Corrigendum to “A validated clinical risk prediction model for lung cancer in smokers of all ages and exposure types: A HUNT study” [EBioMedicine 31 (2018) 36–46]

The authors recently became aware of some minor errors in the algorithms provided in the Supplementary, pages 4 and 5.

In the correct algorithm, to avoid logarithm of zero, the argument of logarithm is incremented by 1 in the risk model equations. Those variables are the following: pack years, quit years, BMI and smoke exposure hours. This increment was mistakenly not written in the Supplementary text.

We also noticed that some of the coefficients reported in the Supplementary were slightly rounded values, probably generated automatically.

In the text, the ln and log were both used. The correct is that all are natural logarithm, and therefore we changed the log to ln and added one line for clarification: “We indicate with ln the natural logarithm.”

Here we provide the full algorithm with non-rounded coefficients and the +1 increment.

Absolute risk estimation in the Cox model for events within 16 years

To estimate the absolute risks per patient, the baseline survival function $S_0\left(t\right)$ was estimated according to van Houwelingen¹⁷: the general Cox model $S\left(t\right|X)=S_0{\left(t\right)}^{\text{exp}\left(X\beta \right)}$ can be rewritten as $\text{ln}(-ln(S\left(t\right|X))=ln(-ln(S_0\left(t\right)))+PI\left(X\right),$ with $PI\left(X\right)=X\beta$ the prognostic index of the model. Using the survfit function, we calculated the baseline survival probability $S_0\left(t\right)$ at corresponding max follow-up time t = 16.4 years for the smokers’ Cox model. A simplification based on the Weibull model specifies that

$$ln\left(-ln\left(S_0\left(t\right)\right)\right)={\beta }_0+{\beta }_1\text{ln}\left(t\right).$$

Baseline survival used in the final model is:

$$S_0\left(t\right)=\text{exp}(-exp\left(-0.929+0.708ln\left(t\right)\right))=0.06$$

We can calculate an individual's estimated risk within max follow-up of t = 16 years given data on the risk factors using the risk model. We indicate with ln the natural logarithm.

The prognostic index Xβ (based on Cox model coefficients in Table 2) is:

$$X\beta =0.12058192\left(male\right)-2.00205575\left[{\left(\frac{age}{100}\right)}^{-1}\right]+1.16301808\left[ln\left(packyrs+1\right)\right]$$

$$-0.02954059\left(cigarettesperday\right)-0.24079981[ln\left(quityrs+1\right)]-1.24626558[ln\left(BMI+1\right)]$$

$$+0.16632008\left[ln\left(smokeexposurehrs+1\right)\right]+0.4059355\left(coughdaily\right)$$

and the calculation of the predicted risk within 16 yrs proceeds as follows:

$$RiskofLungcancer=1-0.{06}^{exp\left(X\beta \right)}.$$

Logistic regression prediction model of lung cancer risk for events within 6 years

We also developed a logistic regression model where risk was the probability of a lung cancer diagnosis within a 6 years period. We indicate with ln the natural logarithm. The prognostic index Xβ_6yrs (based on logistic regression model coefficients) is:

$$X{\beta }_{6\text{yrs}}=1.18203062+0.31573217\left(male\right)-1.98496138\left[{\left(\frac{age}{100}\right)}^{-1}\right]+1.11994217\left[ln\left(packyrs+1\right)\right]$$

$$-0.04002877\left(cigarettesperday\right)-0.24019955[ln\left(quityrs+1\right)]-1.70238304[ln\left(BMI+1\right)]$$

$$+0.0807242\left[ln\left(smokeexposurehrs+1\right)\right]+0.49212668\left(coughdaily\right)$$

and the calculation of the predicted risk within 6 years proceeds as:

$$6\text{yrs}\text{Risk}\text{of}\text{lung}\text{cancer}=\frac{1}{1+exp\left(-X\beta \_\left\{6yrs\right\}\right)}$$

To be totally transparent with the research community, and to allow fellow researchers to easily perform their own calculations and validations, we provide the Excel files for calculating the 6 and 16 years risk.

Another unintentional mistake was noticed in the main manuscript.

In section 3.5. two examples of use of the calculator are shown. Unfortunately, the base 10 logarithm instead of the natural logarithm was used by error. The corrected results for this section are:

«By applying this threshold, using either the HUNT Lung Cancer Model nomogram or the online calculator, a 40-year old person with 15 pack-years and full-score contribution with low BMI (e.g. 22), low smoke intensity (e.g. 10 cigarettes per day), periodical or daily cough, and many hours of indoor smoke exposure (e.g. 10 h, total risk score > 15, LC risk = 1.77% at 16 years and 0.44% in 6 years would be assessed as a medium- or high-risk individual. A 56-year-old with 15 pack-years, high BMI (e.g. 33), high smoke intensity (e.g. 40 cigarettes per day), no periodical or daily cough, and no indoor smoke exposure would be assigned a lower than cut-off risk and would not be eligible for screening (risk score 12.5, LC risk = 0.84% at 16 years and 0.14% at 6 years (Figure 2a, b).»

After correction, the numbers are different but the message is unchanged: compared to an older individual (56 years), a younger person (40 years of age) with the same number of pack-years apparently has a higher risk score within 6 and 16 years if he scores high on the other negative predictive factors.

The correct algorithms were used for all the other results presented in the manuscript. These results, their interpretation and overall conclusions of the manuscript have thus not changed. The authors apologize for any inconvenience caused.