fix format

paper/paper.md

@@ -127,7 +127,7 @@ HL_H_ts, HL_H_p, df = hosmer_lemeshow_test(
     bin_count=bin_counts
 )
 ```
-When performing the HL test on validation sets that are not used in training, the degree of freedom of the HL test changes from $M-2$ to $M$[@hosmer2013applied]. Intuitively, $\frac{(O_{1,m}-E_{1,m})^2}{E_{1,m}(1-\frac{E_{1,m}}{N_m})}$ is the difference squared divided by the variance of a binomial distribution and follows a chi-square distribution with 1 degree of freedom. Hence, the sum of $M$ chi-square distributions with 1 degree of freedom is a chi-square distribution with $M$ degrees of freedom if the data has no effect on the model. The increase in degree of freedom for validation samples has often been overlooked but it is crucial for the test to maintain the correct type 1 error rate. In `calzone`, users can specify the degree of freedom of the HL test by setting the `df` parameter.
+When performing the HL test on validation sets that are not used in training, the degree of freedom of the HL test changes from $M-2$ to $M$ [@hosmer2013applied]. Intuitively, $\frac{(O_{1,m}-E_{1,m})^2}{E_{1,m}(1-\frac{E_{1,m}}{N_m})}$ is the difference squared divided by the variance of a binomial distribution and follows a chi-square distribution with 1 degree of freedom. Hence, the sum of $M$ chi-square distributions with 1 degree of freedom is a chi-square distribution with $M$ degrees of freedom if the data has no effect on the model. The increase in degree of freedom for validation samples has often been overlooked but it is crucial for the test to maintain the correct type 1 error rate. In `calzone`, users can specify the degree of freedom of the HL test by setting the `df` parameter.
 
 ### Cox's calibration slope/intercept
 Cox's calibration slope/intercept is a regression analysis method for assessing the calibration of a probabilistic model [@Cox]. A new logistic regression model is fitted to the data, with the predicted odds ($\frac{p}{1-p}$) as the independent variable and the outcome as the dependent variable. The slope and intercept of the regression line are then used to assess the calibration of the model. A slope of 1 and intercept of 0 indicates perfect calibration. To test whether the model is calibrated, fix the slope to 1 and fit the intercept. If the intercept is significantly different from 0, the model is not calibrated. Then, fix the intercept to 0 and fit the slope. If the slope is significantly different from 1, the model is not calibrated.
@@ -144,15 +144,11 @@ cox_slope, cox_intercept, cox_slope_ci, cox_intercept_ci = cox_regression_analys
     fix_slope=True
 )
 ```
-<<<<<<< HEAD
 The slope and intercept values indicate the type of miscalibration. A slope >1 shows overconfidence at high probabilities and underconfidence at low probabilities (and vice versa). A positive intercept indicates general overconfidence (and vice versa). However, even with ideal slope and intercept values, the model may still be miscalibrated due to non-linear effects that Cox's analysis cannot detect.
 
 ### Integrated calibration index (ICI)
 
 The integrated calibration index (ICI) is very similar to Expected calibration error (ECE). It also tries to measure the average deviation between predicted probability and true probability. However, ICI does not use binning to estimate the true probability of a group of samples with similar predicted probability. Instead, ICI uses curve smoothing techniques to fit the regression curve and uses the regression result as the true probability [@ICI_austin]. The ICI is then calculated using the following formula:
-=======
-The values of the slope and intercept can represent miscalibration throughout the range of probabiliy outputs. The integrated calibration index (ICI) is very similar to Expected calibration error (ECE). It also tries to measure the average deviation between predicted probability and true probability. However, ICI does not use binning to estimate the true probability of a group of samples with similar predicted probability. Instead, ICI uses curve smoothing techniques to fit the regression curve and uses the regression result as the true probability [@ICI_austin]. The ICI is then calculated using the following formula:
->>>>>>> cdd3f8ac405788615653f94d7b4144af7e9acab0
 $$
 \text{ICI} = \frac{1}{n}\sum_{i=1}^{n} |f(p_i)-p_i|
 $$