Detailed explanation of the definition, meaning and calculation of OR value in logistic regression

Logistic regression is a linear model used for classification problems. It is mainly used to predict the probability value in binary classification problems. It converts linear prediction values ​​into probability values ​​by using the sigmoid function and makes classification decisions based on thresholds. In logistic regression, the OR value is an important indicator used to measure the impact of different variables in the model on the results. The OR value represents the multiple change in the probability of the dependent variable occurring for a unit change in the independent variable. By calculating the OR value, we can determine the contribution of a certain variable to the model. The calculation method of the OR value is to take the coefficient of the natural logarithm (ln) of the exponential function (exp), that is, OR = exp(β), where β is the coefficient of the independent variable in the logistic regression model. Specifically, if the OR value is greater than 1, it means that the increase in the independent variable will increase the probability of the dependent variable; if the OR value is less than 1, it means that the increase in the independent variable will reduce the probability of the dependent variable; if the OR value is equal to 1, it means that the independent variable will increase the probability of the dependent variable. The increase has no effect on the probability of the dependent variable. To sum up, logistic regression is a linear model used for classification problems. It uses the sigmoid function to convert linear prediction values ​​into probability values, and uses the OR value to measure the impact of different variables on the results. By calculating the OR value,

1. The concept and meaning of the OR value

The OR value is an indicator used to compare the occurrence ratio of two events , often used to compare the probability of a certain event occurring among different groups or under different conditions. In logistic regression, the OR value is used to measure the impact of two values ​​of an independent variable on the dependent variable. Suppose we face a binary classification problem, in which the dependent variable y has only two values ​​0 and 1, and the independent variable x can take two different values ​​​​x1 and x2. We can define an OR value to compare the probability ratio of y=1 when x takes the value of x1 and x2. Specifically, the OR value can be calculated by the following formula:

OR=\frac{P(y=1|x=x1)}{P(y=0|x=x1 )}\div\frac{P(y=1|x=x2)}{P(y=0|x=x2)}

P(y=1|x= x1) represents the probability that the dependent variable y has a value of 1 when the independent variable x has a value of x1; P(y=0|x=x1) represents that when the independent variable x has a value of x1, the dependent variable y has a value of 0 probability. Similarly, P(y=1|x=x2) and P(y=0|x=x2) represent the probabilities that the dependent variable y takes the value 1 and 0 respectively when the independent variable x takes the value x2.

The meaning of the OR value is to compare the ratio between the ratio of y=1 and y=0 when x takes the value of x1 and x2. If the OR value is greater than 1, it means that x1 is more likely to cause y=1 than x2; if the OR value is less than 1, it means that x2 is more likely to cause y=1 than x1; if the OR value is equal to 1, it means x1 and x2 have the same influence on y.

2. Detailed explanation of OR calculation for logistic regression analysis

In logistic regression, we usually use the maximum likelihood method to estimate model parameters, so that Get the coefficient of each independent variable. After getting the coefficients, we can use the OR value to measure the impact of each independent variable on the dependent variable. Specifically, we can index the coefficient of each independent variable to get an estimate of the OR value, that is:

\hat{OR}=\exp(\hat{\ beta})

Among them, \hat{\beta} represents the coefficient estimate of each independent variable. According to the above definition of OR value, we can rewrite it as:

\hat{OR}=\frac{P(y=1|x=x1)}{P(y =0|x=x1)}\div\frac{P(y=1|x=x2)}{P(y=0|x=x2)}=\exp(\hat{\beta}\cdot\Delta x)

Among them, \Delta x represents the difference between the independent variables x1 and x2. As can be seen from the above formula, if the independent variable x1 is one unit larger than x2, then the OR value will be multiplied by \exp(\hat{\beta}), that is to say, the impact of x1 on the probability of y=1 will Increased by \exp(\hat{\beta}) times than x2. Similarly, if the independent variable x1 is one unit smaller than x2, then the OR value will be divided by\exp(\hat{\beta}), that is, the impact of x1 on the probability of y=1 will be smaller than x2\exp (\hat{\beta}) times.

In logistic regression, the size and direction of the OR value can help us understand the degree and direction of the influence of each independent variable on the result. For example, if the OR value is greater than 1, it means that the independent variable has a positive impact on the probability of y=1; if the OR value is less than 1, it means that the independent variable has a negative impact on the probability of y=1; if the OR value is equal to 1, It means that the influence of the independent variable on y is not significant. In addition, we can also evaluate the reliability of the OR value by calculating the 95% confidence interval.

In short, the OR value is an important indicator in logistic regression to measure the influence of independent variables on dependent variables. Calculating the OR value can help us understand the direction and degree of influence of each independent variable on the result, and its reliability can be evaluated by calculating the confidence interval.

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