Test for Presence of Correlation

It is important to test the successive inter-arrival times for independence by testing them for serial correlation. To check the data for serial correlation, (i - 1)^th TBFs are to be plotted against i^th TBFs of the data. If the plotted points are randomly scattered without any pattern, it can be interpreted that the TBFs are free from serial correlation.

• Testing for the Presence of Serial Correlation

Before modeling the reliability data, it should be tested for the mutual independence by testing it for the presence of serial correlation. The presence of serial correlation can be tested by plotting the i^th TBF say, xi against the (i - 1)^th TBF, x_i-1. If the plotted points are randomly scattered without any pattern, it can be interpreted that the TBFs are free from serial correlation. In case the plot reveals serial correlation then the TBFs should be plotted at greater lags such as xi against x_i-2, x_{i -3}, etc. to search for serial correlation over greater lags (Bendell and Wall 1985).          

• Analysis with Coefficient of Correlation Test

The data can also be analyzed with the Karl Pearson's Coefficient of Correlation Test between i^th v (i-1)^th TBFs and ith v (i-2)^th and so on. The degree of association can be concluded with the value of the coefficient of correlation that ranges between - 1 to 1 via zero. The value nearing to - 1 can be understood as the negative close association, which means that the increase in the first quantity induces the decrease in the second quantity. If the values is near to + 1, it means that there is proportional increasing correlation between the quantities. If the value is near to zero, it indicates poor correlation, which means that the quantities are independent (i.i.d. assumption is not contradicted).

• Analysis of Data Free from Trends and Correlation

When the data are free from the presence of a trend and serial correlation, the next step is to choose a best-fit probability distribution model using "Total Time on Test" (TTT) plots or "goodness of fit" tests to study its statistical characteristics.