# Appendix

**Traumatic Brain Injury Surveillance: Statistical Methods**

**Estimating Age-Adjusted Rates**

Age-adjusted rate estimates were calculated by applying the year 2000 population standard to age-specific crude rate estimates for surveillance year 1997 (Tables 1--5). The year 2000 standard weights (courtesy of the National Center for Health Statistics) are as follows:

Population estimates (by year of age, race, and sex) for 1997 were downloaded for each of the 14 surveillance states from the U.S. Census Bureau website. In any particular analysis, the age-adjusted rate is estimated by

where,

C_{k} denotes the 1997 case count falling into age group k;

P_{k} denotes the estimated 1997 population for age group k; and

W_{k} denotes the year 2000 standard weight for age group k.

This formula applies to rate estimates for the entire surveillance population as well as to rate estimates for
different subpopulations. The values C_{k} and
P_{k} always refer to the population of interest; however, the weights
W_{k} are common across all analyses.

####
**Stability of Rate Estimates**

The stability of each rate estimate (Tables 1--5) is assessed by using the coefficient of variation (C.V.). The C.V. is estimated as

Case counts are assumed to follow a Poisson distribution. Under this assumption, the variance of an estimated crude rate is estimated by

where,

C is the case count used to calculate the rate, and

P is the population used to calculate the rate.

The formula for estimating the C.V. of a crude rate simplifies to

For age-adjusted rate estimates, the formula for estimating the variance is

where C_{k}, P_{k}, and
W_{k} are as defined previously. The square root of this estimated variance is then used when calculating the C.V. as initially defined; no corresponding simplified
formula is available.

####
**Estimation of Case Distributions**

Estimates are provided (Tables 6 and 7) of the distribution of cases across selected one-dimensional classifications. Certain classification variables are core variables, which have a value for every case, whereas other classification variables are extended variables, which have a value only for a sample of cases and involve the use of case weights in the formulation of estimates (see the Sampling Scheme section).

Estimating the distribution of cases across classes associated with a core variable is straightforward because each case implicitly receives a unit weight. The percentage of cases in class k is simply

(N_{k} / N) × 100%

where,

N_{k} denotes the number of cases falling into class k, and

N denotes the total number of cases classified.

Estimating the distribution of cases across classes associated with an extended variable is more involved. Each sampled case is assigned a case weight corresponding to the inverse of the sampling fraction for the stratum from which the case was sampled. Because different surveillance states sampled at markedly different rates (and different strata were occasionally sampled at different rates within a state), the case weights cover a wide range. Let

W_{s,g} denote the case weight assigned to each case sampled from stratum g in state s;

N_{s,g} denote the total number of sampled cases from stratum g in state s; and

N_{s,g,k} denote the number of sampled cases from stratum g in state s falling into class k.

Then, the estimated number of cases from stratum g in state s that can be assigned to some class is W_{s,g} × N_{s,g}, and the estimated number in class k is
W_{s,g} × N_{s,g,k}. An estimate of the overall percentage of cases falling into class k, across all states and strata, is given by

The formulas described in this section apply to percentage estimates for the entire surveillance population as well as to percentage estimates for different subpopulations. The values
N_{s,g} and N_{s,g,k} always refer to the population of interest; however, the case weights
W_{s,g} are common across analyses.

####
**Stability of Percentage Estimates**

The stability of each reported percentage (Tables 6 and 7) is assessed by using the C.V. For a reported percentage, the C.V. is estimated as

The distribution of cases across a set of m > 2 classes is assumed to follow a multinomial distribution; the
marginal distribution of cases falling in or out of any particular class is then binomial. Whether the case weights are unit weights (as with core variables) or nonunit weights (as with extended variables), the variance of an estimated percentage
PCT_{k} of cases falling into class k can be estimated by

where s, g, W_{s,g}, and
N_{s,g} are as defined previously. Note that when all case weights are unit weights, this formula simplifies to the more familiar form

###
**Sampling Scheme for Collecting Data Regarding Extended Variables**

For each state designated to collect data for extended variables by means of medical record abstraction, the goal was to obtain information for approximately 1,000 cases. For states with small traumatic brain injury case populations or
with comprehensive data collection systems in place (Alaska, Nebraska, Oklahoma, and Rhode Island) this involved sampling at
a 100% rate, with abstraction attempted for all hospital discharge cases. Other states (Arizona, Colorado, Louisiana,
Minnesota, New York, South Carolina, and Utah) stratified cases (usually according to hospital size, with strata defined as <100 acute care beds or __>__100 acute care beds) and typically allocated the sample of 1,000 across strata in proportion to stratum size. Multiple states also abstracted data for preadmission death cases, which constituted a separate stratum receiving a proportional share
of the sample. Data for preadmission death cases are not analyzed in this report.
A simple random sample was then selected within each
stratum.

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