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A regression coefficient describes the size and direction of the relationship between a predictor and the response variable. Coefficients are the numbers by which the values of the term are multiplied in a regression equation.
Use the coefficient to determine whether a change in a predictor variable makes the event more likely or less likely. The estimated coefficient for a predictor represents the change in the link function for each unit change in the predictor, while the other predictors in the model are held constant. The relationship between the coefficient and the probability depends on several aspects of the analysis, including the link function, the reference event for the response, and the reference levels for categorical predictors that are in the model. Generally, positive coefficients make the event more likely and negative coefficients make the event less likely. An estimated coefficient near 0 implies that the effect of the predictor is small.
The interpretation of the estimated coefficients for categorical predictors is relative to the reference level of the predictor. Positive coefficients indicate that the event is more likely at that level of the predictor than at the reference level of the factor. Negative coefficients indicate that the event is less likely at that level of the predictor than at the reference level.
The logit link provides the most natural interpretation of the estimated coefficients and is therefore the default link in Minitab. The interpretation uses the fact that the odds of a reference event are P(event)/P(not event) and assumes that the other predictors remain constant. The greater the log odds, the more likely the reference event is. Therefore, positive coefficients indicate that the event becomes more likely and negative coefficients indicate that the event becomes less likely. A summary of interpretations for different types of predictors follows.
The standard error of the coefficient estimates the variability between coefficient estimates that you would obtain if you took samples from the same population again and again. The calculation assumes that the sample size and the coefficients to estimate would remain the same if you sampled again and again.
Use the standard error of the coefficient to measure the precision of the estimate of the coefficient. The smaller the standard error, the more precise the estimate.
These confidence intervals (CI) are ranges of values that are likely to contain the true value of the coefficient for each term in the model. The calculation of the confidence intervals uses the normal distribution. The confidence interval is accurate if the sample size is large enough that the distribution of the sample coefficient follows a normal distribution.
Because samples are random, two samples from a population are unlikely to yield identical confidence intervals. However, if you take many random samples, a certain percentage of the resulting confidence intervals contain the unknown population parameter. The percentage of these confidence intervals that contain the parameter is the confidence level of the interval.
Use the confidence interval to assess the estimate of the population coefficient for each term in the model.
For example, with a 95% confidence level, you can be 95% confident that the confidence interval contains the value of the coefficient for the population. The confidence interval helps you assess the practical significance of your results. Use your specialized knowledge to determine whether the confidence interval includes values that have practical significance for your situation. If the interval is too wide to be useful, consider increasing your sample size.
The Z-value is a test statistic for Wald tests that measures the ratio between the coefficient and its standard error.
Minitab uses the Z-value to calculate the p-value, which you use to make a decision about the statistical significance of the terms and the model. The Wald test is accurate when the sample size is large enough that the distribution of the sample coefficients follows a normal distribution.
A Z-value that is sufficiently far from 0 indicates that the coefficient estimate is both large and precise enough to be statistically different from 0. Conversely, a Z-value that is close to 0 indicates that the coefficient estimate is too small or too imprecise to be certain that the term has an effect on the response.
The tests in the Deviance table are likelihood ratio tests. The test in the expanded display of the Coefficients table are Wald approximation tests. The likelihood ratio tests are more accurate for small samples than the Wald approximation tests.
The p-value is a probability that measures the evidence against the null hypothesis. Lower probabilities provide stronger evidence against the null hypothesis.
The variance inflation factor (VIF) indicates how much the variance of a coefficient is inflated due to multicollinearity.
Use the VIF to describe how much multicollinearity exists in a regression analysis. Multicollinearity is problematic because it can increase the variance of the regression coefficients, making it difficult to evaluate the individual impact that each of the predictors has on the response.
| VIF | Multicollinearity |
|---|---|
| VIF = 1 | None |
| 1 < VIF < 5 | Moderate |
| VIF > 5 | High |
For more information on multicollinearity and how to mitigate the effects of multicollinearity, see Multicollinearity in regression.
When you standardize the continuous variables, the coefficients represent a one-unit change in the standardized variables. Usually, you standardize the continuous predictors to reduce multicollinearity or to put the variables on a common scale.
How you use the coded coefficients depends on the standardization method. The exact interpretation of the coefficients also depends on aspects of the analysis like the link function. Positive coefficients make the event more likely. Negative coefficients make the event less likely. An estimated coefficient near 0 implies that the effect of the predictor is small.
Each coefficient represents the expected change in the mean of the transformed response given that the predictor changes by 1 unit on the coded scale.
For example, a model uses temperature in degrees Celsius and time in seconds. For temperature, the coding makes 0 correspond to 50 degrees Celsius and 1 correspond to 100 degrees Celsius. For time, the coding makes 0 correspond to 30 seconds and 1 correspond to 60 seconds. The coefficient for temperature represents an increase of 50 degrees Celsius. The coefficient for time represents an increase of 30 seconds.
Each coefficient represents the expected change in the mean of the transformed response given that the predictor variable changes by 1 standard deviation.
For example, a model uses temperature in degrees Celsius and time in seconds. The standard deviation of temperature is 3.7 degrees Celsius. The standard deviation of time is 18.3 seconds. The coefficient for temperature represents an increase of 3.7 degrees Celsius. The coefficient for time represents an increase of 18.3 seconds.
Each coefficient represents the expected change in the mean of the transformed response given that the predictor changes by 1.
For example, a model uses temperature in degrees Celsius and time in seconds. The coefficient for temperature represents an increase of 1 degree Celsius. The coefficient for time represents an increase of 1 second.
Each coefficient represents the expected change in the mean of the transformed response given that the predictor variable changes by 1 standard deviation.
For example, a model uses temperature in degrees Celsius and time in seconds. The standard deviation of temperature is 3.7 degrees Celsius. The standard deviation of time is 18.3 seconds. The coefficient for temperature represents an increase of 3.7 degrees Celsius. The coefficient for time represents an increase of 18.3 seconds.
Each coefficient represents the expected change in the mean of the transformed response given that the predictor variable changes by the divisor.
For example, a model uses length in meters and electric current in amperes. The divisor is 1,000. The coefficient for length represents an increase of 1 millimeter. The coefficient for electric current represents an increase of 1 milliampere.
Each coefficient represents the expected change in the mean of the transformed response given that the predictor changes by 1 unit on the coded scale.
For example, a model uses temperature in degrees Celsius. The coding makes 0 correspond to 50 degrees Celsius and 1 correspond to 100 degrees Celsius. The coefficient for temperature represents an increase of 50 degrees Celsius. The coefficient for temperature is 1.8. When temperature increases by 1 coded unit, temperature increases by 50 degrees and the natural log of the odds increase by 1.8.
Each coefficient represents the expected change in the natural log for the odds of the event given that the predictor variable changes by 1 standard deviation.
For example, a model uses temperature in degrees Celsius. The standard deviation of temperature is 3.7 degrees Celsius. The coded coefficient for temperature is 1.4. When temperature increases by 1 coded unit, temperature increases 3.7 degrees Celsius and the natural log of the odds increase by 1.4.
Each coefficient represents the expected change in the natural log for the odds of the event given that the predictor changes by 1.
For example, a model uses temperature in degrees Celsius. The coefficient for temperature represents an increase of 1 degree Celsius. The coefficient for temperature is 2.3. When temperature increases by 1 coded unit, temperature increases 1 degree Celsius and the natural log of the odds increase by 2.3.
Each coefficient represents the expected change in the natural log for the odds of the event given that the predictor variable changes by 1 standard deviation.
For example, a model uses temperature in degrees Celsius. The standard deviation of temperature is 3.7 degrees Celsius. The coefficient for temperature is 1.4. When temperature increases by 1 coded unit, temperature increases 3.7 degrees Celsius and the natural log of the odds increase by 1.4.
Each coefficient represents the expected change in the natural log for the odds of the event given that the predictor variable changes by the divisor.
For example, a model uses length in meters and electric current in amperes. The divisor is 1,000. The coefficient for length represents an increase of 1 millimeter. The coefficient for length is 5.6. When length increases by 1 coded unit, the length increases by 1 millimeter and the natural log of the odds increase by 5.6. The coefficient for electric current represents an increase of 1 milliampere.
For binary logistic regression, Minitab shows two types of regression equations. The first equation relates the probability of the event to the transformed response. The form of the first equation depends on the link function.
The second equation relates the predictors to the transformed response. If the model contains both continuous and categorical predictors, the second equation can be separated for each combination of categories. For more information on how to choose the number of equations to display, go to Select the results to display for Fit Binary Logistic Model.
Use the equations to examine the relationship between the response and the predictor variables.
The first equation shows the relationship between the probability and the transformed response because of the logit link function.
The second equations show how income and whether a customer has children relate to the transformed response. When the customer does not have children, the coefficient for income is about 0.04. When the customer has children, the coefficient is about 0.02. For these equations, the more income a customer has, the more likely they are to buy the product. However, income has a stronger effect on whether the customer buys the product when the customer does not have children.
| P(1) | = | exp(Y')/(1 + exp(Y')) |
|---|
| Children | |||
|---|---|---|---|
| No | Y' | = | -3.549 + 0.04296 Income |
| Yes | Y' | = | -1.076 + 0.01565 Income |
If your model is nonhierarchical and you standardized the continuous predictors, then the regression equation is in coded units. For more information, see the section on Coded Coefficients. For more information about hierarchy, go to What are hierarchical models?.
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