"Logit", "Lod" and Log-Odds

Oxford English Dictionary: Logit (pronounced "low-jit")
"The natural logarithm [ log or ln or loge ] of the quotient of a probability and its complement."

Lod "The logarithm to the base 10 [ log or log10 ] of the quotient of a probability and its complement."

1944. J. Berkson in Jrnl. Amer. Statistical Assoc. 39. 361. "Instead of the observations [1-pi =] qi we deal with their logits li = ln(pi/qi). [Note] I use this term for ln(p/q) following Bliss, who called the analogous function which is linear on x for the normal curve `probit'."

Oxford English Dictionary: Lod
"The logarithm of the odds in favor of or against a given event; also, the logarithm of the ratio of two odds."

1949. G. A. Barnard in Jrnl. R. Statistical Soc. B. XI. 116: "There are great advantages in working in terms of odds rather than in terms of probabilities and greater advantages still in working in terms of logarithms of odds... This suggested the adoption of log odds (or, as they are called below, lods) as the fundamental idea in the theory [of sequential tests], in place of probabilities."

1955. N. E. Morton in Amer. Jrnl. Human Genetics VII. 292: "The scores in a sequential test are lods, or logarithms of the probability ratio... Tables 4-8 give the possible certain matings and the lod scores appropriate to them."

From this it would appear that "lod" and "logit", "lod score" and "logit measure" are synonymous. And they can be, but Barnard's "calculus of lods" is idiosyncratic. He writes "We can look forward to an experiment, or we can look back on it. ... To distinguish between the two senses of likelihood involved here, we shall refer to the first as forward likelihood, or f-likelihood, and to the second as backward likelihood of b-likelihood." This leads to f-lods and b-lods.

As A. W. F. Edwards notes (Nature, 333/26, 308, 1988):

"Lod, log-odds, was first defined by Barnard ... Indeed, the word odds itself is nowhere else used in this backward sense introduced by Barnard. The common name being the likelihood ratio."

"Their [linkage workers'] Lod score is simply the log-likelihood to the base 10, standardized as is customary by the addition of a constant, in this case such as to make the log-likelihood zero at a recombination fraction of one-half."

Barnard G.A. (2001) "Logit", "Lod" and Log-Odds. Rasch Measurement Transactions 14:4 p.785

"Logit", "Lod" and Log-Odds. Barnard G.A. … Rasch Measurement Transactions, 2001, 14:4 p.785

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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
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