We recently posted some correlation statistics on our blog. We believe these statistics are interesting and provide insight into the ways search engines work (a core principle of our mission here at SEOmoz). As we will continue to make similar statistics available, I'd like to discuss why correlations are interesting, refute the math behind recent criticisms, and reflect on how exciting it is to engage in mathematical discussions where critiques can be definitively rebutted.

I've been around SEOmoz for a little while now, but I don't post a lot. So, as a quick reminder, I designed and built the prototype for the SEOmoz's web index, as well as wrote a large portion of the back-end code for the project. We shipped the index with billions of pages nine months after I started on the prototype, and we have continued to improve it since. Recently I made the machine learning models that are used to make Page Authority and Domain Authority, and am working on some fairly exciting stuff that has not yet shipped. As I'm an engineer and not a regular blogger, I'll ask for a bit of empathy for my post - it's a bit technical, but I've tried to make it as accessible as possible.

Why does Correlation Matter?

Correlation helps us find causation by measuring how much variables change together. Correlation does not imply causation; variables can be changing together for reasons other than one affecting the other. However, if two variables are correlated and neither is affecting the other, we can conclude that there must be a third variable that is affecting both. This variable is known as a confounding variable. When we see correlations, we do learn that a cause exists -- it might just be a confounding variable that we have yet to figure out.

How can we make use of correlation data? Let's consider a non-SEO example.

There is evidence that women who occasionally drink alcohol during pregnancy give birth to smarter children with better social skills than women who abstain. The correlation is clear, but the causation is not. If it is causation between the variables, then light drinking will make the child smarter. If it is a confounding variable, light drinking could have no effect or even make the child slightly less intelligent (which is suggested by extrapolating the data that heavy drinking during pregnancy makes children considerably less intelligent).

Although these correlations are interesting, they are not black-and-white proof that behaviors need to change. One needs to consider which explanations are more plausible: the causal ones or the confounding variable ones. To keep the analogy simple, let's suppose there were only two likely explanation - one causal and one confounding. The causal explanation is that alcohol makes a mother less stressed, which helps the unborn baby. The confounding variable explanation is that women with more relaxed personalities are more likely to drink during pregnancy and less likely to negatively impact their child's intelligence with stress. Given this, I probably would be more likely to drink during pregnancy because of the correlation evidence, but there is an even bigger take-away: both likely explanations damn stress. So, because of the correlation evidence about drinking, I would work hard to avoid stressful circumstances. *

Was the analogy clear? I am suggesting that as SEOs we approach correlation statistics like pregnant women considering drinking - cautiously, but without too much stress.

* Even though I am a talented programmer and work in the SEO industry, do not take medical advice from me, and note that I construed the likely explanations for the sake of simplicity :-)

Some notes on data and methodology

We have two goals when selecting a methodology to analyze SERPs:

1. Choose measurements that will communicate the most meaningful data
2. Use techniques that can be easily understood and reproduced by others

These goals sometimes conflict, but we generally choose the most common method still consistent with our problem. Here is a quick rundown of the major options we had, and how we decided between them for our most recent results:

Machine Learning Models vs. Correlation Data: Machine learning can model and account for complex variable interactions. In the past, we have reported derivatives of our machine learning models. However, these results are difficult to create, they are difficult to understand, and they are difficult to verify. Instead we decided to compute simple correlation statistics.

Pearson's Correlation vs. Spearman's Correlation: The most common measure of correlation is Pearson's Correlation, although it only measures linear correlation. This limitation is important: we have no reason to think interesting correlations to ranking will all be linear. Instead we choose to use Spearman's correlation. Spearman's correlation is still pretty common, and it does a reasonable job of measuring any monotonic correlation.

Here is a monotonic example: The count of how many of my coworkers have eaten lunch for the day is perfectly monotonically correlated with the time of day. It is not a straight line and so it isn't linear correlation, but it is never decreasing, so it is monotonic correlation.

Here is a linear example: assuming I read at a constant rate, the amount of pages I can read is linearly correlated with the length of time I spend reading.

Mean Correlation Coefficient vs. Pooled Correlation Coefficient: We collected data for 11,000+ queries. For each query, we can measure the correlation of ranking position with a particular metric by computing a correlation coefficient. However, we don't want to report 11,000+ correlation coefficients; we want to report a single number that reflects how correlated the data was across our dataset, and we want to show how statistically significant that number is. There are two techniques commonly used to do this:

1. Compute the mean of the correlation coefficients. To show statistical significance, we can report the standard error of the mean.
2. Pool the results from all SERPs and compute a global correlation coefficient. To show statistical significance, we can compute standard error through a technique known as bootstrapping.

The mean correlation coefficient and the pooled correlation coefficient would both be meaningful statistics to report. However, the bootstrapping needed to show the standard error of the pooled correlation coefficient is less common than using the standard error of the mean. So we went with #1.

Fisher Transform Vs No Fisher Transform: When averaging a set of correlation coefficients, instead of computing the mean of the correlation coefficients, sometimes one computes the mean of the fisher transforms of the coefficients (before applying the inverse fisher transform). This would not be appropriate for our problem because:

1. It will likely fail. The Fisher transform includes a division by the coefficient minus one, and so explodes when an individual coefficient is near one and outright fails when there is a one. Because we are computing hundreds of thousands of coefficients each with small sample sizes to average over, it is quite likely the Fisher transform will fail for our problem. (Of course, we have a large sample of these coefficients to average over, so our end standard error is not large)
2. It is unnecessary for two reasons. First, the advantage of the transform is that it can make the expect average closer to the expected coefficient. We do nothing that assumes this property. Second, as mean coefficients are near to zero, this property holds without the transform, and our coefficients were not large.

Rebuttals To Recent Criticisms

Two bloggers, Dr. E. Garcia and Ted Dzubia, have published criticisms of our statistics.

Eight months before his current post, Ted Dzubia wrote an enjoyable and jaunty post lamenting that criticism of SEO every six to eight months was an easy way to generate controversy, noting "it's been a solid eight months, and somebody kicked the hornet's nest. Is SEO good or evil? It's good. It's great. I <3 SEO." Furthermore, his twitter feed makes it clear he sometimes trolls for fun. To wit: "Mongrel 2 under the Affero GPL. TROLLED HARD," "Hacker News troll successful," and "mailing lists for different NoSQL servers are ripe for severe trolling." So it is likely we've fallen for trolling...

I am going to respond to both of their posts anyway because they have received a fair amount of attention, and because both posts seek to undermine the credibility of the wider SEO industry. SEOmoz works hard to raise the standards of the SEO industry, and protect it from unfair criticisms (like Garcia's claim that "those conferences are full of speakers promoting a lot of non-sense and SEO myths/hearsays/own crappy ideas" or Dzubia's claim that, besides our statistics, "everything else in the field is either anecdotal hocus-pocus or a decree from Matt Cutts"). We also plan to create more correlation studies (and more sophisticated analyses using my aforementioned ranking models) and thus want to ensure that those who are employing this research data can feel confident in the methodology employed.

Search engine marketing conferences, like SMX, OMS and SES, are essential to the vitality of our industry. They are an opportunity for new SEO consultants to learn, and for experienced SEOs to compare notes. It can be hard to argue against such subjective and unfair criticism of our industry, but we can definitively rebut their math.

To that end, here are rebuttals for the four major mathematical criticisms made by Dr. E. Garcia, and the two made by Dzubia.

1) Rebuttal to Claim That Mean Correlation Coefficients Are Uncomputable

For our charts, we compute a mean correlation coefficient. The claim is that such a value is impossible to compute.

Dr. E. Garcia : "Evidently Ben and Rand don’t understand statistics at all. Correlation coefficients are not additive. So you cannot compute a mean correlation coefficient, nor you can use such 'average' to compute a standard deviation of correlation coefficients."

There are two issues with this claim: a) peer reviewed papers frequently published mean correlation coefficients; b) additivity is relevant for determining if two different meanings of the word "average" will have the same value, not if the mean will be uncomputable. Let's consider each issue in more detail.

a) Peer Reviewed Articles Frequently Compute A Mean Correlation Coefficient

E. Garcia is claiming something is uncomputable that researchers frequently compute and include in peer reviewed articles. Here are three significant papers where the researchers compute a mean correlation coefficient:

"The weighted mean correlation coefficient between fitness and genetic diversity for the 34 data sets was moderate, with a mean of 0.432 +/- 0.0577" (Macquare University - "Correlation between Fitness and Genetic Diversity", Reed, Franklin; Conversation Biology; 2003)

"We observed a progressive change of the mean correlation coefficient over a period of several months as a consequence of the exposure to a viscous force field during each session. The mean correlation coefficient computed during the force-field epochs progressively..." (MIT - F. Gandolfo, et al; "Cortical correlates of learning in monkeys adapting to a new dynamical environment," 2000)

"For the 100 pairs of MT neurons, the mean correlation coefficient was 0.12, a value significantly greater than zero" (Stanford - E Zohary, et al; "Correlated neuronal discharge rate and its implications for psychophysical performance", 1994)

SEOmoz is in a camp with reviewers from the journal Nature, as well as researchers from MIT, Stanford and authors of 2,400 other academic papers that use the mean correlation coefficient. Our camp is being attacked by Dr. E. Garcia's, who argues our camp doesn't "understand statistics at all." It is fine to take positions outside of the scientific mainstream, although when Dr. E. Garcia takes such a position he should offer more support for it. Given how commonly Dr. E. Garcia uses the pejorative "quack," I suspect he does not mean to take positions this far outside of academic consensus.

b) Additivity Relevant For Determining If Different Meanings Of "Average" Are The Same, Not If Mean Is Computable

Although "mean" is quite precise, "average" is less precise. By "average" one might intend the words "mean", "mode", "median," or something else. One of these other things that it could be used as meaning is 'the value of a function on the union of the inputs'. This last definition of average might seem odd, but it is sometimes used. Consider if someone asked "a car travels 1 mile at 20mph, and 1 mile at 40mph, what was the average mph for the entire trip?" The answer they are looking for is not 30mph, which is mean of the two measurements, but ~26mph, which is the mph for the whole 2 mile trip. In this case, the mean of the measurements is different from the colloquial average which is the function for computing mph applied to the union of the inputs (the whole two miles).

This may be what has confused Dr. E. Garcia. Elsewhere he cites Statsweb when repeating this claim. Which makes the point that this other "average" is different than the mean. Additivity is useful in determining if these averages will be different. But even if another interpretation of average is valid for a problem, and even if that other average is different than the mean, it neither makes the mean uncomputable nor meaningless.

2) Rebuttal to Claim About Standard Error of the Mean vs Standard Error of a Correlation Coefficent

Although he has stated unequivocally that one cannot compute a mean correlation coefficient, Garcia is quite opinionated on how we ought to have computed standard error for it. To wit: