Find definitions and interpretations 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.

In General MANOVA, Minitab displays coefficients for the constant and the covariates for each univariate analysis. To assess the categorical factors, see the Analysis of Variance table and the Means table.

The coefficient for a term represents the change in the mean response associated with a change in that term, while the other terms in the model are held constant. The sign of the coefficient indicates the direction of the relationship between the term and the response. The size of the coefficient is usually a good way to assess the practical significance of the effect that a term has on the response variable. However, the size of the coefficient does not indicate whether a term is statistically significant because the calculations for significance also consider the variation in the response data. To determine statistical significance, examine the p-value for the term.

For example, a manager determines that an employee's score on a job skills test can be predicted using the regression model, y = 130 + 4.3x. In the equation, x is the hours of in-house training (from 0 to 20) and y is the test score. The coefficient, or slope, is 4.3, which indicates that, for every hour of training, the mean test score increases by 4.3 points.

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. Dividing the coefficient by its standard error calculates a t-value. If the p-value associated with this t-statistic is less than your significance level, you conclude that the coefficient is statistically significant.

For example, technicians estimate a model for insolation as part of a solar thermal energy test: ### Regression Analysis: Insolation versus South, North, Time of Day

Coefficients
Term Coef SE Coef T-Value P-Value VIF
Constant 809 377 2.14 0.042
South 20.81 8.65 2.41 0.024 2.24
North -23.7 17.4 -1.36 0.186 2.17
Time of Day -30.2 10.8 -2.79 0.010 3.86

In this model, North and South measure the position of a focal point in inches. The coefficients for North and South are similar in magnitude. The standard error of the coefficient for South is smaller than the standard error of the coefficient for North. Therefore, the model is able to estimate the coefficient for South with greater precision.

The standard error of the North coefficient is nearly as large as the value of the coefficient itself. The resulting p-value is greater than common levels of the significance level, so you cannot conclude that the coefficient for North differs from 0.

While the coefficient for South is closer to 0 than the coefficient for North, the standard error of the coefficient for South is also smaller. The resulting p-value is smaller than common significance levels. Because the estimate of the coefficient for South is more precise, you can conclude that the coefficient for South differs from 0.

Statistical significance is one criterion you can use to reduce a model in multiple regression. For more information, go to Model reduction.

The t-value measures the ratio between the coefficient and its standard error.

Minitab uses the t-value to calculate the p-value, which you use to test whether the coefficient is significantly different from 0.

You can use the t-value to determine whether to reject the null hypothesis. However, the p-value is used more often because the threshold for the rejection of the null hypothesis does not depend on the degrees of freedom. For more information on using the t-value, go to Using the t-value to determine whether to reject the null hypothesis.

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 implies 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 can 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 coefficient for a continuous variable is significant, changes in the value of the variable are associated with changes in the mean response value.
- If a coefficient for a categorical level is significant, the mean for that level is different from either the overall mean (-1, 0 ,+1 coding) or the mean for the reference level (0, 1 coding).
- If a coefficient for an interaction term is significant, the relationship between a factor and the response depends on the other factors in the term. In this case, you should not interpret the main effects without considering the interaction effect.
- If a coefficient for a polynomial term is significant, you can conclude that the data contain curvature.