

OR • The chance that your study will “work” given that your hypothesis

is correctDecision H0 True H0 False

Reject H0 Type I Errorp = α

Correct Decision p = 1 - β

Retain H0 Correct Decision p = 1 - α

Type II Error p = β


POWER

Dance of the p Values


Simulation example Cumming, G. (2013) Intro Statistics 9 dance of the p values

ESCI-PPS-p-intervals.xls

Visualizing Power


../../../../Program Files/Wolfram Research/Wolfram CDF Player/10.2/WolframCDFPlayer.exe

../../../../Program Files/Wolfram Research/Wolfram CDF Player/10.2/WolframCDFPlayer.exe

Decreasing the Standard Error


Effect Sizes vs. p Values

• Effect Size: Magnitude of effect (e.g., r, r2, β, η2, d)• Independent of sample size

• Mean difference or association relative to variability

• p Value: Probability of obtaining an effect at least as large as what was actually observed if the null hypothesis is true• Highly dependent on sample size

• Mean difference or association relative to the ratio of variability to sample size

GOAL: Identify minimum sample to have a high likelihood (e.g., 80%) of a low likelihood (e.g., 5%) of obtaining predicted effect if null is true


Why Should I Do a Power Analysis?

• Limited Resources• Studies are effortful, expensive, and time-consuming

• Power analyses are free and less time consuming

• Efficiency• Increases chances of finding significant effects

• Increases chances of replicating prior effects

• Decreases null effects (file drawer)

• Increase confidence in null effects


Why Should I Do a Power Analysis?

• Grants require them

• Good for science• Protects from temptation to use researcher degrees of freedom

• Increases precision of estimates – increasing study-to-study stability (replicability crisis)


Counterarguments

• If the effect is significant with a small sample, it must be real• AKA, why do we care about power in the “replicability crisis” if

power is about Type II errors, not Type I?

• Why don’t we just recruit a ton of people in all of our studies?

• Why don’t we just get rid of null hypothesis significance testing?

• Doesn’t power analysis take a lot of time?• YES


What Information Do I Need?

• Generally working within a regression framework

• Effect sizes and additional sources of variability• Using standardized effects will often reduce the number of

parameters that we need to determine

• Ex. Simple linear regression


δ

zα/2 -zβ

Example: Calculating A Priori Power By Hand for Simple Linear Regression


Standardized slope

1 – β2

Noncentrality parameter

Critical value Critical value

Example: Calculating A Priori Power By Hand for Simple Linear Regression

𝑁 =𝜎𝜀2 𝑧𝛼

2− 𝑧𝛽

2

𝛽2𝜎𝑋2

=1 − 𝛽2 𝑧𝛼

2− 𝑧𝛽

2

𝛽2𝜎𝑋2

=1 − .15 2 × 1.96 − −.84 2

.152× 12= 341


Example: Calculating A Priori Power Using G*Power for Simple Linear Regression


t tests - Linear bivariate regression: One group, size of slopeAnalysis: A priori: Compute required sample size Input: Tail(s) = Two

Slope H1 = 0.1500000α err prob = 0.05Power (1-β err prob) = 0.8Slope H0 = 0Std dev σ_x = 1Std dev σ_y = 1

Output: Noncentrality parameter δ = 2.8098293Critical t = 1.9669451Df = 341Total sample size = 343Actual power = 0.8000912

../../../../Program Files (x86)/GPower 3.1/GPowerNT.exe

../../../../Program Files (x86)/GPower 3.1/GPowerNT.exe

Existing Power Software

• Many models are not supported and even some basic models give limited or imperfect estimates

• Additional resources

Design G*Power Power & Precision

PS PASS

t-tests Yes Yes Yes Yes

Correlation Yes Yes Yes Yes

ANOVA Yes* Yes* Yes* Yes*

Multiple regression Yes* Yes* Yes* Yes*

Multilevel regression No No No No

Path analysis No No No No

Latent variables No No No No


http://power.education.uconn.edu/otherwebsites.htm

http://www.gpower.hhu.de/en.html

http://www.power-analysis.com/

http://biostat.mc.vanderbilt.edu/wiki/Main/PowerSampleSize#Windows

http://www.ncss.com/software/pass/

Early Work

1990s: Closed form or iterative solutions for estimating standard errors and power for two-level models

• Approximation for estimating standard errors for two-level models (Snijders & Bosker, 1993; Liu & Liang, 1997)

• Accommodating CS, AR1, and TOEP residual covariance matrices (Hedeker, Gibbons, & Waternaux, 1999)

• Software for basic nested models [Optimal Design] (Raudenbush & Liu, 2000)


Early Work - Limitations

• Overly restrictive assumptions

• Approximations that had very limited generalizability

• Authors conducted analyses themselves and presented tables of sample sizes• Single papers often consisted of tables for only one type of model

• Did not instruct researchers to conduct power analyses on their own

• Published in statistics and education journals


Early Work (Liu & Liang, 1997 example)


Recent Work2000s-: More flexible simulation approaches, additional software packagesAhn, C., Heo, M., & Zhang, S. (2015). Sample size calculations for clustered and longitudinal

outcomes in clinical research. Boca Raton, FL: CRC Press.

Bolger, N., & Laurenceau, J-P. (2013). Intensive longitudinal methods: An Introduction to diary and experience sampling research. New York: Guilford.

Cools, W., Van den Noortgate, W., & Onghena, P. (2008). ML-DEs: A program for designing efficient multilevel studies. Behavior Research Methods, 40, 236-249.

Gelman, A., & Hill, J. (2006). Data analysis using regression and multilevel/hierarchical models. Cambridge: Cambridge University Press.

Moerbeek, M., & Teerenstra, S. (2016). Power analysis of trials with multilevel data. Boca Raton, FL: CRC Press.

Zhang, Z., & Wang, L. (2009). Statistical power analysis for growth curve models using SAS. Behavior Research Methods, 41, 1083-1094.


Recent Work - Limitations


Design G*Power Power & Precision

PS PASS RMASS PinT Optimal Design

ML-DEs Simulation

t-tests Yes Yes Yes Yes No No No No Yes

Correlation Yes Yes Yes Yes No No No No Yes

ANOVA Yes* Yes* Yes* Yes* No No No No Yes

Multiple regression

Yes* Yes* Yes* Yes* No No No No Yes

Multilevel regression

No No No No Yes* Yes* Yes* Yes* Yes

Pathanalysis

No No No No No No No No Yes

Latent variables

No No No No No No No No Yes





http://hedeker.people.uic.edu/ml.html

https://www.stats.ox.ac.uk/~snijders/multilevel.htm

https://sites.google.com/site/optimaldesignsoftware/home

https://perswww.kuleuven.be/~u0032822/mldes/USE/ML_DEs_introduction.html

Part IIIntroduction to Power Analysis using Simulation


What is Simulation?

• Using a computer to randomly generate data for a hypothetical study that conforms to a specified statistical model

• Simulate 1000s of studies

• Analyze them

• Aggregate them

ID Height Weight

1 -.91 .83

2 -.28 .60

3 -1.86 -1.03

…

98 .98 2.68

99 .07 .29

100 .66 .67

N = 100, r = .15

r1 = .13Power Analysis Using Simulation 22

What is Simulation?

1. Randomly generate data for a hypothetical study • Based on predefined model

2. Repeat 1000s of times to simulate 1000s of studies

3. Analyze each study

4. Record if hypothesized effect is significant

5. Aggregate results from all studies

6. % of significant results = Power



Population Model

Model: Y = b0 +b1*X + e Parameter Values: All means, variances, covariances

Sample Size(s)

Sample 1 Sample 2 Sample 1000…

Analysis 1 Analysis 2 Analysis 1000

p < .05? p < .05? p < .05?

Power = % of significant effects

How do we conduct a simulation?

• Need to know the model we are fitting

• Assume we already we know all of the parameters (e.g. a pilot study)• What to do if you don’t know? (Part V)

• Use random number generators & simple programming to construct data

• Analyze/aggregate it using conventional software modules


Simulate G*Power Example in SAS


Generate the Data


← Declare the dataset we want to create← Create variable for subject id

← Choose a value for the random number generator

← Intercept value← Slope value

← Number of studies to simulate

Define parameters

Simulate studies

← Loops to create data for studies and subjects within studies

← Random predictor value← The model

← Output generated data to dataset

Quick Tangent


• In G*Power only need r (1 parameter)

• Actually 5 parameters• X: M = 0, Var = 1

• Y: M = 1, Var = 1

• Corr(X,Y) = .15

Analyze the Data


← Turn off output

← Turn output back on

The analysis

← Run a regression using our generated data

← Analyze each sample separately

← The model

← Save model parameters

A Little Data Cleaning


A Little Data Cleaning


p-value significant?

Estimate Power


“Same” as G*Power – (Sampling variability)

Check that the data we generated was what we thought

QUESTIONS?


Everything we are doing can be done in SPPS, R, and Mplus


SPSSGenerate Data Analyze Data

10000.



SPSS

Power Results• It runs, but it took over an

hour (SAS < 1 minute)• Clunkier syntax• Can’t turn off output



Mplus

Generate & Analyze Data



Mplus

Power Results• Runs fast• Easily specify wide

variety of models• Can’t loop multiple

sample sizes (re: later)



RGenerate & Analyze Data Power Results

• Runs fast• Easily specify wide variety

of models• Can loop multiple sample

sizes or effect sizes

QUESTIONS?


Interaction Example(s)


Continuous Categorical

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

-2 -1 0 1 2

Y

X1 (SD)

-1 SD X2

0 SD X2

+1 SD X2

0.9

1.0

1.1

1.2

1.3

1.4

1.5

0 1

Y

X1

X2 = 0

X2 = 1

Simulate Interaction Example in SAS (continuous)


← Main effect for first variable← No main effect for second variable

← Interaction effect

← NEW – Run power analyses for 3 different sample sizes

← NEW – Tells loop to reference the array to determine sample size

← Model

Slight complication when adding more variables to get desired effect size.

Clean up dataset so ‘ssize’ corresponds to sample sizeCreate interaction effect for analysis

0.0

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0.8

1.0

1.2

1.4

1.6

1.8

2.0

-2 -1 0 1 2

Y

X1 (SD)

-1 SD X2

0 SD X2

+1 SD X2

← Reduce simulations

Analyze the Data


The analysis← Model with 3 predictors

Clean and Look at All Predictors


Estimate Power – (X1)


Estimate Power – (X1X2)


Simulate Interaction Example in SAS (categorical)


NEW – Create balanced assignment to groups

More complicated algebra, but not to worry…Trial and error can work well

0.9

1.0

1.1

1.2

1.3

1.4

1.5

0 1

Y

X1

X2 = 0

X2 = 1

Analyze the Data (Same Model)


Simple

Effects

← Model with 3 predictors

Save simple effects

Estimate Power – (X1X2)


Simple Effects


Simple Effects


0.9

1.0

1.1

1.2

1.3

1.4

1.5

0 1

YX1

X2 = 0

X2 = 1

QUESTIONS?


Part IIIPower Analysis for Multilevel Models using Simulation


When you can’t use G*Power

• Multilevel modeling (Hierarchical Linear Modeling)

• Structural Equation Modeling• Path Analysis – e.g. Mediation

• Latent Variables – e.g. Factor Analysis, Growth Curves, Latent Classes


σ2 ... 0 … 00 … σ2 … 00 … 0 … σ2

Review Linear Regression and MLM extension

• Basic structure

Yi = b0 + b1*X1i + … + bn*Xni + ei

• Where,• Yi Individual i’s outcome/response (DV)• X1 … Xn n predictor variables (IVs)

• ei Each individual’s residual/error

[ ]ΣR = Errors assumed independent


Multilevel Extension

Yij = (b0 + b0i) + (b1 + b1i)*X1ij + … + (bn + bni)*Xnij + eij

• Where,• Yij Individual i’s response at time j• bni Unique individual effects for each IV

• Can also include predictors that do not vary within person (i.e. between person)• Just bk (no i subscript)



Yij = (b0 + b0i) + (b1 + b1i)*X1ij + … + (bn + bni)*Xnij + eij

• Also have,

σ20 ... σ01 … σ02

σ10 … σ21 … σ12

σ20 … σ21 … σ2n

[ ]ΣG =

σ2 ... 0 … 00 … σ2 … 00 … 0 … σ2[ ]ΣR =



• Without ΣG the off-diagonals of ΣR would likely not be zero• Individuals tend to positively correlate with

themselves

• ΣG accounts for this by incorporating person-specific effects• Removes tendency for individual i to respond in a

particular way so that he/she can more meaningfully be compared to everyone else

σ20 ... σ01 … σ02

σ10 … σ21 … σ12

σ20 … σ21 … σ2n

[ ]ΣG = σ2 ... 0 … 00 … σ2 … 00 … 0 … σ2[ ]ΣR =


Alternative Parameterization

Level 1

Yij = b0i + b1i*X1ij + … + bni*Xnij + eij

Level 2

b0i = γ00 + u0i

b1i = γ10 + u1i

bni = γn0 + uni


Formula for Mixed Model

Person

Level

Compute

Group Mean


Estimated variance of the slope for each person

Average estimated variance of the slope across persons

Key Formula: Sampling Variance of Average Slope

Within

Component

Between

Component


Example using repeated measurements (longitudinal time effect)

• Same model as before but now people are measured multiple times

• Additions• People may differ in their:

1. Initial/Average levels of the outcome (i.e. random intercept)

2. Individual association between in predictor and outcome (i.e. random slope)


Simulate Data in SAS


← Intercept value← Slope value

← Random intercept variance← Random slope variance← Residual variance

← Unique intercept component for each individual← Unique slope component for each individual

Intercept as a combination of the population and person-specific components

Slope as a combination of the population and person-specific components

Check the Data


Analyze the Data


The analysis

Run a multilevel regression

← Analyze each sample separately

← The model

Save the fixed effects

Print the random effects

Save the random effects

Which effects are random and how are they clustered

Data Cleaning (Fixed Effects)


Data Cleaning (Random Effects)


Estimate Power - Time


QUESTIONS?


Another example using repeated assessments

• Previous example• TIME did not vary between person

• Everyone had TIME = [0, 1, 2, 3]

• TIME effect all within-person

• What if predictor varies between person?• X can be associated with Y at the between-person and/or within-

person level

• e.g. people who on average get a lot of social support are more happy, but receiving support decreases happiness relative to own average level


Another example using repeated assessments

• Current example• X varies at the between-person level (different average X’s)

• X varies at the within-person level (same person has different X’s over time)

• Between-person differences across people positive (b1 = .15)

• Within-person change from time to time negative (b2 = -.15)


Simulate Data in SAS


← Between-person slope value← Within-person slope value

← Random intercept variance← Random slope variance← Residual variance

← Unique intercept/slope components for each individual

← Generate unique person-level X for each individual

Between-person component

Within-person component

← Generate unique within-person deviation in X for each time← Actual X is a combination

Check the Data


Analyze


← X_BTW & X_WTH separately

Estimate Power of X (X_BTW & X_WTH)


X_BTW

X_WTH

Part IVPower Analysis for Structural Equation Models using Simulation


Other Software & Other Models

• SPSS, SAS, R use similar syntax structure• Examples 1-3 transferable

• Mplus uses different structure

• Some pluses• Can be easier when complex structures are desired (e.g. path analysis,

latent variables)

• Possible in SPSS, SAS, R environments but need to know (or at least approximate) the algebra

• Mplus – specify what you know you want & it “fills in” the rest• Caveat: May need to adjust values

• 2 Examples Power Analysis Using Simulation 80

Path Analysis (Mediation) Example in MPlus


X

M

Y

a = .15 b = .20

c' = .05(c = .08)

← Candidates for guessing then rerunning

← Estimate indirect effect

Path Analysis (Mediation) Example in MPlus


X

M

Y

a = .15 b = .20

c' = .05(c = .08)

X to Y Latent Variable


Y1

X

Y3

b1 = .15

.40

Y

Y2

.40.40

Candidates for guessing

X to Y Latent Variable


Y1

X

Y3

b1 = .15

.40

Y

Y2

.40.40

Part VDetermining Parameters, Reporting Power Analyses, and Conclusions


Determining Parameters

• Published similar data• Common not to report all statistics (e.g., random effects, residuals)

• May need to contact authors

• Published semi-similar data• Similar variables

• e.g., using a study of change in racial bias post-intervention to estimate parameters for change in sexism post-intervention

• Similar design• e.g., using a baseline|intervention|follow-up|follow-up study of racial bias to estimate

parameters for a baseline|intervention|follow-up|followup study of self-esteem

• Between-person estimates of within-person processes• e.g., using an intervention|control between-subjects lab study of racial bias to estimate

parameters for a baseline|intervention within-subjects study of racial biasPower Analysis Using Simulation 86

Determining Parameters

• Pilot data• Good for all of the usual reasons

• Even very limited pilots can provide useful information about likely ranges of variance components

• Educated guesses• Patterns can be observed across studies of very different

constructs


Educated Guesses

• Fixed Effects• Interaction effects usually smaller than main effects

• Indirect effects (multiplicatively) smaller than direct effects

• Between-person effects usually larger than within-person effects

• Self-report (Actor) effects usually larger than other-report (Partner) effects

• Random Effects• They are usually there (i.e. non-zero)

• Random effect variances usually smaller than residual variance

• If few clusters increasing cluster variance (in intercept and slopes) will have smaller effect

• Residuals• Usually autocorrelated in longitudinal designs

• Usually increase over time in longitudinal designs


Educated Guesses

• Standardized effects: Usually range from -1 to +1

• Unstandardized effects: Can range from −∞ 𝑡𝑜 +∞

• Use possible % of variance explainable as a reference for estimating effect sizes and error variance• Specific to area of research

• Daily life – lots of within-person variability• Upper bound of explainable variance ~10%

• Proportion of variance in random effects relative to residual < 10%

• Individual fixed effects < 1%

• Cognitive experiment – might expect to explain > 50% of the total variance

• Know your area and what is reasonable – use that as a reference


Sensitivity Analysis

• How bad can things be?• Even informed guesses are still guesses

• Maximize likelihood of being well-powered even if you guessed incorrectly

• Systematically vary parameters of interest & recalculate power• Efficiently balance # people and # repeated measurements

• Robustness of hypothesized effects

• Start with most theoretically important parameter• How small can it get?

• Most uncertain parameter• How small/big can it get?

• Usually random effects (if clustered data)Power Analysis Using Simulation 90

Things that affect the sampling variance of the slope…and therefore, power.

Within

Component

Between

Component


Error variance in the outcome[Solution: Increase measurement precision]

Variance in the predictor[Solution: Increase variance by sampling broader range]

Variance of the slope[Solution: Decrease variance of the effect – add moderators]

# repeated measurements[Solution: Increase]

# people[Solution: Increase]

# people > # measurements

Recall Simple Regression Case

Sensitivity Analysis Examples

• N vs. Time (Example 3a) - # people vs. # repeated measurements trade-off• Starting point: N = 100, T = 4

• What buys us more power?

• Time (Example 3a) - Uncertain random effect• Random variance of Time effect was .15

• Recall: More random slope variance decreases power

• How large can it get?


93

Power curves for Example 3a (N vs. T)

Power Analysis Using Simulation

N*T Total Observations

N*4 Adding Subjects

50*T Adding Time Points25*4 50*4 75*4 100*4 125*4 150*4

50*2 50*4 50*6 50*8 50*10 50*12

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

100 200 300 400 500 600

Pow

erN = 50 is never enough…

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1.0

0.00 0.05 0.10 0.15 0.20 0.25

Po

we

r

94

Power curves for Example 3a (Time)

Power Analysis Using Simulation

β1 = .20

β1 = .15

β1 = .10

Variance of Slope

Troubleshooting

• How do I know I did what I was intending to?1. Check means and variances of generated data and population

parameters

2. Check that the model recovers correct coefficients• Assuming, generating model = fit model

• Were my parameter choices reasonable?• In the real world? – up to the researcher

• In the algebra?• Check the algebra

• Trial and error – check standardized and unstandardized effects are equal


Example 1 Revisited


Correct variance was (1-B1**2)

OH NO! Estimates don’t match.

Writing Up a Power Analysis – Grant Example

We conducted a series of simulations in order to estimate the power of observing 1) differences in identity, stress, and well-being over time following the Supreme Court ruling, and 2) an indirect effect of identity on well-being through stress predicted by our mediation model. We assumed a small effect size (d = 0.2) for all change scores over time within a given construct and for all associations between constructs in order to obtain conservative estimates of the necessary sample size and because of a lack of comparable studies that have examined change scores for the majority of our variables of interest. In addition, we modeled a moderate between-person variance in the overall effect (10% of the total variance) and homoskedastic but moderately autocorrelated (ρ = .25) errors. The power of the difference scores (91%) and indirect effect (98%) were both excellent for a sample of 500 Lesbian/Gay participants (and 500 Straight participants).


Writing Up a Power Analysis – Manuscript ExamplePrior to recruitment, we conducted a power analysis in SAS version 9.4 (Lane, Bolger, & Hennes, 2016). To estimate the number of participants needed to have 80% power to detect significant effects on our three primary dependent variables, we simulated data using the results of Study 1. We were agnostic about the relative pattern of effects for the three intervention videos; therefore we estimated the slope post-intervention using the average of the post-intervention slopes in Study 1. Because of the shorter time between measurements (two weeks rather than six months), we estimated no decay, and anticipated less random noise in our faculty sample than in MTurk (50% of the observed residual from Study 1). These simulations estimated that we would need 140 participants to obtain significant effects of the intervention at Time 2. After accounting for potential attrition (~20%), we sought to recruit 175 participants in Study 2.


Necessary Information for Power Analysis

• # participants

• Means and SDs of variables

• Effect sizes of IVs on DV

• # repeated assessments

• Variances for random effects & residual

• Correlations between means, shared effects, and residuals

Cross-sectional designs

Longitudinal designs

Multivariate/Dyadic designs


Conclusions

• Power analysis helps us design studies with high likelihoods of observing effects [big OR small] with a high degree of precision

• Cost-, resource-, and time-effective

• Classic power analysis tools generally not suitable for estimating factorial, multilevel, path, and latent factor models

• Simulation allows for flexible sample size estimates of any model using SPSS, SAS, MPlus, or R • If you can run the analysis, you can conduct the power analysis for it


Conclusions

• Requires a priori hypotheses with educated guesses about effect sizes and variability• Including random effects and residuals

• Similar published papers (even on very different topics) and pilot data can help estimate these parameters

• Sensitivity analysis helps make good decisions when there is uncertainty about the parameters

• Can be integrated into preregistration activities

• Can be adapted for Bayesian or other alternative hypothesis testing techniques Power Analysis Using Simulation 101

Part VIReferences, Appendices, Time for Questions


References – Early Work

Hedeker, D., Gibbons, R. D., & Waternaux, C. (1999). Sample size estimation for longitudinal designs with attrition: Comparing time- related contrasts between two groups. Journal of Educational and Behavior Statistics, 24, 70-93.

Liu, G., & Liang, K.-Y., (1997). Sample size calculations for studies with correlated observations. Biometrics, 53, 937-947.

Raudenbush, S. W., & Liu, X. (2000). Statistical power and optimal design for multisite randomized trials. Psychological Methods, 5, 199-213.

Snijders, T. A. B., & Bosker, R. J. (1993). Standard errors and sample sizes for two-level research. Journal of Educational Statistics, 18, 237-259. Power Analysis Using Simulation 103

References – Recent WorkAhn, C., Heo, M., & Zhang, S. (2015). Sample size calculations for clustered and

longitudinal outcomes in clinical research. Boca Raton, FL: CRC Press.

Bolger, N., & Laurenceau, J-P. (2013). Intensive longitudinal methods: An introduction to diary and experience sampling research. New York: Guilford.

Cools, W., Van den Noortgate, W., & Onghena, P. (2008). ML-DEs: A program for designing efficient multilevel studies. Behavior Research Methods, 40, 236-249.

Gelman, A., & Hill, J. (2006). Data analysis using regression and multilevel/hierarchical models. Cambridge: Cambridge University Press.

Moerbeek, M., & Teerenstra, S. (2016). Power analysis of trials with multilevel data. Boca Raton, FL: CRC Press.

Zhang, Z., & Wang, L. (2009). Statistical power analysis for growth curve models using SAS. Behavior Research Methods, 41, 1083-1094.


Other SoftwareG*Power: http://www.gpower.hhu.de/en.html

Power & Precision: http://www.power-analysis.com/

PS: http://biostat.mc.vanderbilt.edu/wiki/Main/PowerSampleSize#Windows

PASS: http://www.ncss.com/software/pass/

RMASS: http://hedeker.people.uic.edu/ml.html

PINT: https://www.stats.ox.ac.uk/~snijders/multilevel.htm

Optimal Design: https://sites.google.com/site/optimaldesignsoftware/home

ML-Des: https://perswww.kuleuven.be/~u0032822/mldes/USE/ML_DEs_introduction.html

Other resources: http://power.education.uconn.edu/otherwebsites.htm






http://hedeker.people.uic.edu/ml.html

https://www.stats.ox.ac.uk/~snijders/multilevel.htm

https://sites.google.com/site/optimaldesignsoftware/home

https://perswww.kuleuven.be/~u0032822/mldes/USE/ML_DEs_introduction.html

http://power.education.uconn.edu/otherwebsites.htm

How to Cite This Workshop:

• For now:Lane, S. P., Hennes, E. P., & West, T. V. (2016, January). “I’ve got the power”: How

anyone can do a power analysis of any type of data using simulation. Workshop presented at the 17th Annual Convention of the Society for Social and Personality Psychology, San Diego, CA.

• Please request from the authors:Lane, S. P., Hennes, E. P. & Bolger, N. (under review). Power struggles: Estimating

sample size for multilevel relationships research. Journal of Social and Personal and Social Relationships.

• More detailed papers are also in prep. Please contact Sean at [email protected] for working papers or consulting


mailto:[email protected]

Thank You!

Niall Bolger

Pat Shrout

JP Laurenceau

Andy Gelman

Jennifer Hill

Tessa West

Chad Rummel


“I’ve Got the Power”: How Anyone Can Do a Power Analysis...

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