The simplest way to deal with a small significant group (having a significant count of fatalities, such as alcoholics) within the car driver population is to treat the very small group as a stream of negligible size and factor that stream out completely.

This page demonstrates that significant small streams can be discarded without significantly affecting the result.

A PRIORI

1. Known is the 1996 count of car driver fatalities and which were wearing a seat belt and which were alcohol affected.
2. Known is the rate of belt wearing by car drivers in 1996 was found by NOPUS with 95% confidence to be within a band 4.2 % either side of 65.1%. (i.e. there is 95% confidence that the actual wearing rate was between 60.9% and 69.3%)
3. It simplifies the demonstration if we assume the proportion of car drivers that are in the alcohol affected stream as 1%.
4. By assuming that the seat belt is equally effective for alcohol affected and alcohol unaffected streams, it is possible to deduce the effectiveness of the seat belt and to deduce the belt wearing rate of each stream.
5. An important constraint is that if the two streams are merged the resulting population would have the observed belt wearing rate of 65.1% and the observed fatality rate as shown on the Table below.

Columns 1, 2, 3 & 4 contain fatality counts.  Columns 5, 6 & 7 were calculated from columns 1 . . 4 according to the formula given in the column header.

Calculation of the wearing rates and effectiveness using two streams:

Stream A of assumed size 1% of the population.  It can be deduced that alcoholics would be wearing belts 25.3% of the time to produce the observed fatality rate of 30.5%.
Stream B will consist of the remaining 99% of the population.  It can be deduced that people who are not alcohol affected were wearing belts 65.5% to produce the observed rate at fatality of 71.1%.

When these two streams are combined, their combined (average) belt wearing rate would be

        0.01 * 25.3% + 0.99 * 65.5% = 65.1%         (as found by NOPUS)

Effectiveness is given by the expression E = 1-(b/u)*(1-p)/p {where b=belted, u=unbelted, p=proportion wearing belts}

So the belt effectiveness for stream A is          E = 1 - 1,376/3,139*(1-0.253)/0.253         = -29.4%
And the effectiveness for stream B is              E = 1 - 10,254/4,174*(1-0.655)/0.655       = -29.4%

Calculation of the wearing rates and effectiveness using the major stream:

Wearing rate of stream B = 65.1%
Effectiveness of stream B is                         E  = 1 - 10,254/4,174*(1-0.651)/0.651      =  -31.7%

SUMMARY

1. The effectiveness found when the small stream was discarded was -31.7%
2. The effectiveness found (for both streams) when the small stream was assumed to be 1% was -29.4%
3. The belt wearing rate found on the large stream when the small stream was assumed to be 1% was 65.5%
4. The belt wearing rate found on the small stream when the small stream was assumed to be 1% was 25.3%

With increases in the assumed size of the alcohol stream, the calculated value of seat belt Effectiveness becomes more positive.
If the proportion of alcohol affected drivers was assumed to be about 15% of all drivers, the seat belt effectiveness would be zero.

Ignoring the small stream if it was less than 1% does not significantly effect the result.

Tabulation of calculations with group size as the variable is done on the data & calculations document in Tables 6a and 6b.

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