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                    <text>Item ID Number
Author

01700
Carroll, Ray J.

Corporate Author
ROpOrt/ArtlGlO Title Typescript: Report #3, The Effect of Sampling
Battalions Rather than Individuals, August 1982

Journal/Book Title
Year

°ooo

Montn/Oay
Color
Number of Images

D

5

Descrlpton Notes

Monday, June 11, 2001

Page 1701 of 1793

�Mi REPORT n

i'
THE EFFECT OF SAMPLING'BATTALIONS RATHER THAN INDIVIDUALS
j '•

j l.J. Carroll
• JAugust 1982

In my letter of August; 14jto M.E. LeVois, I raised the possibility that sampling:] from; battalions and thehlisampling individuals could have different statis-

i W 1!:V

.

j - ' . i . iil'i

:
tical properties from merely taking a simple random sample of individuals. The
former method is mentioned by,UCLA in their protocol, but it is clear that they
I

•""«.' i '

:

'! ,

intend tojuse the latter as a basis for analysis. I think it is important to
' :•
I
• .".
understand the difference between the two and to investigate the effects of this
difference. This report is a preliminary analysis of this difference.
While the latter method is called simple random sampling (SRS), the former

i
method might best be called two-stage cluster sampling (TSCS). The two methods
j

• '

t

i

are illustrated in Figures tfl-and #2.
j

•: ' . ;

1

'!

: !''

| Suppose there are a total of M battalions and, for simplicity, assume each
! :
i ; - /••!.' j
.i ' i ! :
:ji
battalionjhas m individuals. 'Let battalion #i; (i = 1,2,..., M) have disease

. i '••'•!• : and
';i
rate p.

suppose

' ! overall ! disease
i •!;; ili-!
the:

' I _ •

rate is p. Suppose we randomly select

;
•Mi 1
! '-f1! 1: ! J ! i j
i i:.
N battalions and then select n jindividuals per'.selected battalion. Then the
1

,\

i

estimated probability of disease ,is (for either method) the observed proportion
of diseased individuals. If the probabilities;of disease are all :fairly small
;
, !
: i i "j!|i
: •
(say less I than 3% in every battalion), then the variances are approximately

;!

I

'

•

ill

' „ 'Vanance(SRS) = ^ — P
• ' ' ! ' mnc". • (1-nvn/MN) —
' ; i (|

..

f

,

i

Variance(TSCS) SjiilB/ll)- +
¥ «x j.w.ji'ws* v, &amp; vui~y — . 777 n
r

i
'

i '-S ;' '. I'-i'l',-).^

•

! •' 1 '

': i ;

'*;

:

i

m

' £l ! ' " * •
*'

'
!' i

'''I -i'l :j 't^'ij j i '!

::'

L

M

^ ^
"1 -*1

"] .

: Suppose'there are M = 500 battalions, of whichjwe sample m * 70. ^Suppose that
i : ; ) ' i ji'|;ri «j|.i.j,

&lt;

t

i

I ; j

'! • ' jij' j

'..

j'

j||

�each battalion has N - f&gt;00 individuals, of whom we sample 100, Pui'the?, §V»ppPse
that, the disease probabilities are all less than 3%. Then, to a degree of ap.i
proximation,
,
.
Variance (SRS) = (.0118)2p

i ' 'iir

' »~

Variance(TSCS) '= [.0001149 + |:0030715p]p .
Table #1 compares the ratio of'these two quantities for various values of p.

The exact numbers in Table j#l are not particularly critical, especially
since any real sampling plan wi.ll include some stratification.

Nonetheless, my

calculations indicate the following:

i|

. .

.:*

. .!

(i) As a general strategy, we should sample as many battalions as
•i
i : .'Mi
• •'!•
possible, with appropriate stratification,
i
J
•! | i
'
(ii) For larger sample:sizcs on the order of 6,000 per group, if
the event rates are small the effect of TSCS will not be too
great,
I
,
(iii) If the event rates are large, TSCS will be significantly less
i
•":
efficient than SRS, However, in this instance, we will still
have acceptable statistical powers (see Report #2).
i .,
,
This^report has notjaddressed certain problems, such as confounders and misi
,1 &lt;,

classification. Also, I have;assumed the event rates are fairly homogeneous
' • • ' I i
• ; 'ii iii;
I ;!!:
across battalions; this might\be a questionable assumption, as it is conceivable
:
:I

• j

; '!

;

that a few battalions had extremely high exposure and event rates.
:
•
'•!; ii '
' ••'
study must be guided by the' practical nature of the data set.
i

Further

�Figure _//1
SIMPLH RANDOM SAMPLING OF TRN INDIVIDUALS
FROM A TOTAL OF FOUR BATTALIONS.

,Battalion
#1

Battalion
112

Battalion

Battalion
#4

-X X X X X
X X X X X

J,

o oo

x x

Note:

©

©

On average, we will choose someone from every battalion.
tee this by taking a stratified sample.
;

We could guaran-

*^*1

Figure #2
TWO-STAG!- CLUSTHR SAMPLING OF TRN INDIVIDUALS,
SELECTING AT RANDOM TWO OF FOUR BATTALIONS.

•i,'?!

:

Battalion
til

Battalion

Battalion

ix x x x x

O | O Dip O

X X X X X

o; o bio o

i
* * * * *
•ii
* *' * *, *

Battalion
«f4

© &lt;&amp; © ffl ©
© © e ® ®

Select Battalions #2 and #3
i
j j
i
:•' .
Battalion
;';!'
Battalion
(

x-%.: • ; ;:

,00000 I

fo o o o o I

U i / /
O O O O O

•

;

:!•

' , ! . ! : •;

* * * * *
i

Note: As opposed to simple random or stratified random sampling, in cluster
sampling there is no chance of selecting one or more individuals from
i| I

every battalion.

t

••"i!
• 1$

!

:

i ii

.!

'

�TABLH #1

Approximate Value of
P

.005
,010
.020

Variance *(TSCS)
.95

1.04
1.26

I

!|

�Appendix, Report #3
Variance (SRS) - -^ —'/—.'„ p v( i _lp )
inn

Variance (TSCS) =

'

- , *
in
I

'

nm

c2
2

,
M
S/ = -JT V (p.-p)2
1
n-1 .'•, 1 1 ' '
M
S

2

=

M(¥-lT .? Pi

If the event rates arc all small,

— 2
] —2
(p.-p) &lt; j p

on average.

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                  <text>&lt;p style="margin-top: -1em; line-height: 1.2em;"&gt;The Alvin L. Young Collection on Agent Orange comprises 120 linear feet and spans the late 1800s to 2005; however, the bulk of the coverage is from the 1960s to the 1980s and there are many undated items. The collection was donated to Special Collections of the National Agricultural Library in 1985 by Dr. Alvin L. Young (1942- ). Dr. Young developed the collection as he conducted extensive research on the military defoliant Agent Orange. The collection is in good condition and includes letters, memoranda, books, reports, press releases, journal and newspaper clippings, field logs and notebooks, newsletters, maps, booklets and pamphlets, photographs, memorabilia, and audiotapes of an interview with Dr. Young.&lt;/p&gt;&#13;
&lt;p&gt;For more about this collection, &lt;a href="/exhibits/speccoll/exhibits/show/alvin-l--young-collection-on-a"&gt;view the Agent Orange Exhibit.&lt;/a&gt;&lt;/p&gt;</text>
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                <text>study protocol</text>
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                    <text>Item ID Number

°1699

Author

Carroll, Ray J.

Corporate Author
RflpOrt/ArtlOlO TltlO Typescript: Report #2, False Negatives and their Effect
on Power of Statistical Tests of Differences in Disease
Rates, August 1982

Journal/Book Title
Year

000

°

Month/Day
Color
Number of Images
DOSCrlptOU NOtBS

D

4

See ltem 1773 for a letter

^rom Carro"that mentions

this report.

Monday, June 11, 2001

Page 1700 of 1793

�REPORT 112
FALSE NEGATIVES AND Till:111 EFFECT ON POWER
OF STATISTICAL TF.STS OF DIFFERENCES IN DISH ASF. RATES

R.J. Carroll
i August 1982

In REPORT #1, especially Appendix H3, I calculated statistical power for tests
i'
' '
to see if disease .rates are the-same for exposed and non-exposed soldiers. I concluded that, if the disease rate in the non-exposed population was at all substan|

•

;

"-«
••

tial (say 5%), then samples of 6,000 for each group would give very high statistical power for detecting a relative risk due to exposure of 2.0 or more, even if
1
"
there was substantial misclassifixation of exposed into the non-exposed group
,i
(false negatives). In this report, I will study the effect: of false negatives on
statistical! power when the disease rate in the non-exposed population is fairly
small.

\.i
i;
'To review terminology, recall that
!

: j!

'

'

;;l

. i

p = disease probability;for truly exposed
•
i
!•
q = disease probability;for reported exposed ~ p.
' i "'
i'
p '= disease probability,for truly not exposed
q '= disease probability for reported not exposed
.•'
•••
i 1
r I ~ ep,/p- = true relative risk
true
l*2
.
.
(
r
; i ' = ^i/^o " observed i iRelative risk
OUi?
A
»••
'
6 =ifraction of false negatives
i
N •= sample size for reported exposed
N? = sample size for reported not exposed.
.'i
Before1constructing a test for equality of exposed and not exposed, we need
:

!

!II

two concepts. The level or Type 1 error a of a test is the probability (usually

a

; ,• :

i

I

1
.05 or*! 5%) that we will ' i conclude that theI exposed have different disease proi
. !
'1 i '
in

babilitiesi than do the.non-exposed, but in reality the two are not different.

�The power of a test (V - 3) is the probability we will conclude that the exposed
differ from the non-exposed when they really do differ. Generally, we want small
Type I error but high power; UCLA chooses a = .01, 1-0 = .95.
One point not made clear in the UCLA protocol is that they compute sample
size based upon a one-sided test and not the usual and more conservative twoi

;

In this particular instance 1 think1what they have done is justifi-

sided test.
!

'

able; we can hardly expect exposure to agent orange to have improved health status,
If we assume equal sample sizes so that N I = N? = N, then based upon UCLA's
arcsin transformation, the statistical power of the usual test is given by the
formula
1 - &lt;I»(z..

J. ~0t

where z

- /-=-(2arcsin/ii
v ^

•*•

- 2arcsin/q.J)
£

is the (l-a)100% pcrccntile of a standard normal distribution function

J. ~*vJt

'

$ and

q2 = 6p + (1-&lt;S)P2

(Report #1, Appendix

If we want to detect a true relative risk r
of at least r.
., we will then
:
.
true
detect
p
. I
•
'i
need a minimum sample size of at least
i

M N =

: 2(z

if we want power of 1-3. Note that this is exactly the formula on page 108 of the
UCLA protocal as long as there are no false negatives.

Some idea of the effect of

i

false negatives can be learned from Table #1. Figure #1 shows this effect even
i
,
.
more dramatically. Note that if we desire a Type I error of a = .01 and power
i !'

1-3 a .95|to detect a true relative risk of 2.0 or more, then we need about 4,500
soldiers inieach group if there are no false negatives; we will need over 7,600

�soldiers in each group if there are 20'-i false negatives.
This report shows that for diseases which are fairly rare, the effect of misclassification can be severe.

EFFECT OF FALSK NEGATIVES ON CALCULATION OF SAMPLE SIZE

a

• 1-3

r. . .
detect

lp~

2

6

UCLA
N

Actua

9,124
11,721
15,397
20,828
29,305
4,528
5,815
7,637
10,328
14,527
2 , 230
2,863
3,757
5,078
7,138

.01

.95

2.0

.01

0.0
.1
.2
.3
.4

9,124
9,124
9,1 24
9,124
9,124
4,528
4,528
4,528
4,528
4,528

.01

.95

2.0

.02

0.0
.1
.2
.3
.4

2,230
2,230
2,230
:
2,230
2,230

.01

.95

2.0

.005

0.0
.1
.2
.3

.4

N__

�«4

l
or

\
\

f r

\

u

\

--f-O!

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                  <text>&lt;p style="margin-top: -1em; line-height: 1.2em;"&gt;The Alvin L. Young Collection on Agent Orange comprises 120 linear feet and spans the late 1800s to 2005; however, the bulk of the coverage is from the 1960s to the 1980s and there are many undated items. The collection was donated to Special Collections of the National Agricultural Library in 1985 by Dr. Alvin L. Young (1942- ). Dr. Young developed the collection as he conducted extensive research on the military defoliant Agent Orange. The collection is in good condition and includes letters, memoranda, books, reports, press releases, journal and newspaper clippings, field logs and notebooks, newsletters, maps, booklets and pamphlets, photographs, memorabilia, and audiotapes of an interview with Dr. Young.&lt;/p&gt;&#13;
&lt;p&gt;For more about this collection, &lt;a href="/exhibits/speccoll/exhibits/show/alvin-l--young-collection-on-a"&gt;view the Agent Orange Exhibit.&lt;/a&gt;&lt;/p&gt;</text>
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01698

AllthOT

Carroll, Ray J.

Corporate Author
RepOrt/ArtldB TltlQ Typescript: Report #1 (with typographical corrections),
False Negatives and their Effect on Estimates of the
Risk of Exposure to Agent Orange, August 1982

Journal/Book Titlo
Year

000

°

Month/Day
Color
Number of Images

D

9

Doscripton Notes

Monday, June 11,2001

Page 1699 of 1793

�IMPORT #1
(wil.li typographical corrections)
FALSE NEGATIVES AND THEIR EFFECT ON ESTIMATES
OF Till- RISK OF EXPOSURE TO AGF.NT ORANGE

R.J. Carroll
August 1982

I am going to assume that a large random sample is taken from individuals
known to be at risk (e.g. combat troops) and that another large random sample is
taken from those thought not to he at risk (rear-echelon troops)'.

As explained

to me, this is not really possible because only battalions and not individuals
can easily be sampled for their exposure levels.
I will also assume that all those reported to have been exposed actually were
exposed:

no false positives.

A certain fraction 6 are false negatives, i.e., if

&lt;5 ~ .25, 25% of those thought not; to be at: risk (exposed) actually were.

These

false negatives do have an effect on estimating risk of disease due to exposure.
Let p. = true probability of disease for exposed individuals, and let p« =
true probability of disease for non-exposed individuals.

The relative risk is

defined to be
r

true = VP2 '

In practice we night get the following table:
Reported Exposed
"sick"
"well"

N

N

E

N N

r n
N,
1

The sample sizes N

Reported Not Exposed

N

NE

N

2" NE
N,
2

and N? of those reported exposed and reported not exposed (re-

spectively) have been fixed in advance.

N.. represents the number of "sick" indiLi

viduals among those reported exposed, while N .,, is the number of "sick" individuals

�among those reported not exposed. The observed relative risk can be computed by
A
i&gt;

(N,/N )

__ „. ._.,* _____
observed ~ (NIN ti /N«)
^
=

Because of i the false negatives, the observed relative risk estimates not the true
!

value r
but rather
true
observed

„_____.
____
&lt;5 rtrue + (l-o)

For example, if the true relative risk is 2.0 (exposed are twice as likely to be
"sick" as non-exposed) and if we have a false negative rate of '25%, then we would
report a relative risk of only
A
^
Observed * (

'

2.
'

= 1&lt;6

•

Another way to look at this example is as follows. Suppose 10% of the truly
.i
exposed become "sick." Since the true relative risk is 2.0, only 5% = 10% * 2.0
of the truly non-exposed become "sick."

However, because of the false negatives,

we will announce that 10% f 1.6,= 6.25% of the reported non-exposed become "sick."
TABLE J (Selected Values)
True R e 1 at i v e JR. is k _ {lalse__Negati.ve_ % _ Observed Rej.ati.ye Risk
,:
1.0
0
1.0
•'
1.0
;!
50
1.0
1
2.0
,
0
2.0
,;
'
10%
1.82
:
25%
1.60
50%
1.33
3.0
0
3.0
10%
2.5
25%
2.0
50%
1. 5
4.0

i

!
i

;

0
10%
25%
50%

4.0
3.08
2 . 29
1.60

�THE
EFFECT
A/EGA r s V f i S

..

_

_ ^) v
_ 70'

i.i

-

,:j. U

i •

:•!!

OF
OM

30%-

�A l ' I M - N O I X "1 (8/14/82)

p, ~ P (Sick/Exposed)
p- - P (Sick/Not Exposed)
True relative risk of exposure is
' ' r truc

:i
- P (Sick/Reported Exposed)
1
r
i
i
q = P (Sick/Reported Not Rxposcd)

q

fc»

J-

!

i

Suppose there arc n6"' false positives, i.e.,
. i
'•
:
I
Pr (Exposed/Reported Exposed) = 1 .
Further, suppose a certain percentage of false negatives is possible,
Pr (Exposed/Reported Not Exposed) = 6.
Then, the probabilities q 1 ,q ? solve
1

I

«£

q2 = Pr(Sick/Hxposcd, Report Not Exposed)6
+ Pr(S.ick/Nbt Exposed, Report Not Exposed) (1-6) .

If we fui'ther assume that whether the person becomes sick depends only on exposure
and not on reported exposure, we get
c

l&lt;-&gt;
Z

=

^PiI * (l-^)P-i£.

Thus, by using the misclassified table, you will be estimating an observed relative risk of exposure as

obs

q
q?

p
3!p.+(l-6)p-

(Pi/Pi)
[S'l)-*)!

'i
= * /(6r
* (1-6)} '.
true
true
&gt; i!
r

�In this instance, the observed relative risk will underestimate the true relative
risk, and this bias depends heavily on the false positive rate &lt;S.
What about the excess number of cases?

Per 1,000 individuals,

1,000 p

truly exposed become sick

1,000 p

truly not exposed become sick

True excess number/1,000 is
1,000 (prp2) = 1,000 (rtrue-l)p2 .
Thus if 1% of the trtfly not exposed become sick and r

"C3TLIG

= 2.0, then from 1,000

truly exposed individuals we can expect 10 more sick persons than we can from
1,000 truly not exposed.
However, if we have a false negative rate of .50 (50%) , then our estimated
excess number will not be 10 but will, be
1 0 0 Cr
0

'

observed~1)p2 * 1.000(1.33-1)(.01) = 3.3 .
(see Table 2)

This is a rather dramatic difference.
A technical note. My definition of false negative is as on page 4 of Fleiss,
not as on his page 135 (Section 11.2).

His calculations in Section 11.2 are for
s

a retrospective study (it of sicks and wells fixed in advance) .
TABLE 2 (More Details than Table 1)
True
Relative
Risk

False
Negative
Rate (Not %)
0.000
0.025
0,050
0.075
0.100
0. 125
0.150
0.175
0. 200
0.225
0.2*0
0.275
0.3CO
O.J25

Observed
Relative
Risk
.00
.00
.00
.00
.00
.CO
.00
.CG
.00
.00
.00
.00
.00
.CO

�TABU: 2 (More D e t a i l s t h a n Table 1)
True
R e l a t i ve
Risk
1
i
l
i
1
1
1
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
3
3
3
3

3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
'*
4
'4
14
&lt;4
4
1
&lt;i
4
4
4
4

4
&lt;i
D
4
&lt;l
4
4
4
4

False
Ncgati ve
Rate [Not Hi)
O..J50
0.315
0.400
0.425
0.450
0.475
0.5CC
O.OCO
0.025
0.05C
0.075
0.100
0. 1i5
0. 150
0.175
0.200
0.225
0.2SC
0.275
C.3CC
0.315
0.350
0.375
0.400
0.425
0.450
0.475
0.5CO
O.OOU
0.025
0.05C
0.075
0.1CO
0. 115
0.150
0. 175

0.200
0.225
0.250
0.275
0.3 CO
0.315
0.350
0.375
0.400
0.425
0.4SO
0.475
0.5CO
0.000
0.025
0.05C
0.075
C.1CC

o. i;s
0.150
0. 17!;
0.200
0.225
0.250
0.275
0.3CC
0.325
C.35C
0.375
0.400
0.425
0.450
0.475
0.500

Observed
Relative
Risk
1.00
1.00
1.00
1.00
1.CO
1,00
1.CO
2. CO
1.95
1.90
1.86
1.81
1.77
1.73
1.70
1.66
1.63
1.6C
1.56
1.53
1.50
1.48
1.45
1.42
1.40
1.37
1.35
1.33
3.00
2.85
2.72
2,60
2.50
2.40
2.30
2.22
2.14
2.06
2. CO
1.93
1.87
1.81
1.76
1.71
1.66
1.62
1.57
1.53
1.50
4.00
3.72
3.47
3.26
3.07
2.90
2,75
2.62
2.50
2.38
2.28
2.19
2.10
2.02
1.95
1.68
1.81
1.75
1.70
1.64

i.eo

cont'd

�API'liNUIX 112 J8/J4/_82)
Suppose we sample N
posed.

who arc reported exposed and N? wlio are reported unex-

What, is the probability ol" detecting a relative risk different from 1 if

we ignore the effects of misclassification?

For this prospective study, the usual

test says (assuming a fairly larj'c percentage (7.5°d) are diseased) says that the
that the relative risk differs s i g n i f i c a n t l y from 1 i r

,&gt;. r. 1 &gt; . — r.--.....* —T
'e olis
- / I
= 1.96 o(q1,q2,N1,N2) .
If the normal probability function is called &lt;I&gt;, the statistical power is

- $(1.96 +

Now,
lop,

c

r

obs

= loy

e

r

true

- log {6r
+(1-6")}
e
true

If N I = N , then

2

0 (q

1

q 2 , N l f N 2 ) = ^-[(1-pj) + 1 - fipj - (l-i$)p 2 ]

=

1[2 - p
1 O

- fip - (1-5)P 2 1

'5

= Ti-f— - (l + &lt;5) r true - ( 1 - ^ ) 1
N l p_
'
2

Hence the power of the usual test depends not only on the relative risk but also
on the probability of disease in the truly unexposed population.

�AIM'J-NDIX_ ff3_ (8
Suppose we observed a table such as
Reported Exposed
N

sick
well

Reported Not Exposed

.;

N

VNK
N

NE

N

V NE

i

1

N

2

How can we estimate the true relative risk, and how much power do we have for
detecting relative risks different from one?
large.

Recall, I am assuming q

is fairly

From Appendix #1 (8.14.82), page 4 ,

_
true ~ TT - ~~iT"r~"~jT"
L
obs
Since an estimate of q

is

«i = VNi
and an estimate of q? is
*2
the estimate of r

= N

NE/N2 '

is
A
A

^i

*obs " *1
°2

and the estimate of rtrue is

A

It turns out that log

and variance

o

r _
is almost normally distributed with mean log
t.i*Lio

�The power of the tost for r e l a t i v e ri.sk d i f f e r e n t from one is
2 - 'K1.96 -

- *&lt;+l'96

(Highly technical, but to keep all the details written down) I want to show that
(*) is true.

We have
A

log r
h

true

- log r
b
true

A

- log r o b s ]

obs

-

obs

...
r obs
.

_
-_
1 - o i obs
.

A

_olKS~ "ob_s_
-—--- -

x

l
f_
}
!-_ Tr~~

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