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handout 4: sampling theory

ok, we've concluded the preliminaries, we've done the first experiment, 
we're shifting gears now into some new stuff.

essentially, in what's done in medicine as well as other endeavors 
involving making decisions under uncertainty, there are two categories of 
things:
one, clinical trial type thing, where experimenter has complete control 
over experiment - can fully control everything, determine what's being 
tested, assign people to different groups, etc. and this is a whole realm 
of research - fully controlled clinical trials - and mostly this is what 
we read about when looking at the journals.

another whole separate area - doesn't exclude clinical research but is 
more epidemiological, looking at diseases, infections, retrospective 
studies, etc. you no longer have control. this isn't randomized. you 
can't ASSIGN soeone to have a dz - either they have it or they do not. so 
you're in an observational mode measuring attributes that already exist 
in the population. you have no control over it. so you can't measure 
EVERYONE OUT THERE, right? i mean, you can't find ALL diabetics on the 
face of the earth and test them....so just as in randomized design where 
you use small number and extrapolate, you do the same thing here. you 
draw a sample. eg, on election day, when some votes are in, you make a 
prediction over who will win, and the next day you see if you were right. 
now, in epidemiology, you NEVER KNOW for sure that you're 100% right.

how do you draw conclusions w/o ever seeing everyone or getting a hard 
return of facts?

by drawing a small sample from a very large pool - eg, millions of votes 
are being cast, but you only have access to a thousand or 1100 out of 
those millions - with these thousand votes or 1100 votes, you can obtain 
with a near exact probability basis what the results will be. so if you 
want to draw a sample the night of the election that is 99% accurate, it 
can be done. how? well, 99% of the time you draw the sample - doesn't 
matter where - if many samples done in different venues, 99% will be 
right, and one percent will err. eg, on election night, some anchor will 
say they're calling the election - or they will say it is too close to 
call, due to the 'accuracy phenomenon'

if you generalize this to research, there are multiple populations of 
people, etc, and there are millions, and you can't measure them all...but 
the mechanism is the same, the theory is the same. so what we'll do now 
is he'll show us how if you have to estimate to true value of something 
in apopulation, be it of votes or diseased people, you can do 
exceptionally well by taking a small sample and you never need to examine 
everyone. 

some woman was upset that she had never been asked for her opinion in an 
opinion poll, and how could these polls be valid if no one ever asked 
her? well, said the expert, when we sample for things, there are usually 
only two or three possible answers. if it's only yes/no/maybe, it isn't 
necessary to contact EVERYONE. you draw a sample....and the individuals 
are represented within the sample. so...how many people are there out 
there with a given dz? millions? but we don't have to observe them all. 
just some.so...

periodontal dz in animals example...
(note: handouts 4-9 are like a textbook, not like lecture outline. so 
take notes) anyway.... you start out in descriptive mode and introduce a 
concept like periodontal dz in dogs.

pop1: young

pop2: middle age

pop3: old

the theory is at least epidemiologically, re: is this a problem in dogs, 
at least from a scientific standpoint to go out in the beginning and see 
how it is affected by age, it's not surprising that it follows the human 
trend....so you would expect as the age increases, periodontal dz would 
get worse, right?

so it shouldn't surprise anyone if the graph of results for each group 
was a bell curve shifted to the right as age increases.(graphing the 
distance bet CEJ and gumline). so distance would increase with age, on 
average.

in general this is probably not a bad depiction. but, let us consider for 
the sake of discussion a different possibility, just to give us some 
idea, however contrived, of how things can be misleading in some 
situations.
say you have bell curve w/avg distance = zero for young dogs. 
then for middle aged dogs, you have an asymmetric curve, with peak 
shifted to left, but with longer extension to the right - eg a skew to 
the right with a hump to the left, and then for old dogs a BIG hump to 
the left, and a huge skew to the right (eg, very few animals with very 
SEVERE dz. but in all cases the MEDIAN is the same.  in young dogs, mean 
and median are same, as is mode. but in middle aged animals, you have a 
skewed high point, so it's monotonic, nonsymmetric, single mode...so 
average value moves toward the extreme value.
so the median is less than the average. median depicts EXACT CENTRAL 
TENDENCY. the AVERAGE accounts in for extreme values. mode, recall, is 
most frequently occuring value. 

so in a distribution of disease, the average value under a symmetric view 
is superimposed over the median, but when there are outlyers, the mean is 
pulled toward the extreme value, eg there is a distortion of central 
tendency. central tendency is supposed to represent to the veiwers of the 
data what the central tendency is. the median shows central tendency 
regardless of bizarre extremes. always half in front and half behind it. 
eg, a single value best describing a larg number of values for LOCATION, 
not spread. now the AVERAGE or MEAN is easily distorted by a few extreme 
values - when you lose symmetry, you get a distorted view of the data. 

eg if median income is 30K/family, the average might be 45k/family. now, 
50% of people are making 30K or less or more, but a few people are making 
a million or more, so the AVERAGE is higher. this is the point of central 
tendency. but usually, the media give you the average, not the median.

so in the dog thing, the median stays the same, but because a few dogs 
get bad dz, so the average would lead you to believe that OVERALL as dogs 
age they get worse dz - but in our depiction, on median, dogs stay same, 
but a select few get dz. (in fact, this depiction is bogus, but it's just 
an example. in real life, whole bell curve shifts to right.)
so data can be misleading, and using the average and not the median can 
be a mistake.

suffice it to say that it is reasonable to compare distributions as long 
as they are comparable. if you ahve to use averages or to compare skewed 
distributions, it's ok as long as populations are relatively similarly 
skewed. however, in extreme cases eg if population one is skewed with a 
right hump and a left extreme, and population two is skewed with a left 
hump and a right extreme, your averages would be mistakenly calculated to 
be very far apart, but that would be wrong.

another view of comparative distributions: differential variation as 
opposed to central tendency or location.

can draw two distributions on top of each other. same average value but 
pop1 is wide bell curve and pop2 is narrow bell curve. by inspection 5% 
of observations are small areas under edges of wide curve and small areas 
under edges of narrow curve. therefore 90% are in middle. in pop1 it's 
wide, dispersed, and in pop2 it's very narrow. so it would seem that in 
pop1 there are more lows and more highs - why? because it is so 
dispersed, that only very far from the mean do you start getting to that 
last 5%, whereas in pop2 the 5% part starts much earlier. 

now, for each curve you could calculate as in handout 3 - the VARIANCE, 
the square root of which is standard deviation. (see handout 3 for 
equation).
in the distribution which is more dispersed the variance will be much 
larger than in the non-dispersed distribution. so the standard deviation 
of the narrow distribution will be smaller. seems logical. 

now.
let's take another step and investigate another handout3 concept. these 
are preliminaries to handout 4 which we shoudl get to next time.

remember from handout 3 our friend the Z value? what was it for? for 
individual scores varying about a mean, you could calculate a Z value 
which if the distribution were normal would allow us to calculate areas. 
this will be revisited in handout 4, but we won't look at individual 
values. before that, we need to get some notation and a concept....

when we say a distribution is normal...there is notation used to write 
that down. by convention, to say whether or not a system of scores is 
normal, we say that Xi is distributed as normal curve with mean mu and 
variance sigma squared.  sorry, can't write the equation with this text 
editor, hope you wrote it down :)

Some polish mathemetician pointed out that P[(Xi-µ)„ k*sigma ¾ 1/k^2
irrespective of shape of distribution, if you consider departure of score 
from mean relative to a number of standard deviations, can guarantee 
probability of particular score deviating from mean being no greater than 
1/k^2

i have no bloody buggering idea what he's talking about.

say you're two standard deviations from mean. K=2. 1/2^2 = .25, so you 
can't be farther than that from the mean. [what?] 
without knowing anything and excluding the normal, this is guaranteed as 
an upper bound. you can do no worse. 

conceptually, the analogy to z is k. the nubmer of standard deviations a 
score is from the mean. if it isn't normal, you don't know what it is, 
you can use this equation that was developed by Chebyshev.



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