The use of punched card tabulating equipment in multiple correlation problems, collected and prepared for the use of sta...

MISSING IMAGE

Material Information

Title:
The use of punched card tabulating equipment in multiple correlation problems, collected and prepared for the use of statisticians of the Bureau
Physical Description:
Unknown
Creator:
Smith, Bradford Bixby, 1899-
United States -- Bureau of Agricultural Economics
Publisher:
United States Dept. of Agriculture, Bureau of Agricultural Economics ( Washington, D.C )
Publication Date:

Record Information

Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 028420962
oclc - 25680395
System ID:
AA00017354:00001


This item is only available as the following downloads:


Full Text







13 UN or PUN= CARD TAnJLAL!IN EQUIRPMJN
3 MuQimn cwsnoTIu n0oxms a-


U S DEPOSITORY
a77III i


Col lected and prepared for the use
Iatimtilcans of the Bureau.


Bradford B. tith, In Charge
M iahis hbulation and Computing Section


Washington, D. C.
October, 1923.


It:

S i;,.:









V














C


The use of pinched card tabulating e equipment is not new: I t
mae used a number of years ago in the Bureau of the Census for the
preparation of Correlatioa tables. More recently it "has been used
in at least two of the, Bureaus of the Department of Agriculture in
a similar way. The writer in no sense claims to be the sole author
of the mthods herein set forth except insofar as the special tech-
nique Is devised to suit the equipment and problems of the Bureau of
Agricultural Economic s.

Especial credit is due to Messrs. Tolley and Ezekie. of this
BTeau for devising the least square method (page 15 et seq.) of
approaching correlation problems. The writer also wishes to express
his appreciation to them for going over the following text, their
suggestions and helpful criticism.

The coefficient of Multiple Correlation, R, is a measure of _
the degree of agreement between a given series and the estimated
(generally forecasted) values of the same series, when these esti-
mated values are determined mathematically from two or more differ-
ent (independent) series.* The following pages describe the forms
and arithmetic used in computing the value of B and in deriving the
"regression equation" or forecasting formula. A five variable (four
independent) example is given.


*The coefficient of Multiple Correlation (B) may be defined as the
coefficient of correlation (Pearsontan) between the dependent vari-
able and corresponding estimates thereof, computed from the inde-
pendent variables, i. e. if X', the estimated value of X, equals
blA plus b2B and b3C....... plus a constant, where A. B, and C, are
* *em tbe3dSpendent *varlblt -n bl, b2, b3, are-the net regression
coefficients, B equals the mean product of the deviations of X and
X' from their respective averages divided by the product of the
standard deviations of X and X1. B is also equal to the standard
deviation of X' divided by the standard deviation of X. Also, the
standard deviation of XI squared is equal to the mean product of
the deviations of X and X' from their respective averages.


-1-










is esao here between ten and thirty, depeting upon t iscretiom

9f the person conducting thi investigation. The seller the vars-0

tion, the simpler is the process of performing the calculatUos, for

the whole success of using Hollerith iachines* lies in the grouping

of like values.*" On the other hand, -tod geat a grouping--or re-

duction of variatJon--remults in reduced accuracy. TAo coding my

be accomplished either by subtraction or by division or both: Ius|

if the series under consideration varies between 120 and 175, Snb-

tract 120 from all items, and divide the differences by 5 thus re-

ducing the variation to from 0 to 11. This coding of the series in

no way invalidates the calculation of R. Coding by division slightly

reduces accuracy, but subtracting in no way affects accuracy.,

On the following pages are given the coded series used tn the

example. A, B, C, & D are the independent variables. X is the de-

pendent variable (the values of which it is desired to estimate, or

"predict".) The series should be so arranged that the dependent

variable is placed in a column to the right of the independents. After

the coded series have been listed, cross-add the values for A, B, C,

CHECK and D, & X for each observation and list the sum in the "Check Su"

Sum cnlun on tne right, Col. S in the tabulation. This check sum column

is rnot essential to the solution but is of considerable aid in check-

ing the arithmetic.
---------------------------------------------------------------------
l inhre are several types of punched card sorting and tabulating Ia-
chines. Those used in this bureau are for the most part Hollerith
Machines.

SMrrer;var, if the extensions are to be made by hand, figures that
can be readily multiplied mentally greatly facilitate the process.




w3

ea~eea~..- -. -------- -
S10 14 7 00 10 31
2 3 10 11 3 11 38
10 5 12 00 6 33
.10 3 11 2 10 36
5 10 6 10 3 5 34
6 6 9 1 7 10 33
7 3 13 10 4 0 20
6 0 7 5 1 5 1i
9 3 6 10 81 28
to10 3 1 1 5 4 25
11 6 6 a o 6 26
12 10 6 2 4 6 28
|3 1 3 5 2 6 4 20
li4 12 3 5 0 2 22
15 12 6 0 12 2 32
16 12 4 5 4 6 31
17 6 3 11 3 3 26
18 10 12 8 11 11 52
19 6 8 8 3 25
I0 10 3 4 3 5 25
21 .10 3 11 12 12 48
22 10 3 10 3 7 33
23 3 5 00 10 4 22
24 6 3 8 00 7 24
25 0 1 11 7 5 24
26 10 5 8 0 5 25
27 10 4 0 9 4 27
28 10 6 12 6 4 35
29 12 3 10 12 1 38
3 10 5 12 2 0 29
31 12 9 7 1 4 33
32 10 5 11 3 2 31
33. 7 6 4 4 11 32
34 6 4 10 0 12 32
35 7 4 12 6 12 41
36 3 5 0 10 a 26
37 3 5 5 4 2 19
35 3 1 0 12 4 20
3 0 1 8 1 13
12 3 11 9 4 39
41 10 1 8 0 12 31
42 10 3 10 1 12 36
6 6 5o 5 0 16
:110 3 11 1 5 30
'45 12 5 54 i123
'67 5 5 13 21
417 12 7 10 0 0 29
48 10 2 7 3 4 26
19 10 3 7 0 6 26
*50 10 3 4 4 U 3
51 12 4 5 1 2 24
52 3 4 4 3 '4 1
53 10 4 5 3 0 22
54 3 9 10 0 1 23
62 10O 6 1 27







Wco'raU O. A Ci E i x U
----------------- ------------------ ---------
56 6 3 7 1 5 2a
57 10 3 11 2 11 37
58 6 6 5 0 3 20
59 10 3 5 0 12 33
60 12 3 4 0 23
61 10 7 5 0 2 a4
62 10 3 20
63 6 6 8 2 22
64 12 3 10 0 12 37
65 10 1 4 7 26
66 12 6 5 1 10 3
67 3 2 5 3 0 13
65 12 S 11 11 12 54
69 6 2 10 7 2 27
70 12 6 11 1 3 33
71 10 7 8 0 5 30
72 12 4 7 3 2 2A
73 10 3 10 4 3 30
74 12 4 0 0 24
75 12 5 7 3 12 39
7O 10 1 7 1 7 26
77 3 6 8 b 2 25
7 3 11 5 0 27
79 12 6 08 2 36
so 3 3 11 2 8 27
a1 10 6 7 b 1 30
32 10 4 1 6 27
83 b 3 8 5 3 30
84 10 3 11 7 2 33
65 10 4 2 7 5 28.
66 10 4 11 1 .1 27
37 6 3 51 6 21
M8 12 6 0 0 26
39 12 2 0 12 12 38
90 o 2 8 11 12 39
91 0 0 12 8 28
9p 10 42 b 2 N
93 3 10 3 a 22
U 6 3 1 7 2 19
95 3 1 7 1 8 20
90 10 3 3 8 12 41
97 o 0 0 4 11 21
93 10 6 0 4 25
39 3 7 7 3 4 24
100 7 S 3 12 0 31
101 u 3 3 23
J 10T3 3 1 2 21
103 0 3 0 3 1l
104 0 b 2 10
1:rj 0 46 0 1 2 7
100 3 b 0 7 6 22
1C7 10 1 10 1 0 22
100 10 o 3 2 25
Sf4 7 0 b 27




5 -
-5-
--s -ees -e -e s sea ----------- as a- -5 -
Sbod Ibo. A B C D Z S
asno ---se---- ------ as-----s----s""- Q ----
110 0 1 0 3 0 4
111 0 5 2 3 0 10
112 10 0 11 2 2 25
i 6 4 0 7 0 13
1 0 5 7 1 3
115 3 3 8 0 7 21
116 10 5 11 1 0 27
117 10 1 0 7 2 20
118 10 a 5 3 5 25
119 o 6 12 8 9 35
120 6 3 11 1 9 30
121 12 3 5 4 0 24
122 10 5 11 3 0 29
123 10 3 8 0 4 25
a 0 4o 5 3 4 26
125 10 6 11 10 1 38
1.26 3 6 11 4 4 28
127 10 6 10 12 a 40
12 6 4 0 7 2 19
129 10 3 11 3 7 34
130 3 S 8 0 1 20
131 7 6 4 0 4 17
132 3 6 4 4 3 20
133 12 4 4 4 6 30
134 10 9 8 0 5 32
135 6 3 5 4 22
136 6 5 2 0 1 14
137 10 6 0 9 6 31
138 10 3 7 1 4 25
19 3 2 11 4 0 20
lUo 0 4 7 0 4 15
141 12 0 12 4 12 40
142 6 3 5 12 0 26
143 0 10 11 1 28
144 10 3 11 6 7 37
145 3 b 11 1 11 32
146 10 10 4 4 31
147 6 3 8 6 5 28
148 10 6 8 1 6 31
149 10 4 12 4 S 38
150 3 3 0 10 3 19
151 10 3 7 8 2 30
152 6 1 2 3 12 24
153 10 4 11 1 4 30
154 10 3 10 7 0 30
155 10 2 4 3 11 30
156 12 a 10 1 2 33
157 12 7 7 3 1 30
155 10 3 4 4 2 23
159 3 6 5 1 2 17
160 10 3 7 3 4 27
161 10 3 7 1 3 24
162 0 2 6 2 0 12
163 o10 4 11 1 2 2a
16& 10 3 10 4 0 27





-- ( -
e cordt, No A ......B ... c D.5"."

165 12 4 0 6 4 a6
166 6 4 5 a4 4 a6
167 10 4 S 3 2 27
165 10 4 0 7 4 25
169 3 4 11 3 5 26
170 3 1 4 3 11 22
171 7 0 0 11 5 23

---- -- ---------------- ----- ---------------
(1) Original series minus 35 and reminder divided by 3

(2) I 350 4

(3) t2 6

(5) 2" 935 -53
(5) 'II 935 aiI 33





~- 7-


SAfter the data has been coded and listed as on the preceding

HPtf the values are then punched on punch cards, the record number

i also being punched for the purpose of identification. There are

9bi .rty-five columns on a punch card. In our example the record number
vas punched in colums 31-2-3; A in columns 34-5; B in 36-7, C in 38-90,

D in 40-1; x in 42-3; and S in 44-5. One card is used for each line--

for each observation, that is.

After the cards have been punched, they should be sunaed:
The sumn of A, of B, of C, of D, of X and of S should be obtained; also

the number of cards should be counted. The results should be recorded

: in some such form as follows:
!Form 1

-- Suams and Means of the Variables.
| Item : A : B : C : D : X : 3

171 1302 705 1149 680o 774 4610
i: ~Means: ::::i.23: 99
Seas7.6140 4.1 : 6.7193 : 3.9766 4.5b3 6.9591

The use of the Check Sum first becomes apparent here: Evidently

the s- of the sums of A, B, C, D, & X. should equal the Sumn of S; which

is the case. The same is true of the means (averages). This checks

the first additions used in building up the check sum itself; -1lt also -

checks the accuracy of the punching; and also of the division in securing

the averages. The values filled into the form above are for our example.

*It is sometimes feasible to do the coding by punching the orilgal values
upon the punch cards. Then sort the cards on the variable to be coded,
group the arrayed values into the determined upon classes and gang punch
each group in a new column with the assigned class value--such as 0, 8, &c..
The check sum for the individual record then can be prepared by showing
each card separately in the tabulator, adding across, and subsequently
Ching upon the card, after which the procedure is as given above.








The next step is to sort the cards upon the first variable, A;.,-
the form for recording the tabulation .is given below: U' j





SVar iable No.
Class of A B C D
Valug Items Sum&xt SuM- Ext. aum Ext. SBn Ext. Sum Ext. &i Sm
r) 5 ( .Lk ilL 4L (o 20.12) 1i i23 (1_i W51 iL


.17 112







I"-i



__________________ ___ I ___________ ________ ____-__ -_ i_ _- -*









lb. card@ being sorted nn the first variable, A. group them
i+!



S "into packs--all the cards ;f fthe lowest value of A in the first pack,

of the next value of A itx the second pack Ac. ac.. Li st in column

|: (l)-IbYr 2--the value of A in the first pack. Tabulate this pack.

1 Okn the first line in column (2) write the number of cards in the

|: Iack; in column (5) write the sum of the values of A in this first
I+ pack; in coluam (7) write the sun of the values of B in this first

Ip: pack; in column (9) the sum of the values of C in this pack; in
|colna (11), D; in colum (1i), 1; in column (1i) 5. Take the

... second pack, list the value of A in this pack on the second line of

:, ...the form; and list the corresponding sum values as for the first

I. pack. Repeat until all packs have been so treated. When this is

completed mkJ the extensions for columns 6, 8, 10, 12, 14, & 16 as

follows:

Multiply the values listed respectively in columns 5, 7, 9,

11, 13 & 15 on any line each timesthe value listed on the same line

in colam 1. List the products so obtained in columns 6, 8, 10, 12,

141, & 16 respectively. Do this for all lines.

Add coluans 2, 6, 8, 10, 12,. 14 & 16.

Take the cards and sort them again; this time on the second

variable, B. Take a second sheet (Form 2; Sh.2); divide the sorted

cards into packs, according to the values of B and list these values

of successive packs in column 1. Tabulate, list and extend in a
+ manner exactly similar to that when the cards were sorted on A, ex-

cept no figures need appear in the two A columns 5 & 6.
H!l









: :i iiitil l
Sort n C and proceed as for B on a n &heet (Form 2 S M. 2

No figures need appear in the A, or B colms; C0olums 5, 6, l7, A 0!7

Sort on D and repeat (Form 2; Ah 4): No. figures in the A, ,
or C colums: Columns to 10 inclusive.

Sort on X andt repeat (Form 2, Sh 5). No figure in the 3*
C, or D columns; Coluns 5 to 12 inclusive.

(Note: The reason that an increasing. number of columns can
be omitted is that to make the extensions and mn them would give fig-
ures already computed: Thus if we sort on C and extend its valtS
times D. adding the extensions, we arrive at the same figure as If wk
had sorted on D and extended its values times C.) In case difficulty
i a encountered in making the figures check to the check sm -- e-
plained later in connection with Form 3--it may be advisable to O
the extensions here directed to be omitted, for the sake of comprl-
sons and to help locate the errors.)

Following are the tabulations of the five sorting*s ad In per-
forming the above steps for our example. Note that in each case a

check is afforded by adding up column 2. This should add to the toatl

number of cards in the problem as shown by the data on form 1. A

further check may be afforded by adding the sum columns for each vari-

able--coluans 5, 7, 9, 11, 13 & 15. These should on every sheet add

to the same corresponding figures given on form 1.







lajg ,l*J


(5)
0
90
16s
42
690go
312


0
2034

1155
20470
=a
37M55


Sinr ted nn U


nitm.A n .1.


5
13
12
45
30
19
.31
6
4
4
1

(171)


Form 2 Sheet 0.


0

2
135
120
95
1l6
42
32
36
10
12


0
13
~4s
405
480
475
iin6
294
256
324
100
144
3655


24
70
68
353
180
121
209
42
37
26
11

(1149)


0
70
136
1059
720
605
1254
294
296
234
110
96487
4_74


29
50
74
169
o109
59
144
7
17
9
3
11
(680)


0
50
148
507
436
295
864
49
136
72
30
129
2719


31
73
57
228
119
66
123
17
18
20
11
11
(774)


0
7
684
476
330
738
119
144
180
110
-132
3100


122
284
295
1273
777
468
588
155
137
121
38
15)


0
284
590.
3108
23W0w
5325-
1085*
1096
1089
350
S19743
19743


Sorted on C -_ Form 2 Sheet M.


21
3
7
13
25
19
26
21
25

(17%j


0

14
52
125
133
208
210
308


0
3

625
205

2100
33899
-1102-
10099


166
22


57
31
93
109
.35)
(6so)


0
22
5g
176
285
273
LBE
930
1199
42051
4051


89
13
30
79
82
90
127
6s
114i

(774)


0
13
6o
316
41o
630
1016
ioi6
680o
1551
53366
5336


454
65
148
333
Z^9
748
627
894
-an-
(4610)


0

2-96
1332
2515
;34
5984
6270
9$34

33437


k I )


190
67s
659
165
2047

(61)


12
30
28
6
69

(171)


0
270
1008
294
69o00
i744
12-216-


46
134
107
25
270
(71235)
(705)


0
402
642
175
2700
14765395
5395


67
194
162
29
522

(1149)


0
582
972
203
5220
2-109
9077


143
136
119
35
236

(6o80)


0
408
714
266
2360
1a96
5044


la&.;


34
124
133
31
329
It2l


0
372
798
217
3290
6153


(3) (9) (10) (111) ()


g










(1)








-13-


The next step is to transcribe figures from the shouta (1 to

5) of Tun 2, to another sheet of the forum shown on the following

peae (orn 3).

On line A-i (lbzz 3) colwma A list the figure taken from Forrn

2; Sh. 1 Ooi. 6. last (total) line. the other figures on line A-l aro

taken from the sane shoot (lForn 2; Sh. 1) last line, coluzzms 8, 10.

12, 14 & 16. the figures filled into form 3 apply to our cxar aplo, so

they ray be traced through the various forms.

The data for line B-1 comes from Form 2, Sh. 2, last linu,

columns 8. 10, 12, 14, & 16. (TBits was the sheet used when the cards

were sorted on B.)

'The data for line C-1 cones from Form 2, Sh- 3, last line,

coluMns 10, 12. 14, & 16. (This was the sheet used when the cards

were sorted on C.)

The data for line D-1 comes from Form 2, Sh. 4, last line,

colmus 12, 14, & 16. (This was the sheet used when the cards were

sorted on D.)

The data for line X-I comes from Form 2, Sh. 5, last line,

''lu.2.s 14, & 16. (Thlis was the sheet used when the cards were sorted

on X) W U have nr.. 'filled i. rll '-t has to go on thu lines ending

13. "1" (i.e.: A-., 3-I &c &c) on Form 3.

Tb obtain the figures to go on the lines ending in "'-" on Forn

3. we cnke computations fror the data on Form 1. The suLi of the vari-

able A 1'3iG in our example) is put into a coiputif mnachinu and mul-

tiplied s-acccssivoly by the mean of A. 9f B. of C &c. *c.









Transcripton of Sm of Ixtencionm.
Line
# A B ,
A-i 12a6.o 5395.0 9077.0 5044.0 6153.0 37885.0
A2 9913.4 5367.9 5714a.5 5177.5 5893.2 35100.5
A-3 2302.6 7.1 3258.5 -133.5 259.5 2784.5

B-1 3655.0 4874.0 271 9.0 3100.0 19743.0
B-2 2906.6 4737.1 2803.5 3191.0 19006.1
B-3 748.4 136.9 54.5 -91.0 736.9

C-i 10099.0 4051.0 5336.o 3J1437.0
c-2 7720.5 45bo0.l 5200.7 30975.9
c-3 2378.5 -518.1 135.3 2461.1

D-1 4786.0 3094.0 19694.0
D-2 2704.1 3077.9 i5332:1
D-3 2051.9 16.i 1361.9

x-1 5590.0 23573.0
x-2 3503.4 2866.2
x-3 2386.6 27o6.8


V






15-


the products so obtained are listed respectively in Columns A, B, C &c.,

on Form 3. line A-2.

Next the sum of the second Variable, (B in our example; or 705)

is put into the computing machine and multiplied successively by the

Man of B. of C. of D &c. &e.. The products so obtained are listed

respectively in Columns B, C. D, &c. of Form 3, line 3-2.

Next the aum of the third Variable. (C in our example; or 1149)

is put into the computing machine and multiplied successively by the

mean of C, of D &c. &c.. The products so obtained are listed respec-

tlvtely in Columns C. D, &c. of Form 3, line C-2.

In a similar manner the computations are made for the other

lines ending in "2-, Form 3.

Note:- In practice it is most convenient to prepare Form 3 on
a asheet of paper which also carries Form 1 at the top. The figures
for making the extension for lines of designation ending in "2" are
then before the operator.

Every item an a line ending in n2". is now subtracted from the

figure directly above it on a line ending in "I1." (Note: naturally

should the minuend be greater than the subtrahend the difference will

be a negative value.) These differences are listed on the lines of

designation ending in "3" These lines are then transcribed to Form 4.

The differences thich have Just been secured are the product

moments and squared standard deviations (times I1); and are the neces-

sary data for any solution of multiple, and partial correlation coef-

ficients, or gross and net regression coefficients. The usual solution

may be found in Yule: "Introduction to Statistical Method." The solu-

tion given in the following., however. Is a nLeast Square" method, first

conceived of and developed by Wssrs. Trolley and Ezekiel of the Bureau









of Agricultural SEconomics. An aruticl ascribing the theory of 1W S

method is published by them in the Journal of the Am. Stat. Asoo., ::..


for December, 1923.


Note:- In cas it is decided to mako the extensions by head
rather than to use punch cards and tabulating machine s--frequsntll :
the case when shabort series. such as time series, are being analy.,--
a multi-columnar form should be used. In the six left-hand colua)
list the data; column beadingo s would b1 B, C D, X.& 8. The
remaining co Lims should be headed: A, XAJV AC, AD# AX. AS; Bic, SGI
ED, BX, B5; Cc, CD. CX, CS; D', DX, DS; X, X5. Inthe A column
write the squares of the values in the A Colum. In the AB column
write the products of the A items times their corresponding B items,
Ac Sc. When all columns have been extended, add them, listing t1e
totals below, in their respective colbms.
SFind the means (averages) of the.A, B, C. D, X, & &aolunls.
Multiply the sum of the A column times each of the meane of the A,
B C D, X, & S columns and write the products below the 8us of the
A. AB, AC, AD, Al, & AS columns respectively. Multiply the sum of
the B column times the means of the B, C, D, X, & S columns and write
the products below the sums of the 32 BC, BD, BX .'& BS33, colums. In
a simi lar manner extend the sum of the C colum times the means of
C, D, X, & S; and inscribe the products in the C2, CD, CX, & CS
columns. Also the sums of the D, & X colums. It is not nt&V1 -
To multiply the sun of the S column times anything.
Now subtract the last values listed in the A2 column and in
columns to the right thereof from the figures just above them. If the
minuend should be greater than the subtrahend the. -difference would-
naturally be a negative value. These differences are now to be trnse-
ferred to a new sheet of the arrangement shown in form 1. The dif-
ferences in the columns comnencing with an "A" in their designation
are transferred to the first line-of form 4, designated as line A-).
The differences in the columns commencing with a "B" in their desig-
nation (this of course includes the B2 col.) are transferred to line
B-3 of form 4. The remaining differences are transferred in a sim-
ilar manner. The so arranged differences constitute the Normal EVLua-
t ,o"Bor a least square solution for the value of the net rg4s.tO --.
of A, B, C, & D, on X.


F-


* -










At this point the ue or tMe ,imAcE 1am IeL. Q a I EO 6-- a.

the work to this point may be shown: On Line A-i (Forn 3) the Sm of

the items in columme A. B, C. D & X should equal the figure in Column

8 (Line A-l), thus chcking the extension and addition of all the

figures used in connection with d4oviag these valuese.

Line Colunm
The Sa of the following' A-1 B
B-1 B
.. C
D
H 7 D
X
Should check to:
B-1 S

The Sum of the following: A-1 C
B-1 C
C-1 C
D
n X
Should check to:
C-i S

The SaB ofthe following:
A-i D
B-1 D
C-i D
D-1 D
P X
Should check to:
D-1 S

The sum of the following:
A-1 X
B-i X
C-I X
D-1 X
X-1 X
Should check to:
X-1 S.

By substituting "2"' for "I"; and also "3" for "1" In the above

schedule, a further check my be secured. It is essential that the

values should all check, before the work is carried to a further stage.











J,


2302.6


I


27.1
748.4


328.5
136.9
2375.5


A D


-133.-5
&r.5
-518.1
2081.9.


I
259.5
-91.0
1)5.3
1.6.1
21afi-6
9'86.6


2IRS..6 flt~F .8
mttiYInV .


14
7361.9
1361.9
2706-9g


2302.6
-1.0000


27.1
-.0115
-.3
7148.1
-1.0000


1
2
3
4
5
6
7
5
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25


33.5

1369.9-
e39-

-3.9'
133.0
-.1778
237s.5
-46.9.
-23.7
2307.9
-1.0000


6


-l


-J
-1.


.0495
-.0097
-.0078


248.45 3.60 17.9
Square root of 2386.6


133.5 259.9
,0q -.11-2
-8W.5 --91.0
1.6 -3.1
-82.9 -94.1,
.1108 .lc1
58..1 135.3
19.0 .-37.1
14.8 16.7
4.5 3 114.9
.20O8 -.049
)58.9 16.1
-7.7 15.1
-9.2 -10.4
L0o.6 j24..1
163.4 *44:9
0000 -.o229
D : .0o29
.o048 : .0546
,0025 -.1330
0013 : .1079


a


2784.5
-1.20NO
736 .9
-32.8
704.1


-397.2
-125.2
1935.7
-.8400
1361.9
161.4
76.0
406.6
2006.1
-1.0228
z 16.1;-
2 135.3 :
x -91.0 :
x 259.8 :
P.M.
Sq. Root :


-3.05 : 250.79
(See Line X-3; Col.r
equals 6.94 -- 45.8


: o.8
or .142


14=t
A-3
3-3
C-3
D-3
X-3


SC:
B: S -.1258
A .1128 .0016


a


1,L


6i.:
6$9


m


mm


maim


7


w~ Ai
MCM













tion is given on lines 1 to 25, as described below.*


On Line 1 write the first normal equation, i.e., copy Line A-3.

now, divide every item on Line I by the first Item of Line 1, reverse

the algebraic sips and list the quotients on Line 2. In our emanple,

we divide by 2302.6. The algebraic sum of the items on Line 2 Colunns

Al B, C D, and I. should equal the quotient appearing in Column S on

the ama line. This will not always check to the last digit, owing to

the dropping of places in the division. Now, draw a line under the

figures Just written in. On Line 3 copy in the second normal equation,

that is, copy Line B-3. Now put the figure on Line 2, Column B (i.e.

-.0118) into the multiplying machine and multiply it consecutively by

the items on Line 1 in Columns B, C, D, X, and S, listing the products

in the respective columns on Line 4. In our example, we multiply -.0118

by 27.1 by 328.5 by :133.5 by 259.9 and by 2734.5, giving as quotients

the items appearing on Line 4, i.e., giving -.3, -3.9; l.6; -3.1; -32.5.

How add the items on Line 4 to the items immediately above on Line 3,

giving Line 5. Careful attention must be given to the algebraic signs.

sow, divide the figures on Line 5 by the first figure on Line 5, and

-reverse the algebraic signs, listing the quotients on Line 6. In our

exampl, -e divide by 748.1. The algebraic -m of the first four itdes

on Line 5 should check to the last Item of the line, 704.1, in the S

Column. In lik mnmwr, the algebraic sum of the first four items on
-aac aee----e-e-------------------
'This is the tDoolitt$1 Method" See Oscar 5. Adams "Geodesy Appli-
cation of the Tbooty of Least Sqmure .to the Adjustment of Triangula-
tion.a 1912. special publication 52, Geodetic Survey.












machine and multiply it consecutively by the items in the C, D), X
6 col=as of Line 1, listing the product s in the corresponling cop
on LIve 8, giving careful attention to the algebaic algps. INt,.I".^
the item in the C colm= of Line 6, -.1778, into ,the multiplying af.i t
.."ii ........ ..
and multiply it consecutively by the C, D. 1, and 8 colv= figures e9 ii
Line 5, listing the products in their respective colnsm on Limn 9,s
.:: *iliipi

giving careful attention to algebraic signs. Now, add together fo|
each column the values in Lines 7, 5, aad 9, giving Limw 10. the ftr"s :i
items of this line should check to the last, simlar to the cae for
Lines 5 andM 1. Divide each of the items of this line by the first its
in the line; that Is, divide by 2307.9, reverse the sigmas and list tM
quotient. on Line 11. In a msanmer mimllar to Line .6, the first items
on this line should check to the last when added together. mDrw mothW
line. On Line 121 write the fourth normal equation, that Is coy Line
D-3. Put into the multiplying machine the value on Lined 2, CoLa D,
and multiply it consecutively by the D, 1, and S column values of Lim
1, listing the products on Line 13 in their respective columns, givibT
careful attention to the algebraic signs. Next, place the value on
Line 6 colunn D into the multiplying machine. .1108, and multiple' con-
secutively by the values on Line 5, collins D, azd S, listing the
products in their respective column on Line 14, giving careful atten-
tion to the algebraic signs. Next, put the figure on Line 11, column
D. .2.09, into the multiplying machine q4d multiply consecutively by
the values on Line 10 colms D, X, and.S,8 listing the products in their
I









Ii reaCstive *ohnm on AA 16. giving careful attention to algebraic siM.

Aitt, ndtld or colwzis 3) -nDA S the vrlucs of Lines 12, 13, 14, rnd 15,

S 11sting the algebraic sis on Lino 16. CGruful uttontion should to Given
**ii" iiE:: ** "i
to al brnic sinjs. The flfobrmic am: of the first itc:.a on LUnc 16. should

check to the last itor, as was the caso for Lines 10, 5, rnL 1. Divi'.o

the ities on Line 16 by the first ite:. on LLUe 11, 19r3.,. rovcrsc tho

alj-braic sits. a M list the qrotntions on Linu 17. *hu first itcom oi

Lino 17 shoul. choc: to the Thst ito-. of Line '17, ms fl-s thL. creo for

ses 11., 6, and 2. Te have now finished the 'forward!' solution for the

4 worral omations, and. by this tinc the rc-dor prosuarbly undcrstPnda tho

mtbod so tkr.t the eZtcnsion of tho solution to r. rrt;.'tor nttbor of v'ri-

abless ill to n co:.:preratively si..plc zatter. 1e are now ready for the

'beck solution, which in iven on Lines 1, ct s-c.


On Lino 1., coluL- X writo the vplu or.n Liae 17 colmI2n X; rcvorse

the sign. This value is tLe net repression coefficient of the variable

D on X. Bext in Column % ULaas 18 tV 21, inclusive, list in inverse

order the values in colun X,. Lines A-3. -3 -3 -3, r-3. Ncxt on Line 19

Colua C. write the vqluo on Linc 11 Colu.n X, rovarsinr the sign. :n

Line 20, clzim 7, writc the vrluc. on Line G, column X, revereir; the

sim. On Lim; 2J1. colum A. t.Tit- the vnluc on Linm. ;l cnlu.n X. rt.vcrsin,'-

the suin. Mt thc. vrluc o, Lin.; 18. cluL- X., into the :.xltiplylat rw-

chine -nd iultip.ly it tL..s th- vlIuc on t-c s. ii. in colI.;.n 5, r

list the 'rojuct in Coluim 7.. (Protuct Rn-r' t). Th en Lultiply it tir.,S

the' vrl'2.3 ia colu.-i D), LUes 11, 6 "n. C. l.stin" the proAucts r-swTc-

tively on tL-an 19. 20 n's 21 in c'lw.n '. l.ultiply it "Iso ti:zsa the

value in colnum D on Line A-3, listing the product in Colurm v) on Li.n, 2/.










nm in 0colvM A, Line 19. Ths least sum is the net regression co.fiii|
of the variable C on X. f lL:IX

Put the net regression coefficient of the variable C on.X into tO:.h:u

multiplying machine, and, having a care for algebraic sigas, multiple itl

times the value listedL beside It in column', writing tie product on to I

same line in column PHM. Then, multiply it IF the values In column Co .an

Lines 6,2 and A-3, writing the products on Lines 20, 21 aMd 24, reqxpe-

tively of the same column. Now, add the values on Line 20, colume J, C

and D together writing the sum in column X on the sai line, having a p-

ticular regard to algebraic signs. This last am written on Line 20 in

column X is the net regression coefficient of the variable B on X. Place

it in the multiplying machine, and, having a care to algebraic signs, mul-

tiply it times the value listed beside it in column S, writing the prodnt

on the same line in column PM. Then, multiply it by the values in colan

B on lines 2 and A-3, listing the products respectively on Lines 21, as

24 of the sme column. Now, add the 'Values on Line 21, and in colums

A, B, C, and D, writing the sum in Column X on the same line. This aml

is the net regression coefficient of A on X. Place it in the mualtiplying

machine and multiply it times the value listed beside it in column S, hrv-

ing a care for algebraic signs, and list the product in column S on the

same line. Also, multiply it times the value on Line A-3' column A, liAst-

ing the product in the ama column on Line 24. Now, the values on Lime a

In columns A, B. C, D added together algebraically, should equal the value'

listed on Line A-3, column X., which serves as a check upon the derivationU

nf other net regression coefficients than D on X. There is a difference





- 23 -


tof 9 beblsa the We values in our seample. A greater accuracy may be a -

cred Marryin the arithmetic to a greater number of places throughout the

entire solution. It as deemed e*pedIent to make the example as simple as



We have now secured the net regression coefficients, which are es-

eatial& to the forming of the regression equation for predicting or esti-

mtftgte values of S. To ascertain to how great a degree these predic-

tions conform to the actual values, it is necessary to obtain some measure

of agreement between them, the predicted-and the actual. This measure of

aunenat is the coefficient of multiple correlation, 5, defined on Page 1.

o secure this coefficient of multiple.correlation, add the values in the

FN cola, listing the sum on Line 22. Next, secure the square root of

Sthis aM, given on Line 23. Finally, secure the square root of the value

listed on Line 1-3, colsum X, listing this in the PM column on Line 25.

This value divided into the value Jmlidiately above it on Line 23, gives

the coefficient of multiple correlation.

There are certain aids and other checks in the solution which can

be applied to help in locating errors. The diagonal terms of the normal

equations (2j02.6, 714S.4, 2378.5 *\are always positive in sign. In mak-

Ing the solution the figures listed Immediately below these figures (to be

added to them in the course of the solution) are always negative in sign.

Me sms appearing above the -1.00000 terms are always positive, (i.e.

74.1, 2307.9, 196314).

accuracy is increased if comparatively small diapnonal terms are

avoided. (This can be controlled by controlling the original coding.)

%e Product Moent, Line 22, Is always positive in sign.




uNIVERSiTY OF FLORIDA


q C Cd UWd U UI l 191


S.... .....
vs nintszou wasass. :

sbe fial step in the arithmetic i1s to write t nr.grq"e.

or "preticitire or Uestimting.R equation as it is various ly ealls Av

first write down the retgrossion coefficient of A on X1 la-r in -o

this is .1079. Beside this value write tbat wae donL*..w | '. .,a 4....

simpltfying the A series: thi- XUl b" .ot,4f fom. A 3 5.( =

6, note (1)). Then further write the subtraction of the &nflg a("4i

the A series as taken from farm 1. enclosing all in pare#tbhes s.1

algebr- c for. so far will look like this: .1079 (A I
... 3 ,.?:.:. ]mtt
3
In an exactly similar manner treat the B, C, D. & X series. (Dlikq.

the S series) There trill be no regression coefficient for the X ....

The algebraic sum of. the A. B. C, & D expressions hsuld be equated .

the X expression as follows: ; .

C ) ( : ) (y
.1079(L25 7.614) .1330(%M2 4.1226) + .o00A- 6
(3) (41 ) (6^::


+ -0229(D 5-3-97f6 6 I z..2 4
*E '" l
) ::.:



This is the "rai" regression equation. It is only necessary now to..

evaluate for X, involving only elementary algebra, to put the eqatis

in its most useful form:

X- .1079 A- .0998 B' .0273 c 0 .0137 D 99-055

This is the predictingg" equation mentioned in the definition,..

of R in the note on page 1.




Full Text
xml version 1.0 encoding UTF-8
REPORT xmlns http:www.fcla.edudlsmddaitss xmlns:xsi http:www.w3.org2001XMLSchema-instance xsi:schemaLocation http:www.fcla.edudlsmddaitssdaitssReport.xsd
INGEST IEID EZGUL9NGX_JBX9XS INGEST_TIME 2014-07-22T21:23:13Z PACKAGE AA00017354_00001
AGREEMENT_INFO ACCOUNT UF PROJECT UFDC
FILES