Normal Distribution - Hazard Plotting

	Normal Distribution - Hazard Plotting

The following section describes hazard plotting for the normal distribution using the Reliability & Maintenance Analyst. The manual method is located here.

Hazard plotting is an excellent method for determining goodness-of-fit. To determine the goodness-of-fit click the "Plot" button. If the plotted points form a straight line, the distribution provides a good time to fail model for the data. The "R-Squared" value is a measure of how well the data forms a straight line. An R-Squared value of 1.0 indicates a perfectly straight line. R-Squared is also known as the coefficient of determination. To estimate the parameters of the normal distribution using hazard plotting, follow these steps:

Enter the data using one of the data entry grids, or connect to a database.
Select the "Parameter Estimation"
Select "Normal"
Select "Hazard Plot"

The figure below shows the normal hazard plotting screen using the data in the file "Demo2.dat".

Clicking the "Plot" button gives a hazard plot. If the plotted points do not follow a straight line, the normal distribution with the estimated parameters does not provide an adequate time to fail model. The untransformed option plots the cumulative distribution function against time. This is useful for determining the probability of failure at a given time. The title of the graphs can be changed by editing the text in the Graph Title frame. To check the spelling of the title, click the "Spell Check" button.

To predict reliability or time-to-fail using the estimated parameters use the Predicting Module.

Manual Hazard Plotting

The normal cumulative hazard function is

where is the standard normal cumulative distribution function.

By rearranging the equation above, the survival time can be represented as a function of the cumulative hazard function.

where is the inverse of the standard normal cumulative distribution function.

It can be seen that by plotting x versus the resulting y-intercept equals m and the resulting slope equals s. The hazard function, h(x), is estimated from the inverse of the reverse rank of the ordered failures, and H(x) is the cumulative of the values of h(x). Censored data points are used to compute ranks, but are not included in hazard plots.

An alternative to plotting x versus on conventional graph paper is to plot x versus H(x) on specialized hazard paper. The advantage of hazard paper is that the values of do not have to be computed. Computers have made this technique obsolete.

Example

Use hazard plotting to determine the parameters of the normal distribution given the following multiple censored data set. A "c" following an entry indicates the censoring.

Time to Fail

150 c

183 c

235

157 c

209

235

167 c

216 c

248 c

179

217 c

257

Solution
The table below is constructed to obtain the necessary plotting data.

Time to
Fail

Reverse
Rank

150 c

12

157 c

11

167 c

10

179

9

0.1111

0.1111

0.1052

-1.2527

183 c

8

209

7

0.1429

0.2540

0.2243

-0.7578

216 c

6

217 c

5

235

4

0.2500

0.5040

0.3959

-0.2640

235

3

0.3333

0.8373

0.5671

0.1691

248 c

2

257

1

1.0000

1.8373

0.8408

0.9976

By plotting the last column of the data above on the x-axis, and the time to fail on the y-axis, the resulting line provides parameter estimates. The best fit straight line through the data points is found using linear regression. The y-intercept of the best fit straight line through the five points provides an estimate of m, 230.3 in this case, and the slope of the line provides an estimate of s, 32.9 in this case.

Click here to download this example in Microsoft Excel.