Find definitions and interpretation guidance for every statistic that is provided for random effect predictions.

The Best Linear Unbiased Predictor (BLUP) for a specific level of a random factor term describes the effect of the term level on the response. Minitab uses these values to calculate the conditional fitted values for the given levels of the random factors.

Use the BLUP to evaluate how different the effects of the random factor at the given levels are on the response. The value and the sign of the BLUP for a specific level describe the direction and size of the effect.

The standard error of the best linear unbiased prediction (BLUP) for a specific level represents the uncertainty in the predicted effect on the response.

Standard error of BLUP measures the uncertainty in the predictor. The standard error of the BLUP is used to calculate the t-value and then to construct the test on whether the effect at a specific level is significantly different from 0. If the associated p-value is less than your significance level (*α)*, you conclude that the effect for the specific level is different from 0.

The degrees of freedom represent the amount of information in the data to estimate the confidence interval and to construct the test for the Best Unbiased Linear Prediction (BLUP).

Use the DF to compare how much information is available about the BLUPs. Generally, more degrees of freedom make the confidence interval for the BLUP narrower than an interval with less degrees of freedom.

These confidence intervals (CI) are ranges of values that are likely to contain the true values of the Best Linear Unbiased Prediction (BLUP) for the random terms 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.

Use the confidence interval to assess the specific level effect of a random term on the response. An interval that does not contain 0 indicates a statistically significant effect. If the interval is strictly greater than 0, the specific level has a positive effect on the response. An interval that is strictly less than 0 indicates a negative effect on the response. An interval that contains 0 does not support a significant level effect of the random term on the response.

The t-value measures the ratio between the Best Linear Unbiased Prediction (BLUP) and its standard error.

Minitab uses the t-value to calculate the p-value, which you use to make a decision about the statistical significance of the BLUP values.

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 rejection is the same no matter what the degrees of freedom are.

The p-value is a probability that measures the evidence against the null hypothesis. The null hypothesis is that the specific level effect of a random factor on the response is 0. Lower probabilities provide stronger evidence against the null hypothesis.

To determine whether the Best Linear Unbiased Prediction (BLUP) for a specific level of a random factor is different from 0, compare the p-value for the BLUP to the significance level.

- P-value ≤ α: The effect is statistically different from 0
- If the p-value is less than or equal to the significance level, you can conclude that the specific level effect of the random factor term on the response is significantly different from 0.
- P-value > α: The effect is not statistically different from 0
- If the p-value is greater than the significance level, you cannot conclude that the specific level effect of the random factor term on the response is significantly different from 0.