Agenda for Aug 25 to Sept 08, 2023

[updated August 24, 2023 --please notify JH if you encounter any glitches]
Topic: The Quality of Measurements and the Effects of Measurement Error.
 
Preamble: This topic of measurement is probably new for you, as it was for JH when he began in cancer clinical trials in 1973, and oncologists (cancer doctors) were judging responses of advanced cancer to chemotherapy by measuring tumours by 'palpation'.

Re Q1 and Q2: In early measurement courses JH gave, students measured readability manually by counting the lengths of words and sentences, and the number of syllables in words. Early on in bios601, the task became much easier using online (and online textxs) and online tools, or those in Microsoft Word. From 3 measurements of readability, they calculated the standard error of measurement as the SD of the 3. The CV is the SD divided by the mean of the 3, expressed as a percentage.
standard error of measurementsince everyone knows what 0.7 of a grade is. But if the scale is arbitrary (running from say 0 to 70) a SEE of 9 'points' is more difficult to judge, unless one knows well what a '40' or a '20' is. In this case the ICC is more useful, but it requires that one has > 1 measurement each on each of several texts of different difficulty .. so one can judge how much is genuine 'between-text' and within-text' variation.

For validity students went to the library (or looked around at home, or online) for books of different difficulty so that we could see if the measurements agreed well with what difficulty experts though the difficulty of each book was. It was not like the study in Q16 where 500 meant 500 or 1500 meant 1550 and all would agree on this 'gold standard'. Unlike in physical measurements, this issue of an independent gold standard' is a challenging one in psychometric measurement. It's not like you can order 'a grade 6' book from the US National Institute of Standards and Technology (NIST) the way you can order a substance with a known cholesterol concentration, or a 1Kg weight.

One strategy we used more recently was to look online for lists of books recommended by teachers for children in different grades, and our job was made easier if we could find the texts themselves online, and simply cut and paste samples of them into MS Word or an online 'readability' tool. Some years, we used the full range from 'the Cat in the Hat' to university texts, and plotted the measurements against the grade or age level. If you measure a text with different instruments or tools, and if they have a common scale (e.g. grade level) then you could use a linear model to estimate how systematically (if any) they vary from one to another. If you think of these tools as the only ones available (like iPhone vs Android) then you should treat them as 'fixed' effects. If one the other hand they are a sample of the many tools 'out there' then a random effects model might be more appropriate.

Even though we will use examples involving the measurement of physical quantities such as activity and biochemical parameters, JH kept the readability example in Q1 and Q2 as a reminder of the early psychometric focus on the quality of measurements. The 2002 article, referred to in Q29, very much brings us into the computer era

Week of Aug 26- Sept 1 (optional, but strongly recommended)

Upload (via MyCourses) your answers to however much/little you are able to do of

    Q3, Q4, Q5, Q6, Q7[PhD]

by Friday Sept 1.

JH will go over some of these in class in Wed Aug 30. You can also look/listen back ot audio/videos from past years, where some of this material has been covered. {Qs 3, 4, 5 abd 7 are 'just algebra', while Q6 is asking you to think about how the 'numbers' came to be.} [Even if you cannot get to Qs 3-5, please do look at page 16, and results 1 and 2 on page 22 of the Notes before the 1st class on Aug 31.]

Q3 [ 'm-s' is JH shortand for 'math-stat' ].

The point of asking you to derive the link is to emphasize that the Standard Error of Measurement and R are not ENTIRELY separate. Yes, the Standard Error of Measurement is more limited, because it does not tell you how much variation there is from person to person (or object to object). But you can think of R or the ICC as the proportion of the OBSERVED variance that is 'real' (i.e. due to genuine person to person variation), and think of the remainder, 1 - R, as the square of the Standard Error of Measurement.

(1) The square of the Standard Error of Measurement is ONE of the TWO components in the observed variance. (2) The square of the genuine between-person variance is the other. The ICC is (2) as a fraction of (1)+(2).

A good example of the 2 concepts can be found in the blurb from the Educational Testing Service called INTERPRETING YOUR GRE SCORES, page 9 of the Notes.

Q4

Relationship between test-retest correlation and ICC(X). The point here is to see the same concept from two different perspectives.

Q5

Relationship between correlation(X,X') and ICC(X): Some people like this explanation of the ICC, since it echoes what was said above about the ICC as the proportion of the variance we observe in an imperfectly measured characteristic that is 'real'. Think of a correlation as another way to measure how strongly an imperfectly measured characteristic correlates (agrees) with the perfectly measured version.

If you were trying to explain the ICC to a lay person, you would probably have better success using 'correlation' than 'variance'. To explain correlation you don't have to get into as many details as you would have to if you take the 'variance' route. If we are willing to cheat a little bit (and tell people that the SD is like the typical or average absolute deviation), you might get away with using the concept of a SD, but the concept of a typical or mean squared deviation will for sure lose more people.

Q6

Galton's data: Have a look at the Record of Family Faculties (Family album) respondents used to report the family heights.

Who, if anyone, did the measuring? Who did the 'reporting'?

Do you know how tall your parents and grandparents are (were?)

Q7

'Increasing Reliability by averaging several measurements'

This is a very topical and 'charged' issue at funding agencies, such as the Canadian Institues of Health Research, where each grant application used to be reviewed by 2 primary reviewers, and then an average is made of the scores of up to 20 panel members (incl. the 2) who heard and discussed the 2 reviews, and had also looked through the application themselves.

The new system uses 5 reviewers who do not meet/communicate, and their scores are averaged.

If in the old system, where the ICC was say 0.4, what would be the ICC if we used the average of 2 raters? 3 raters? 4 raters?

You can manipulate the algebra as you wish, but you might also think of it as follows:

if we average m raters, the true sigma-sq-between is not affected, but the true sigma-sq-within now gets reduced from sigma-sq-within / 1 if 1 random rater, to sigma-sq-within / 2 if we average 2, ... sigma-sq-within / m if we average the scores of m raters.

So with an average of m raters, the observed variance of these averages is now
sigma-sq-between + sigma-sq-within / m
The fraction that this that is signal is

sigma-sq-between / [ sigma-sq-between + sigma-sq-within / m]

SO what the question is asking is what if we use N*m raters

so we have fractions

sigma-sq-between / [ sigma-sq-between + sigma-sq-within / N*m]

and

sigma-sq-between / [ sigma-sq-between + sigma-sq-within / 1*m]

The algebra is a matter of manipulation this ratio, so as ro remove the 'm' that is there to start with, and end with the basic ICC[1] (ie what if m=1) and the scaling factor N.

Another example, if a 3 hour GRE exam, done by a paper and pencil, has a reliability of 0.9, what reliability would a 6-hour or 12-hour exam have? Taking 3 hours as the unit of effort, it is

0.9/ (0.9 + 0.1 ) for 3 hours
0.9/ (0.9 + 0.1/2) for 6 hours
0.9/ (0.9 + 0.1/4) for 12 hours
etc.

Geoff Norman was part of a group who developed McMaster's 'Multiple Mini Interview' system. McMaster, and many other schools since then have abandoned the traditional interview and use this instead

see Med Educ. 2004 Mar;38(3):314-26. An admissions OSCE: the multiple mini-interview. Eva KW, Rosenfeld J, Reiter HI, Norman GR.

and subsequent publications that evaluated its measurement properties.

Week of Sept 1-8

Upload your answers to however much/little you are able to do of

Q08,
Q09a,
Q17 parts a and b,
Q20,
        Q26 (PhD),
        Q27 [PhD optional],
        Q28 [PhD optional],
Q29,
Q30 (3 teams, each led by a PhD student),
Q31

by Friday September 8.

Q8

Just because (random) measurement errors tend to cancel out in averages doesn't mean that errors in measurement can be ignored. For example, how comfortable would you be in measuring how much physical activity JH does by having him wear a 'step-counter' for a randomly selected week of the year, and using that 1-week measurement as an 'x' in a multiple or logistic or Cox regression? See slides 7 and 8 from part of JH's "Scientific reasoning, statistical thinking, measurement issues, and use of graphics: examples from research on children" at Royal Children's Hospital in Melbourne, some years ago. pdf

Some of the the terminology will be new to you, and so (as you will discover when you do run the simulations in Q8 of how well you can estimate the conversion factors between degrees F and degrees C) will some of the consequences of measurement error. The "animation (in R) of effects of errors in X on slope of Y on X" might be of interest, as might the java applet accompanying "Random measurement error and regression dilution bias". These consequences are rarely touched on, yet alone emphasized, in theoretical courses on regression, where all 'x' values are assumed to be measured without error! Welcome to the REAL world.

Q9

The point is to 'smooth' the decay curve. But (as the hint says) its broad form should not be a big surprise : it was the subject of a question earlier on in the math-stat. questions. But, as JH recently realizes, and addresses in part [b] the functional form is not as simple or universal as he thought: he had been 'over-selling this formula in his teaching for the last 15 years.

Q17

* Measuring Heart Rate is more challenging     Link   Link

* Measuring Environmental Noise: Link

* Measuring Physical Activity: interpreting the statistical REPORT is challenging!   Link

Q20

!! This is a good illustration of result 2 on page 22 {BUT, we might actually need futher consitions on the distribution of X !}.

Q21

A real example to convince you that MEASUREMENT ERROR MATTERS.

By the way, section 4.7.4 Measurement error in regression, of the Cox-Donnelly book cited in question 21 gives a nice visual (rather than algebraic) explanation for the flattening of the slope.

Its the same visual explanation that JH's co-authors give in their BMJ tutorial, here J Hutcheon, A Chiolero and J Hanley Random measurement error and regression dilution bias The AIDS example in Cox and Donnelly is a very nice illustration of how measurement errors led to a delay in figuring out how HIV was transmitted. This re-inforces the message that measurement errors can cause quite big but subtle distortions.

This Q, new in 2019, is meant to show that the dilutions you have calculated/seen are not just in toy math examples, but also present in just about all research involving regression -- since it is near impossible to measure all X's perfectly. Clearly, some authors like the ICC, while others make it a bit more complicated by using a ratio of the 2 components in the ICC. In the end, it comes to the same thing. JH prefers the ICC version, since it is more easily remembered.

Q23

Unlike the ICC or R, or the variance-ratio used in Q21, the magnitude of the errors in this Q, new in 2020, is not naturally quantified by a variance, or a fraction of the overall variance. But the effect is actually easier to explain to a lay person than the variance-based attenuations we have focused on. The 1856 publication is not well known, and the description of the effects of the errors in the addresses is even less well known. Thus, a short note to a modern day Epidemiology journal, explaining this early attention to measurement , and recounting Snow's clear description of the consequences of errors, would make a nice contribution to the teaching of epidemiology and its history! JH is hoping that your answers to this Q can be the start of such a note... authored by the class!

Q24

Will be interested to read your suggestions.

Q25

A hand-drawn version of the diagram will be fine.

Q26

No need to go beyond a 2 point X, or a 2-point errors. If you understand this case, you understand the more general case. [see page 23 and the link on page 24 -=- to p 6 of his notes for another course]

Q27

This Q, new in 2021. it might be easier to do the Blood Pressure example, and the even age distribution, since it should give more or less the same answer as the log of the 28-day mortality proportions as a linear function of age.

[The point of showing you that complex model was to make it a bit more 'real' and to introduce you to Gompertz law; he was dealing with all-cause mortality in a general population, and he needed 3 straight lines to show the U -shaped 'force of mortality' function. See here for a slightly more complex version.

We don't have enough really young people in the UK data, to test if the same 3-part piece-wise log-linear model applies with 28-day mortality in persons who have tested positive for COVID.

Even though the Blood Pressure example may look more like a 'toy' math example where it is easier to 'see' what is going on, it is not THAT far-fetched. For example, see here The '100 + your age' equation was commonly used when JH was young. There was another rule, which his doctor gave him regarding weight-gain: "after you get married (JH did at 26, and weighted 130 lbs or so) you are allowed to put on a pound a year". Until COVID, he has more or less stayed within this guideline!

Q28

This Q was also new in 2021, so the wording has not been extensively beta-tested, unlike those from earlier years.

The difference between it and earlier Qs re. error is that that the real 'X' (here VOC+ or VOC-) is BINARY, and (especially early on) was not perfectly measured. And when Y (1=dead, 0=survived) is also binary, things get a bit more complicated. This example is easier than the typical one, since there are only false positives, and no false negatives. This simplest error context may allow you to come up with a simpler algebraic formula than would apply when both types of error are present (as in the John Snow example in Q23, where there are mistakes in both sides.)

Q29

This Q is new in 2022. JH finds it to be a nice modern intro to readability. He is sad to find that the SMOG index (which we used to use for manual 'measurements' way back when) did not make it to the list. Maybe when Microsoft put some of the other tools into MS Word, that was the end of the SMOG index. Some of the SMOG links no longer work, but you could probably find new ones. JH always liked the name!