2003 Motor Vehicle Occupant Safety Survey: Air Bags
 

APPENDIX A: PRECISION OF SAMPLING ESTIMATES*

Precision of Sample Estimates

The objective of the sampling procedures used on this study was to produce a random sample of the target population. A random sample shares the same properties and characteristics of the total population from which it is drawn, subject to a certain level of sampling error. This means that with a properly drawn sample we can make statements about the properties and characteristics of the total population within certain specified limits of certainty and sampling variability.

The confidence interval for sample estimates of population proportions, using simple random sampling without replacement, is calculated by the following formula:


equation 1 - click [d] for long description [d]

Where:

se (x)
=
 the standard error of the sample estimate for a proportion;
p
=
 some proportion of the sample displaying a certain characteristic or attribute;
q
=
 (1 p);
n
=
 the size of the sample;
z
=
 the standardized normal variable, given a specified confidence level (1.96 for samples of this size).

The sample sizes for the surveys are large enough to permit estimates for sub-samples of particular interest. Table 56, on the next page, presents the expected size of the sampling error for specified sample sizes of 12,000 and less, at different response distributions on a categorical variable. As the table shows, larger samples produce smaller expected sampling variances, but there is a constantly declining marginal utility of variance reduction per sample size increase.

TABLE 19

Expected Sampling Error (Plus or Minus)
At the 95% Confidence Level
(Simple Random Sample)

 Percentage of the Sample or Sub-Sample Giving
A Certain Response or Displaying a
Certain Characteristic for Percentages Near:    

Size of Sample
or Sub-Sample 
10 or 90  
20 or 80  
30 or 70 
40 or 60
50
6,000       
0.8 
1.0 
1.2 
1.2
1.3
4,500       
0.9
1.2
1.3
1.4
1.5
4,000       
0.9
1.2
1.4 
1.5
1.5
3,000       
1.1
1.4 
1.6
1.8
1.8
2,000       
1.3
1.8
2.0
2.1
2.2
1,500       
1.5
2.0
2.3
2.5
2.5
1,300       
1.6
2.2
2.5
2.7
2.7
1,200       
1.7
2.3
2.6
2.8
2.8
1,100       
1.8
2.4
2.7
2.9
3.0
1,000       
1.9
2.5
2.8
3.0
3.1
900       
2.0
2.6
3.0
3.2
3.3
800       
2.1
2.8
3.2
3.4
3.5
700       
2.2
3.0
3.4
3.6
3.7
600       
2.4
3.2
3.7
3.9
4.0
  500       
2.6
3.5
4.0
4.3
4.4
400       
2.9
3.9
4.5
4.8
4.9
300       
3.4
4.5
5.2
5.6
5.7
200       
4.2
5.6
6.4
6.8
6.9
150       
4.8
6.4
7.4
7.9
8.0
100       
5.9
7.9
9.0
9.7
9.8
75       
6.8 
9.1
10.4
11.2
11.4
50       
8.4
11.2
12.8
13.7
14.0

_______________________________________________________________________

  NOTE:  Entries are expressed as percentage points (+ or -)

However, the sampling design for this study included a separate, concurrently administered over-sample of youth and young adults (age 16-39). Both the cross-sectional sample and the over-sample of the youth/younger adult population were drawn as simple random samples; however, the disproportionate sampling of the age 16-39 population introduces a design effect that makes it inappropriate to assume that the sampling error for total sample estimates will be identical to those of a simple random sample.

In order to calculate a specific interval for estimates from a sample, the appropriate statistical formula for calculating the allowance for sampling error (at a 95% confidence interval) in a stratified sample with a disproportionate design is:

equation 2 - click [d] for long description[d]

where:

ASE
=
 allowance for sampling error at the 95% confidence level;
h
=
 a sample stratum;
g
=
 number of sample strata;
Wh
=
 stratum h as a proportion of total population;
fh
=
 the sampling fraction for group h the number in the sample divided by the number in the universe;
s2h
=
 the variance in the stratum h for proportions this is equal to ph (1.0 ph);
nh
=
 the sample size for the stratum h.

Although Table 19 provides a useful approximation of the magnitude of expected sampling error, precise calculation of allowances for sampling error requires the use of this formula. To assess the design effect for sample estimates, we calculated sampling errors for the disproportionate sample for a number of key variables using the above formula. These estimates were then compared to the sampling errors for the same variables, assuming a simple random sample of the same size. The two strata (h1 and h2) in the disproportionate sample were all respondents age 16-39 and all respondents age 40 and over, respectively. The proportion for the 16-39 year old stratum (w1) was 53.0 percent while the proportion for the 40 and over stratum (w2) was 47.0 percent.

As shown in Table 20, the disproportionate sampling increases the confidence interval by an average of 0.7 percent, compared to a simple random sample of the same size. This means the sample design slightly decreases the sampling precision for total population estimates, while increasing the precision of sampling estimates for the sub-sample aged 16-39 years old. Since the average difference in the confidence interval between the stratified disproportionate sample and a simple random sample is less than one percentage point, the sampling error table for a simple random sample will provide a reasonable approximation of the precision of sampling estimates in the survey.

TABLE 20
Design Effect on Confidence Intervals for Sample Estimates
Between Disproportionate Sample Used in Occupant Protection Survey
and a Proportionate Sample of Same Size
    
CONFIDENCE INTERVALS
PERCENTAGE POINTS + AT 95% CONFIDENCE LEVEL
p=
HYPOTHETICAL
PROPORTIONATE
SAMPLING
CURRENT
DISPROPORTIONATE
SAMPLING
DIFFERENCE IN CONFIDENCE
INTERVALS ABOUT ESTIMATES
VARIABLE (Version 1 only)
 
 
 
 
Driven in the past year
89.2%
0.77
0.78
1.3%
Drank alcohol in past year
63.4%
1.21
1.23
1.7%
Always use safety belt (N=5502)
85.1%
0.94
0.94
----
Dislike seat belts (N=5505)
33.1%
1.24
1.26
1.6%
Always use passenger belt (N=5655)
82.7%
0.98
0.98
----
Favor (a lot) seat belt laws
69.3%
1.15
1.16
.9%
Should be primary enforcement
63.9%
1.20
1.22
.9%
Ever ticketed by police for seatbelt
9.3%
0.73
0.72
-1.4%
Ever injured in vehicle accident
23.6%
1.06
1.08
1.9%
Drives a car for work almost every day
17.2%
0.94
0.96
2.1%
Set a good example for others (N=5413) (reason for using seat belts)
74.1%
1.17
1.19
1.7%
Driver-side air bag in vehicle (N=5551)
76.5%
1.12
1.14
1.8%
Race: Black/African American
8.6%
0.70
0.70
----
Ethnicity: Hispanic
13.2%
0.84
0.81
-3.6%
Gender: Male
48.0%
1.24
1.27
2.4%
AVERAGE DIFFERENCE IN CONFIDENCE INTERVALS
 
 
 
0.7%
*Total sample proportions using SRS formula
Unless specified otherwise N=6180



Estimating Statistical Significance

The estimates of sampling precision presented in the previous section yield confidence bands around the sample estimates, within which the true population value should lie. This type of sampling estimate is appropriate when the goal of the research is to estimate a population distribution parameter. However, the purpose of some surveys is to provide a comparison of population parameters estimated from independent samples (e.g. annual tracking surveys) or between subsets of the same sample. In such instances, the question is not simply whether or not there is any difference in the sample statistics that estimate the population parameter, but rather is the difference between the sample estimates statistically significant (i.e., beyond the expected limits of sampling error for both sample estimates).

To test whether or not a difference between two sample proportions is statistically significant, a rather simple calculation can be made. The maximum expected sampling error (i.e., confidence interval in the previous formula) of the first sample is designated s1 and the maximum expected sampling error of the second sample is s2. The sampling error of the difference between these estimates is sd and is calculated as:

equation 3 - click [d] for long description [d]

Any difference between observed proportions that exceeds sd is a statistically significant difference at the specified confidence interval. Note that this technique is mathematically equivalent to generating standardized tests of the difference between proportions.

An illustration of the pooled sampling error between sub-samples for various sizes is presented in Table 58. This table can be used to determine the size of the difference in proportions between drivers and non-drivers or other sub-samples that would be statistically significant.

TABLE 21
Pooled Sampling Error Expressed as Percentages for Given Sample Sizes (Assuming P=Q)
Sample Size
4000 14.1 10.0 7.1 5.9 5.1 4.7 4.3 4.0 3.8 3.6 3.5 3.0 2.7 2.5 2.4 2.3 2.2
3500 14.1 10.0 7.1 5.9 5.2 4.7 4.3 4.1 3.8 3.7 3.5 3.0 2.7 2.6 2.4 2.3  
3000 14.1 10.0 7.2 5.9 5.2 4.7 4.4 4.1 3.9 3.7 3.6 3.1 2,8 2.7 2.5    
2500 14.1 10.0 7.2 6.0 5.3 4.8 4.5 4.2 4.0 3.8 3.7 3.2 2.9 2.8      
2000 14.2 10.1 7.3 6.1 5.4 4.9 4.6 4.3 4.1 3.9 3.8 3.3 3.1        
1500 14.2 10.2 7.4 6.2 5.5 5.1 4.7 4.5 4.3 4.1 4.0 3.6          
1000 14.3 10.3 7.6 6.5 5.8 5.4 5.1 4.8 4.7 4.5 4.4            
900 14.4 10.4 7.7 6.5 5.9 5.5 5.2 4.9 4.8 4.6              
800 14.4 10.4 7.8 6.6 6.0 5.6 5.3 5.1 4.9                
700 14.5 10.5 7.9 6.8 6.1 5.7 5.5 5.2                  
600 14.6 10.6 8.0 6.9 6.3 5.9 5.7                    
500 14.7 10.8 8.2 7.2 6.6 6.2                      
400 14.8 11.0 8.5 7.5 6.9                        
300 15.1 11.4 9.0 8.0                          
200 15.6 12.1 9.8                            
100 17.1 13.9                              
50 19.8                                
  50 100 200 300 400 500 600 700 800 900 1000 1500 2000 2500 3000 3500 4000
Sample Size
*Reprinted from:
Boyle, J. and P. Vanderwolf. 2003 Motor Vehicle Occupant Safety Survey. Volume I. Methodology Report . Washington DC: U.S. Department of Transportation, National Highway Traffic Safety Administration
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