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FSIS Estimation of Pathogen Rates in the Population Represented by the 

Samples Collected in Each Sampling Project for Fiscal Year 2015

Introduction

FSIS created a SAS program to calculate aggregate sampling pathogen rates. This 

document describes the methods. 

Types of Calculations

FSIS computed one of three types of calculations for a sampling project: 

• Prevalence – a calculation for sampling projects in which the selected samples 

represented all product produced by all establishments that produced the product

(except for establishments that were exempted for religious purposes or because 

their production volume did not meet the threshold for inclusion). 

• Volume-weighted percent positive – a calculation applied to sampling projects 

that did not include all establishments; inference can be made only to the entire

product produced by the included establishments.

• Percent positive –calculated as the percent of samples that tested positive. No 

weighting was done to account for the volume of product produced by the 

sampled establishments. This type of calculation was generated for sampling 

projects with small sample sizes or for exploratory sampling projects.

Sampling Projects

Table 1 contains the list of sampling projects covered by this report. 

Table 1: FSIS Sampling Projects Included in This Report

Project Code Project Name

EM31 Egg Products Sampling - Pasteurized - Egg Whites – Salmonella

EM32 Egg Products Sampling - Pasteurized - Whole Egg or Yolks – Salmonella

EM33 Egg Products Sampling - Pasteurized - Whole Eggs with Added Yolks or Whole Egg Blends –

Salmonella

EM34 Egg Product Sampling - Pasteurized - Whole Eggs or Yolks – Salmonella

EM34 Egg Product Sampling - Pasteurized - Whole Eggs or Yolks with > 2% salt or sugar added -

Salmonella

EM35 Egg Products Sampling - Pasteurized - Dried Yellow Egg Products – Salmonella

EM36 Egg Products Sampling - Pasteurized - Dried Egg Whites – Salmonella

EM37 Egg Products Sampling - Pasteurized - Pan Dried Egg Whites – Salmonella

EXP_CH_MSK01 Exploratory Sampling for Mechanically Separated Chicken

EXP_TU_MSK01 Exploratory Sampling for Mechanically Separated Turkey

HC_CH_CARC01 HACCP Verification for Young Chicken Carcasses

HC_CH_COM01 Sampling for Ground and Other Comminuted Chicken (not Mechanically Separated)

HC_CPT_LBW01 Sampling for Chicken Parts – Legs, Breasts, and Wings

HC_TU_CARC01 HACCP Verification for Young Turkey Carcasses

HC_TU_COM01 Sampling for Ground and Other Comminuted Turkey (not Mechanically Separated)

MT43 Risk-based Sampling of Raw Ground Beef or Veal Products - E.coli O157:H7 & Salmonella
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Project Code Project Name

MT54 Sampling of Raw Ground Beef or Beef Patty Components (other than trim)

MT55 Sampling of Bench Trim for further use in ANY raw, non-intact beef products

MT60 Sampling of Beef Manufacturing Trimmings

MT64 Sampling of Raw Ground Beef or Beef Patty Components (other than trim)

MT65 Sampling of Bench Trim for further use in ANY raw, non-intact beef products

RTEPROD_RAND RTEPROD Sampling - Random RTE Products

RTEPROD_RISK RTEPROD Sampling - Risk-based RTE Products

Data Extracted from FSIS Databases

For each sampling project in Table 1, FSIS extracted all samples collected in fiscal year 

2015 (October 1, 2014 to September 30, 2015) with lab result “Negative”, “Positive” or 

“Confirmed Negative of a Previous Presumptive”. FSIS restricted the samples to those 

with test codes for pathogens Salmonella, Campylobacter, Listeria monocytogenes

(Lm), Escherichia coli (E. coli) O157:H7, or non-O157 Shiga Toxin-producing E. coli

(non-O157 STEC). 

In addition to lab sample results data, FSIS extracted the slaughtered totals for chicken 

and turkey carcasses (sampling projects HC_CH_CARC01 and HC_TU_CARC01) for 

each establishment that produced these products. These data were used to statistically 

weight the samples in the computation of prevalence for these two sampling projects. 

The slaughter totals were for the months May to September of 2015 since these 

sampling projects went into effect in May of 2015.

FSIS also extracted data on establishment product volume and production days from its 

databases. These data were used to compute statistical weights to estimate prevalence 

or volume-weighted percent positive for all project/pathogen pairs for which either of

these two types of estimates was computed (other than chicken or turkey carcasses). 

As noted in the preceding paragraph, slaughtered totals were used for these two poultry 

products.

FSIS updates its databases regularly so sample data can change. Fiscal year 2015 

data for this report were as of December 11, 2015.

Statistical Methods

FSIS used the SAS Institute Inc., Cary, North Carolina software SAS 9.4 (TS1M2) to 

extract the data from the PHIS and DW databases and for all computations. 

Data Preparation and Consolidation of Sampling Projects

Since FSIS extracted sample data from two databases (Public Health Information 

System [PHIS] and Data Warehouse [DW]), FSIS prioritized inclusion of samples from 

PHIS as it is the contemporary FSIS database. Duplicated samples from the DW were 

removed. FSIS then grouped all egg projects into a single project. FSIS also combined 

MT54 with its replacement project MT64, and combined MT55 with its replacement 

project MT65. 
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Computation of Sampling Weights

Projects for which FSIS calculated prevalence or volume-weighted percent positive first 

required an estimate of each establishment’s total product volume for the year. Each 

establishment’s estimated annual production volume for its relevant products was 

calculated from the following two components from establishment product profiles in 

PHIS: 

1) The average daily production volume for the relevant product groups within a 

given establishment. (Each product group is assigned a mean daily value of one 

of seven volume group ranges, as per FSIS PHIS Directive 5300.1). 

2) The maximum number of production days in a month for the relevant products in

each establishment. 

The product of these two values yielded a monthly estimated volume which was then 

multiplied by 12 to get the estimated annual production volume. Another step followed 

for projects such as MT43 and MT60 where samples of a specific weight (325 grams for 

these two projects) were tested. The estimated annual volume was converted to the 

number of 325-gram units of the particular product, as 325-gram portions of the product 

were sampled. 

As an example of the computation of the total annual volume and the statistical 

sampling weights, suppose an establishment’s estimated mean daily volume was 

50,000 pounds of ground beef subject to sampling under the MT43 project, and its 

maximum number of production days in a month was 20. Since 325 grams of the 

product was tested for each sample, FSIS computed the total number of 325-gram units 

of the product in the population as follows: 

16748307.6 grams 325 of units 9 325

000,50 1220 16.453 ≈ ×× × × = gramssampleunit

pound

grams

year

months

month

days

day

pounds Total volume

If 50 samples were tested for the establishment in the year, each sample was assigned 

a statistical sampling weight of: 16,748,307.69/50 = 334,966.15. No rounding was done 

in the programs.

FSIS computed the sampling weights for projects HC_CH_CARC01 and 

HC_TU_CARC01 (chicken and turkey carcasses, respectively) by dividing the 

slaughtered total by the number of samples for each establishment covered by these 

two projects. This was done separately for each of these two projects. The samples 

and slaughter totals for these projects were for the months May to September of 2015 

as these projects went into effect in May 2015. The sampling weights for these two 

projects reflected the actual production volumes for the establishments.

Estimation of Percentages, Standard Errors (Variances) and Confidence Intervals

All sampling units (carcasses, parts, or 325-gram portions of a product) in all 

establishments did not have an equal probability of selection because the volume of 

product varied greatly across establishments. While the selected samples for an 

establishment (for a given project) were equally weighted because they had an equal 
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chance of selection within the establishment, the statistical weights varied across 

establishments. Unequal weighting of samples applied to each project for which

prevalence or volume-weighted percent positive was estimated.

FSIS employed a SAS procedure designed to produce calculations and standard errors 

of the calculations by taking into account the complexities of the sample design. FSIS

estimated the variances, hence the standard errors, of the percentages with the 

Jackknife, a statistical method commonly used to get estimates of standard errors for 

complex sample designs.

The commonly used linear confidence interval for a proportion or percent can result in a 

confidence limit that is not between 0 and 100 percent for estimates near 0 or 100, 

leading to artificial truncation of the confidence interval. Logit-transformed confidence 

intervals are guaranteed mathematically to be between 0 and 1 for a proportion so there 

is never any artificial truncation of a confidence interval; no limit would need to be 

adjusted because it is negative or exceeds 1 (100%).

Logit-transformed confidence intervals are obtained by first computing the confidence 

limits on the logit scale and then transforming them back to a proportion or percent 

scale. Obtaining the confidence interval on the logit scale and transforming the limits to 

get the confidence interval for a proportion is more accurate than the common linear 

interval, especially for a proportion far from one-half (Fleiss et al., p. 295). As the 

percent of samples that tested positive for a pathogen in FSIS sampling projects 

generally fell below 10 percent, FSIS decided to use logit-transformed confidence 

intervals for the estimated percentages.

The logit is the natural logarithm of the odds of a positive test. The odds of a positive 

test is the ratio of the probability of a positive test to the probability of a non-positive 

test. If p is the estimated proportion positive of a given pathogen for a particular 

product, the logit of p is defined as:

p

p it p − = 1 log ( ) ln and log ( )

log ( )

1 it p

it p

e

e

p

+ =

where e is the exponential function. Fleiss et al. (p. 295) showed that the estimated 

standard error of p is:

se( p) = p(1− p)se[logit( p)]

From the last equation, it follows that the standard error of the logit of p is:

se[logit( p)] = se( p)/( p(1− p))

On page 195 of their book, Fleiss et al. provided the following formula for the limits of an 

approximate 100(1-α) percent confidence interval for the logit of p :

log ( ) (log ( )) 1 / 2 it p z se it p ± −α
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The level of confidence for confidence interval estimation is 100(1-α) percent. To get 95 

percent confidence intervals, α=.05. The inverse transformation of the confidence limits 

for the logit yields the following formulas for the confidence limits for a proportion (m in 

the formulas is the logit of p): 

Lower confidence limit: 

1

)1(

)( 2/1

1

−

− +−

















+ − pp

se p zm

e α

Upper confidence limit:

1

)1(

)( 2/1

1

−

− −−

















+ − pp

se p zm

e α

Proportions and their confidence limits would then be multiplied by 100 to transform 

them to the percent scale. The SAS procedure that we utilized for estimation provided 

logit confidence limits as an option that FSIS selected. 

Reference

Fleiss, J. L., Levin, B., & Paik, M. C. (2003). Statistical Methods for Rates and 

Proportions (3rd ed.). Hoboken, New Jersey: John Wiley & Sons.