We read the article by Tabaei and Herman (1) in the November 2002 issue of Diabetes Care with great interest and feel that it covers an exciting and vital area of diabetes research. We applaud the authors’ attempts to improve current standards of diabetes screening and the journal’s willingness to publish important new findings in the field. However, when we reviewed the article, we found the results difficult to reproduce, especially as there were no numerical examples provided in the text. Using the precise formula given in the article, our calculations produced negative probability values. We would therefore like to take this opportunity to point out a source of possible error in the equation and clarify the logistic regression model.

The logistic regression model and the “logit” model can be written as:

\[ln{[}P/(1{-}P){]}{=}{\beta}_{0}{+}{\beta}_{\mathrm{i}}\mathrm{X}_{\mathrm{i}}{+}e\ (2)\]

where X = β0 + βiXi are the coefficients (parameters) for the linear predictor and Xi represents the values of the explanatory variables (we have assumed that these coefficients have been correctly estimated in the article); ln is the natural logarithm, logexp, where exp = 2.71828… ; P is the probability that the event “previously undiagnosed diabetes” occurs; P/(1 − P) is the “odds ratio”; and ln[P/(1 − P)] is the log odds ratio, or “logit.”

In contrast to the simple linear probability model (Y = a + bX + e), the logit distribution constrains the estimated probabilities to lie between 0 and 1. For instance, the estimated probability is:

\[P{=}{[}\mathrm{exp}({\beta}_{0}{+}{\beta}_{\mathrm{i}}\mathrm{X}_{\mathrm{i}}){]}/{[}1{+}\mathrm{exp}({\beta}_{0}{+}{\beta}_{\mathrm{i}}\mathrm{X}_{\mathrm{i}}){]}\]

or

\[P{=}1/{[}1{+}\mathrm{exp}({-}{\beta}_{0}{-}{\beta}_{\mathrm{i}}\mathrm{X}_{\mathrm{i}}){]}\]

For a 45-year-old man with a BMI of 29 kg/m2, a plasma glucose level of 125 mg/dl, and a postprandial time of 3 h, the probability calculated with the published formula would be P = −0.0555 (i.e. P < 0). This result does not match the requirement for probabilities to lie between 0 and 1. For the same case, but with the corrected formula, the probability of being previously undiagnosed for diabetes is P = 0.04996.

\[P{=}1/{[}1{-}\mathrm{exp}({-}\mathrm{X}){]}\]

X = −10.0382 + 0.0331*(age = 45 years) + 0.0308*(plasma glucose = 125 mg/dl) + 0.25*(postprandial time = 3 h) + 0.562*(sex = 0) + 0.0346*(BMI = 29 kg/m2). X = −2.9453 and P = 1/[1 − exp(−X)] = 1/(1 − exp(2.9453) = 1/(1 − 19.01636644) = 1/−18.01636644. P = −0.0555.

\[P{=}1/{[}1{+}\mathrm{exp}({-}\mathrm{X}){]}\]

X = −2.9453; P = 1/[1 + exp(−X)] = 1/(1 + exp[2.9453]) = 1/(1 + 19.01636644) = 1/20.01636644; P = 0.04996.

Therefore, we would suggest that the following equations should be used:

\begin{eqnarray*}&&\mathrm{X}{=}{-}10.0382{+}0.0331{\ast}(\mathrm{age}){+}0.0308{\ast}\ (\mathrm{plasma\ glucose}){+}0.25{\ast}\ (\mathrm{postprandial\ time})\ \\&&\ {+}0.562\ (\mathrm{if\ female}){+}0.0346{\ast}\ (\mathrm{BMI})\ \mathrm{and}\ P{=}1/{[}1{+}\mathrm{exp}({-}\mathrm{X}){]}\end{eqnarray*}

We hope these comments serve to clarify this logistic regression equation to screen for diabetes and, in conjunction with the valuable work of our colleagues Drs. Tabaei and Herman, help expand the knowledge base in this key area of diabetes research.

1.
Tabaei BP, Herman WH: A multivariate logistic regression equation to screen for diabetes.
Diabetes Care
 
25
:
1999
–2003,
2002
2.
Hosmer D, Lemeshow S:
Applied Logistic Regression
. New York, Wiley,
1989

Address correspondence to Stéphane Roze, CORE: Center for Outcomes Research, St. Johanns-Ring 139, 4056 Basel, Switzerland. E-mail: [email protected].

DIABETES CARE
2003