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.
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.
The coefficient of the term represents the change in the mean response for one unit of change in that term. If the coefficient is negative, as the term increases, the mean value of the response decreases. If the coefficient is positive, as the term increases, the mean value of the response increases.
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_{1} + 10.1x_{2}. In the equation, x_{1} is the hours of in-house training (from 0 to 20). The variable x_{2} is a categorical variable that equals 1 if the employee has a mentor and 0 if the employee does not have a mentor. The response is y and is the test score. The coefficient for the continuous variable of training hours, is 4.3, which indicates that, for every hour of training, the mean test score increases by 4.3 points. Using the (0, 1) coding scheme, the coefficient for the categorical variable of mentoring indicates that employees with mentors have scores that are an average of 10.1 points greater than employees without mentors.
Minitab can fit linear models using a variety of coding schemes for the continuous variables in the model. These coding schemes can improve the estimation process and the interpretation of the results. In addition, coded units can change the results of the statistical tests used to determine whether each term is a significant predictor of the response. When a model uses coded units, the analysis produces coded coefficients.
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:
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.
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.
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 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.
The variance inflation factor (VIF) indicates how much the variance of a coefficient is inflated due to the correlations among the predictors in the model.
Use the VIF to describe how much multicollinearity (which is correlation between predictors) 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 correlated predictors has on the response.
VIF | Status of predictor |
---|---|
VIF = 1 | Not correlated |
1 < VIF < 5 | Moderately correlated |
VIF > 5 | Highly correlated |
For more information on multicollinearity and how to mitigate the effects of multicollinearity, see Multicollinearity in regression.