# Repeated measures ANOVA with R (functions and tutorials)

Repeated measures ANOVA is a common task for the data analyst.

There are (at least) two ways of performing “repeated measures ANOVA” using R but none is really trivial, and each way has it’s own complication/pitfalls (explanation/solution to which I was usually able to find through searching in the R-help mailing list).

So for future reference, I am starting this page to document links I find to tutorials, explanations (and troubleshooting) of “repeated measure ANOVA” done with R

### Functions and packages

(I suggest using the tutorials supplied bellow for how to use these functions)

• aov {stats} – offers SS type I repeated measures anova, by a call to lm for each stratum. A short example is given in the ?aov help file
• Anova {car} – Calculates type-II or type-III analysis-of-variance tables for model objects produced by lm, and for various other object. The ?Anova help file offers an example for how to use this for repeated measures
• ezANOVA {ez} – This function provides easy analysis of data from factorial experiments, including purely within-Ss designs (a.k.a. “repeated measures”), purely between-Ss designs, and mixed within-and-between-Ss designs, yielding ANOVA results and assumption checks. It is a wrapper of the Anova {car} function, and is easier to use. The ez package also offers the functions ezPlot and ezStats to give plot and statistics of the ANOVA analysis. The ?ezANOVA help file gives a good demonstration for the functions use (My thanks goes to Matthew Finkbe for letting me know about this cool package)
• friedman.test {stats} – Performs a Friedman rank sum test with unreplicated blocked data. That is, a non-parametric one-way repeated measures anova. I also wrote a wrapper function to perform and plot a post-hoc analysis on the friedman test results
• Non parametric multi way repeated measures anova – I believe such a function could be developed based on the Proportional Odds Model, maybe using the {repolr} or the {ordinal} packages. But I still didn’t come across any function that implements these models (if you do – please let me know in the comments).
• Repeated measures, non-parametric, multivariate analysis of variance – as far as I know, such a method is not currently available in R.  There is, however, the Analysis of similarities (ANOSIM) analysis which provides a way to test statistically whether there is a signiﬁcantdifference between two or more groups of sampling units.  Is is available in the {vegan} package through the “anosim” function.  There is also a tutorial and a relevant published paper.

### Troubelshooting

Unbalanced design
Unbalanced design doesn’t work when doing repeated measures ANOVA with aov, it just doesn’t. This situation occurs if there are missing values in the data or that the data is not from a fully balanced design. The way this will show up in your output is that you will see the between subject section showing withing subject variables.

A solution for this might be to use the Anova function from library car with parameter type=”III”. But before doing that, first make sure you understand the difference between SS type I, II and III. Here is a good tutorial for helping you out with that.
By the way, these links are also useful in case you want to do a simple two way ANOVA for unbalanced design

I will “later” add R-help mailing list discussions that I found helpful on the subject.

# Quantile LOESS – Combining a moving quantile window with LOESS (R function)

In this post I will provide R code that implement’s the combination of repeated running quantile with the LOESS smoother to create a type of “quantile LOESS” (e.g: “Local Quantile Regression”).

This method is useful when the need arise to fit robust and resistant (Need to be verified) a smoothed line for a quantile (an example for such a case is provided at the end of this post).

If you wish to use the function in your own code, simply run inside your R console the following line:

### Background

I came a cross this idea in an article titled “High throughput data analysis in behavioral genetics” by Anat Sakov, Ilan Golani, Dina Lipkind and my advisor Yoav Benjamini. From the abstract:

In recent years, a growing need has arisen in different fields, for the development of computational systems for automated analysis of large amounts of data (high-throughput). Dealing with non-standard noise structure and outliers, that could have been detected and corrected in manual analysis, must now be built into the system with the aid of robust methods. [...] we use a non-standard mix of robust and resistant methods: LOWESS and repeated running median.

The motivation for this technique came from “Path data” (of mice) which is

prone to suffer from noise and outliers. During progression a tracking system might lose track of the animal, inserting (occasionally very large) outliers into the data. During lingering, and even more so during arrests, outliers are rare, but the recording noise is large relative to the actual size of the movement. The statistical implications are that the two types of behavior require different degrees of smoothing and resistance. An additional complication is that the two interchange many times throughout a session. As a result, the statistical solution adopted needs not only to smooth the data, but also to recognize, adaptively, when there are arrests. To the best of our knowledge, no single existing smoothing technique has yet been able to fulfill this dual task. We elaborate on the sources of noise, and propose a mix of LOWESS (Cleveland, 1977) and the repeated running median (RRM; Tukey, 1977) to cope with these challenges

If all we wanted to do was to perform moving average (running average) on the data, using R, we could simply use the rollmean function from the zoo package.
But since we wanted also to allow quantile smoothing, we turned to use the rollapply function.

### R function for performing Quantile LOESS

Here is the R function that implements the LOESS smoothed repeated running quantile (with implementation for using this with a simple implementation for using average instead of quantile):

# Nutritional supplements efficacy score – Graphing plots of current studies results (using R)

In this post I showcase a nice bar-plot and a balloon-plot listing recommended Nutritional supplements , according to how much evidence exists for thier benefits, scroll down to see it(and click here for the data behind it)
* * * *
The gorgeous blog “Information Is Beautiful” recently publish an eye candy post showing a “balloon race” image (see a static version of the image here) illustrating how much evidence exists for the benefits of various Nutritional supplements (such as: green tea, vitamins, herbs, pills and so on) . The higher the bubble in the Y axis score (e.g: the bubble size) for the supplement the greater the evidence there is for its effectiveness (But only for the conditions listed along side the supplement).

There are two reasons this should be of interest to us:

1. This shows a fun plot, that R currently doesn’t know how to do (at least I wasn’t able to find an implementation for it). So if anyone thinks of an easy way for making one – please let me know.
2. The data for the graph is openly (and freely) provided to all of us on this Google Doc.

The advantage of having the data on a google doc means that we can see when the data will be updated. But more then that, it means we can easily extract the data into R and have our way with it (Thanks to David Smith’s post on the subject)

For example, I was wondering what are ALL of the top recommended Nutritional supplements, an answer that is not trivial to get from the plot that was in the original post.

In this post I will supply two plots that present the data: A barplot (that in retrospect didn’t prove to be good enough) and a balloon-plot for a table (that seems to me to be much better).

Barplot
(You can click the image to enlarge it)

The R code to produce the barplot of Nutritional supplements efficacy score (by evidence for its effectiveness on the listed condition).

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 # loading the data supplements.data.0 < - read.csv("http://spreadsheets.google.com/pub?key=0Aqe2P9sYhZ2ndFRKaU1FaWVvOEJiV2NwZ0JHck12X1E&output=csv") supplements.data <- supplements.data.0[supplements.data.0[,2] >2,] # let's only look at "good" supplements supplements.data < - supplements.data[!is.na(supplements.data[,2]),] # and we don't want any missing data   supplement.score <- supplements.data[, 2] ss <- order(supplement.score, decreasing = F) # sort our data supplement.score <- supplement.score[ss] supplement.name <- supplements.data[ss, 1] supplement.benefits <- supplements.data[ss, 4] supplement.score.col <- factor(as.character(supplement.score)) levels(supplement.score.col) <- c("red", "orange", "blue", "dark green") supplement.score.col <- as.character(supplement.score.col)   # mar: c(bottom, left, top, right) The default is c(5, 4, 4, 2) + 0.1. par(mar = c(5,9,4,13)) # taking care of the plot margins bar.y <- barplot(supplement.score, names.arg= supplement.name, las = 1, horiz = T, col = supplement.score.col, xlim = c(0,6.2), main = c("Nutritional supplements efficacy score","(by evidence for its effectiveness on the listed condition)", "(2010)")) axis(4, labels = supplement.benefits, at = bar.y, las = 1) # Add right axis abline(h = bar.y, col = supplement.score.col , lty = 2) # add some lines so to easily follow each bar

Also, the nice things is that if the guys at Information Is Beautiful will update there data, we could easily run the code and see the updated list of recommended supplements.

Balloon plot
So after some web surfing I came around an implementation of a balloon plot in R (Thanks to R graph gallery)
There where two problems with using the command out of the box. The first one was that the colors where non informative (easily fixed), the second one was that the X labels where overlapping one another. Since there is no “las” parameter in the function, I just opened the function up, found where this was plotted and changed it manually (a bit messy, but that’s what you have to do sometimes…)

Here are the result (you can click the image for a larger image):

And here is The R code to produce the Balloon plot of Nutritional supplements efficacy score (by evidence for its effectiveness on the listed condition).
(it’s just the copy of the function with a tiny bit of editing in line 146, and then using it)

# Siegel-Tukey: a Non-parametric test for equality in variability (R code)

Daniel Malter just shared on the R mailing list (link to the thread) his code for performing the Siegel-Tukey (Nonparametric) test for equality in variability.
Excited about the find, I contacted Daniel asking if I could republish his code here, and he kindly replied “yes”.
From here on I copy his note at full.

Corrections and remarks can be added in the comments bellow, or on the github code page.

* * * *

# Post hoc analysis for Friedman’s Test (R code)

My goal in this post is to give an overview of Friedman’s Test and then offer R code to perform post hoc analysis on Friedman’s Test results. (The R function can be downloaded from here)

### Preface: What is Friedman’s Test

Friedman test is a non-parametric randomized block analysis of variance. Which is to say it is a non-parametric version of a one way ANOVA with repeated measures. That means that while a simple ANOVA test requires the assumptions of a normal distribution and equal variances (of the residuals), the Friedman test is free from those restriction. The price of this parametric freedom is the loss of power (of Friedman’s test compared to the parametric ANOVa versions).

The hypotheses for the comparison across repeated measures are:

• H0: The distributions (whatever they are) are the same across repeated measures
• H1: The distributions across repeated measures are different

The test statistic for the Friedman’s test is a Chi-square with [(number of repeated measures)-1] degrees of freedom. A detailed explanation of the method for computing the Friedman test is available on Wikipedia.

Performing Friedman’s Test in R is very simple, and is by using the “friedman.test” command.

### Post hoc analysis for the Friedman’s Test

Assuming you performed Friedman’s Test and found a significant P value, that means that some of the groups in your data have different distribution from one another, but you don’t (yet) know which. Therefor, our next step will be to try and find out which pairs of our groups are significantly different then each other. But when we have N groups, checking all of their pairs will be to perform [n over 2] comparisons, thus the need to correct for multiple comparisons arise.
Our first task will be to perform a post hoc analysis of our results (using R) – in the hope of finding out which of our groups are responsible that we found that the null hypothesis was rejected. While in the simple case of ANOVA, an R command is readily available (“TukeyHSD”), in the case of friedman’s test (until now) the code to perform the post hoc test was not as easily accessible.
Our second task will be to visualize our results. While in the case of simple ANOVA, a boxplot of each group is sufficient, in the case of a repeated measures – a boxplot approach will be misleading to the viewer. Instead, we will offer two plots: one of parallel coordinates, and the other will be boxplots of the differences between all pairs of groups (in this respect, the post hoc analysis can be thought of as performing paired wilcox.test with correction for multiplicity).

### R code for Post hoc analysis for the Friedman’s Test

The analysis will be performed using the function (I wrote) called “friedman.test.with.post.hoc”, based on the packages “coin” and “multcomp”. Just a few words about it’s arguments:

• formu – is a formula object of the shape: Y ~ X | block (where Y is the ordered (numeric) responce, X is a group indicator (factor), and block is the block (or subject) indicator (factor)
• data – is a data frame with columns of Y, X and block (the names could be different, of course, as long as the formula given in “formu” represent that)
• All the other parameters are to allow or suppress plotting of the results.
 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 friedman.test.with.post.hoc <- function(formu, data, to.print.friedman = T, to.post.hoc.if.signif = T, to.plot.parallel = T, to.plot.boxplot = T, signif.P = .05, color.blocks.in.cor.plot = T, jitter.Y.in.cor.plot =F) { # formu is a formula of the shape: Y ~ X | block # data is a long data.frame with three columns: [[ Y (numeric), X (factor), block (factor) ]]   # Note: This function doesn't handle NA's! In case of NA in Y in one of the blocks, then that entire block should be removed.     # Loading needed packages if(!require(coin)) { print("You are missing the package 'coin', we will now try to install it...") install.packages("coin") library(coin) }   if(!require(multcomp)) { print("You are missing the package 'multcomp', we will now try to install it...") install.packages("multcomp") library(multcomp) }   if(!require(colorspace)) { print("You are missing the package 'colorspace', we will now try to install it...") install.packages("colorspace") library(colorspace) }     # get the names out of the formula formu.names <- all.vars(formu) Y.name <- formu.names[1] X.name <- formu.names[2] block.name <- formu.names[3]   if(dim(data)[2] >3) data <- data[,c(Y.name,X.name,block.name)] # In case we have a "data" data frame with more then the three columns we need. This code will clean it from them...   # Note: the function doesn't handle NA's. In case of NA in one of the block T outcomes, that entire block should be removed.   # stopping in case there is NA in the Y vector if(sum(is.na(data[,Y.name])) > 0) stop("Function stopped: This function doesn't handle NA's. In case of NA in Y in one of the blocks, then that entire block should be removed.")   # make sure that the number of factors goes with the actual values present in the data: data[,X.name ] <- factor(data[,X.name ]) data[,block.name ] <- factor(data[,block.name ]) number.of.X.levels <- length(levels(data[,X.name ])) if(number.of.X.levels == 2) { warning(paste("'",X.name,"'", "has only two levels. Consider using paired wilcox.test instead of friedman test"))}   # making the object that will hold the friedman test and the other. the.sym.test <- symmetry_test(formu, data = data, ### all pairwise comparisons teststat = "max", xtrafo = function(Y.data) { trafo( Y.data, factor_trafo = function(x) { model.matrix(~ x - 1) %*% t(contrMat(table(x), "Tukey")) } ) }, ytrafo = function(Y.data){ trafo(Y.data, numeric_trafo = rank, block = data[,block.name] ) } ) # if(to.print.friedman) { print(the.sym.test) }     if(to.post.hoc.if.signif) { if(pvalue(the.sym.test) < signif.P) { # the post hoc test The.post.hoc.P.values <- pvalue(the.sym.test, method = "single-step") # this is the post hoc of the friedman test     # plotting if(to.plot.parallel & to.plot.boxplot) par(mfrow = c(1,2)) # if we are plotting two plots, let's make sure we'll be able to see both   if(to.plot.parallel) { X.names <- levels(data[, X.name]) X.for.plot <- seq_along(X.names) plot.xlim <- c(.7 , length(X.for.plot)+.3) # adding some spacing from both sides of the plot   if(color.blocks.in.cor.plot) { blocks.col <- rainbow_hcl(length(levels(data[,block.name]))) } else { blocks.col <- 1 # black }   data2 <- data if(jitter.Y.in.cor.plot) { data2[,Y.name] <- jitter(data2[,Y.name]) par.cor.plot.text <- "Parallel coordinates plot (with Jitter)" } else { par.cor.plot.text <- "Parallel coordinates plot" }   # adding a Parallel coordinates plot matplot(as.matrix(reshape(data2, idvar=X.name, timevar=block.name, direction="wide")[,-1]) , type = "l", lty = 1, axes = FALSE, ylab = Y.name, xlim = plot.xlim, col = blocks.col, main = par.cor.plot.text) axis(1, at = X.for.plot , labels = X.names) # plot X axis axis(2) # plot Y axis points(tapply(data[,Y.name], data[,X.name], median) ~ X.for.plot, col = "red",pch = 4, cex = 2, lwd = 5) }   if(to.plot.boxplot) { # first we create a function to create a new Y, by substracting different combinations of X levels from each other. subtract.a.from.b <- function(a.b , the.data) { the.data[,a.b[2]] - the.data[,a.b[1]] }   temp.wide <- reshape(data, idvar=X.name, timevar=block.name, direction="wide") #[,-1] wide.data <- as.matrix(t(temp.wide[,-1])) colnames(wide.data) <- temp.wide[,1]   Y.b.minus.a.combos <- apply(with(data,combn(levels(data[,X.name]), 2)), 2, subtract.a.from.b, the.data =wide.data) names.b.minus.a.combos <- apply(with(data,combn(levels(data[,X.name]), 2)), 2, function(a.b) {paste(a.b[2],a.b[1],sep=" - ")})   the.ylim <- range(Y.b.minus.a.combos) the.ylim[2] <- the.ylim[2] + max(sd(Y.b.minus.a.combos)) # adding some space for the labels is.signif.color <- ifelse(The.post.hoc.P.values < .05 , "green", "grey")   boxplot(Y.b.minus.a.combos, names = names.b.minus.a.combos , col = is.signif.color, main = "Boxplots (of the differences)", ylim = the.ylim ) legend("topright", legend = paste(names.b.minus.a.combos, rep(" ; PostHoc P.value:", number.of.X.levels),round(The.post.hoc.P.values,5)) , fill = is.signif.color ) abline(h = 0, col = "blue")   }   list.to.return <- list(Friedman.Test = the.sym.test, PostHoc.Test = The.post.hoc.P.values) if(to.print.friedman) {print(list.to.return)} return(list.to.return)   } else { print("The results where not significant, There is no need for a post hoc test") return(the.sym.test) } }   # Original credit (for linking online, to the package that performs the post hoc test) goes to "David Winsemius", see: # http://tolstoy.newcastle.edu.au/R/e8/help/09/10/1416.html }

### Example

(The code for the example is given at the end of the post)

Let’s make up a little story: let’s say we have three types of wine (A, B and C), and we would like to know which one is the best one (in a scale of 1 to 7). We asked 22 friends to taste each of the three wines (in a blind fold fashion), and then to give a grade of 1 till 7 (for example sake, let’s say we asked them to rate the wines 5 times each, and then averaged their results to give a number for a persons preference for each wine. This number which is now an average of several numbers, will not necessarily be an integer).

After getting the results, we started by performing a simple boxplot of the ratings each wine got. Here it is:

The plot shows us two things: 1) that the assumption of equal variances here might not hold. 2) That if we are to ignore the “within subjects” data that we have, we have no chance of finding any difference between the wines.

So we move to using the function “friedman.test.with.post.hoc” on our data, and we get the following output:

\$Friedman.Test
Asymptotic General Independence Test
data:  Taste by
Wine (Wine A, Wine B, Wine C)
stratified by Taster
maxT = 3.2404, p-value = 0.003421
\$PostHoc.Test
Wine B – Wine A 0.623935139
Wine C – Wine A 0.003325929
Wine C – Wine B 0.053772757

The conclusion is that once we take into account the within subject variable, we discover that there is a significant difference between our three wines (significant P value of about  0.0034). And the posthoc analysis shows us that the difference is due to the difference in tastes between Wine C and Wine A (P value 0.003). and maybe also with the difference between Wine C and Wine B (the P value is 0.053, which is just borderline significant).

Plotting our analysis will also show us the direction of the results, and the connected answers of each of our friends answers:

Here is the code for the example:

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 source("http://www.r-statistics.com/wp-content/uploads/2010/02/Friedman-Test-with-Post-Hoc.r.txt") # loading the friedman.test.with.post.hoc function from the internet   ### Comparison of three Wine ("Wine A", "Wine B", and ### "Wine C") for rounding first base. WineTasting <- data.frame( Taste = c(5.40, 5.50, 5.55, 5.85, 5.70, 5.75, 5.20, 5.60, 5.50, 5.55, 5.50, 5.40, 5.90, 5.85, 5.70, 5.45, 5.55, 5.60, 5.40, 5.40, 5.35, 5.45, 5.50, 5.35, 5.25, 5.15, 5.00, 5.85, 5.80, 5.70, 5.25, 5.20, 5.10, 5.65, 5.55, 5.45, 5.60, 5.35, 5.45, 5.05, 5.00, 4.95, 5.50, 5.50, 5.40, 5.45, 5.55, 5.50, 5.55, 5.55, 5.35, 5.45, 5.50, 5.55, 5.50, 5.45, 5.25, 5.65, 5.60, 5.40, 5.70, 5.65, 5.55, 6.30, 6.30, 6.25), Wine = factor(rep(c("Wine A", "Wine B", "Wine C"), 22)), Taster = factor(rep(1:22, rep(3, 22))))   with(WineTasting , boxplot( Taste ~ Wine )) # boxploting friedman.test.with.post.hoc(Taste ~ Wine | Taster ,WineTasting) # the same with our function. With post hoc, and cool plots

If you find this code useful, please let me know (in the comments) so I will know there is a point in publishing more such code snippets…

# Is it harder to advertise to the more educated? Correlation in US States data will not be enough to answer us…

“Chitika research” published today a fun small dataset (you can download it from here) in a post titled “The Educated are Harder to Advertise To”.

In this post I have three goals in mind:

1. Suggesting another plot instead of the one used in the original post.
2. Emphasizing the “Correlation does not imply causation” rule.
3. Inviting other R lovers (as myself) to find fun things to do with this (and similar) dataset.

### The Data

The data set is comprised of 51 rows, one for each US states with the two variables (columns):

• CTR - The CTR means “Click Through Rate” and is from chitika data base and collected from over two random days in January (a total of 31,667,158 total impressions), and is from the full range of Internet users (they don’t have traditional demographic data – every impression is completely anonymous).
• Percent of the population who graduated college.

### Super basic analysis and plot

This data presents a stunning -0.63 correlation between the two measurements. Hinting that “The Educated are Harder to Advertise To” (as the original post suggested). The data can be easily visualized using a scatter plot:

Created using just a few lines of R code:

 aa <- read.table("http://www.r-statistics.com/wp-content/uploads/2010/02/State_CTR_Date.txt", sep = "\t", header = T) aa[,2:3] <- aa[,2:3] * 100 plot(aa[,2] ~ aa[,3], sub = paste("Correlation: ", round(cor(aa[,2], aa[,3]), 2)), main = "Scatter plot of %CTR VS %College_Grad per State", xlab = "%College_Grad per State", ylab = "%CTR per State" ) abline(lm(aa[,2] ~ aa[,3]), col = "blue")

### My conclusion from the analysis

I was asked in the comments (by Eyal) to add my own conclusions to the analysis. Does higher intelligence imply lower chances of clicking ads, my answer (under the present data) is simple “I don’t know”. The only real conclusion I can make of the data is that there might be a point in checking this effect in a more rigorous way (which I am sure is already being done).

What should we have done in order to know? When doing scientific research, we often ask ourselves how sure are we of our results. The rule of thumb for this type of question is called “the pyramid of evidence“. It is a way to organize various ways of getting “information” about the world, in an hierarchy of reliability. Here is a picture of this pyramid:

(Credit: image source)

We can see that the most reliable source is a systematic review of randomized controlled trials. In our case, that would mean having controlled experiments where you take groups of people with different levels of “intelligence” (how would you measure that?), and check their CTR (click through rates) on banner ads. This should be done in various ways, correcting for various confounders , and later the results and conclusions (from several such experiments) should be systematically reviewed by experts on the subject.

All of this should be done in order to make a real assessment of the underlying question – how does smarts effects banner clicking.
And the reason we need all of this work is because of what is said in the title of the next section:

### Correlation does not imply causation

As is written in the article on wikipedia:

“Correlation does not imply causation” is a phrase used in science and statistics to emphasize that correlation between two variables does not automatically imply that one causes the other (though it does not remove the fact that correlation can still be a hint, whether powerful or otherwise). The opposite belief, correlation proves causation, is a logical fallacy by which two events that occur together are claimed to have a cause-and-effect relationship.

But a much clearer explenation of it was given by the following XKCD comic strip:

### Next step: other resources to play with

The motivation for my post is based on this digg post trying to hint how Religiousness is connected to “negative” things such as crimes, poverty and so on. That post was based on the following links:

• http://www.gallup.com/poll/114022/state-states-importance-religion.aspx#2
• http://www.top50states.com/average-iq-score.html
• http://www.census.gov/cgi-bin/saipe/national.cgi?year=2008&ascii=
• http://www.census.gov/compendia/statab/cats/law_enforcement_courts_prisons/crimes_and_crime_rates.html
• http://www.fraserinstitute.org/researchandpublications/publications/7071.aspx
• http://www.gallup.com/poll/122333/political-ideologt-conservative-label-prevails-south.aspx#2
• http://www.ahiphiwire.org/wellbeing/display.aspx?doc_code=RWBStateRanks

If someone is motivated, he/she can extract that data and combine it with the current provided data.

In conclusion: this simplistic dataset, combined with other data resources, provides opportunity for various fun demonstrations of pairs correlation plots and of nice spatial plots (of states colored by their matching variable). It is a good opportunity to emphasize (to students, friends and the like) that “Correlation does not imply causation!”.
And finally – If you are an R lover/blogger and feel like playing with this – please let me know .

# Barnard’s exact test – a powerful alternative for Fisher’s exact test (implemented in R)

(The R code for Barnard’s exact test is at the end of the article, and you could also just download it from here, or from github)

Barnards exact test - p-value based on the nuisance parameter

About half a year ago, I was studying various statistical methods to employ on contingency tables. I came across a promising method for 2×2 contingency tables called “Barnard’s exact test“. Barnard’s test is a non-parametric alternative to Fisher’s exact test which can be more powerful (for 2×2 tables) but is also more time-consuming to compute (References can be found in the Wikipedia article on the subject).

The test was first published by George Alfred Barnard (1945) (link to the original paper in Nature). Mehta and Senchaudhuri (2003) explain why Barnard’s test can be more powerful than Fisher’s under certain conditions:

When comparing Fisher’s and Barnard’s exact tests, the loss of power due to the greater discreteness of the Fisher statistic is somewhat offset by the requirement that Barnard’s exact test must maximize over all possible p-values, by choice of the nuisance parameter, π. For 2 × 2 tables the loss of power due to the discreteness dominates over the loss of power due to the maximization, resulting in greater power for Barnard’s exact test. But as the number of rows and columns of the observed table increase, the maximizing factor will tend to dominate, and Fisher’s exact test will achieve greater power than Barnard’s.

### About the R implementation of Barnard’s exact test

After finding about Barnard’s test I was sad to discover that (at the time) there had been no R implementation of it. But last week, I received a surprising e-mail with good news. The sender, Peter Calhoun, currently a graduate student at the University of Florida, had implemented the algorithm in R. Peter had  found my posting on the R mailing list (from almost half a year ago) and was so kind as to share with me (and the rest of the R community) his R code for computing Barnard’s exact test. Here is some of what Peter wrote to me about his code:

On a side note, I believe there are more efficient codes than this one.  For example, I’ve seen codes in Matlab that run faster and display nicer-looking graphs.  However, this code will still provide accurate results and a plot that gives the p-value based on the nuisance parameter.  I did not come up with the idea of this code, I simply translated Matlab code into R, occasionally using different methods to get the same result.  The code was translated from:

Trujillo-Ortiz, A., R. Hernandez-Walls, A. Castro-Perez, L. Rodriguez-Cardozo. Probability Test.  A MATLAB file. URL

My goal was to make this test accessible to everyone.  Although there are many ways to run this test through Matlab, I hadn’t seen any code to implement this test in R.  I hope it is useful for you, and if you have any questions or ways to improve this code, please contact me at calhoun.peter@gmail.com

# Statistics plugins for WordPress

Today I came across a post named “24 Noble WordPress Plugins To Determine The Performance of your Blog” through Weblog Tools Collection (one of my favorite places to stay updates on wordpress). The post provided a good solid list of statistics plugins for wordpress. Some of them are too old to count (pun intended), others are much more recent and relevant.

As a statistics (and WordPress) lover myself, I was inspired to extend the list of wordpress statistics plugins for the hope of benefiting the community:
Blog Metrics
This plugin is based on ideas in an excellent post by Avinash Kaushik (Whom I consider a Web analytics guru and a brilliant blogger!).

it calculates:

• Raw Author Contribution:
• average number of posts per month
• average number of words per post
• Conversation Rate:
• average number of words used in comments to posts

Both for all the time you’ve been blogging, and for the last month, it then adds these values in a page on your WordPress dashboard.

Search Meter

This plugin is a must for any blogger. Period.

If you have a Search box on your blog, Search Meter automatically records what people are searching for — and whether they are finding what they are looking for. Search Meter’s admin interface shows you what people have been searching for in the last couple of days, and in the last week or month. It also shows you which searches have been unsuccessful. If people search your blog and get no results, they’ll probably go elsewhere. With Search Meter, you’ll be able to find out what people are searching for, and give them what they want by creating new posts on those topics.  [...]

Google Analytics Dashboard gives you the ability to view your Google Analytics data in your WordPress dashboard. You can also alow other users to see the same dashboard information when they are logged in or embed parts of the data into posts or as part of your theme.

The biggest advantage of this plugin in my view is that it adds sparklines in the “posts -> edit” page in the admin area.

Analytics360
I don’t use this one much. But one feature it has that I find interesting is that is adds information of when you posted something with the trend line of the google analytics traffic data. It also mixes data from MailChimp’s, which I don’t use.

MailChimp’s Analytics360 plugin allows you to pull Google Analytics and MailChimp data directly into your dashboard, so you can access robust analytics tools without leaving WordPress.

This plugin is also a must.

This plugin will monitor your blog looking for broken links and let you know if any are found.

• Monitors links in your posts, pages, the blogroll, and custom fields (optional).
• Detects links that don’t work and missing images.
• Notifies you on the Dashboard if any are found.
• Makes broken links display differently in posts (optional).
• Link checking intervals can be configured.
• New/modified posts are checked ASAP.
• You view broken links, redirects, and a complete list of links used on your site, in the Tools -> Broken Links tab.
• Searching and filtering links by URL, anchor text and so on is also possible.
• Each link can be edited or unlinked directly via the plugin’s page, without manually editing each post.

Piwik + WP-Piwik

This plugin adds a Piwik stats site to your WordPress dashboard. It’s also able to add the Piwik tracking code to your blog.
Piwik is an open source (GPL licensed) web analytics software program. It provides you with detailed real time reports on your website visitors: the search engines and keywords they used, the language they speak, your popular pages and so on…

You can install Piwik more or less like you install WordPress, and then you are left to integrate it into your blog. The only real down side of it for me (compared to google analytics) is the advanced segmentation and pivoting. But in general it is a free, great (and growing!) Web analytics solution.

Woopra Analytics Plugin
I have been using Woopra since their release thanks to lorelle. I enjoy the ability to follow the live actions that are happening inside the blog. Although since woopra went from BETA to GOLD, I lost most interest because the total blogs I track have more traffic volume then woopra allow tracking in their free account. But small bloggers could find the service gratifying.

Woopra is the world’s most comprehensive, information rich, easy to use, real-time Web tracking and analysis application.

Features include:

• Live Tracking and Web Statistics
• A rich user interface and client monitoring application
• Real-time Analytics
• Manage Multiple Blogs and Websites
• Deep analytic and search capabilities
• Click-to-chat
• Visitor and member tagging
• Easy Installation and Update Notification

## Final notes

If you are into web analytics, I also encourage you to give the following a try: Nuconomy,ClickTale, Crazy Egg. And of course, Google analytics. Each of them (and also Woopra) strips you and your visitors a bit more from their privacy. But that is the ultimate price we pay for the strong Web analytics solutions that exists out there.
If you got any more statistics plugins I missed, feel encouraged to share them with me in the comments