# Equality of parameters for Parametric Distribution Analysis (Right Censoring)

## Test for equal scale and location parameters

You can test whether two or more data sets come from the same distribution (population). If the data sets are from the same distribution, then they should have equal parameters.

A simultaneous chi-square test determines whether the distribution parameters for the two data sets are significantly different from each other. Compare the p-value with your pre-determined α-value.
• If the p-value is less than the α-value, then you can conclude that at least one of the distribution parameters for the data sets is significantly different.
• If the p-value is greater than the α-value, then you cannot conclude that the distribution parameters for the data sets are significantly different.

If the data sets come from different distributions (the p-value is less than the α-value), then examine the results of the individual tests for equal shape (or equal location) and equal scale parameters. Using the results from the individual tests, you can determine whether the differences between the distributions occur in the scale parameter (shape for Weibull distribution), the location parameter (scale for Weibull distribution), or both parameters.

### Interpretation

For the engine windings data, the test is whether the time until failure at 80° C and the time until failure at 100° C come from the same distribution.

Because the p-value of 0.000 for the simultaneous test is less than the α-value of 0.05, you can conclude that at least one of the parameters for the distribution for 80° C is significantly different than the parameters of the distribution for 100° C. Therefore, the two data sets do not come from the same distribution.

## Test for equal scale parameters

If the simultaneous test for equal scale and location parameters indicates a statistically significant difference, then the test for equal scale parameters can help you determine whether the differences between the distributions occur within the scale parameters.

A chi-square test determines whether the scale parameters for the data sets are significantly different from each other. Compare the p-value with your pre-determined α-value. If you are testing more than one parameter from a distribution, such as the location and the scale, adjust the α-value to account for multiple tests. In this example, two parameters are tested, so the α-value for each test is 0.05/2=0.025.
• If the p-value is less than the α-value, then you can conclude that the scale parameters for the data sets are significantly different. When there is a significant difference, examine the Bonferroni confidence intervals for the parameters to identify the magnitude of the differences in the parameter between the distributions.
• If the p-value is greater than the α-value, then you cannot conclude that the scale parameters for the data sets are significantly different.

### Interpretation

For the engine windings data, the test is whether or not the time to failure at 80° C has the same scale parameter as the time to failure at 100° C.

Because the p-value of 0.021 is less than the α-value of 0.025, you can conclude that the scale parameters for the distribution of time to failure at 80°C and at 100° C are significantly different. Examine the Bonferroni confidence intervals for the scale parameters to identify the magnitude of the differences in the scale parameters between the two distributions.

## Test for equal location parameters

If the simultaneous test for equal scale and location parameters indicates a statistically significant difference, examine the test for equal location parameters to determine whether the differences between the distributions occur within the location parameters.

A chi-square test determines whether the location parameters for the two data sets are significantly different from each other. Compare the p-value with your predetermined α-value. If you are testing more than one parameter from a distribution, such as the location and the scale, adjust the α-value to account for multiple tests. In this example, two parameters are tested, so the α-value for each test is 0.05/2=0.025.
• If the p-value is less than the α-value, then you can conclude that the location parameters for the data sets are significantly different. If there is a significant difference, then examine the Bonferroni confidence intervals for the parameters to identify the magnitude of the differences in the parameter between the distributions.
• If the p-value is greater than the α-value, then you cannot conclude that the location parameters for the data sets are significantly different.

### Interpretation

For the engine windings data, the test is whether the time to failure at 80°C has the same location parameter as the time to failure at 100° C.

Because the p-value of 0.001 is less than the α-value of 0.025, you can conclude that the location parameters for the distribution of time to failure at 80°C and at 100° C are significantly different. Examine the Bonferroni confidence intervals for the location parameters to identify the magnitude of the differences in the location parameters between the two distributions.

## Bonferroni confidence intervals for shape or scale parameters

If a test for equal scale parameters or equal shape parameters indicates a statistically significant difference, then examine the Bonferroni confidence intervals to determine the magnitude of the difference.

You can also compare the intervals for multiple samples to see which parameters are different. If the confidence interval for the ratio of two parameters contains 1, then you cannot conclude that the two parameters are different.

### Interpretation

For the engine windings data, likely values for the scale parameter of Temp100 range from 1.011 to 2.236 times that of the scale parameter for Temp80, with the estimated ratio being 1.503.

## Bonferroni confidence intervals for location parameter

If a test for equal location parameters indicates a statistically significant difference, then examine the Bonferroni confidence intervals to determine the magnitude of the difference.

You can also compare the intervals for multiple samples to see which parameters are different. If the confidence interval for the ratio of two parameters contains 1, then you cannot conclude that the two parameters are different.

### Interpretation

For the engine windings data, likely values for the location parameter of Temp80 range from 0.1546 to 0.7734 greater than the location parameter for Temp100, with the estimated difference being 0.4640.

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