Variance Estimation Requirements for Summary Totals and Rates for the Annual SOII

Variance Estimation Requirements for
Summary Totals and Rates for the
Annual Survey of Occupational Injuries and Illnesses

Notation
Let
h
i
j
k
K
ˆ
E
H
R
V
%RSE
W
Fkij

= reported size class (1, 2, 3, 4, 5, all)
= original size class (1, 2, 3, 4, 5)
= establishment
= TEI
= aggregate industry
= an estimate
= reported employment
= hours worked
= rate
= variance
= percent relative standard error
= original weight
= Dkij ∗ Akij ∗ Pkij ∗ Bkij
Where
D = Reaggregation Factor
A = Unit Nonresponse Adjustment Factor
P = Outlier Adjustment Factor
B = State Benchmark
Factor
q
W

−1

kij
= Wkij Fkij
Wkij
(Note: if Wkij = 0, then Zkij = 0)
nki = total sample size for ith original size class
mki = usable sample size for ith original size class
δkhij = 1 if reported size class for the j th unit is equal to original size class (i = h)
= 0 otherwise
µ = value from Table of Means for Hours, Employment, and the 22 Variables of
Interest. The SMG will provide these values.
Yg = total of g th variable of interest, where g is one of the following:
Injuries and Illnesses Combined
Total recordable cases (TRC)
Lost workday cases (LWDC)
Cases with days away from work (DAFWC)
Cases with days of restricted work activity (DRWAC)
Nonfatal cases without lost workdays (CWOLWD)
Injuries Only
Total recordable cases (TRC)
Lost workday cases (LWDC)
Cases with days away from work (DAFWC)
Cases with days of restricted work activity (DRWAC)
Nonfatal cases without lost workdays (CWOLWD)
Illness Only
Total recordable cases (TRC)
Lost workday cases (LWDC)
Cases with days away from work (DAFWC)
Cases with days of restricted work activity (DRWAC)
Nonfatal cases without lost workdays (CWOLWD)
Types of Illnesses
Skin diseases
Dust diseases of the lung
Respiratory conditions due to toxic agents
Poisoning
Disorders due to physical agents
Disorders associated with repeated trauma
All other trauma
Note: if any calculated value for a variance (denoted by V ) is less than zero, set the variance equal to zero.

Zkij

2

1. For each area and for each TEI, k,
(a) For each reported size class h and for each original size class i = 1 to 4:
i. Compute:
SZkhi =

m
khi
X

Zkhij δkhij

j=1

ii. For Reported Employment E and Hours Worked H, compute:
SEkhi =

SEEkhi =

SHkhi =

SHHkhi =

SHEkhi =

m
khi
X
j=1
m
khi
X
j=1
m
khi
X
j=1
m
khi
X
j=1
m
khi
X

(Zkhij Ekhij δkhij )
2

(Zkhij Ekhij δkhij )

(Zkhij Hkhij δkhij )
2

(Zkhij Hkhij δkhij )

2
Zkhij
Hkhij Ekhij δkhij

=

j=1

m
khi
X

(Zkhij Hkhij δkhij ) (Zkhij Ekhij δkhij )

j=1

iii. For each variable of interest g, compute:
SYkghi =

SY Ykghi =

SY Ekghi =

SY Hkghi =

m
khi
X
j=1
m
khi
X
j=1
m
khi
X
j=1
m
khi
X

(Zkhij Ykghij δkhij )
2

(Zkhij Ykghij δkhij )

khi
X

m
2
Zkhij
Ykghij Ekhij δkhij =
(Zkhij Ykghij δkhij ) (Zkhij Ekhij δkhij )

2
Zkhij
Ykghij Hkhij δkhij =

j=1

j=1
m
khi
X

(Zkhij Ykghij δkhij ) (Zkhij Hkhij δkhij )

j=1

(b) For each reported size class h:
i. For Hours Worked H, compute:
P5
LkHh =

Pmkhi
j=1 Wkhij Fkhij Hkhij δkhij
i=1
P5 Pmki
i=1
j=1 Wkhij Fkhij Ekhij

ii. For each variable of interest g, compute:
P5 Pmki
Lkgh =

i=1

Wkhij Fkhij Ykghij δkhij
Pmki
i=1
j=1 Wkij Fkij Ekij

P5

3

j=1

(c) For each reported size class h and for each original size class i = 1 to 4:
i. If nki = 0 or mki = 0, then set


ˆkhi = 0
V E


ˆ khi = 0
V H


ˆE
ˆkhi = 0
C H

For all variables of interest g, then set


V Yˆkghi = 0


ˆkghi = 0
C Yˆ E


ˆ kghi = 0
C Yˆ H

ii. For Reported Employment E:
A. If nki > 1 and mki > 0, compute:


ˆkhi =
V E

h
i
1
2
nki SEEkhi − (SEkhi )
(nki − 1)

B. If nki = 1 and mki = 1, compute:


ˆkhi = (SEkhi − SZkhi µi,employment )2
V E
iii. For Hours Worked H:
A. If nki > 1 and mki > 0, compute:
h
i
1
2
nki SHHkhi − (SHkhi )
(nki − 1)


1
ˆE
ˆkhi =
C H
[nki SHEkhi − SHkhi SEkhi ]
(nki − 1)


ˆ khi =
V H

B. If nki = 1 and mki = 1, compute:


ˆ khi = (SHkhi − SZkhi µi,hours )2
V H


ˆE
ˆkhi = SHEkhi − (SZkhi )2 µi,hours µi,employment
C H
iv. For each variable of interest Yg
A. If nki > 1 and mki > 0, compute:
1
2
[nki SY Ykghi − (SYkghi ) ]
(nki − 1)


1
ˆ kghi =
C Yˆ H
[nki SY Hkghi − SYkghi SHkhi ]
(nki − 1)


1
ˆkghi =
C Yˆ E
[nki SY Ekghi − SYkghi SEkhi ]
(nki − 1)


V Yˆkghi =

4

B. If nki = 1 and mki = 1, compute:


2
V Yˆkghi = (SYkghi − SZkhi µi,g )


ˆ kghi = SY Hkghi − (SZkhi )2 µi,g µi,hours
C Yˆ H


ˆkghi = SY Ekghi − (SZkhi )2 µi,g µi,employment
C Yˆ E

(d) For each reported size class h, compute:
i.
4

 X


ˆkh =
ˆkhi
V E
V E
i=1





ˆ kgh =
C Yˆ H

4
X



ˆ kghi
V Yˆ H

i=1





4
X





4
X

ˆ kh =
V UH



ˆ khi
V H

i=1

V U Yˆkgh =



V Yˆkghi

i=1

ii. If TEI k’s BMF FLAG(a, o, t) = NO or PRORATE FLAG = YES, then
4 h

 X
i
ˆ kh =
ˆ khi , and for all variables of interest g,
V H
VH
i=1





V Yˆkgh =

4 h 
i
X
V Yˆkghi
i=1

2. Within each Grantee State
(a) For Reported Employment, E:

 X 

ˆKh =
ˆkh
V E
V E
k

(b) For Hours Worked, H:

 X 

ˆ Kh =
ˆ kh
V H
V H
k





ˆ Kh =
V UH

X
k

5



ˆ kh
V UH

(c) For each variable of interest, g:

 X 

V YˆKgh =
V Yˆkgh
k





X





X

ˆ Kgh =
C Yˆ H



ˆ kgh
C Yˆ H

k

V U YˆKgh =



V U Yˆkgh

k

(d) For each reported size class h, compute:
i.
4

 X


ˆkh =
ˆkhi
V E
V E
i=1





ˆ kgh =
C Yˆ H

4
X



ˆ kghi
C Yˆ H

i=1

ii. If TEI k’s BMF FLAG(a, o, t) = NO or PRORATE FLAG = YES, then
4 h 
 X
i
ˆ
ˆ khi , and for all variables of interest,Yg
V Hkh =
V H



i=1



V Yˆkgh



4 h 
i
X
=
V Yˆkghi
i=1

otherwise compute:
4 h 



 X


i
ˆ kh = L2kHh V E
ˆk,all +
ˆ khi − 2LkHh C H
ˆE
ˆkhi , and for all variables of interest,g
V H
V H
i=1







ˆk,all
V Yˆkgh = L2kgh V E



4 h 


i
X
ˆkghi
+
V Yˆkhi − 2Lkgh C Yˆ E
i=1

3. Within each area and for each aggregate industry K and for each reported size class h, compute (where
summation is over all the TEIs contained in K):
(a) For Reported Employment, E:

 X 

ˆKh =
ˆkh
V E
V E
k

(b) For Hours Worked, H:

 X 

ˆ Kh =
ˆ kh
V H
V H
k

(c) For each variable of interest, g:

 X 

V YˆKgh =
V Yˆkgh
k





ˆ Kgh =
C Yˆ H

X
k

6



ˆ kgh
C Yˆ H



ˆ H,
ˆ Yˆ , R
ˆ in the
4. Requirements for Database of Standard Estimates: With each estimate E,
standard database, associate the estimate’s Percent Relative Standard error as calculated below. These
values should be available through the Extract System. The Percent Relative Standard Errors should
have four integer places and be rounded to four decimal places. NOTE: If an estimate is zero, then its
Percent Relative Standard Error is zero.
(a) For TEI k, compute:



ˆkh =
%RSE E



ˆ kh =
%RSE H



%RSE Yˆkgh =

r 

ˆkh
V E
∗ 100

ˆ
E
r kh 
ˆ kh
V H
ˆ
H
r kh 
V Yˆkgh

∗ 100

∗ 100
ˆkgh
E
v 





u
ˆ
ˆ
ˆ
ˆ

 u
Y
H
V
U
H
C
V
U
Y
kgh 
kh
kgh
u
ˆ kgh = t
 ∗ 100
+
−2
%RSE R

2
2
ˆ
ˆ
ˆ
ˆ
Ykgh
Hkh
Ykgh Hkh 

(b) For aggregate industry K, compute:
r 

ˆKh
V E


ˆKh =
%RSE E
∗ 100
ˆKh
E
r 

ˆ
H
V
Kh


ˆ Kh =
%RSE H
∗ 100
ˆ Kh
H
r 

V YˆKgh


∗ 100
%RSE YˆKgh =
ˆKgh
E
v 





u
u
ˆ
ˆ
ˆ
ˆ

 u V U YKgh
V U HKh
C Y HKgh 
ˆ Kgh = t
 ∗ 100
%RSE R
+
−
2

2
ˆ2
ˆ Kh 
YˆKgh
H
YˆKgh H
Kh

7

5. Requirements for Abbreviated Sampling Errors Report: For each area, generate a report
containing the following:
(a) Area Identification
(b) TEI or Aggregate Industry Identification
For each TEI and Aggregation Industry:
(c) Total Employment %RSE
(d) Total Hours Worked (in thousands) %RSE
(e) Total TRC for Injuries and Illnesses Combined %RSE
(f) TRC Rate for Injuries and Illnesses Combined %RSE
(g) Total CWOLWD for Injuries and Illnesses Combined %RSE
(h) Total LWDC for Injuries and Illnesses Combined %RSE
(i) LWDC Rate for Injuries and Illnesses Combined %RSE
(j) Total DAFWC for Injuries and Illnesses Combined %RSE
(k) DAFWC Rate for Injuries and Illnesses Combined %RSE

8