Super Useful Sampling Distributions Applet

Similar to the applets I used for my P211 research methods class, there is an online program which allows the user to specify a population distribution, and then build a sampling distribution of statistics such as mean, median, and variance. When I was first starting out I had a difficult time grasping what exactly a sampling distribution was, or what it meant, exactly; but tools like this are great for visualizing the process and building an intuition about what's really going on. The result is, I still don't understand it - like, at all - but I sure as hell feel more confident. And that's what is really important.

Stats Videos (Why do you divide samples by n-1?)

Because FMRI analysis requires a strong statistical background, I've added a couple videos going over the basics of statistical inference, and I use both R and Excel to show the output of certain procedures. In this demo, I go over why the sums of squares of sample populations are divided by n-1; a concept not covered in many statistical textbooks, but an important topic for understanding both statistical inference and where degrees of freedom come from. This isn't a rigorous proof, just a demonstration of why dividing by n-1 is a unbiased estimation of sample variance.

Bootstrapping

As I am covering bootstrapping and resampling in one of my lab sections right now, I felt I should share a delicious little applet that we have been using. (Doesn't that word just sound delicious? As though you could take a juicy bite into it. Try it!)

I admit that, before teaching this, I had little idea of what bootstrapping was. It seemed a recondite term only used by statistical nerds and computational modelers; and whenever it was mentioned in my presence, I merely nodded and hoped nobody else noticed my burning shame - while in my most private moments I would curse the name of bootstrapping, and shed tears of blood.

However, while I find that the concept of bootstrapping still surpasses all understanding, I now have a faint idea of what it does. And as it has rescued me from the abyss of ignorance and impotent fury, so shall this applet show you the way.

Bootstrapping is a resampling technique that can be used when there are few or no parametric assumptions  - such as a normal distribution of the population - or when the sample size is relatively small. (The size of your sample is to be neither a source of pride nor shame. If you have been endowed with a large sample, do not go waving it in the faces of others; likewise, should your sample be small and puny, do not hide it under a bushel.) Say that we have a sample of eight subjects, and we wish to generalize these results to a larger population. Resampling allows us to use any of those subjects in a new sample by randomly sampling with replacement; in other words we can sample one of our subjects more than once. If we assume that each original subject was randomly sampled from the population, then each subject can be used as a surrogate for another subject in the population - as if we had randomly sampled again.

After doing this resampling with replacement thousands or tens of thousands of times, we can then calculate the mean across all of those samples, plot them, and see whether 95% of the resampled means contains or excludes zero - in other words, whether our observed mean is statistically significant or not. (Here I realize that, as we are not calculating a critical value, the usual meaning of a p-value or 95% confidence interval is not entirely accurate; however, for the moment just try to sweep this minor annoyance under the rug. There, all better.)

The applet can be downloaded here. I have also made a brief tutorial about how to use the applet; if you ever happen to teach this in your own class, just tell the students that if the blue thing is in the gray thing, then your result fails to reach significance; likewise, if the blue thing is outside of the gray thing, then your result is significant, and should be celebrated with a continuous bacchanalia.

Computational Modeling: A Confession

In a desperate attempt to make myself look cool and connected, on my lab webpage I wrote that my research

...focuses on the application of fMRI and computational modeling in order to further understand prediction and evaluation mechanisms in the medial prefrontal cortex and associated cortical and subcortical areas...

Lies. By God, lies. I know as much about computational modeling as I do about how Band-Aids work or what is up an elephant's trunk. I had hoped that I would grow into the description I wrote for myself; but alas, as with my pathetic attempts to wake up every morning before ten o'clock, or my resolution to eat vegetables at least once a week, this also has proved too ambitious a goal; and slowly, steadily, I find myself engulfed in a blackened pit of despair.

Computational modeling - mystery of mysteries. In my academic youth I observed how cognitive neuroscientists outlined computational models of how certain parts of the brain work; I took notice that their work was received with plaudits and the feverish adoration of my fellow nerds; I then burned with jealousy upon seeing these modelers at conferences, mobs of slack-jawed science junkies surrounding their posters, trains of odalisques in their wake as they made their way back to their hotel chambers at the local Motel 6 and then proceeded to sink into the ocean of their own lust. For me, learning the secrets of this dark art meant unlocking the mysteries of the universe; I was convinced it would expand my consciousness a thousandfold.

I work with a computational modeler in my lab - he is the paragon of happiness. He goes about his work with zest and vigor, modeling anything and everything with confidence; not for a moment does self-doubt cast its shadow upon his soul. He is the envy of the entire psychology department; he has a spring in his step and a knowing wink in his eye; the very mention of his name is enough to make the ladies' heads turn. He has it all, because he knows the secrets, the joys, the unbounded ecstasies of computational modeling.

Desiring to have this knowledge for myself, I enrolled in a class about computational modeling. I hoped to gain some insight; some clarity. So far I have only found myself entangled in a confused mess. I hold onto the hope that through perseverance something will eventually stick.

However, the class has provided useful resources to get the beginner started. A working knowledge of the electrochemical properties of neurons is essential, as is modeling their effects through software such as Matlab. The Book of Genesis is a good place to get started with sample code and to catch up on the modeling argot; likewise, the CCN wiki over at Colorado is a well-written introduction to the concepts of modeling and how it applies to different cognitive domains.

I hope that you get more out of them than I have so far; I will post more about my journey as the semester goes on.