Data Mining for Clinical Trials

I’ll be this June 7, speaking before Fundación de Investigación (http://www.fundaciondeinvestigacion.com/), research arm of University of PR’s School of Medicine.

I’ll be presenting on Data Mining in Clinical Trials and how we can use the Self-Weighting Model for modeling the extracted data. The audience will be medical doctors, medical technologists, and chemists.

As we can see, data mining is not just about crunching numbers for search engines and search marketing. It is all about using the same models, but for solving different problems.

Puerto Rico’s Science and Technology Trust Fund: Innovation Island Blast II

I was last night at the Innovation Island Blast II event from Puerto Rico Science & Technology Trust Fund. The event was held at the new Hotel Sheraton & Convention Center next to PR Convention Center. I have the opportunity of meeting personally great scientists and entrepreneurs. We chat for a while about science, technololgy, and research.

Unlike other so-called “entrepreneur experts” the speakers were the real deal. I’m glad I attended the event and learned about great success stories such as CDI-Lab, Fundacion de Investigacion, JumpStart, and other movers and shakers at the intersection of Science and Entrepreneurship.

Hats off to the Trust’s director, speakers, and organizers. BTW, great red wine and “sangria”.

The L’Hôpital Rule: Deriving the Geometric Mean

In two previous posts, we discussed the L’Hôpital Rule:

http://irthoughts.wordpress.com/2012/05/22/understanding-the-lhopital-rule/

http://irthoughts.wordpress.com/2012/01/23/lhopitals-rule-and-the-00-power-controversy/

This time we show how to use the L’Hôpital Rule to derive the geometric mean from the power mean.

Deriving the Geometric Mean

The power mean from a set of n numbers of x values is defined as

Power mean = [(sum of x^p)/n]^1/p

where p is any real number.

Taking natural logs,

ln(Power mean) = [ln(sum of x^p) – ln(n)]/p

With respect to p, the right side of this expression is of the form

f(p)/g(p)

where

f(p) = ln(sum of x^p) – ln(n)

g(p) = p

So, for p = 0,

f(p)/g(p) = 0/0

This indeterminate form asks for applying the L’Hôpital Rule, which reduces to evaluating

f(p)/g(p) = f’(p)/g’(p) in the limit where p vanishes (approaches zero).

Proceeding with the derivation,

f’(p) = (1/sum of x^p) [sum of (x^p)ln(x)]

g’(p) = 1

f’(p)/g’(p) = (1/sum of x^p) [sum of (x^p)ln(x)]

Now when p = 0,

f’(p)/g’(p) = (1/n) [sum of ln(x)]

However, since

sum of ln(x) = ln(product of x)

then

f’(p)/g’(p) = (1/n) [ln(product of x)] = ln(product of x)^1/n

Therefore, taking the antilog, we obtain the geometric mean,

Geometric mean = (product of x)^1/n

Limitations of the geometric mean

Evidently, to compute the geometric mean from n numbers of x values these must be positive, greater than zero, and not written as percentages.

In the next section, we describe workarounds for circumventing these limitations.

Circumventing the limitations of the geometric mean

For a dataset with a zero value, we may add a positive value, k, calculate the geometric mean and substract k to the result. We may use a similar approach for a dataset with a negative, but insuring that k is greater than the largest negative value of the set.

For example, to calculate the geometric mean of the values +12%, -8%, and +2%, we may add 1 to remove the negative value and calculate the geometric mean of their decimal multiplier equivalents of 1.12, 0.92, and 1.02, to compute a geometric mean of 1.0167. Subtracting 1 from this value gives the geometric mean of +1.67%.

Note that for a dataset consisting of percents, we divide each percent by 100, compute the geometric mean, and multiply the result by 100.

Additional workarounds are found in:

http://www.buzzardsbay.org/geomean.htm

http://www.waterboards.ca.gov/water_issues/programs/swamp/docs/cwt/guidance/3413.pdf

Have a mean day!

PS. Sorry. I fixed the last section to add an example describing the workaround.

Understanding the L’Hôpital Rule

Understanding the L’Hôpital Rule

If you are struggling trying to understand the L’Hôpital Rule, also called the L’Hospital Rule (with the ‘s’ silent), this post if for you. I assume you know basic calculus—at least what is a derivative.

Let f(x) and g(x) be two different functions of x, where f’(x) and g’(x) are their corresponding first derivatives.

Assume that we want to evaluate the f(x)/g(x) ratio when x = c, where c can be any real number. If direct substitution of c for x yields an indeterminate form, this obstacle is circumvented by evaluating the f’(x)/g’(x) ratio when x = c.

This is the so-called L’Hôpital Rule.

If the result still is an indeterminate form, we keep applying the L’Hôpital Rule to the result until said forms no longer are. To illustrate, consider the following example, where f(x) = (1 – cos x) and g(x) = x^2. If we evaluate the ratio f(x)/g(x) when x = c = 0,

f(x)/g(x) = 0/0

and this result asks for applying the rule. It can be shown that f’(x)/g’(x) when x = c = 0 yields

f’(x)/g’(x) =  (sin x)/(2x) = 0/0

So, we keep applying the rule. Then, f’’(x)/g’’(x) when x = c = 0 yields

f’’(x)/g’’(x) = (cos x)/2 = ½.

In an upcoming post, we will explain how to use the L’Hôpital Rule to derive the geometric mean from the power mean.

How to Create Windows Metro Style Apps with JavaScript

Neat:

How to Create Windows Metro Style Apps with JavaScript

More and more those that resist adopting JavaScript are deemed to fade away.