# Random effect predictions for Stability Study

Find definitions and interpretation guidance for the random effect predictions.

## BLUP

The Best Linear Unbiased Predictions (BLUP) are the estimated coefficients for levels of a random batch term. With these coefficients, you can determine the intercept and the slope for the conditional fitted equations, which predict the fitted values for the specific batches. You can view the conditional equations using Predict for Stability Study.

### Interpretation

Use the BLUP to estimate how different the batches are. Larger BLUP values at for the batch factor indicate that the shelf lives for the batches in the data are more different at time 0. If the time by batch interaction is not in the model, the shelf lives for the batches in the data are the same distance apart for all times. If the time by batch interaction is in the model, then the BLUP values show how the batches degrade at different rates.

For the batch terms, the most positive BLUP is for Batch 1, approximately 1.36. The BLUP for Batch 7 is closer to zero, approximately 0.05. The conditional fit for Batch 1 at time 0 is approximately 100.6 + 1.36 = 101.42. The conditional fit for Batch 7 is approximately 100.06 + 0.05 = 100.11.

Because the month by batch interaction is also in the model, the BLUP values for the interactions describe differences in how fast the different batches degrade. The most positive interaction BLUP is for Batch 2, approximately 0.02. Thus, the conditional fits for Batch 2 show the slowest degradation.

Coefficients Term Coef SE Coef DF T-Value P-Value Constant 100.060247 0.268706 7.22 372.378347 0.000 Month -0.138766 0.005794 7.22 -23.950196 0.000
Random Effect Predictions Term BLUP StDev DF T-Value P-Value Batch 1 1.359433 0.313988 12.45 4.329567 0.001 2 0.395375 0.313988 12.45 1.259203 0.231 3 0.109151 0.313988 12.45 0.347629 0.734 4 -0.409322 0.313988 12.45 -1.303623 0.216 5 -0.135643 0.313988 12.45 -0.432001 0.673 6 -1.064736 0.313988 12.45 -3.391006 0.005 7 0.049420 0.313988 12.45 0.157394 0.877 8 -0.303678 0.313988 12.45 -0.967164 0.352 Month*Batch 1 0.006281 0.008581 10.49 0.731925 0.480 2 0.019905 0.008581 10.49 2.319537 0.042 3 -0.013831 0.008581 10.49 -1.611742 0.137 4 0.003468 0.008581 10.49 0.404173 0.694 5 0.001240 0.008581 10.49 0.144455 0.888 6 0.000276 0.008581 10.49 0.032144 0.975 7 -0.010961 0.008581 10.49 -1.277272 0.229 8 -0.006378 0.008581 10.49 -0.743220 0.474

## StDev for BLUP

The standard deviation of the best linear unbiased prediction (BLUP) estimates the uncertainty from estimating the BLUP from sample data.

### Interpretation

Use the standard deviation of the BLUP to measure the precision of the estimate of the BLUP. The smaller the standard deviation, the more precise the estimate. Dividing the BLUP by its standard deviation calculates a t-value. If the p-value associated with this t-statistic is less than your significance level (denoted as alpha or α), you conclude that the difference between the BLUP and 0 is statistically significant.

## DF for BLUP

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).

### Interpretation

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.

## Confidence interval for BLUP (95%CI)

These confidence intervals (CI) are ranges of values that are likely to contain the true value of the Best Linear Unbiased Prediction (BLUP) for each randomly-selected batch in the data.

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

Use the confidence interval to assess the estimate of the population BLUP for each batch.

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.

## T-value

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

### Interpretation

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.

## P-Value for BLUP

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 the Best Linear Unbiased Prediction (BLUP) is different from zero, compare the p-value for the BLUP to the significance level. The null hypothesis is that the BLUP is zero, which implies that the prediction for that particular batch is not different from the prediction for a randomly-selected batch.

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 difference between the BLUP and zero.
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 difference between the BLUP and zero.
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