BIOS601 AGENDA: Tuesday September 08, 2015

Agenda for Tuesday Sept 08, 2015

• Discussion of computing and statistical inference issues in the assignment on sampling of space (surface of the Earth) and time (w.r.t. JH activity)
  
  answers to be handed in for ...
  
  Q1 [not required to do the R programming and sampling for part (i): use a made-up but realistic estimate],
  Q2 [again, not required to do the R programming and sampling for part (i): use made-up but realistic statistics],
  Q3,
  Q4,
  Q5,
  Q6
  
  The first (general) computing issue is (if need be) to get up to speed in the use of R. See the R links on the main course page. If you run into problems, let JH know asap.
  
  A statistical/computing issue might be how to come up with a way to randomly sample locations on the surface of a sphere, using latitude and longitude co-ordinates. See the notes at the bottom of the file containing the 2 R functions inside the Oceanography link (on the height of the land and the depth of the ocean) inside the resources for surveys. JH thinks of the problem by visualizing the segments of a peeled orange!
  
  Like the others, Q6 is blend of the theoretical and the practical. Here you are asked to use R to read in the .csv file, and to produce some summary statistics, and calculate some standard errors. You probably have not worked out the SE [sqrt(Variance)] for a ratio, but this is a good example of something often used in applied work. Hint: the log of a ratio is the difference of its components; the approx. variance of a log of a positive rv is (by the Delta method) the original variance times the square of the Jacobian or scaling factor, evaluated at/near the centre of the old scale. Think of the variance of September temperatures in F as the variance of September temperatures in C, multiplied by the square of the scaling factor... the scale of F is 9/5 ths larger than the scale of C. If the scaling is not linear (eg an elastic band) use the scale factor at the centre.
  
  Reference is made to the Finite Population Correction (FPC) for the sampling variance. This would apply in cases where you sample n (< N) of the N members of Canadian or U.S. Senate.
  
  It is written in slightly different version by different people, but JH tends to think of it's approx. value as (1 - n/N). To get the proper variance (assuming in our case that the target is just these 2 years, nothing else), the variance computed under an infinite population or sampling with replacement assumption needs to be multiplied by this less-than-unity factor, so that in the limit, if we sampled all n of the N, so that the FPC = 1 - n/N =0, the variance for our (census) estimate is 0.
  
  The form of the FPC can be derived for the binary response case using the ratio of the hypergeometric to Binomial variance: the binomial is for samples from an infinite population -- or a finite one but sampling with replacement-- whereas the hypergeometric is for samples from a finite one, but without replacement.
  
  Your other choice of method to sample the days from the scanned logs can be based on what factors you think most affect activity, and can be used in the sampling design. There is no one best one a priori (indeed the computerized data from 2010-2011 could be used to test out various designs/estimators but this is not required for the exercise, which is designed just to get you thinking about the issues.
  
  When using R (or another random number generator or random number table) keep track of how exactly you started the sequence (see set.seed in R).
  
  One of the savings to think about when entering data is discarding digits... what would be the effect if you only entered thousands of steps (rounded or truncated to an integer number of thousands) or hundreds of steps. Can you anticipate how much 'damage' is done by such approximations?
  
  Remarks: Some of the conceptual and practical statistical issues raised by this assignment include the distinction between standard deviation and standard error; the concept of a margin of error; when it is appropriate to use the Normal (Gaussian) approximation to the binomial distribution; the (often under-appreciated) centrality of the Central Limit Theorem (CLT) in applied statistical work, not just for the sampling distribution of a sample proportion, but also for that of a sample mean.
  
  Other points the exercise tries to make are that most of sampling theory involves calculation of variances. These derivations bring you back to fundamentals, and its good to be able to work them out from scratch rather than consult a textbook or the internet and copy the formulae blindly without having an understanding of what the formula should look like.
  
  You will notice that we have already starting calling the sqrt of the variance of a STATISTIC the STANDARD ERROR of that statistic. One key point about a standard error is that is referes to the variability of a STATISTIc, not of individual observations. Some writers reserve the term for cases where one uses a plug-in estimate into a variance formula. For example, they would call sigma/sqrt(n) the standard deviation of the sample mean, but they would call s/sqrt(n) the estimated standard deviation, or standard error for short. Bear in mind that this terminology is not standardized across the profession.
  
  One of the points of asking you to think about various sampling options to get at the 'census' or 100% answer is to get you to think of sampling as a measurement tool. Usually, the fancier and costlier the measuring instrument, the better the measurement. But we cant always afford the million-dollar answer and have to live with the hundred dollar answer.
  
  I once encountered a high-up Air Canada executive who didn't like the fact that the Canadian long-form portion of the census involved only a 10% sample, and he wasn's sampled. So he did not trust the results, since not everyone was surveyed. I asked him whether when a doctor drew a blood sample from him, he should give 100% of his blood so as to get an accurate concentration value.
  
  The Harper government has since done away with a the 1 in 10 random sample that was a compulsory component of the Canadian census, and replaced t with a 1 in 3 sample, so that the large refusal rate bigger sample size would still leave about the same n. What you you think of estimates based on this scheme, which has had only about a 65% participation rate? The participation rate in the old 1-in-10 sample was 99.999% and (despite Harper's claims to the contrary) only a handful complained or refused.
  
  A lot of psychological or psychometric measurement also involves sampling -- of items to use in a time-limited questionnaire or exam that can only contain a sample of the items that might be asked about (think of the format of the old paper-based GRE exams described in the Measurement Statistics Notes). The measurement model
                  observed estimate = TRUTH + error
  is the same whether the error comes for sampling of items or of persons or of time. In one case we might call it measurement error, and in another sampling error. But it could include both!
  
  Is it worth extracting and entering all of the step counts in all of their digits to have an answer that has too many decimal places for what one wants? And in so doing, we overlook other SEs -- ie statistical errors than do not decrease in magnitude as we increase the sqrt(n)! You can probably think of some in the case of the StepCounter!
  
  JH likes to say that besides standard error, SE could stand for many other types of error. It could be SAMPLING error, or STATISTICAL error, of STUPID error. Sadly, statistical theory is only good at quantifying sampling error, where sqrt(n) is always in the underside of the formula. BUT IT DOES NOT KNOW HOW TO JUDGE OTHER TYPES OF ERROR, and SOMETIMES THESE CAN BE A LOT LARGER THAN THE SAMPLING ERROR, AND THESE NON-SAMPLING ERRORS CANNOT BE MADE SMALLER BY INCREASING n. INCREASING n will just make the answer MORE PECISELY WRONG