A logistic regression analysis was run using Stata relating...

A logistic regression analysis was run using Stata relating disease status to lifetime acetaminophen intake divided by $100(1 \text { unit }=100 \mathrm{g})$ denoted by ace_100. The results are given in Table 13.46 CAN'T COPY THE TABLE Suppose a cutoff for excessive lifetime acetaminophen intake (L) is a level such that the risk of at least a $30 \%$ decline in GFR over 11 years is $\geq 10 \% .$ What is your best estimate of L?

A logistic regression analysis was run using Stata relating disease status to lifetime acetaminophen intake divided by $100(1 \text { unit }=100 \mathrm{g})$ denoted by ace_100. The results are given in Table 13.46
CAN'T COPY THE TABLE
Suppose a cutoff for excessive lifetime acetaminophen intake (L) is a level such that the risk of at least a $30 \%$ decline in GFR over 11 years is $\geq 10 \% .$ What is your best estimate of L?

Key Concepts

Logistic Regression Analysis

Logistic regression is a statistical method used to model the relationship between a binary outcome and one or more independent variables. It estimates the probability of an event (such as a disease status) by modeling the log-odds of the probability as a linear combination of the predictors. This approach is widely used in risk assessment and medical research to understand how changes in predictors affect the likelihood of an event.

Logit Function and Transformation

The logit function is the natural logarithm of the odds, which linearizes the relationship between the probability of an event and the predictor variable. Specifically, it is expressed as log(p/(1-p)) = ?0 + ?1*X, allowing one to solve for the predictor value X that yields a specific probability threshold. This transformation is a core element of logistic regression analysis and is used to determine cutoff values corresponding to particular risk levels.

Probability Threshold Estimation

Determining a cutoff value using a logistic regression model involves setting the predicted probability (e.g., 10%) equal to the model’s probability function and solving for the predictor variable. This process provides an estimate of the level of the predictor at which the outcome risk meets or exceeds the specified threshold. The method is crucial in applications where risk stratification is needed to identify individuals at higher risk based on their exposure levels.

Unit Transformation of Predictors

Often in regression analysis, predictors are transformed (for example, scaling by a constant) to simplify interpretation or stabilize the estimation process. Understanding such transformations is essential because the coefficients relate to the transformed unit of the predictor. Correct interpretation and back-transformation help in applying the results to real-world decision making, such as determining clinically relevant exposure cutoffs.

Transcript

00:01 All right, for part a, we're just plugging and interpreting.

00:03 So a would be f of 25 equal to 0 .9 over 1 plus 271e, 271e negative 0 .122 times 25, and that equals 0 .065.

00:30 Once you plug into your calculator.

00:32 Oh, that's just an equal.

00:33 And what this represents is it's the fraction of 25 -year -olds who have the disease is 0 .65 or 6 .5%.

00:41 And for part b, this 25 becomes a 65.

00:47 So let's do that.

00:51 65, 65.

00:54 And that equates to 0 .82.

00:58 So the percentage, not the percentage, the fraction of, 65 -year -olds who have the disease is 0 .82 or 82%.

01:10 Now, for part b, let me erase this.

01:14 We want to set the left side to half, because that's what we're solving for, 50 % half.

01:20 So we go to part b.

01:22 So 0 .5 is equal to 0 .9 over 1 plus 271e to negative 0 .122x, we're going to solve for x we want to multiply both sides by the denominator and then i'm going to do two steps in one because i don't want to rewrite this 100 times but basically we're going to multiply the both sides by this and then once this this this whole equation will be on this side we can divide by 0 .5 so i'll do that on one step i'll show you what that result looks like that just becomes 1 plus 271 e negative 0 .122x is equal to 0 .9 divided by 0 .5, which is 1 .8.

02:17 And then we want to subtract 1 from each size...

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