What is a regression coefficient?

Coefficients are the numbers by which the variables in an equation are multiplied. For example, in the equation y = -3.6 + 5.0X1 - 1.8X2, the variables X1 and X2 are multiplied by 5.0 and -1.8, respectively, so the coefficients are 5.0 and -1.8.

The size and sign of a coefficient in an equation affect its graph. In a simple linear equation (contains only one x variable), the coefficient is the slope of the line.

The coefficient (and slope) is positive 5.

The coefficients are 2 and -3.

When calculating a regression equation to model data, Minitab estimates the coefficients for each predictor variable based on your sample and displays these estimates in a coefficients table. For example, the following coefficients table is shown in the output for a regression equation:

Regression Equation Heat Flux = 325.4 + 2.55 East + 3.80 South - 22.95 North + 0.0675 Insolation + 2.42 Time of Day

This equation predicts the heat flux in a home based on the position of its focal points, the insolation, and the time of day. Minitab displays the coefficient values for the equation in the second column:

Coefficients Term Coef SE Coef T-Value P-Value VIF Constant 325.4 96.1 3.39 0.003 East 2.55 1.25 2.04 0.053 1.36 South 3.80 1.46 2.60 0.016 3.18 North -22.95 2.70 -8.49 0.000 2.61 Insolation 0.0675 0.0290 2.33 0.029 2.32 Time of Day 2.42 1.81 1.34 0.194 5.37

Each coefficient estimates the change in the mean response per unit increase in X when all other predictors are held constant. For example, in the regression equation, if the North variable increases by 1 and the other variables remain the same, heat flux decreases by about 22.95 on average.

If the p-value of a coefficient is less than the chosen significance level, such as 0.05, the relationship between the predictor and the response is statistically significant. Minitab also includes a value for the constant in the equation in the Coef column.

Note

The term coefficient can also be used to denote a calculated numerical value used as an index, such as a coefficient of correlation, a coefficient of determination, or Kendall's coefficient.

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