More Correlations in Data

About 75 years ago, Astronomers used the simple technique of correlation to discover the Universe was expanding. For nearby galaxies they measured a redshift and plotted that against the distance to the galaxy. Here is the data:

### The line through the data is a "best fit" linear relationship which shows that there is a linear relationship between the the velocity at which a galaxy moves away from us and its distance. This linear relatinship is consistent with a model of uniform expansion for the Universe.

Returning to Salmon:

For the Bonneville Dam data:

• Is their a correlation between Chinook salmon counts and year (i.e. have the counts been steadily decreasing?
• Is there a correlation between various species decline?

Here is the data. We want to mine this data statisically and visually in order to see what we might be able to infer. In other words, we don't just want to use a machine but our brains to help interpret the data. This is the only way to understand and appreciate the true complexity of a particular environmental problem.

There is no correlation in this case over the whole time period. In fact there appears to be:

• quasi-cyclical behavior with a period of 15--20 years between peaks and valleys (note there is not enough data to really say this)
• There seems to be a rapid decline since the last peak (1984) --> could it mean that these Salmon counts are anomolously high

• Recent Salmon Levels are consistent with those in 1980, 1960 and 1940

Formally there is a very little correlation. The correlation coefficient, r, is 0.31. But look at the data closer to notice that its kind of odd.

There are 9 distinct occurences where the Steelhead Count is significantly above average (this corresponds to counts above 250,000). If we ignore those 9 points (years) out of the total of 57 years worth of data, the average Steelhead count is

143,000 +/- 32,000 (N=48)

The mean count for those 9 higher years is

306,000 +/- 35,000 (N=9)

Is the difference in these means significant?

• E1 = 32,000/7 = 4500
• E2 = 35,000/3 = 12000
• E2 > 2*E1
• so use (M1-M2)/E2 (306-143)/12 = 13 !!

One can therefore to conclude that something produces very high Steelhead Counts. Examining the data in time shows that the high Steelhead Counts occured in 1952--1953 and again in 1984-1989 and 1991-1992. High Steelhead count, however, does not mean high chinook count (nor does it correlate with anyother species)

For the whole data set, the weak correlation (r = 0.31) is shown below:

While a social scientist might argue that a correlation exists, you should be able to do better than that.

• The formal correlation is Y = 0.25*X + 76 but the scatter around that correlation is 66,000 Steelhead. Since the average Steelhead count is 143,000 then using Chinook as the tracer of the Steelhead poulation only predicts Steelhead to an accuracy of 66,000/143,000 or around 45% (pretty lousy)

• But look at the data and notice that the weak correlation is almost entirely driven by the two points with the highest X-values (highest Chinook counts). If we eliminate those two points, then r lowers from 0.31 to 0.10 which is no correlation at all.

• Removing the 9 periods of high Steelhead Counts from the data shows no correlation at all. In fact, the average Steelhead counts is the same over a range of Chinook counts from 200--500 (thousand).

Okay, what about using just the chinook counts as a tracer of the entire salmon population. How well does that work? Here is the data:

Your eye sees a correlation and indeed r = 0.79 for this data set. Of course, some trend is expected since roughly 30--40% of the total Salmon Population is chinook; the question is, what is the dispersion in total salmon counts that results from using chinook as the tracer?

The formal fit is:

Y = 1.50X + 106 with a dispersion of 97

This means that chinook counts can be used to predict the total Salmon counts to an accuracy of 97,000. Since the Salmon count ranges from 500,000 to 1 million, that means an accuracy of 10-20%. This suggests that, if you are only interested in total Salmon, you can use chinook as a reliable tracer, provided that you don't require accuracy better than 20%.

The fit as applied to the data is shown here. In this case, r =0.79 and the fit is a good fit. There are no strongly abberant data points.

Some Final Remarks

And so after this evolution we arrive at a crossroads, strongly driven by non-equilibrium growth, and we look for solutions about how to better manage the planet.

Much of the current dialogue in environmental studies or management needs to shift away from belief to a position of knowledge. The acquisition of knowledge requires gathering good data, analyzing it correctly, and then forming new questions on the basis of the data.

The Data Commandments: (Apply them often)

1. Always, always ALWAYS plot your data.

2. Never, never NEVER put data through some blackbox reduction routine without examining the data themselves.

3. The average of some distribution is not very meaningful unless you also know the dispersion. Always calculate the dispersion.

4. Always exam correlation data for points that could be rejected. Never reject them just because they are "too far from the line" but rather examine if poor measurement or some other error is responsible for these peculiar data values.

5. Always present and plot data without any compression in the axis so that you don't distort the data by fostering an unfair visual impression.

6. Always compute the level of significance when comparing two distributions. Just because they might have different mean values doesn't necessarily mean they are significantly different.

7. Always know your measuring errors.

8. Always require someone to back up their "belief statments" with data

9. Always calculate the dispersion in any correlative analysis and always look to see if the residuals correlate with another parameter

10. Always remember that unambiguous data resolves conflict.

GOOD LUCK