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

## Test for equal shape and scale 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 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 t 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 muffler data, the test is whether the amount of miles before failure for the new type of mufflers and the amount of miles before failure for the old type of mufflers 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 the new type of mufflers is significantly different than the parameters for the old type of mufflers. Therefore, the two data sets do not come from the same distribution.

## Test for equal shape parameters

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

A chi-square test determines whether the shape parameters for the two or more 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 the shape 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 shape 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 shape parameters for the data sets are significantly different.

### Interpretation

For the muffler data, the test is whether the miles driven until failure for the new type of muffler and the miles driven before failure for the old type of muffler come from a distribution with the same shape parameter.

Because the p-value of 0.000 is less than the α-value of 0.025, you can conclude the shape parameters of the distribution significantly differ for the two types of mufflers. Examine the Bonferroni confidence intervals for the shape parameters to identify the magnitude of the differences in the shape parameters between the two distributions.

## Test for equal scale parameters

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

A chi-square test determines whether the scale parameters for the two 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 of a distribution, such as the shape 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 statistically 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 muffler data, the test is whether the distribution of the miles driven until failure for the new type of muffler and the miles driven until failure for the old type of muffler have the same scale parameter.

Because the p-value of 0.000 is less than the α-value of 0.025, you can conclude that the scale parameters of the distribution for the two types of muffler are significantly different.

## Bonferroni confidence intervals for shape parameter

If the results of the test for equal shape (or location) parameters indicate 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 muffler data, likely values for the shape parameter of the old type of mufflers range from 0.5954 (59.54%) to 0.7133 (71.33%) of the shape parameter for the new type of mufflers. The estimated ratio of the shape parameter is 0.6517 or 65.17%.

## Bonferroni confidence intervals for scale parameter

If the results of the test for equal scale 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 muffler data, likely values of the scale parameter for the old type of mufflers range from 0.8225 (82.25%) to 0.8631 (86.31%) of the scale parameter for the new type of mufflers. The estimated ratio of the scale parameter is 0.8426 or 84.26%.

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