Coefficients table for Fit Mixed Effects Model

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

Coefficients

The coefficients table provides coefficients for individual levels of fixed factor terms and coefficients for covariate terms. A coefficient for a fixed factor term at a specific level describes the effect of the factor level on the response, compared with the rest of the factor levels. A coefficient for a covariate term represents the size and direction of the linear relationship between the term and the response.

Interpretation

The interpretation of each coefficient depends on whether the coefficient is for a fixed factor term or for a covariate term:
Fixed factor term

The coefficients for a fixed factor term display how the level means for the term differ. You can also perform a multiple comparisons analysis for the term to further classify the level effects into groups that are statistically the same or statistically different.

Covariate term

The coefficient for a covariate term represents the change in the mean response associated with a 1-unit change in that term, while everything else in the model is the same. The sign of the coefficient indicates the direction of the relationship between the term and the response. The size of the coefficient usually provides a good way to assess the practical significance of the term on the response variable.

SE Coef

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 model terms and the sample size would remain the same if you sampled again and again.

Interpretation

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 significantly different from 0.

DF

The degrees of freedom (DF) are the amount of information in your data. Minitab uses the degrees of freedom to construct the t-test for the coefficient.

Confidence Interval for coefficient (95% CI)

These confidence intervals (CI) are ranges of values that are likely to contain the true values of the coefficients for a fixed effect 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.

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.

Interpretation

If the confidence level is 95%, you can be 95% confident that the confidence interval contains the true value of the corresponding coefficient. 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.

T-value

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

Interpretation

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.

P-Value – Coefficient

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

Interpretation

To determine whether a coefficient is significantly different from 0, compare the p-value for the coefficient to the significance level. 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 affect exists when there is no actual affect.

P-value ≤ α: The coefficient is statistically different from 0

If the p-value is less than or equal to the significance level, you can conclude that the coefficient is significantly different from 0..

P-value > α: The coefficient is not statistically different from 0
If the p-value is greater than the significance level, you cannot conclude that the coefficient is significantly different from 0.
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