Correlation

The Basics of Correlation

Correlation refers to a value, \(r\), that provides insight into the relationship between two variables. This mathematical formula goes by many names including Pearson product moment correlation, bivariate correlation, Pearson’s r, and others.

Mathematically, the correlation coefficient is determined by

1. Summing the product of standardized score for \(X_1\) variable and standardized score for a corresponding \(X_2\) variable, and then
2. Dividing by the number of paired \(X_1X_2\). Most statistical packages, including Excel, Minitab, R, SAS, and SPSS quickly, easily, and painlessly perform this calculation.

At this point, we will stop discussing the mathematical formula and instead focus on the output and the ensuing interpretation. However, we will come back to the correlation coefficient, \(r\), when we discuss regression.

In this example, we will consider five metric variables, including:

1. Average amount spent per transactions
2. Number of transactions
3. Number of pages viewed
4. Number of unique visitors by week
5. Size of discount (e.g., 50% off, $10 off)

All of these numbers are fake and are used for only illustrative purposes. Proceed with that caveat.

All the correlation coefficient values shown in Table 1 appear positive. As \(X_1\) moves in a positive direction (out from the origin on the X axis), another variable (e.g., number of unique pages visited or \(X_3\)) moves in a positive (up from the origin on the Y axis) direction. Conversely, negative correlation coefficient values would be interpreted as \(X_1\) moves in a positive direction (out from the origin on the X axis), \(X_3\) moves in a negative direction (down toward the origin on the Y axis).

Correlation coefficient values range from -1.00 to 1.00 with 0.00 interpreted perfectly unrelated, -1.00 perfectly negatively related, and 1.00 perfectly positively related. Most analysts will not observe a 0, -1, and/or 1 ( the world is messy!).

“Crud factor” comes from philosopher Paul Mehl, who noted that everything is correlated with everything at least 0.2. In our example, the number of pages viewed and size of discount appears as a crud relationship.

Collinearity refers to measuring the same variable twice. It presents problems in regression analysis. In our example, the number of users and size of discount seems collinear. The analyst should investigate this issue later in the regression analysis.

The Null Hypothesis (\(H_0\))

The null hypothesis, or testable statement, is there there is no relationship between these two variables. This null hypothesis appears similar to the null hypothesis from cross tabulation with a \(\chi^2\) test for independence. However, unlike a cross tabulation with a \(\chi^2\) test for independence, the correlation test is rho (\(\rho\)) equals 0. Rho refers to the coefficient of the population.

Other Types of Correlation

There are a number of different types of correlation:

• Point Biserial correlation refers to including a dichotomous or binary variable (e.g., yes/no, on/off) (i.e., nonmetric data) with an interval or ratio variable (i.e., metric data). The interpretation changes to: as we move from no (or off) to yes (or on) then the other variable moves in a positive (or negative) direction.

• Spearman Rho correlation is used to find a relationship between two ordinal variables. For examples, a web analyst wants to compare page ranks from Google to Bing. Spearman Rho correlation could explain how related the two ranks are.

• Phi correlation is applied to understand the relationship between two dichotomous or binary variables. Instead of a Phi correlation, the digital analyst is better off setting up a cross tabulation with a chi square test for independence.

Performing correlation analysis in R

The base function cor() will perform correlations on a data.frame. Let’s give this a go with some data.

Let’s correlate!

Correlations will only work with numeric data, so we subset to just those columns and then run the base R function cor() to see a correlation table:

web_data_metrics <- web_data[,c("sessions","pageviews","entrances","bounces")]
## see correlation between all metrics
kable(cor(web_data_metrics))

The table is mirrored in the diagonal and provides the correlation coefficient (aka, “\(r\)”) between each pair of values that intersect in the cell. 1 means a perfect correlation, 0 means no correlation and -1 means a perfect negative correlation.

Does the R that we’re working on learning today have anything to do with the correlation coefficient \(r\)? Well…no. Or, at least, only to the extent that you can use R-the-platform to calculate r-the-correlation-coefficient. Good question, though!

When working with correlations, its always a good idea to view an exploratory plot. A handy function for this is pairs() which creates a scatter plot of all the metrics passed in combination:

pairs(web_data_metrics)

Here you can see the correlation numbers in graphical form. For instance, the high correlation of 0.9999923 between sessions and entrances results in an almost perfect straight line. Since a session starts with an entrance, this makes perfect sense! A correlation of less than 1 may be a quick diagnostic that something is wrong with the tracking.

How do web channels correlate?

One useful piece of analysis is seeing how web channels possibly interact.

## Use tidyverse to pivot the data
library(dplyr)
library(tidyr)

## Get only desktop rows, and the date, channelGrouping and sessions columns
pivoted <- web_data %>% 
  filter(deviceCategory == "desktop") %>% 
  select(date, channelGrouping, sessions) %>%
  spread(channelGrouping, sessions)

## Get rid of any NA's and replace with 0
pivoted[is.na(pivoted)] <- 0

kable(head(pivoted))

## can't include the date as its not numeric, so remove
cor_data <- pivoted[, -1]
## not including first column, so -1 subset
cor_table <- round(cor(cor_data),2)
kable(cor_table)

pairs(cor_data)

library(ggplot2)
gg <- ggplot(data = pivoted) + 
      theme_minimal() + 
      ggtitle("Paid (blue) vs Organic (green) search")
gg <- gg + 
      geom_line(aes(x = as.Date(date), y = `Paid Search`), col = "blue")

gg + geom_line(aes(x = as.Date(date), y = `Organic Search`), col = "green")

library(ggplot2)
gg <- ggplot(data = pivoted) + 
              theme_minimal() + 
              ggtitle("Social (red) vs Video (orange)")
gg <- gg + 
      geom_line(aes(x = as.Date(date), y = Social), col = "red")
gg + geom_line(aes(x = as.Date(date), y = Video), col = "orange")

Cross correlation

The correlations, above all, compare the same date point, but what if you expect a lagged effect? Perhaps the video traffic drove social traffic later on due to client advocacy?

Cross correlations are useful when dealing with time-series data and can examine if a metric has an influence on itself or another after some time has passed.

This can be a powerful way to find if, say, a TV or Display campaign increased SEO traffic over the course of the few weeks following the campaign.

The below compares paid search on SEO using the ccf() function. The result is the correlation for different lags of days. We can see a correlation at 0 lag at around 0.5, but the correlation increases if you lag the Social trend up to 10 days before.

ccf(pivoted$Social, pivoted$Video)

You could then conclude that Video was having a lagged effect on Social traffic up to 10 days beforehand. But, beware! The nature of cross-correlation is that if both datasets have a similar looking spike, cross-correlation will highlight it. Careful examination of the raw data trends should be performed to verify it. In some cases a smoother line will help get rid of spikes that affect the data (e.g., do the analysis on weekly or monthly data instead of daily). And, of course, there is no substitue for rational thought: if you find a relationship like this, can you explain it rationally? And, if so, can you conduct further analysis to validate that rationalization? (If only R had an Easy Button to do the actual thinking, too. Alas!)