Find definitions and interpretation guidance for every statistic in the Coefficients table.

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 reference event for the response and the reference levels for categorical predictors. 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.

Binary logistic regression in Minitab Express uses the logit link function, which provides the most natural interpretation of the estimated coefficients. 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.

- Continuous predictors
- The coefficient of a continuous predictor is the estimated change in the natural log of the odds for the reference event for each unit increase in the predictor. For example, if the coefficient for time in seconds is 1.4, then the natural log of the odds increase by 1.4 for each additional second.
- Estimated coefficients can also be used to calculate the odds ratios, or the ratio between two odds. To calculate the odds ratio, exponentiate the coefficient for a predictor. The result is the odds ratio for when the predictor is x+1, compared to when the predictor is x. For example, if the odds ratio for mass in kilograms is 0.95, then for each additional kilogram, the probability of the event decreases by about 5%.
- For continuous predictors, the interpretation of the odds can be more meaningful than the interpretation of the odds ratio.
- Categorical predictors
- The interpretation of the estimated coefficients for categorical predictors is relative to the reference level of the predictor. In Minitab Express, the reference level for a numeric categorical predictor is the level with the lowest value or for a text categorical predictor, is the level that is first in alphabetical order. Positive coefficients indicate that the event is more likely at that level of the predictor than at the reference level. Negative coefficients indicate that the event is less likely at that level of the predictor than at the reference level.
- The coefficient is the estimated change in the natural log of the odds when you change from the reference level to the level of the coefficient. For example, a categorical variable has the levels Fast and Slow, and the reference level is Slow. If the coefficient for Fast is 1.3, then a change in the variable from Slow to Fast increases the natural log of the odds of the event by 1.3.
- Estimated coefficients can also be used to calculate the odds ratio, or the ratio between two odds. To calculate the odds ratio, exponentiate the coefficient for a level. The result is the odds ratio for the level compared to the reference level. For example, a categorical variable has the levels Hard and Soft, and Soft is the reference level. If the odds ratio for Hard is 0.5, then the change from Soft to Hard decreases the odds of the event by 50%.

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.

The confidence interval is composed of the following two parts:

- Point estimate
- This single value estimates a population parameter by using your sample data. The confidence interval is centered around the point estimate.
- Margin of error
- The margin of error defines the width of the confidence interval and is determined by the observed variability in the sample, the sample size, and the confidence level. To calculate the upper limit of the confidence interval, the margin of error is added to the point estimate. To calculate the lower limit of the confidence interval, the margin of error is subtracted from the point estimate.

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 p-value is a probability that measures the evidence against the null hypothesis. Lower probabilities provide stronger evidence against the null hypothesis.

To determine whether the association between the response and each term in the model is statistically significant, compare the p-value for the term to your significance level to assess the null hypothesis. The null hypothesis is that the term's coefficient is equal to zero, which indicates that there is no association between the term and the response. Usually, a significance level (denoted as α or alpha) of 0.05 works well. A significance level of 0.05 indicates a 5% risk of concluding that an association exists when there is no actual association.

- P-value ≤ α: The association is statistically significant
- If the p-value is less than or equal to the significance level, you can conclude that there is a statistically significant association between the response variable and the term.
- P-value > α: The association is not statistically significant
- If the p-value is greater than the significance level, you cannot conclude that there is a statistically significant association between the response variable and the term. You may want to refit the model without the term.
- If there are multiple predictors without a statistically significant association with the response, you must reduce the model by removing terms one at a time. For more information on removing terms from the model, go to Model reduction.

If a model term is statistically significant, the interpretation depends on the type of term. The interpretations are as follows:

- If a continuous predictor is significant, you can conclude that the coefficient for the predictor does not equal zero.
- If a categorical predictor is significant, you can conclude that the probability for that level is different from the probability for the reference level. In Minitab Express, the reference level for a numeric categorical predictor is the level with the lowest value or for a text categorical predictor, is the level that is first in alphabetical order.