About Rasch scores

from a letter to me by Grady Towers dated February 25, 1999
Used with permission from the author

You're going to learn to hate me before long. People in super-high IQ societies loathe being condescended to, and all the lectures I've been sending your way must seem like disrespect. Actually, they're just the opposite.

There's something I need to tell you about Rasch scores that you probably don't know. But before I can tell you what it is, I'm going to have to give you a lecture on some fundamental ideas about statistics.

As an engineer, you've probably been exposed to the idea of measurement error, and even the laws governing the propagation of errors, so you probably think you have some appreciation for what error is. Actually, you don't. Engineers and physical scientists are rarely exposed to the concept of levels of measurement, and since they deal almost exclusively with ratio strength scales (Fahrenheit and Celsius temperature scales are exceptions), they may never have been given the opportunity to learn otherwise.

I've done statistical problems almost everyday for 40 years, and there are two overriding principles that govern how I solve a problem. One is level of measurement, which I'm about to describe to you, and the other whether or not the data is from matched or unmatched samples. The last is easy to discover. If you can calculate a correlation coefficient on the data, you have a matched sample. If not, you have an independent sample.

There are four levels of measurement generally acknowledged by statisticians. From weakest to strongest, these are nominal, ordinal, interval and ratio. These are important because they determine what kind of statistical procedure can be used. Any statistical procedure using a given level of measurement can be used only on that level. But tests of lesser strength can also be used for the same data. Nominal strength data, for example, can use only tests and procedures appropriate for nominal data. Interval strength data can be tested with interval level tests, but they can also be tested with ordinal level tests and nominal level tests. There's a tradeoff. The lower the level of statistical test used, the fewer assumptions need to be made about the data (normality, symmetry, homoscedasticity, etc), but the larger the sample has to be to reject the null hypothesis.

Nominal scale: numbers are used to name, identify or classify. Telephone numbers are a nominal scale. The correct/incorrect responses used on the items from mental ability tests are also on a nominal scale. Only the statistical techniques based on counting are permitted.

Ordinal scale: numbers represent rank or order. The numbers used to represent the hardness of minerals, from diamond as 10 and talc as 1, represent an ordinal scale. Some people believe that mental abilities represent at most an ordinal scale. Only statistical procedures based on counting, and on greater than or less than are permitted.

Interval scale: intervals between numbers are presumed to be equal. IQ tests are thought to be approximately on an interval scale. They have been described as rubber rulers. Only statistical techniques based on counting, and greater than and less than, and addition and subtraction are permitted.

Ratio scale: all numbers are thought to represent a distance from zero. Weight and distance are ratio scales. All statistical (arithmetic) procedures are permitted, including multiplication and division. This is called the ratio scale because it's permitted to say that one measuremement is twice as large as another. Ten feet is twice as long as five feet. This is not permitted on an interval scale. It is not permitted to say that an IQ of 140 is twice as great as an IQ of 70.

Do you get the idea?

Rasch scores are not rubber rulers! They are on a rigid interval scale. But what is truly apocalyptic about them is that there is a mathematical transformation that will put them on a ratio scale. For the first time in history, it is possible to say that one person is twice as intelligent as another. For the first time in history, it's possible to construct an intelligence scale with amoebas at one end and Jehovah at the other.

When Rasch scores are expressed as logarithms, then

(e^Rasch1) / (e^Rasch2)

is a meaningful ratio.

You now have the means to find out how much more intelligent Prometheus Society members are from Triple Nine members, or the average human being. Take your time and do this right. You are about to become an historical figure.

By the way, because the error variance for the Mega Test gets so very large at super-high levels, I hope you've drawn the same conclusion I have. There's no reason at all to believe that Mega Society members are one whit more intelligent than Prometheus members.

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