### Power Analysis for mixed-effect models in R

The power of a statistical test is the probability that a null hypothesis will be rejected when the alternative hypothesis is true. In lay terms, power is your ability to refine or "prove" your expectations from the data you collect. The most frequent motivation for estimating the power of a study is to figure out what sample size will be needed to observe a treatment effect. Given a set of pilot data or some other estimate of the variation in a sample, we can use power analysis to inform how much additional data we should collect.

I recently did a power analysis on a set of pilot data for a long-term
monitoring study of the US National Park Service. I thought I would
share some of the things I learned and a bit of R code for others that
might need to do something like this. If you aren't into power
analysis, the code below may still be useful as examples of how to use
the error handling functions in R (`withCallingHandlers`

,
`withRestarts`

), parallel programming using the `snow`

package, and linear mixed effect regression using `nlme`

. If you
have any suggestions for improvement or if I got something wrong on
the analysis, I'd love to hear from you.

## 1 The Study

The study system was cobblebars along the Cumberland river in Big South Fork National Park (Kentucky and Tennessee, United States). Cobblebars are typically dominated by grassy vegetation that include disjunct tall-grass prairie species. It is hypothesized that woody species will encroach onto cobblebars if they are not seasonally scoured by floods. The purpose of the NPS sampling was to observe changes in woody cover through time. The study design consisted of two-stages of clustering: the first being cobblebars, and the second being transects within cobblebars. The response variable was the percentage of the transect that was woody vegetation. Because of the clustered design, the inferential model for this study design has mixed-effects. I used a simple varying intercept model:

where *y* is the percent of each transect in woody vegetation sampled
*n* times within *J* cobblebars, each with *K* transects. The
parameter of inference for the purpose of monitoring change in woody
vegetation through time is *β*, the rate at which cover changes as
a function of time. *α*, *γ*, *σ ^{2}_{γ}*, and

*σ*are hyper-parameters that describe the hierarchical variance structure inherent in the clustered sampling design.

^{2}_{y}Below is the function code used I used to regress the pilot data. It should be noted that with this function you can log or logit transform the response variable (percentage of transect that is woody). I put this in because the responses are proportions (0,1) and errors should technically follow a beta-distribution. Log and logit transforms with Gaussian errors could approximate this. I ran all the models with transformed and untransformed response, and the results did not vary at all. So, I stuck with untransformed responses:

Model <- function(x = cobblebars, type = c("normal","log","logit")){ ## Transforms if (type[1] == "log") x$prop.woody <- log(x$prop.woody) else if (type[1] == "logit") x$prop.woody <- log(x$prop.woody / (1 - x$prop.woody)) mod <- lme(prop.woody ~ year, data = x, random = ~ 1 | cobblebar/transect, na.action = na.omit, control = lmeControl(opt = "optim", maxIter = 800, msMaxIter = 800) ) mod$type <- type[1] return(mod) }

Here are the results from this regression of the pilot data:

Linear mixed-effects model fit by REML Data: x AIC BIC logLik -134.4319 -124.1297 72.21595 Random effects: Formula: ~1 | cobblebar (Intercept) StdDev: 0.03668416 Formula: ~1 | transect %in% cobblebar (Intercept) Residual StdDev: 0.02625062 0.05663784 Fixed effects: prop.woody ~ year Value Std.Error DF t-value p-value (Intercept) 0.12966667 0.01881983 29 6.889896 0.0000 year -0.00704598 0.01462383 29 -0.481815 0.6336 Correlation: (Intr) year -0.389 Number of Observations: 60 Number of Groups: cobblebar transect %in% cobblebar 6 30

## 2 We don't learn about power analysis and complex models

When I decided upon the inferential model the first thing that occurred to me was that I never learned in any statistics course I had taken how to do such a power analysis on a multi-level model. I've taken more statistics courses than I'd like to count and taught my own statistics courses for undergrads and graduate students, and the only exposure to power analysis that I had was in the context of simple t-tests or ANOVA. You learn about it in your first 2 statistics courses, then it rarely if ever comes up again until you actually need it.

I was, however, able to find a great resource on power analysis from a Bayesian perspective in the excellent book "Data Analysis Using Regression and Multilevel/Hierarchical Models" by Andrew Gelman and Jennifer Hill. Andrew Gelman has thought and debated about power analysis and you can get more from his blog. The approach in the book is a simulation-based one and I have adopted it for this analysis.

## 3 Analysis Procedure

For the current analysis we needed to know three things: effect
size, sample size, and estimates of population variance. We set
effect size beforehand. In this context, the parameter of interest
is the rate of change in woody cover through time *β*, and
effect size is simply how large or small a value of *β* you want
to distinguish with a regression. Sample size is also set *a priori*. In the analysis we want to vary sample size by varying the
number of cobblebars, the number of transects per cobblebar or the
number of years the study is conducted.

The population variance cannot be known precisely, and this is where
the pilot data come in. By regressing the pilot data using the
model we can obtain estimates of all the different components of the
variance (cobblebars, transects within cobblebars, and the residual
variance). Below is the R function that will return all the
hyperparameters (and *β*) from the regression:

GetHyperparam<-function(x,b=NULL){ ## Get the hyperparameters from the mixed effect model fe <- fixef(x) if(is.null(b)) b<-fe[2] # use the data effect size if not supplied mu.a <- fe[1] vc <- VarCorr(x) sigma.y <- as.numeric(vc[5, 2]) # Residual StdDev sigma.a <- as.numeric(vc[2, 2]) # Cobblebar StdDev sigma.g <- as.numeric(vc[4, 2]) # Cobblebar:transect StdDev hp<-c(b, mu.a, sigma.y, sigma.a, sigma.g) names(hp)<-c("b", "mu.a", "sigma.y", "sigma.a", "sigma.g") return(hp) }

To calculate power we to regress the simulated data in the same way we
did the pilot data, and check for a significant *β*. Since
optimization is done using numeric methods there is always the chance
that the optimization will not work. So, we make sure the regression
on the fake data catches and recovers from all errors. The solution
for error recovery is to simply try the regression on a new set of
fake data. This function is a pretty good example of using the R
error handling function `withCallingHandlers`

and
`withRestarts`

.

fakeModWithRestarts <- function(m.o, n = 100, ...){ ## A Fake Model withCallingHandlers({ i <- 0 mod <- NULL while (i < n & is.null(mod)){ mod <- withRestarts({ f <- fake(m.orig = m.o, transform = F, ...) return(update(m.o, data = f)) }, rs = function(){ i <<- i + 1 return(NULL) }) } if(is.null(mod)) warning("ExceededIterations") return(mod) }, error = function(e){ invokeRestart("rs") }, warning = function(w){ if(w$message == "ExceededIterations") cat("\n", w$message, "\n") else invokeRestart("rs") }) }

To calculate the power of a particular design we run
`fakeModWithRestarts`

1000 times and look at the proportion of
significant *β* values:

dt.power <- function (m, n.sims = 1000, alpha=0.05, ...){ ## Calculate power for a particular sampling design signif<-rep(NA, n.sims) for(i in 1:n.sims){ lme.power <- fakeModWithRestarts(m.o = m, ...) if(!is.null(lme.power)) signif[i] <- summary(lme.power)$tTable[2, 5] < alpha } power <- mean(signif, na.rm = T) return(power) }

Finally, we want to perform this analysis on many different sampling designs. In my case I did all combinations of set of effect sizes, cobblebars, transects, and years. So, I generated the appropriate designs:

factoredDesign <- function(Elevs = 0.2/c(1,5,10,20), Nlevs = seq(2, 10, by = 2), Jlevs = seq(4, 10, by = 2), Klevs = c(3, 5, 7), ...){ ## Generates factored series of sampling designs for simulation ## of data that follow a particular model. ## Inputs: ## Elevs - vector of effect sizes for the slope parameter. ## Nlevs - vector of number of years to sample. ## Jlevs - vector of number of cobblebars to sample. ## Klevs - vector of number of transects to sample. ## Results: ## Data frame with where columns are the factors and ## rows are the designs. # Level lengths lE <- length(Elevs) lN <- length(Nlevs) lJ <- length(Jlevs) lK <- length(Klevs) # Generate repeated vectors for each factor E <- rep(Elevs, each = lN*lJ*lK) N <- rep(rep(Nlevs, each = lJ*lK), times = lE) J <- rep(rep(Jlevs, each = lK), times = lE*lN) K <- rep(Klevs, times = lE*lN*lJ) return(data.frame(E, N, J, K)) }

Once we know our effect sizes, the different sample sizes we want,
and the estimates of population variance we can generate simulated
dataset that are similar to the pilot data. To calculate power we
simply simulate a large number of dataset and calculate the
proportion of slopes, *β* that are significantly different from
zero (p-value < 0.05). This procedure is repeated for all the
effect sizes and sample sizes of interest. Here is the code for
generating a simulated dataset. It also does the work of doing the
inverse transform of the response variables if necessary.

fake <- function(N = 2, J = 6, K = 5, b = NULL, m.orig = mod, transform = TRUE, ...){ ## Simulated Data for power analysis ## N = Number of years ## J = Number of cobblebars ## K = Number of transects within cobblebars year <- rep(0:(N-1), each = J*K) cobblebar <- factor(rep(rep(1:J, each = K), times = N)) transect <- factor(rep(1:K, times = N*J)) ## Simulated parameters hp<-GetHyperparam(x=m.orig) if(is.null(b)) b <- hp['b'] g <- rnorm(J*K, 0, hp['sigma.g']) a <- rnorm(J*K, hp['mu.a'] + g, hp['sigma.a']) ## Simulated responses eta <- rnorm(J*K*N, a + b * year, hp['sigma.y']) if (transform){ if (m.orig$type == "normal"){ y <- eta y[y > 1] <- 1 # Fix any boundary problems. y[y < 0] <- 0 } else if (m.orig$type == "log"){ y <- exp(eta) y[y > 1] <- 1 } else if (m.orig$type == "logit") y <- exp(eta) / (1 + exp(eta)) } else{ y <- eta } return(data.frame(prop.woody = y, year, transect, cobblebar)) }

Then I performed the power calculations on each of these designs. This could take a long time, so I set this procedure to use parallel processing if needed. Note that I had to re-~source~ the file with all the necessary functions for each processor.

powerAnalysis <- function(parallel = T, ...){ ## Full Power Analysis ## Parallel if(parallel){ closeAllConnections() cl <- makeCluster(7, type = "SOCK") on.exit(closeAllConnections()) clusterEvalQ(cl, source("cobblebars2.r")) } ## The simulations dat <- factoredDesign(...) if (parallel){ dat$power <- parRapply(cl, dat, function(x,...){ dt.power(N = x[2], J = x[3], K = x[4], b = x[1], ...) }, ...) } else { dat$power <- apply(dat, 1, function(x, ...){ dt.power(N = x[2], J = x[3], K = x[4], b = x[1], ...) }, ...) } return(dat) }

The output of the `powerAnalysis`

function is a data frame with
columns for the power and all the sample design settings. So, I wrote
a custom plotting function for this data frame:

plotPower <- function(dt){ xyplot(power~N|J*K, data = dt, groups = E, panel = function(...){panel.xyplot(...) panel.abline(h = 0.8, lty = 2)}, type = c("p", "l"), xlab = "sampling years", ylab = "power", strip = strip.custom(var.name = c("C", "T"), strip.levels = c(T, T)), auto.key = T ) }

Below is the figure for the cobblebar power analysis. I won't go into detail on what the results mean since I am concerned here with illustrating the technique and the R code. Obviously, as the number of cobblebars and transects per year increase, so does power. And, as the effect size increases, observing it with a test is easier.

Date: 2009-09-18 Fri

HTML generated by org-mode 6.30trans in emacs 22

Labels: error handling, mixed-effect models, parallel processing, power analysis, R, simulation

## 31 Comments:

This is awesome Todd! I'm trying to adapt your parallel processing code for a randomization procedure I'd like to speed up.

Thanks!

Dan McGlinn

this is really complex

Online Tutoring

This is remarkable work

Online Tutoring,Home Tutor,Private Tutor

This is really very complicated and comlex, I don't how people can solve this.

bioinformatics training india

This blog is nice and amazing. I love your post! It's also nice to see someone who does a lot of research and has a great knack for ting, which is pretty rare from bloggers these days.

Thanks a lot!

Pilot license

Thanks for sharing very nice info. your post is very informative and awesome and

its very helpful for the reader.I like your post.Keep it

up.http://www.technologyexplores.com

You have some really good ideas in this article . I am glad I read this. I agree with much of what you state in this article. Your information is thought-provoking, interesting and well-written. Thank you.

Term Papers Essay Services

I would like to thank for the efforts you have put in writing this blog. I am hoping the same high-grade blog post from you in the upcoming as well. In fact your creative writing abilities has inspired me to get my own blog now. Really the blogging is spreading its wings quickly. Your write up is a good example of it.

Best security system for home in Bangalore

great blog man, really helped with some of my research.

See secret documents about Nikola Tesla

http://fbi-about-tesla.blogspot.com

I love the way you are teaching us. Waiting to read more from you. I love to design websites.

http://electricfireplaceheater.org/component/k2/item/66-10-best-electric-fireplace-heaters-by-user-reviews.html - find best electric fireplace heaters ...

This is not spam. :) Unfortunately links to figures don't work any more. Can you look at this?

The Apache® Hadoop project is a framework enabling distributed processing of Hadoop Online Training large data sets through a network of computers using simple programming models.

It is born to scale single servers up to thousands of machines, each that can offer local computation and storage. Hadoop Online Training To deliver high availability and uptime of 99% and more rather than relying on hardware, the library can itself detect and handle failures at the application layer.

So delivering a value based service on top of a network of computers Hadoop Online Training which may be prone to failures is the objective that is attained with the Hadoop project.

Collect and learn them". William Penn, founder of the State of Pennsylvania.makita impact driver

Given a set of pilot

data or some other estimate of the variation in a sample, we can use

power analysis to inform how much additional data we should collectc10fce2 review

Thank you for posting it will be helpful by knowing more about GDAL. Thank you and please keep update like this with this site. Definitely it will be useful for all.

SQL DBA Training in Chennai

Hi Todd,

Thanks for this post, it's very helpful. A couple of the images are currently unavailable though. I was wondering if you could possibly fix them?

Excellent info here, I am currently doing some research and found exactly what I was looing for. my review here

Really good job on the site, Thanks for guide! best 14 bandsaw

All are saying the same thing repeatedly, but in your blog I had a chance to get some useful and unique information, I love your writing style very much, I would like to suggest your blog in my dude circle, so keep on updates.

Peridot Systems Adyar Contact Number

Really Nice post Todd, you just made my problem solved. Thanks again. web design

The image with your model structure doesn't load in my browser. I am not sure if this is a problem with my firewall or a broken link.

great post

Hadoop training in coimbatore

Java training in coimbatore

Oracle training in coimbatore

Informatica training in coimbatore

Oracle training in coimbatore

Informatica training in coimbatore

Your post is really great. Thank you for taking time to provide us some of the useful and exclusive information with us. Keep on blogging!!

Best SharePoint Training Institute in Chennai

TS DSC District wise Vacancies list 2017

TS Forest Recruitment 2017

TSTRANSCO AE, AEE Recruitment 2017

TSGENCO AE, AEE Recruitment 2017

TS DSC 2017

Telangana DSC 2017

TS DSC Notification 2017

TS DSC Model Papers 2017

Sakshi TS DSC Model Papers 2017

Namasthe Telangana TS DSC Model Papers 2017

TS DSC District wise Vacancies list 2017

Telangana DSC Recruitment 2017 Notification is announced in last week of December with District wise Vacancies lists TS DSC 2017 , Telangana DSC 2017, TS DSC Notification 2017, TS DSC Model Papers 2017, Sakshi TS DSC Model Papers 2017, Namasthe Telangana TS DSC Model Papers 2017, TS DSC District wise Vacancies list 2017 Government Primary and Secondary Schools for Secondary Grade Teacher (SGT),Language Pandit (LP),School Assistant(SA)and Physical Education Teacher (PET) Posts with the TS DSC 2017.

Really Good blog post.provided a helpful information about power analysis of mixed model.keep updating...

Digital marketing company in Chennai

All the board students can Download Subject wise new Syllabus for all state and central board schools with model papers 2018 for all board of secondary education boards and students can apply scholarship 2018 for ekalyan epass to online application and status check

SSC JHT Paper II Exam Results

UPTET Recruitment Notification

Best Dyson Vacuums

## Post a Comment

Subscribe to Post Comments [Atom]

<< Home