On the Additivity, or Lack of it, of Variables

Ignoring the additivity nature, or lack of it, of a variable can invalidate any statistical treatment. This is part of a research paper on the Self-Weighting Model (SWM) that I’m writing. I presented a sneak preview on this during a recent seminar before Fundación de Investigación, a local research company dedicated to clinical trials.

In general, if x is a non-additive variable, we cannot:

• obtain a mean (arithmetic average) x value from a set of x’s.
• calculate a standard deviation from a set of x’s.
• mean-center a set of x’s by subtracting a mean score from each x.
• standardize a set of x’s by subtracting a mean score from each x and dividing each by a standard deviation score (i.e., convert each x into a z score) .
• compute the L1-norm (Manhattan, taxicab distance) from a vector whose elements are a set of x’s.

[Added on 6-21-2012] In addition for said variable, we cannot:

• take the difference between any two pairs of x values.
• compute a coefficient of variation (standard deviation/mean), from a set of x values.
• compute a mean difference from any two sets of x values.
• compute a pooled standard deviation from any two sets of x values.
• compute a Cohen’s d (mean difference/pooled standard deviation) from any two sets of x values.

In a math/statistics scenario, the following are non-additive: slopes, cosines, sines, tangents, and ratios (e.g., standard deviations, correlation coefficients, beta coefficients, coefficients of variations, Cohen’s d, etc).

In an experimental sciences scenario, the following are non-additive: any intensive property, any ratio of extensive properties (e.g., density = mass/volume), any dissimilar ratio, etc.

The information presented in the aforementioned seminar is applicable to many dissimilar fields, including web analytics, data mining, information retrieval, and almost any research field that requires of numerical analysis of experimental variables.

Unfortunately, from time to time we see research articles published wherein the additivity/non-additivity nature of variables is ignored and data crunching and analysis is arbitrarily carried out. The result: sloppy approximations, invalid models, and erroneous forecasts.

PS. There might be counterexamples to the notion of classifying properties as intensive or extensive. Very few properties are neither one. There are also some cases wherein instrumental lectures (not properties) are subtracted in order to compute signal responses or signal-to-noise ratios, derivatives, numerical analysis, etc, but for the most part the above holds in the physical world.

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What is the more effective way of writing scientific research articles?

Over the years, I’ve been asked about the more effective way of writing peer-reviewed articles for scientific journals.

My response is always the same: Think like a referee/editor. Here is a list of items that they want to see accomplished:

Referees/editors like to see that the content and format of the title, abstract, document body, tables, images, graphics, appendices, and references follow their journal guidelines.

In general, referees/editors like to see in the first page of the printed version of an article:

1. Statement of the problem – what is the problem to be solved.
2. Purpose of the article – how the present research solves the problem.
3. Organization – how the article is organized and what is covered in each section.

This is a general practice across scientific journals. So, whenever possible, I try to accomplish 1 – 3 in the first three paragraphs of the first page of the printed article. To do this, you need to avoid lengthy introductions and wordiness. Be concise and ‘go the point’.

Referees/editors also like to see the article as a whole semantic unit. So they like to see:

Transitional statements; i.e., sections ending as an introduction to the next section.

1. One paragraph, one idea; i.e., each paragraph discussing one main idea.
2. Short paragraphs; i.e., each paragraph of about five sentences or less, where sentences are of appropriate length. This provides a natural stop to the reading. In general, short paragraphs and sentences are easier to read than the long ones. Use compound sentences with caution.
3. Facts supported by pertinent references.
4. Opinion written as opinions, not as facts.

Of course, there are other tips to think about, but in my opinion, the above can make a difference… well, in my opinion 🙂

When Big Data leads to Big Errors

The hilarious picture above shows how some SEOs look when playing to be scientists. This often occurs when interpreting big data.

Few specific scenarios:

1. Applying the statistical theory of small samples to extremely large samples, like …
2. …using large amount of data to force very small correlation coefficients to become statistically significant.
3. Trying to arithmetically average ratios (like correlation coefficients, standard deviations, slopes, and cosine similarities).
4. Mistaking Cauchy Distributions for Normal Distributions.
5. Adding together intensive properties.

Fortunately, I know of good folks that are doing a great job at educating their search marketing peers (Mike Grehan, Bruce Clay, Danny Sullivan, etc) without playing to be scientists.

Cauchy Distributions: Why you should never arithmetically average ratios

I’m putting the final touches to the second article on SWM to be submitted soon to a statistical journal.

In it I expand on the mathematics behind SWM to show why one should never try to compute arithmetic averages from ratios of the a/b form and similar forms. There are many reasons against doing this: Cauchy Distributions, additivity arguments, Hölder’s Inequality concepts, etc.

The paper kills for good the notion of arithmetically averaging correlation coefficients and other type of ratios frequently encountered in statistical analysis.

Part of this will be part of my presentation on Data Mining for Clinical Trials before Fundación de Investigación.