Measures, Correlations, and Explained Variances

Rosenthal and Rubin (1979) deliberately challenge conventional statistical wisdom by means of a paradoxical interpretation of a simple data set. They suppose that 100 patients in a medical study are randomly assigned to new treatment N and 100 to standard treatment S. At the end of one year, 70% of the patients under treatment N are still alive, but only 30% under treatment S. Here is their data matrix:

Treatment            Survival after
Assignment    Dead   one year
New            30     70
Standard       70     30

How much better is Treatment N than Treatment S?
1) The Pearson product-moment correlation, r, between surviving and receiving treatment N is 0.4. Not an impressive value.

2) The variance in "survival explained" by assignment to treatment is r^2 = 0.4^2 = 0.16, i.e., only 16%. This is less than 1/6th of the total variance. Obviously a small and, therefore, obviously an "unimportant" amount.

But, when considering one's own medical treatment, Treatment N is clearly preferable to Treatment S. Rosenthal and Rubin (1979) stop here, having made their point that the relationship between statistics and substance is not always obvious. Let us continue.

3) The standard error of r for testing whether r=0 is approximately 1/sqrt(200) = 0.07. t = 0.4/0.07 = 5.7. Thus the probability that there is no correlation between treatment and survival is <.0001. This is statistically very significant, but it enables us only to declare that N and S are "not the same". It does not say how much better N is than S.

4) The logit distance between the effects of treatments N and S is the difference between the log-odds for survival with N and the log-odds for survival with S = loge(70/30) - loge (30/70) = 1.7 logits, with a standard error of sqrt(200/(30*70)) = 0.3 logits. Testing the hypothesis that the distance is zero, t = 1.7/0.3 = 5.7 (as expected).

5) Let us take this extremely simple logit analysis a step further. The analysis provides a useful 95% confidence interval for the survival advantage of the New treatment over the Standard:

(New - Standard) = 1.7 +- (2 * 0.3) = 1.7 +- 0.6 logits

Now we can conclude, with 95% confidence, that the survival advantage of New over Standard is:

1.1 logits < (New - Standard) < 2.3 logits

6) Finally, what are these advantages of 1.1, 1.7 and 2.3 logits on New treatment benefit worth to you, a prospective patient? The advantage to you depends on your Standard probability of survival.

Your               Your New survival probability
Standard           for various logit advantages
Survival           At least    Average     At most
Probability         1.1         1.7         2.3
.10                 .25         .38         .50
.20                 .43         .58         .71
.30 (in example)    .56         .70         .81
.40                 .67         .78         .87
.50                 .75         .85         .91

Despite the low correlation of 0.4 and low "explained variance" of only 16%, are you in any doubt as to which treatment is best for you? These data, when understood from the Rasch measurement point of view, are in no way paradoxical. They provide an entirely decisive answer to your important question.

By choosing to think with measures rather than correlations and variances, the conflict between the substantive and the statistical findings disappears.

Rosenthal R, Rubin DR (1979) A note on percent variance explained as a measure of the importance of effects. Journal of Applied Social Psychology 9:5, 395-396.

Measures, correlations and explained variances. Rosentahl R, Rubin DR, Linacre Jm, Wright BD. … Rasch Measurement Transactions, 1995, 9:2 p.435

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Rating Scale Analysis, Wright & Masters Introduction to Rasch Measurement, E. Smith & R. Smith Introduction to Many-Facet Rasch Measurement, Thomas Eckes Invariant Measurement: Using Rasch Models in the Social, Behavioral, and Health Sciences, George Engelhard, Jr. Statistical Analyses for Language Testers, Rita Green
Rasch Models: Foundations, Recent Developments, and Applications, Fischer & Molenaar Journal of Applied Measurement Rasch models for measurement, David Andrich Constructing Measures, Mark Wilson Rasch Analysis in the Human Sciences, Boone, Stave, Yale
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