1998 SURVEY RESULTS

APPENDIX A

*PRECISION OF SAMPLE ESTIMATES


*Reprinted from:

Boyle, J. and K. Sharp. 1998 Motor Vehicle Occupant Safety Survey: Methodology Report. DOT-HS-809-029. Washington DC: U.S. Department of Transportation, National Highway Traffic Safety Administration.



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:

FigureA-1
Where:


The sample sizes for the surveys are large enough to permit estimates for subsamples of particular interest. Table 5, on the next page, presents the expected size of the sampling error for specified sample sizes of 8,000 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 5
Expected Sampling Error (Plus or Minus)
At the 95% Confidence Level
(Simple Random Sample)

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

Size of
Sample or
Subsample
10 or 90 20 or 80 30 or 70 40 or 60 50
8,000 0.7 0.9 1.0 1.1 1.1
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 oversample of youth and young adults (age 16-39). Both the cross-sectional sample and the oversample 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:

FigureA-2

where:

Although Table 5 above 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 45.7 percent while the proportion for the 40 and over stratum (w2) was 54.3 percent.

As shown in Table 6 below, the disproportionate sampling increases the confidence interval by about 2 percent, compared to a simple random sample of the same size. This means that sample design introduces almost no measurable loss in sampling precision for total population estimates, while increasing the precision of sampling estimates for the target population aged 16-39 years old. Since the difference in sampling precision between the stratified disproportion sample and a simple random sample is less than one tenth of percentage point in each case, the sampling error table for a simple random sample will provide a reasonable approximation of the precision of sampling estimates in the survey.


TABLE 6
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
HYPOTHETICAL
PROPORTIONATE
SAMPLING*
CURRENT DIS-
PROPORTIONATE
SAMPLING
DIFFERENCE IN
CONFIDENCE
INTERVALS ABOUT
ESTIMATES
USE NEW VARIABLES
Driven in the past year .61 .63 3.2%
Drunk alcohol in past year 1.39 1.37 -1.3%
Always use safety belt .93 .94 0.7%
Dislike seat belts 1.55 1.61 3.4%
Always use passenger belt (front) 1.40 1.40 0.0%
Favor (a lot) seat belt laws 1.45 1.48 2.0%
Secondary enforcement 1.41 1.44 2.0%
Ever ticketed by police for seatbelt .85 .83 -2.6%
Recall Crash dummies 1.11 1.17 5.0%
Ever injured in vehicle accident .94 .97 2.9%
Drives a car for work almost every day 2.64 2.76 4.3%
Set a good example for others (reason for using seat belts) 1.43 1.47 2.6%
Driver-side only Air Bag in vehicle 2.04 2.08 1.6%
Race: Black/African American 0.66 0.65 -0.5%
Ethnicity: Hispanic 0.63 0.61 -4.0%
Male/Female 1.08 1.10 2.2%
AVERAGE DIFFERENCE IN CONFIDENCE INTERVALS 1.94%
* Total sample proportions using SRS formula


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 which 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. Call the total sampling error (i.e., var (x) in the previous formula) of the first sample s1 and the total sampling error of the second sample s2. Then, the sampling error of the difference between these estimates is sd which is calculated as:

FigureA-3

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 subsamples for various sizes is presented in Table 7. This table can be used to indicate the size of difference in proportions between drivers and non-drivers or other subsamples that would be statistically significant.



TABLE 7. 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