# Trend tests (also called goodness-of-fit tests)

## What is a trend test?

Use the tests for trend to determine whether a homogeneous Poisson process or a nonhomogeneous Poisson process is the appropriate model.

Regardless of the model you choose, the hypotheses for the tests for trend test are generally:
• H0: No trend in data (homogeneous Poisson process)
• H1: Trend in data (nonhomogeneous Poisson process)

If you reject the null hypothesis, you can conclude that there is some trend in your data and you should model your data with a nonhomogeneous Poisson process such as the power-law process.

If you fail to reject the null hypothesis, there is not enough evidence to reject the homogeneous Poisson process model. Although the power-law process may still be appropriate, the homogeneous Poisson process is a simpler model and thus a better choice.

## Which trend tests are included in Minitab?

With exact data, Minitab provides three trend tests:
• MIL-Hdbk-189 (The military handbook test)
• Laplace
• Anderson-Darling
With exact data from multiple systems, Minitab provides five trend tests:
• MIL-Hdbk-189 (pooled)
• MIL-Hdbk-189 (TTT-based)
• Laplace (pooled)
• Laplace (TTT-based)
• Anderson-Darling

With interval data, Minitab provides only the MIL-Hdbk-189 test. Minitab uses the pooled version of the MIL-Hdbk-189 test when the data for different systems are in one column and another column provides system identifiers. When the data are in one column, Minitab assumes that the different systems are from identical processes. Minitab uses the TTT-based version of the MIL-Hdbk-189 test when the data for different systems are in different columns. When the data are in different columns, Minitab assumes that different systems are from different processes.

## A comparison of trend tests

Minitab's trend tests behave differently depending on two conditions::
• Whether the data follow a non-monotonic trend
• Whether the data are from heterogeneous systems

### Monotonic and non-monotonic trends

If the times change in a systematic way, a trend exists in the pattern of times between failures. Trends can be monotonic or non-monotonic.

Monotonic trends
Times between failures are getting either consistently longer (decreasing trend) or consistently shorter (increasing trend).
Non-monotonic trends
Times between failures alternate between increasing and decreasing trend (cyclic) or have a decreasing trend, no trend, and then increasing trend (bathtub).
The Anderson-Darling test rejects the null hypothesis in the presence of both monotonic and non-monotonic trends. The other tests generally only detect monotonic trends. While the Anderson-Darling test is useful if you suspect the existence of a cyclic or other non-monotonic trend, the other tests are more powerful in the case of a monotonic trend.

### Homogeneous and heterogeneous systems

The null hypothesis of no trend differs slightly for each test:
• The null hypothesis for the pooled tests (MIL-hdbk-189 and Laplace's) is that the data come from a homogeneous Poisson process (HPP) with a possibly different mean-time-between-failures (MTBF) for each system. Thus, rejecting the null hypothesis means that you can conclude that a trend exists in your data.
• The null hypothesis for the TTT-based tests (MIL-hdbk-189, Laplace's, and Anderson-Darling) is that the data come from a homogeneous Poisson process (HPP) with the same MTBF for each system. Thus, rejecting the null hypothesis could mean that either a trend exists in your data or your data come from heterogeneous systems. Therefore, you should use TTT-based tests only when you are confident that your systems are homogeneous.

### Conclusions from trend tests

This table summarizes the conclusions you can make for each test.
Test Null hypothesis Rejecting H0 means
MIL-Hdbk-189 (Pooled)

Laplace's (Pooled)

HPP (possibly different MTBFs) Monotonic trend
MIL-Hdbk-189 (TTT-based)

Laplace's (TTT-based)

HPP (same MTBFs) Monotonic trend or systems are heterogeneous
Anderson-Darling HPP (possibly different MTBFs) Monotonic or non-monotonic trend or systems are heterogeneous
A relatively large difference in p-values between TTT-based tests (including the Anderson-Darling test) and the pooled tests may indicate heterogeneity between systems. You may need to analyze the data separately for each system.
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