So sometimes we care about competing risks and sometimes we don’t. Sometimes we just want to see if Anton will ever get caught, either by Randy or someone else. So what we can do is look at a Kaplan-Meier curve. Now, lets say we focus on ten dimensions, and in two of them, Anton gets caught. He gets caught at 25 months in one dimension and at 40 months in another dimension. So we can plot something called a Kaplan-Meier curve that looks like this.
And we can see that if we follow the dimensions for 50 months, Anton has about a 55% percent chance of getting away scot free. Now, I know what you’re thinking. It’s Black Friday and Cyber Monday is coming out so of course, I’m pimping my trilogy again. No! You’re thinking if Anton only gets caught in two of the ten dimensions, why are his chances of getting away only 55% and not 80%? Well, that’s because some dimensions are followed only 10 months or 20 or 30 or 35 or so on. These dimensions are referred to as censored, meaning we do not know what happens in those realms after we stop following them, so we drop them out from our final probability at 50 months. Of course, Anton could have gotten away in all ten dimensions, so that our curve could have just looked like a straight line at 100%. Is that the case? Well, you know Black Friday and Cyber Monday are here so … okay, I’m stopping. Until next time.
Sorry — that’s it. I promise.
So today I’m keeping my post short and sweet when discussing competing risks. So what does that entail, you ask. I will tell you, I say! Basically, it’s when we want to look at when an event occurs but we may never know because another event occurs first.
Like we may want to know when Randy captures spy Anton in a particular dimension, only to find out he was caught by another agent or by mafia Anton or by whoever. So does Randy ever catch spy Anton in any dimension? Are you intrigued yet? Ready to get the trilogy yet? No? Not yet? Okay then — I’ll try again to sell you next time. Until then, just another glimpse of what else is in the trilogy.
Yes, sirree! It’s a reindeer ride in the Finnish country side! Now, are you intrigued? No? Okay, okay … I’m stopping … for now!
So after we stratify the data, who can we randomize the subjects? Well, that’s kinda the fun part! So going back to my trilogy (and yes, I just heard you groan — and I’m not amused!), say we divide the justified realms into east (Atlanta, Detroit, and Philadelphia), and west (Houston, Indianapolis, and St. Louis), regions and sample them 5 times to see where each of our 5 characters and their alters end up. So doing that, I got 5 east region dimensions and 5 west region dimensions.
[,1] [,2] [,3] [,4] [,5]
[1,] “Philadelphia” “Atlanta” “Detroit” “Detroit” “Atlanta”
[2,] “Houston” “Houston” “St. Louis” “Indianapolis” “Houston”
Which you can see are pretty balanced as opposed to say getting a sample of 10 dimensions all together like below, where we see that the east region dimension can be grossly over-represented.
[,1] [,2] [,3] [,4] [,5]
[1,] “Atlanta” “St. Louis” “St. Louis” “Detroit” “Atlanta”
[2,] “St. Louis” “Philadelphia” “Philadelphia” “Detroit” “Atlanta”
Isn’t that fun? Well, I thought it was fun! You know what is fun? Asking my friend, Erika, my publicist, Jessi, or my colleague, Mary, to randomly pick a number between 1 through 6 so I can choose that radio preset to listen to for the day. Yes, I do realize that my definition of fun can be indeed very odd. But hey, they did make some superb choices. Like the other day, Jessi picked a number corresponding to a station playing Taio Cruz’ Dynamite, which always makes me think of this meme, which is such a fun meme, I must add.
Yeah, again, I might have to re-examine my definition of fun. I know that. But join me next time, for some really really real fun! (Okay, what my and 0.02% of world’s population of fun is at least.)
So often times in my line of work or in any statistics-related stuff, we want to divide a sample population into two or more groups evenly and make sure the groups are the same in every aspect except for the outcome we are studying. So that is why we often randomly assign people to different groups. But we also might want to stratify by gender and then randomize to make sure both groups are have about the same number of boys and girls in them.
Or like in my trilogy, some characters can be stratified geographically before we allot them to a good or bad (justified or unjustified) dimension. So in a good dimension, a character may in Atlanta but in a bad dimension, they’re in Tampa. Or in a good dimension, a character may in Philadelphia and in NYC in a bad dimension. In either case, I guess we could say the South and Northeast are equally divided between good and bad dimensions then. Although why would you want to be in a bad dimension? Well, again, you know what to do if you wanna find out.
So then how can we randomize after we stratify? Well, we can cover that next time. So I’ll have a topic to cover next time, won’t I? Till then, I leave you with this image.
As I did find a lot of big dog vs. little dog images when googling stratified random allocation.
Out of This Dimension 