Tag Archives: maximum likelihood estimation

TWSB: Happy birthday, Sir Ronald Fisher!

Happy birthday to one of the greatest statisticians ever: Sir Ronald Fisher!

Fisher (1890 – 1962) was an English statistician/biologist/geneticist who did a few cool things…you know…like CREATING FREAKING ANALYSIS OF VARIANCE.

Yes, that’s right kids. Fisher’s the guy that came up with ANOVA. In fact, he’s known as the father of modern statistics. Apart from ANOVA, he’s also responsible for coining the term “null hypothesis”, the F-distribution (F for “Fisher!”), and maximum likelihood.

Seriously. This guy was like a bundle of statistical genius. What would it be like to be the dude who popularized maximum likelihood? “Oh hey guys, I’ve got this idea for parameter estimation in a statistical model. All you do is select the values of the parameters in the model such that the likelihood function is maximized. No big deal or anything, it just maximizes the probability of the observed data under the distribution.”

I dealt with ML quite a bit for my thesis and I’m still kinda shaky with it.

I would love to get into the heads of these incredibly smart individuals who come up with this stuff. Very, very cool.

Data, data everywhere and not a model to fit

Things a normal person does to relax:
– sleeps
– hangs out with friends
– copious amounts of alcohol
– screws around

Things Claudia does to relax:
– ignores sleep
– locks herself in her apartment
– copious amounts of Red Bull
– fits a structural equation model to her music data

Yeah.

I’ve spent a cumulative 60+ hours solely on my thesis writing this week, and considering all the other crap I had to finish, what with the semester ending and all, that’s a pretty large amount of time.
Despite that, it’s pretty sad that I spent my first few hours of free time this week fitting an SEM to my music.
BUT IT HAPPENED, so here it is.

With “number of stars” the variable I was most interested in, I wanted to fit what I considered to be a reasonable model that showed the relationships between the number of stars a song eventually received from me (I rarely if ever change the number after I’ve assigned the stars) and other variables, such as play count and date acquired. Note: structural equation modeling is like doing a bunch of regressions at once, allowing you to fit more complicated models and to see where misfit most likely occurs.

Cool? Cool.

Onward.

This is the initial model I proposed. The one-way arrows indicate causal relationships (e.g., there is a causal relationship in my proposed model between the genre of a song and the number of stars it has), the double-headed arrow indicates a general correlation without direction. Oh, and “genre” was coded with numbers 1 through 11, with lower numbers indicating my least favorite genres and higher numbers indicating my favorite genres. Important for later.

Using robust maximum likelihood estimation (because of severe nonnormality), I tested this model in terms of its ability to describe the covariance structure evident in the sample (which, in this case, is the 365 songs I downloaded last year).

So here’s what we got!
Satorra-Bentler scaled χ2(7) = 9.68, p = 0.207
Robust CFI: .992
Robust RMSEA: .032
Average absolute standardized residual: 0.0190

All these stats indicate a pretty awesome fit of the model to the data. This is shocking, considering ridiculous non-normality in the data itself and the fact that this is the first model I tried.

Here are the standardized pathway values (analogous to regression coefficients, so if you know what those mean, you can interpret these), with the significant values marked with asterisks:

So what’s this all mean? Well, in general, the relationships I’ve suggested with this model are, according to the stats, a good representation of the actual relationships existing among the variables in real life. Specifically:
– There is a significant positive relationship between genre and play count, which makes sense. Songs from my more preferred genres are played more often.
– There is a strong positive relationship between play count and stars, which also obviously makes a lot of sense.
– The significant negative relationship between date added and play count makes sense as well; the more recently downloaded songs (those with high “date added” numbers) have been played less frequently than older songs.
– There is no significant correlation between genre and song length, which surprises me.
– Genre, length, and play count all have significant, direct effects on how many stars I give a song.
– Another interesting finding is the positive relationship between stars and skips, which suggests that the higher number of stars a song has, the more often it is skipped. Perhaps this is just due to the sheer number of times I play the higher-starred songs. Who knows?

Yay! Fun times indeed.