Reader Ron D. was not pleased to see this dual-axis chart purporting to show a cause-effect between the decline in union membership and the drop in the proportion of income earned by middle-class households (defined as the middle 60% of households). Click here to read the original article. They credit CAP's David Madland and Karla Waters for this chart.

Using dual axes is a well-tested way of creating correlation where there may be none. Playing with the scales will do that for you. I wrote about this issue here.

However, the correlation in this data cannot be denied, as the scatter plot below shows. Note that the scatter plot is much better at revealing correlational patterns than a chart with multiple time-series lines. (Here's an example of two lines that display a spurious correlation.)

If one were to ask for a linear regression line, one will obtain a very high R-squared indeed (over 0.9). The problem is with the interpretation of this correlation. Any two data series that move with time will be highly correlated with each other, just because each series is highly correlated with time. Despite what you might believe after reading *Freakonomics*, regression -- especially in social science data -- cannot prove causation.

The writers at Think Progress show no such restraint, from the title "The American middle class was built by unions and will decline without them." to the sentences "these assaults have successfully decreased union membership over time... this has had a detrimental effect on the American middle class."

Note: these statements may in fact be true; I'm just pointing out that the chart does not buttress the assertions.

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It's often hard to elevate a correlation to a causal effect. We have to try different tests. One such test for this data set is: if a change in union membership causes a change in middle-class incomes, then we'd expect that the annual changes of one to be correlated with the annual changes of the other (at least in direction, better in magnitude).

So, in a year in which union membership declined a lot, one should expect to see middle-class incomes also drop substantially.

The next scatter plot contrasting these annual differences suggests that causation is probably absent. At this smaller time scale, one just doesn't see any correlation at all. Annual declines in the proportion of union membership has been around 2-4% for most of this period but shifts in middle-class incomes have been ranged widely in terms of direction as well as magnitude.

P.S. Andrew suggested connecting the lines. Here are the charts with the lines:

What appears to be a very strong correlation on the left chart does not look that well-coordinated on the right chart! (The lines connect the dots in chronological order.)

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