Death Of Soldiers In 
The field of reliability is concerned with identifying, predicting, and preventing failures. One way to make reliability obvious is to prepare Crow/AMSAA plots of cumulative failures versus cumulative time.
Crow/AMSAA methodology is useful for mixed failure modes, which means the cause for the failures can be by many different reasons. When improvements have occurred, a cusp will appear on the trend line signifying failures are coming more slowly (if improvements are being made). Or the failures may be coming more quickly (if the situation deteriorates to a less favorable condition).
The interesting thing about Crow/AMSAA plots is stable processes give straight lines when plotted on loglog paper. The straight line can be regressed using simple curve fit techniques to forecast future failures. The slope (the beta value from the regression) of the Crow/AMSAA trend line is an important statistic as it tells if failures are increasing, decreasing, or floundering along with no deterioration or no improvement.
The website http://icasualties.org has been
publishing fatality statistics during Gulf War II. Obviously, deaths of soldiers represent
failures. War deaths have mixed modes of
failures. Crow/AMSAA plots should
produce straight line plots on loglog paper and give a way to determine if:
1)
failures are reducing (b<1), deteriorating (b>1), or
without changes (b≈1),
2)
a cusp has formed on the trend line signifying improvement or deterioration,
3)
predict future failures (deaths) of soldiers.
Here are the statistics from Icasualties (which I have accepted as “fact” although that is frequently a risky thing to do with data from the Internet). Table 1 shows the December 25, 2008 Christmas Day score card with incomplete data for December ‘08. Revisions to earlier data continues.
A Crow/AMSAA plot of the data in Figure 1 (including incomplete month of December ‘08) with a single trend line for USA Data. The line slope shows casualties continue to increase:

Figure
1: Total 
Zooming in on the top right hand portion of the trendline we see substantial reductions in fatalities following the troop surge announced January 10, 2007:

Figure
2: Recent 
The clear cusp on the current data is favorable and the decision for where the cusp is marked is made by engineering judgment. The post surge trend line moves horizontally to the right with fewer deaths over a long time interval! Compare the favorable line slope beta = 0.419 with NASA’s favorable trend line displayed for the space shuttle failures and it’s beta = 0.459 or compare the favorable cusps on accidents associated with Houston, TX light railroad before and after improvements the line slope beta = 0.844 which clearly in not as good as shown in Iraq.
Here’s the sad forecast from Figure 2, we’re going to loose more soldiers (but at a declining rate) in the Iraqi war.
The goodness of fit criteria for the
Please note, I’m not causing the forecasted fatalities (so don’t send hysterical Email!), I’m only predicting the outcome based on the data. I’ll be happy to forecast fewer failures if I can see objective evidence of meaningful improvements—in short, show me the improvements don’t tell me about the good things that could or might happen.
We desperately needed a favorable cusp on the trend line to prevent the deaths of USA soldiers and we got it, now we need to even make more improvements to further reduce casualty rate in Iraq (by the way, I don’t have a magic solution for how to accomplish this feat)!!! How many more deaths of soldiers do I want to see?—the answer is ZERO, and how many more deaths will we have—the answer is too many. Wars are dirty business to meet the objectives of peace.
Where can you learn more about Crow/AMSAA plots:
1) The New Weibull Handbook by Dr. Robert B. Abernethy
2) WinSMITH Visual software by Fulton Findings for making reliability growth plots
3) Crow/AMSAA Reliability Growth Plots Problem Of The Month
4) MILHDBK189 Reliability Growth Management
5) TR652 AMSAA Reliability Growth Guide available for download from the November ‘02 Problem Of The Month
6) IEC 61164 (2004) Reliability growth – Statistical test and estimation methods (55 pages (~US$190). Describes procedures to estimate the parameters of the power law model along with confidence intervals for failure intensity, and prediction intervals for the time to future failures along with a test of goodness of fit to data from repaired items.
You can download a PDF copy of this by clicking here.
The bottom line:
Crow/AMSAA plots show cusps when
failures decline to push the trend line to lower failure rates. Likewise, a cusp forms when failures increase
and the trend predicts more failures.
The main issue is to decrease failures.
Comments:
Refer to the caveats on the Problem Of The Month Page about the limitations of the following solution. Maybe you have a better idea on how to solve the problem. Maybe you find where I've screwedup the solution and you can point out my errors as you check my calculations. Email your comments, criticism, and corrections to: Paul Barringer by clicking here. Return to top of page.
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