When analysing many real datasets, I have noticed that the most of correlations is positive. For example, here is a visualization of Pearson's correlation matrix for my current data. Is there any reason for this?

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EDIT: Matrix is computed from 500*140 dataset, where the most of variables are Likert items. Here is the distribution of correlations.

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And here is example from another area. Vitamin and mineral contents were measured in different food samples (of the same weight).

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    $\begingroup$ They are supported by the Dark Energy. $\endgroup$ – ttnphns Jan 19 '14 at 10:49
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    $\begingroup$ Because people tend to collect data on variables that are positively correlated. $\endgroup$ – Peter Flom Jan 19 '14 at 11:51
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    $\begingroup$ It would be easier to see the distribution of the correlations if you just plotted a histogram of them (of course the correlations on the diagonal are 1 - and if they were slightly positively skewed it would make it appear double by portraying both above and below the diagonal at the same time). Of course if you actually said what your data represent you would likely get more tailored answers as well. $\endgroup$ – Andy W Jan 19 '14 at 13:57
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    $\begingroup$ (1) To slightly rephrase @Peter, people tend to direct their measuring scales so that the majority of the measured phenomena correlate positively. (2) People understand positive values and magnitudes more easily than negative. (3) In psychometry (e.g. when asked to rate smth.) there often shows up the so called g(eneral)-factor which is one of the basic cognitive context-appraisal facility. $\endgroup$ – ttnphns Jan 19 '14 at 14:40
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    $\begingroup$ Similar in spirit to this question. The meaning of the notion that "things in general" are positively correlated is hard to pin down. $\endgroup$ – Scortchi - Reinstate Monica Jan 19 '14 at 16:06

Although ttnphns's comment is slightly in jest - it actually has bearing on your question. We may consider different phenomenon as being caused by a set of related factors (which may or may not be measured). So for example say we have a latent factor of $\lambda$ that affects responses to a set of Likert items on a survey.

$$\begin{align*} y_1 = 0.5\lambda + e \\ y_2 = 0.7\lambda + e \\ y_3 = 0.6\lambda + e \end{align*}$$

In this example $y_1$, $y_2$ and $y_3$ will all have a positive correlation because they are all related the same way through $\lambda$. For many datasets it may be that many of the items have some variable that is underlying in common. For example in the vitamin and mineral contents if the food samples are of different size I would expect more vitamins and minerals for larger food samples, making the marginal correlations of each positively correlated. Another explanation might be producers that intentionally increase vitamin content also increase mineral content (as they aren't really competing with one another and may be marketed as healthy foods).

In the case of Likert items, as Peter Flom stated in a comment, we typically construct the survey to identify these underlying latent factors, so it is by construction that many items are positively correlated. Also the anchors are somewhat arbitrary, but questions stated positively (e.g. "Do you support the death penalty?") tend to be measured more accurately than negated questions (e.g. "Do you not support the death penalty?"). It is also the case that you could assign different numeric values to the Likert items, but it is typical to have a scale of $1$ to $n$ (with $n$ being the different potential responses) as the default for coding the values.

Note you could arbitrarily flip this coding though, so if all of the correlations in the sample were positive, you could flip half the variables so the correlations were equal. Often times there is an arbitrariness in how we represent values, e.g. if you have a nominal category of men and women you could set $\text{men} = 1$ and $\text{women} = 0$ or you could do it the obverse way. Again people may make these arbitrary coding decisions to make items appear to have positive correlations.

  • $\begingroup$ To be more precise, all food samples have the same weight, so values are relative. $\endgroup$ – Miroslav Sabo Jan 19 '14 at 15:12
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    $\begingroup$ @Miroslav: Water content, higher level of metabolic activity in the particular tissue? $\endgroup$ – Scortchi - Reinstate Monica Jan 19 '14 at 15:54
  • $\begingroup$ @Scortchi, sorry, I do not understand. I have used data like this static.happycow.net/images/table_fruits_vegetables.jpg $\endgroup$ – Miroslav Sabo Jan 19 '14 at 16:27
  • $\begingroup$ @Miroslav: I meant that you'd expect some foodstuffs to have higher concentrations of vitamins & minerals across the board. Cheese vs milk (as cheese is concentrated milk), or pork liver vs pork belly (as liver has a higher level of metabolic activity than fat, & many vitamins & minerals are enzymes' cofactors). $\endgroup$ – Scortchi - Reinstate Monica Jan 19 '14 at 23:28
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    $\begingroup$ What happens in long surveys is people fill them out rote, so if you stick a question in there that a respondent would answer the opposite way of the majority of questions they might miss it more often. It is somewhat a more general problem of getting people to fill out long surveys to a certain extent - as well as evaluating negatively stated questions are more difficult IMO. $\endgroup$ – Andy W Jan 21 '14 at 17:02

To expand on Scortchi's/AndyW's point of confounding factors:

For the food stuff I think the water content is an extremely important confounding variable. In particular, if you mix in fruits that are naturally eaten with very high water content (tomatoes, cucumbers) with fruit where the "natural" state is already dried (raisins) and which therefore contain more of about everything, the huge difference in the water content can actually influence the correlation.
The effect becomes very clear if you consider a small table that just lists raisins and grapes...

Note that water is not listed in the table, so the negative correlations are just not shown. So another reason (in addition to @Peter Flom's comment) is that the way people tabulate data can also emphasize positive correlations: if you want to know the water content, you just have to subtract the proteins, lipids, carbohydrates (depending on the way carbohydrates are listed also fiber) from the 100 g raw weight - the information is redundant. But because the water content is for these tables of less interest that the other nutrient contents, the subtraction is left to the reader.

And then, we actually know certain (co)relations in the data, e.g.

  • the energy content for proteins and non-fiber carbohydrates (both 17 kJ/g) and lipids (37 kJ/g) etc. is well known, and the total energy is usually just calculated as the sum of all those contributions
  • Na⁺ to K⁺ concentrations are similar among plants and among animals (much higher difference between plant and animals: plants have comparably more K⁺)
  • These tables sometimes list subcategories which then obviously have an upper bound. Consider

    • carbohydrates,
      • thereof mono- and disacharides
    • lipids
      • thereof saturated lipids

    This relation tend to produce positive correlations as well, which is again caused by the way we group and tabulate our data.


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