- Discussion of issues in the
Assignment on measurement

Q1 and Q2 (measuring 'Readability'): answers need not be handed in; just think about the issues; If there is time, we might discuss and do some 'measuring' in class.

Q3, Q4, Q5, Q6, Q7, Q8, Q9, Q18: Answers to be handed in.

Q10, Q11, Q12, Q13, Q14, Q15, Q16, Q17: answers need not be handed in. If there's time, you and we will think about what the answers to them might have looked like.

**Remarks**: 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'.

**Q1**

'Back then' ('BC') students had to measure the readability*manually*by counting the lengths of words and sentences, and the number of syllables in words. Today that is made much easier using online tools, and those in Microsoft Word.

From 3 measurements of readability, you can calculate 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.

If the scale is a natural one, like a grade level, then the SEE makes sense, since 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 you have > 1 measurement each on each of several texts of different difficulty .. so you can judge how much is genuine 'between-text' and within-text' variation.

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.

**Q2**

Again, 'back then' we went to the library (or looked around at home) 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.

**Q3**

[ 'm-s' is short for 'math-stat' ]. The point of asking you to derive the link is to emphasize that the SEE and R are nor ENTIRELY separate. Yes, the SEE 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 SEE.

(1) The square of the SEE 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 in the same piece is the explanation from the Educational Testing Service called INTERPRETING YOUR GRE SCORES, contained on page 7 of JH's 'Introduction to Measurement Statistics' Notes (available under Measurement -- Lecture Notes, etc in the resources).

**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 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 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.

**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, earlier this year. 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.

For this exercise, and the topics it addresses, the most relevant portions of the 'surveys' resources are Measurement: Reliability and Validity and Effects of Measurement Error

[last year: Computing issues that may arise in Q14: Dates are a pain, even in R. If you get stuck, use some of the R code supplied, to compute week and day of week. Incidentally, whereas the exercise makes reference to 104 weeks, there are a few weeks with some missing data, so best keep them out of the calculations for now (in practice JH would try to use all the data, but the imbalanced data have a messier EMS structure that -- for now -- distracts us from the main point) ].

**Q9**

The point is to 'smooth' the decay curve. But (as the hint says) its form should not be a big surprise : it was the subject of a question earlier on in the math-stat. questions.

**Q18**

JH did some pilot testing of the variability to expect in subjects of your age. He has wondered why 100 per day were used in the study of the effects of sleep deprivation.

IF you want to achieve shorter reaction times, JH's pilot testers tell him that its better to use a computer than a phone or tablet, and also to use the space bar rather than the return or enter key or the mouse. (The 'Hints' tell you you can use any key, rather than click with the mouse or trackbar.)