Now each of these is an estimate of 02, which, The above calculations can be summarized in an
it wi 11 be rememhered, is assumed constant for analysis-of-variance table, whose form is
all treatments. Hence, to find a bettet· esti- shovm in Table 12.3. The 'mean squares' in the
mate for 02, it is legi t·illlatteu combine the penultiwat0 column are obtained by dividing
st values, giving the entl i '.:s in the' 5um of squares' column by
the corresponding entries under 'Degrees of
c freedom'. The last row of the table shows the
L s~/c sum of ~quares and degrees of freedom that
i= 1 vlOuld be calculated if the variance of all
kn observations were being found, the fact that
on substituting' for s ~ from Equation (12.4). they are cl assified according to treatment
This is called the variance 'within treat- being ignored. It is quite easy to show that the
ments' . 'between-treatments' and 'within-treatments'
sums of squares add up to the total sum of
The second estimate of o? is obtained by using squares, and the same is true for the degre2s
the treatment means x·i. If Ho is true, the of freeriom.
data can be reg3rded as constituting c samples
from the same population. This is because Ho Equations (12.5) and (12.6) for s~ and ss aloe,
states that the treatment means are identical, however, not the best for computational pur~o-
and we have assumed that the treatment vari- se s. In ~ract ice, it is better to proceed a,;
ances are the same. Now, we know, if samples follows.
of size n are chosen at random from a popula-
tion with variance 02, the variance of the Let
sample means is 02/n (I; 6.5). Thus the vari-
ance of the sample averajes, namely, 5, = sum of squares of all individual
c , observations: LLX .. ' ,
= L (x-x)'/(c-l) i j lJ
i=1 I
sum of squares of tr~atment totals
is an estimate of o2/n. Another esti,oIateof 02 number of observations in each total
is therefore,
square of grand total
tot" 1 number of observations
nsB' c
= n L (xi-x)'/(c-l) .
i= 1
This is called the variance 'between treat-
ments' .
What we have shown lS that, if Ho is true, s2 2: 2:(x .. -x.)'
ij lJ 1
--- W
and sfi are both estimates of 02, and therefore
the ratio
will have an F distribution (I; 7.9) with 50urce of Sum of Degrees of Mean F
(c-1) and c(n-1) degrees of freedom. On the variation squares freedom square
other hand, if Ho is not true, s0 I,i11 sti 11 ,2
be an estimate of02 b~s ~ will be inflated Between n 2: (Xi-X)' c-l , sB/sW
by the differences betweeR treatments. In treatments
fact, it can be shown that, if the experiment i sB
were repeated many times, the average value of : 52-53
s~ wou 1d be ,
Within L "j- (xU-xi )2 c(n-1 ) Sw
0' "+ _n_ 2c :,' treatments i
c-1 i:l 1 : 5,-5,
"hich is always greater than 02 unless all the Total 2.: L (xij-x)2 cn-l ,.
"i = O. Hence the F ratio would be expected to i j
be significantly large if Ho is not true. The
test for differences between more than two : 5,-53
treatment means has thus been reduced to a
comparison of two variances.
2:. 2:. (x.-x): = $,-), J
i j 1J
Carrying out thesE c .Iculations for the data The f,rst case arises when one of the treat-
in Table 12.1, "e find lIIetns may be though t of as a control or stand-
ard, '.-lit\h'Ihichthe other tt"eatiiieiiTaSretolJe
compal"ed. This lIIightarlse, for example, if
Fabri~: I in Example 12.1 "ere of our own man 1-
facture while the other fabrics had been p'o-
duced by competitors. The compal isons of i 1-
teres~ in such C3ses are with the control,
i.e., the difference" X2-Xl, ](3-3(1,and :<4-)("1.
In th,=se circumstances, it is, appropriate to
calc~late the least significant difference,
whic~ is based on a conslderatlon of the con-
fidence limits for the difference between two
means. It will be recalled from (I; 7.6) that
such limits for the difference between trea:-
ment i (f;) and the control treatment 1 are
given by
and the ANOVA table i~ sho"n below. The test where ni and n1 are the sample sizes. When
of significance is carried out by comparing ni ' nl (=n), the 1imi ts become
the observed ratio in the final column,
Fa = 6.69, with the values tabulated in x,1-'<,±tk s JIn2
Table A5. Entering that table, vie find that
values for 3 and 16 degrees of freedom are ,a
not ~iven. The values for 3 and 15 degrees of
freedom are shown, howe'ler, and tLis is good and any difference
enough for our purpose, since the variation greAter than
of F with k2 is slow "hen k2 is as large
as 16. We find that, for CI = 0.01, iftk,ns
and, since Fa is greater than this, the is significantly greater t~an zero, since the
significance level for the experiment is less confidence interval would ~n exclude the
than 0.01 (1%). We therefore confidently va 1:,e zero. The 1ast expre ssiona.liCiVethere-
reject the null hypothesis and conclude that fore gives the least significant difference.
there are real differences among the makes of
showerproof fabrics tested. The value of s to be used in evaluating this
expi"ession is the square root of the 'within-
Having decided that there are significant dif- tre~tments' mean square, i.e., s = sw, and k
ferences among the treatments, we next have to
ascertain which of the differences are of real is the number of degrees of freedom associated
importance. In an experiment like Wat of wit~ that mean square.
Example 12.1, there are 4C2 = 6 pairs of
means, and not all of these pairs are neces- We shall imagine that Fabric 1 is of our own
sarily different; we need to know which are of ma~ufacture, to be regarded as a standard, tte
significance. How this question is dealt with other fabrics having been produced by competi-
depends on the particular context of the ex-
periment, and there are, in fact, two impor- to:"s. From Table 12.4, we find
tant cases that need to be considered.
s = Sw = ~ = 12.76 ,
Tc evaluate 95% confidence limits, we put
a = 0.025 (I; 7.2), and from Table A4 we then
fi nd
Source of Sun, of Degrees of r~ean F The least significant difference is therefore
variation squa res freedom squa res
T'k ,as If!nf = 2.12 x 12.76 x V!Sf = 17.1
Between 3270 3 1090 6.69
trea tmen ts any deviation from the control greater than
2606 16 162.9 .:his is thus ~ignif';cant.
Wlthin ','hemean values for each fabric are shO'llnin
treatments 5876 19 Table 12.1 and are repeated bel.ow.
Total lIean penetration time
(less 100)
We therefore see that Fabrics II and III are We see from this that Treatment I is certainly
different from Treatments II and III but not
significantly better than the standard Fabric significantly different from IV. The compari-
I. The diffrrence betwern Fabr,cs I and IV is son betwern IV and III approaches significance
not greate" thun the le,1:;tsignifiCunt <Iif(er- (the difffrence is 21 cnmpared with a least
~nce and is thus not significant. *** significart difference of 23.1), but all othe,'
comparisons are not significant. ***
12.2.4.2 Case (b) Global Comparisons:
-- ~ey' s Procedu,C,'C ~(c._~_R..ando!T.0..~~c_!E.x:p:e.r':i.mle,noctks,
The ether situation that occurs is when it is It \olaspointed out in Section 12.1 that it is
essential whenever possible to randomize the
required to make comparisons between all pairs order in Ilhich experi menta 1 runs an conducted
so as to overcome the effects of any system-
of means ..This case would arise, for example, atic trenr.s in experimental conditions. How-
ever, consider an experiment carried out to
if in Example 12.1 the faorics had been bought compare five treatments, and suppose it is
estimated that the experiment can be completed
in from four suppliers by a retailer who sim- in five d',ys. It would obviously be foolish to
arrange t~e experiment so that treatment A was
ply wished to compare th,~m \olith each other. tested un day 1, treatment B on day 2, and so
on, because any differences found might be due
The same basic approach is used as for case to variation b~tween days rather tnanlDetween
treatments. And the same would be true even if
(a) but now the least significant difference the treatments were assigned to the days at
random. Evidently some further thought needs
has to be greater, since all comparisons are to be givp.n to the experimental design.
being made. The appropria'felimits have been Fortunately, the solution is fairly easy. If
we assume that the conditions during a single
derived by J.W. Tukey and are day will ne reasonably stable, it seems sen-
-_ - sible to test all five treatments in a day,
/0xh -x'q c,k 11 1 the order of testing within the day being ran-
1 domi zed. 'h is procedure wou 1d then be repeated
± S -2 (-nh + -n) i , on each of the five days, which would give an
experiment of the same total size as the pre-
where hand i are the treatment numbers being vious one, This arrangement ensures that the
compared, c is the total number of treatments comparisons between treatments are valid and
in the experiment, k is the number of degrees unaffected by any differences between days.
of freedom associated with s, nh and ni are
This arrangement is called a randomized-block
the sample sizes, and qC,k is the studentized experiment, the 'blocks' in the ,bove example
range, \olhose numerical values are given in 5elng the days. Another example 'llows.
TaIiTe' A6,
This experiment was carried out to compare the
The least significant difference is now there- eff ec tivene ss of four shri nk -resi st treatments
fore A, B, C, 3nd D, on a particular fabric. Treat-
ment A Wi,S a standard treatment, currently in
Assuming now that all comparisons between use, and it was decided to run this treatment
treatments are of interest, we have, for this twice. We shall denote the two A treatments by
exampl e, A1 and AZ; thus effectively there were five
treatments being compared in the experiment.
To find 95% confidence limits, we must use the
value of Q4, 16 in Table A6, which gives A large roll of fabric was available for the
tests. NOI'I, there is sometimes a tendency fer
and any difference greater than this is signi- the inhe"ent shrinkage of a roll of fabric to
ficant at the 5% level. vary from one end to the other, possibly be'
A convenient way of. illustrating the applica- cause the inner layers have been stored undl'r
tion of this rule is shown below. The treat- different stress conditions from those at the
ment means are arranged in ascending order of Jutside of the roll. To allow for this poss'-
magnitude, and the horizontal lines cover the bility, the experiment was therefore arranged
treatments that are not significantly dif- in randOl.liz~dblocks. The roll of fabric wa';
ferent. di vided into ten equal lengths, which const'-
tuted the 'bloGks'. Each block was then sub-
divided ~nto five sub-blocks or 'plots', and
each of these plots was assigned to a treat-
ment at random. The arrangement is shown dia-
grammatically in Fig. 12.2.
A furthe,' factor had to be taken into account.
It is well-known that different runs of the
same was:ling machine can produce different
shrinkag~s on similar fabrics. £onsequently,
all pieces of fabric from'one block were
washed in the same run of the washing machine,
the blocks being assigned at random to the
runs. By doing this, it was ensured that the
comparisons between treatments were not affec-
12. Analysis of Variance
ted, or confounded, by any differences between An examination of the blcck me'os in the last
runs of the wash1ng machine or by variations column of Table 12.5 will confl, that it was
in inherent shrinkage along the roll of fabric. wise to "rrange the experiment in randomized
blocks. there is a distinct upward trend in
I\rter .,ash ill~, till: dl'C" ,;ilrink"~,,,; ul' a II tc~t th" blu', II1eM'5 from [,lock I to block X, indi-
specimcns \;e,-c calculdted, alill aol,-' 12.0 CJLi/ll] d sysL~lI1utic Vu(-\dtion in inherenl
shows the results. The data recorded are shrinkage from one end of the roll to the .
other. This trend has effectively been removed
from th, comparisons between treatments, .,hose
fIIean va lues are shown in the table. ***
It is neal-lyall-lays advisaok to transform the
data in this way, by using a linear transfor-
mation ('J; 3.5), even .,hen a calculator is
available, so that only reasonably small num-
bers are involved in the calculations.
i
Treatment I Block
Block
Block IA, A, B C 0 totals means
T..
I x'j
II IJ
III 2.2
IV I2 11 3.8
V 6 -5 08 5.2
VI 2 11 I 19 • 5.8
VI I 3 7 -4 8.4
V! I I 13 4 -3 -1 13 I 26 10.0
IX 11.0
X 11 7 0 9 2 29 12.0
I10 11 6 4 11 14.4
42 15.4
16 10 3 8 13 1 50 8.82
I8 19 3 6 1g
55
9 17 10 11 13 60
10 21 7 15 19 72
19 15 7 11 25 77
Treatment 101 117 24 65 134 441
totals T.1. 10.1 11. 7 2.4
I
Treatment
means X.1. 6.5 13.4
Grand total = T= 2:i:T. l. = 2:: T. j = 441
j
Grand mean x- = T/5 x 10 = 8.82
I B 0 A,
A, C
A, B lA, A, B
C 0 A, C 0
'-
a of the form ill u s t r J t ed by Tab 1e 12. 5 c" n and this is re~lect('c1 in the ilssumed model of
the dilta. This is s',,; lar in form to that for
atdenoted generally by Xij, ,Iherc i is the Example 12.1 (Eqp.·' .. ", 12.3) but is extended
to include ilb;',c. effect, i.e., we suppose
e atment numbel' and j the block number. inus that
re is the observati on on the thi rd treatment
where i' is the over-illl mean, Tithe treatment
'J\he fiftl1 block. The general layout of the efl'eet. Ilj the block ('Heet, anel ('ij random
cr ror.
:\a 15 shOI·m in Table 12.G, in IVhich there
, r treatments and b blocks. 8eciluse the
·e ',1I'c clilssified ilC"rclH19 to two main
(1, !l<Jl1lcly, tXt'dtll1cnls imd blocks ill
instance, th"is farlll of expcl'llllcnt is knO\'!rl
tl-JQ-vlay c i a~"::2.f.i.<:~2,~'
useful notation for totals and means is in- The analysis-or-variance table reflects the
oduced in Table 12.G, in which summation is
fact that there are three sources of variation
cJted by dots. Thus the sum of the b ob-
Jtions made on the ith treatment is in a tlVo-way classification, but the calcula-
tions involved are similar in kind to those
for J one-way classification. Once the treat-
ment, block, and grand totals have been found,
the following 'iums of squares are computed.
e dot denoting surnmation over all values of
s~ffix j. Similarly the jth block total is
c 2:2:
i.; 2: " 1J j
i= 1
\lIe dot in the i position indicating surrmation sum of squares of treatment totals
iVef all values of i. The grand total is number of observations in each treatment total
Ireatrnent and block means are given a simil ar sum of squares of block totals
notation. The numerical values of all these
~antities are shovJn in Table 12.5.
llie value of any observation will be affected number of observations in each block total
tv
(grand to_t_a_l}_2 _
total number of observations
;able 12.6 A Two-llay Classi~'ication
Block Treatment c Block Block
----- i tl1tals means
· xc1
x11 x21 Xi 1 · xc2 T'l x'l
x12 x22 xi2 1. x'2
2
x1j x2j x .. x cj T .. X'j
1J J
.' b x1b x2b xib · \b
' 'b x'b
Treatment TL T 2. T. T c. Grand total T
tota 1 s XL x2. 1.
Treatment x. x c. Grand mean x ,.
means 1.
The ANOVAtable is then as shown in Table !.?-.:_l.:.~L~L.fJi f~~~~<:!~._~?~g Treatments
~2. 7. Note that, as before, tile source Surll5 The null hypothesis being tested is that all
treatll",nt means are identical or, Vlhat iJl10unt
of squar'es add up to tire total SUIll of to the Silrlle thing, that all t.,·eat.lllent. effeet.s s
dI'·' ll' I'll, i. I'. >
;quan's, ilnd the same is tfue for the de-
NO\-!, the residual mean square s2 lS an esti-
q,'ees of fn'edom. I he c,'I'o" snlll'ce is oft.en mate of the error' vari ance >J 2, and so is the
bel"lel:n-t,'eallrrl:n ts lIIean square s~ .i ~ Ho ~
CJllvd llil~ 'n""du,li', ',Ir1IT II I{, tht, cOldr] true. However, lf Ho lS not true, tlien it can
!JUL10I1 lo !.lIt' luldl Vdj-ldliolll"1I1Jlflill~J \·/IJCII 'be'~hO\;n that s~, estimates
the contniJutions 01 the rrrain effects have 02 + _b_
e -1
been accounted for'.
\'Ihich is greater than a2if any t itO. The F
For this example, b=10, c=5, and the treatmentJ ratio s~/s2 is therefore expected to be signi-
block and grand totals are shown in Table ficantly large if Ho is not true.
12.5. The necessary sums of squares are:
1012+1172+ ... +1342 46647 4664.7
52 = 10 = -1-0-
112+192+ ... +722+77.2 = 24001 = 4800.2 In this case, the null hypothesis is that all
53 = block effects ar'e zero, i.e.,
55
S. _ 4412 = 194481 = 3889.62
10 x 5 50 and. when Ho is not true, the between-blocks
mean square s~ emmates
The tests of si9nificance are simila, in kind L: 82
to that explained when dealing with Example jJ
12.1, but now we must consider both differ-
ences among treatments and differences among Henr:e s~/s2 will be significantly large if
blocks.
there are real differences among blocks.
Example 12.2 (continued)
The ratios s~/s2 and s2/s2 are shown in the
TaDle i2.7 ANOVATable for a Two-way Fi~al column of Table Q2.8. For differences
Classification amcng treatments, we have sr1s2 = 13.6, with 4
and 36 degrees of freedom. Reference to Table
AS shows that the significance level for this
Source of 5um'Df Degrees of Mean comparison is less than 0.01 (1%), since the
" F tabular value is much smaller than the ob-
variation squares freedom squares sT21 s 2 served ratio 13.6.Consequently, we conclurle
sB21 s 2
8etween , c-l 52 thJt there are real differences among the
tre~tments T treatments. .
';'2 -5,
So far as blocks are concerned, s~/s2 = 7.1 >.. ~
Between 53-5, 2 wi'.h 9 and 36 degrees of freedom. Again the
blocks
b-' sB si£nificance level is less than 0.01. There
were real differences among the blocks, i.e.,
Residual 5,.52-53+5, (c-' )(b-l) S2 th2 trend in inherent shrinkage noted earlier
(error)
di.1 exi st. WhiIe this is interesting, and pro-
vi:es a justification for arranging the expe-
Total 5,-5, bc-l i'irent in randomized blocks, it is of no great
importance so far as the main point of the ex-
Table 12.8 ANOVATable for Example 12.2 perime~t was concerned, which was the detec-
tion of differences among treatments. ***
Source of Sums of Degre es of Mean 12.3.4 Discussion of the Treatment Means in
variation squares freed om squa r.es Example 12.2
Between 775.08 4 Having shown that signif~.cant differences ex-
treatments
910.58 9 isc among the treatments, we myst examine the
Between 511.72 36
blocks 2197.38 49 da~a further to discover just where the dif-
Residual ferences are. It will be recalled that the
Total standard treatment A was applied twice, the
two applications being denoted by A'1 and A2'
It is obviously of interest to test first
wrether the difference between the me?ns of
th2 two applications was significant, i.e., to
examine the comparison xAl - xA2.
This can be tested by the least-significant- where we have replaced 02 by its estim;te s2,
difference 1110tllOd./\ccot"(lirlt~ol Equation
(,2.8), the least significJrlt difference bet- Jnd Co is the value of C observed in the ex-
ween the means of t\·,osamples of size n is periment.
\~'Iwrc s 2 is the rcsidutd IllCdll ~qutlre tllld k is The least significant difference at the 200nX
tileresidual degrees of freedom. ~.!.'is case,
i"om Table 12.8, k = 36 and 0 = ·J14.21 = 3.77. level is therefore
working at the 5% significance level, we put
" _ 0.025, and "Ie find that T"ble M does not ----
give t values for as many as 36 degrees of JL e J
~ freedom. However, It IS apparent from the Zb
table that the required value is about 2, and k _,,"
~hiS is good enough for our purpose *. Hence For the pr"sent example, k = 36, s = 3.77, b =
the least significant diffe"ence is 10, and t3p,0.025=2. The 5% least significant
difference IS therefore
2x3.77x ~ = 3,37 .
Jr-
No", Tab Ie 12.5 g ives x A 1 = 1C .1 and x A2 .11 .7 ,
2 x 3. 77 x 2 x\ 0 = 2.92 ,
50 thatlxA1 - XA21= 1.6. This is smaller than
the least significant difference, and we con- This is larger in absolute valUE than the
clude that the two repeats of the standard least signIficant difference, so we may con-
tceatment did not differ significantly. The clude that treatment B gave a significantly
~ta for both applications can therefore be lower mean shrinkage than the standard treat-
~embined to form a better estimate of the mean ment A.
Of the standard treatment, and thus
Comparisons of treatments C and D with A fol-
1.,\ = '21 (x- A1+xA-2) = '21 (10.1+ 11.7) = 10.9. low the same pattern and have the same least
significant difference. Thus we find the fol-
Treatment A being a standard, one of the main lowing:
purposes of the experiment would be the com-
parison of the other treatments with A, i.e., .Lo_
we should be interested in comparisons of the
type -8.5
Now, in (I; 6.3, Equation (6.7)) ,it was shown We therefore. reach the final conclusion that
that, if treatment D was neither better nor worse than
the standard treatment A. Treatments Band C
0' 1 were sign' ficantly better than A; this is es-
= - +- pecially 'rue of treatment B. ***
b4 The randomized-block experiment considered in
the last section was a particular example of
since each mean like xB' xA1 and xA2 is the the general two-way classification. The data
average of b observations. Hence 100 (1-2a) ",ere cla~,3ified according to 'treatments' and
confidence limits for the comparison C are (by 'blocks', but an analysis of exactly the same
using an argument similar to that of (I; 7.6)) kind would be carried out if 'blocks' were re-
placed by another kind of treatment. This is
C± tk Jvadc) = C ± t s f(ZTb demonstrated by the next example, which al so
O,a 0 k,a illustrates a point made in Section 12.1, that
whenever possible it is best to carry out re-
--------------_._-- peat observations on all treatment combina-
* A more extended set of tables gives the tions in order to obtain a good estimate of
the experimental error.
value t36,0.025= 2.02.
Example ~£:l
This expEriment was carried out to examine the
effect of two 'treatments' on the strength of
sewn joints in parachute webbing. The.two
'treatments' were the length of the overlap of
the two pieces of "Iebbing being jo1'ned (either
short or long) and the stitch density, i.e.,
the numbpr of sewing stitches per cm (low, me-
dium, or high). Note lhat the word 'treiltment' ar" <,iven in the 1,ISt line of Table 12.10
and ;,how Urat the effect of increasing overlap
is belng used in iln extended sense. The is greatest at medium stitch densities,
This dependence of the effect of one factor
'treatments' in this example ilre not applica- on the value of another is called an
inter'dction betVleen Lhe L"IOfacLors and is
tions of SOllie kind of finish, like a shrink- ar'ladcfltl·o·n'a I, and ililportilnt, sour-ce of
r, si st trealilielll, but Me phy ..ical parameters, vdri<'cion in the data, vlhich replication al-
10Vis US to separate from experimental er-rnr.
11"5 usage is .:Gllilian ill the context of experi-
mental design; anolher "lord sometimes used is
factor,
Since there were two levels of length of over-
lap and three levels ofsTllch density, there
were six treat'liJentcombinations altogether.
Two joints were sewn at each of these treat- When data are classified according to tViO
ment combinations and their strengths tested.
The data shown in lable 12.9 are the strengths trecClilents, with n repl ications carried out at
of the joints expressed as a percentage of the each treatment combination, the general laYout
of ti,e data.is as in Table 12.11, in which
mean strength of the webbing itself. In order
to eliminate as far as possible umvanted ther.? are a level s of treatment A and b levels
trends, the lests "Ier-e carried out in a of treatllent D. Any observation is denoted by
random or-der. Xijk, whLre i ("1,2, .. '. ,al represents the
lel/els of treatment A, j ("1,2, ... ,b) the
In an experiment of this kind, it would be levels of treatment S, and k ("1,2, ... ,n)
expected that the strength of the joints would the replications. Various totals and means are
increase Vlith length of overlap and Vlith shawn in the table, the notation being an
stitch density. That this is so can be seen obv'ous extension of that used in the last
sec;.ion.
from Table 12.10, which shows the mean value
for each overlap/density combination. These
averages are plotted in Fig. 12.3, in which Each observation is subject to four sources of
var~ation, namely,
the upward trend as stitch density increases
(il which level of treatment A was used,
is apparent, as is the fact that the results
(W Ivhich level of treatment S was used,
for long overlaps are greater than those for
short overlaps at the same stitch-density
level. However, it is interesting to note that
the size of the increase in strength brought
about by increasi ng the overl ap depends on
which sti tch density is used. These increases
The experimental model is, "erefore ,i
xijk 0 ~ + ai + Bj + Yij + eijk ' .,
Table 12.9 Relative Strengths of Sewn Joints whei"e "i are the effects of treatment A, Bj
~JebblOg are the effects of treatment S, Yij are the
effects of the interaction of treatment Ai
Length of Stitch density Overlap wi1 h treatment Sj and eijk is random error.
overlap Low Medium High totals
Short 32 41 68 Fi~. 12.3 density combination
Long 28 38 71 Melns of each overlap / stitch
- --
60 79 139 278
- --
,37 61 77
42 64 79
-- -
79 125 156 360
-- -
Stitch- 139 204 295 638
density
totals
Length of Stitch density
overlap LOVI Medium High
Short 30.0 39.5 69.5
Long 39.5 62.5 78.0
Mean i nc rease 9.5 23.0 8.5
in strength
due to
inueasing
overl ap
The following sums of squares must be calcu- sum of squares of treatment A totals
I ated. number of observations in each treatment A total
LiLLXjookk'· lJ 2: ri·:/bn .
stm of squares of interaction totals i
number of replications
sum of squares of treatment B totals
2:: 2:: T i//n . number of observations in each treatment B total
ij
Table 12.11 General Two-way Classification
wlth Repllcatlons
'_evels of Levels of treatment A Trt B Trt B
treatment B totals means
2 ..• i ...
rllC IINOVII tJok i, Llll'II d, ,;1101"" ill Idl,I,'
12. 12.
In this example, a ; 3" b ? and n ; 2. The
necessary totals are shown in Table 12.9, and
the ;U1IlSof squaf'es arc as follows.
rll",e IIdV" t Il'IIO\y-fdmillJI'101'111I.t Cdll be 60'+79'+ ... +156' 75364 37682
sllQlm tlldt tile residual medII square is an es- 52
timdte of the error variance 02. Furthermore,
the interaction mean square is an estimate of 22
LII LY" 53 139'+204'+295' 147962 36990'.5
2 x2 4
i j 1J
5, 278'+360' 206884 34480.7
+ --~--_.-.- 3x2 6
(a-1 ) (b-l ) 55; 638' ; 407044 = 33920 3
3x2x2 12 ••
so that, if any interaction effect is non-
zero, the irteraction mean squa,'e v'ould be ex- The cnalysis vI' variance is therefore as in
pected to be significantly large. The ratio Table 12,13.
SAn/s2 therefore tests the significance of the
interaction,
If it is concluded that non-zero interaction
effects exist, this implies that the effccL,
of tr"eatment II depend on tile level of treat-
ment B. Thus each level must be examined sepa-
rately, and the ratios sA/52 and s~/s2 shol'lO
in Table 12.12 vlould not be calculilted.
However, if the interaction effect is not sig-
nificant, it is legitimate to test the signi-
ficance of the A and B treatments by comparing
the A and B mean squares with s2, as shown in
Table 12.12.
Tab le 12. 12 ANOVA Table for Two-way The F ratio for the interaction between stitch
tlassification with Replicatiolls density and length of overlap was 10.9, VIHh 2
and 6 degrees of freedom. Comparison of tris
Source of Sums of Degl'ees of Mean F value with Table A5 shows that the interaction
Variation squares is significant, the level of significance
freedom squares beinc about 0.01 (1%). We are thus confidEnt
Treatment A S3-S5 in asserting that the effect of increasin,
a-1 s' s'A/s' overlap is dependent on which level of stitch
Treatment B S,-S5 A density is used, and there is therefore no
, point in evaluating the F ratios for treat-
Interaction S2-S3-5,+55 b-1 sB s~/s' ments A and B in Table 12.13.
(A x B)
(a -1) (b-1) sAB 5A~/s' To investigate the interaction further, we re-
Residual turn to Table 12.10 and examine the mean in-
51-5, ab(n-l ) s' creases in strength due to changing overlap,
I Total 51-55 abn-1 shown in the last row of the table. The least
significant difference between two means is
given by Equation (12.8), i.e.,
least significant difference = tk,as ~ '
where s2 is the residual mean square and k the
associated de9rees of freedom. From Table
12.13, lye find s = 116.0O; 2.45 and k; 6.
Working at the 5% level of significance,
Table A4 gives
Source of 5ums of Degrees of Mean F least significant difference between means of
variation squa res freedom squares
- samples of size 2 6.00
Stitch 3070.2 2 1535.1 -
densities 560.4 1 560.4 If= 2.45 x 2.45 x
131.1 2 65.55 10.9
Overl aps 36.0 6 All tile average increases in the last row of
11 6.00 Table 12.10 are greater than this and are
Densities x 3797.7 there;ore significant. Thus, whatever level of
Overlap stite:) density is used, it is worthwhi le
incredsing the length of the overlap, but the
Residual effect of doing so is greatest at medium
overl ups. ***
Total
In the last example, we studied the effects of Example 12.4
two factors, or treatments, at all combina-
tions of the levels of the factors, and exper- 'A 24 exper iment ~Ias performed to " .;.tigate
iments of this kind are often called complete the effects of varying four knit ;',g-machine
parameters on the dimensions of ,111 to-di -Roma
factorial designs. The same general technique fabrics knitted from polyester-f bre yarn. The
can be used to design and analyse exper'iments factors WEre
in which three or more factors are examined
simultaneously, However, there is a special 0: needle-timing delay,
class cf complete factorial designs in which
each of n factors is studied at t~1O levels on- and they ~ere all applied at two levels, which
ly. These are called 2n factoriaTO'esigns, and we shall call 'high' and' low'. After knit-
ting, the fabrics were allowed to relax and
they are often very useful in the ear,y stages their dimelsions ~Iere measured. Table 12.14
of an investigation when it is necessary to shows the lengths of fabric knitted by 100
revo1utiors of the knitting machine. The ac-
discover wh ich are the most important of a tual measurements have been transformed to
number of factors that could affect the re- make the arithmetic easier.
sponse variable. This enables unimportant
factors to be omitt~d from later, more de- There is a quick method, known as Yates's
tailed, investigations, The designs have also algorithm, for analysing data in a 2n facto-
found application in the discovery of the op- rial experiment. The method relies on the re-
sults of the experiment being written down in
timum conditions for running a manufacturing a standard order, which is shown in Table
plant. 12.15. In this table, the high and low levels
of each factor are denoted by + and -, respec-
Revolutions of a Knitting Machine tively. The first four columns of the table
(Actual Length in cm- 26) x 10 are headed by the letters denoting the fac-
tors. In the A column, - and + signs aternate;
I 0- A- A+ under B, pairs of - and + signs alternate; in
B- B+ B - B+ the C co1u~ four + signs alternate with
C- 20 22 four - signs; ana-;-in the D columns, eight -
a 30 signs are followed by eight + signs. ~
0+ 9 28 system can obviously by extended if more than
2 35 four factors are involved, and, in general,
C+ D- 11 19 the kth column has 2k-1 minus signs
16 25 13 17 alternating with 2k-1 plus signs.
0+
23 24
Row A B C O' (x) (1) (2) (3) Factor Factor sum
number (4) effect of squares Fac tor
1 ---- 0 20 72 143 294 18.375 - mean
2 A
3 +--- 20 52 71 151 -16 -2.00 16.00 B
4 -+- - 702.25 AB
5 30 27 74 1 106 13.25 100.00 C
6 ++--
7 22 44 77 -17 -40 -5.00 0.25 AC
8 --+- 100.00 BC
9 16 11 12 49 2 0.25 240.25 ABC
10 +-+- 121.00
11 11 63 -11 57 -40 -5.00 0
12 -++- 4.00 AD
13 25 36 0 -29 -62 -7.75 20.25 BD
14 +++- 4.00 ABD
15 19 41 -17 -11 44 5.50 20.25 CD
16 ---+ ACD
2 20 32 - 1 8 1.00 1.00
+--+ , BCD
9 -8 17 3 -18 -2.25 2.25
-+ -+ 64.00 ABCD
35 -5 52 -23 B 1.00 6.25
++-+
28 -6 5 -17 18 2.25
--++
23 7 -28 -15 4 0.50
+- ++
-+++ 13 -7 - 1 -47 6 0.75
4.00
++++ 24 -10 -14 27 _ 32
17 -7 3 17 -10 -1.25
~[----
12. Analysis of Variance Th~ final column identifies the factor cOmbi-
natiuns correspunding to the sums of squares
The combinations of signs ill each row define a just calculated. They are found by.writin1
treatment combination. TIlus rOli number 1 (----) down the factor letters correspondlng to the +
is the observation obtained by using all factors signs in -the first columns of the Table. From
at the 10vi ] eve 1. ROli number 4 (++ --) is the this C01UIIIII, it vii11 be seen that all the
ubservat ion obtained by usi ng A and U at lIle nlai,1factor'S (l1,e,C,OI uppea,', together'vlith
high level, C and 0 ilt the 101'1 icvl:l; ilnd so their interaction:; (Ae,AC,eO,etc.l. Higher-
on. The values curresponding to these order interactions, like ACO, ABC, also "ppear
combinations are then copied frum the but these are often difficult to interpret ard'
original Table 12.14 and entered in the column will be used to provide an estimate of experi-
headed (x). This produces the standard order mental error.
necessary for the application of Yates's algo-
r ithln. From the information of Table 12.15., an ANOVA
tab:e can be drawn up,-and this is shown in
12.5.2 Yates's Algorithm table 12.16. The sums of squares in this table
are taken from Table 12.15, and, since each
The calculations of Yates's algorithm occupy factor is at two levels. all the main effects
the remaining columns of Table 12.15. The ob- and interactions have one degree of freedom
served values (x) are considered in pairs. The each.
first eight entries in the column headed (1)
are obtained by adding the pairs; thus 20 = 0 Sir~e the experiment was not replicated, there
+ 20, 52 = 30 + 22, 27 = 16 + 11, etc. The is ~o unbiassed estimate of the experimental
last eight entries in (1) are obtained by sub- error. HOIiever, it is usually assumed that the
tracting the top member of each pair from the three- and four-factor interactions are of no
bottom member; thus 20 = 20 - 0, -8 = 22 - 30, practical significance, and they are combined
-5 = 11 - 16, etc. Just as the entries in (1) in the I'layshovlllin Table 12.16 to provide a
are obtained from (x), the entries in (2) are 'residual' or error mean square, equal to
obtained from those in (1) in the same vJay, 42.75 in this case. All the other mean squares
those in (3) from those in (2), and so on. are divided by this to provide F values, used
This process proceeds until n columns headed to test the significance of the main effects
(1),(2), ... , (n) have been generated, vlhere n and two-factor interactions, by comparing them
is the number of factors in the experiment. with the corresponding entries in Table A5.
Since n = 4 in Example 12.4, the process stops From that table we find that
at (4).
we therefore conclude that the B factor, i.e.,
The entries in the 'Factor ef,ect' ~olumn are take-down tension, is significant at the 1%
found next, by dividing the first entry in (n) level, and that the BC interaction is approa-
by 2n and the remaining entries by 2"-1. In ching significance at the 5% level. None of
Example 12.4, since n = 4, these divisors are the other sources of variation is significant.
16 and 8, respectively. Thus we get in the
'Factor effect' column 1B.375 = 294/16, -2.00
= -16/8, 13.25 = 106/8, etc. The factor sums
of squares in the penultimate column are found
by squaring the 2ntries in (n) and dividing by
2n, though no sum of squares is attributable
to the first row, which represents the over-
all mean of the data. Thus we find 16 = 162/16
702.25 = 1062/16, 100 = (-40)2/16, etc.
Source Sums of Degrees or Mean F
squa res freedom squares
A 0.374
B 16.00 1 16.00 16.4
C 702.25 1 702.25 0.006
0 1 0.094
AB 0.25 1 0.25 2.34
AC 4.00 1 4.00 2.34
AD 100.00 1 100.UO· 0.474
BC 100.00 1 100.00 5.61
BO 20.25 1 20.25 0.094
CO 240.25 1 240.25 0 .0.23
ABC 4.00 1 4.00
ABO 1.00 1. 00
ACO 1}5
BCD 11LOG} 42.75
ABCD 15
-- 20.25
2.25 213.75
Total
64.00
6.25
1401.75
I
It is instructive to examine the BC interac- NoL ,111 C'xperimcn ts Dre Cdrri ed out to compare
tion, which is nearly si91\ificilnt. Tilhle 12.17
Llle effectiveness of several l''''0]sof treat-
c,lIows t1H' me,11\ ,; or tll<' flltll' C'xl"','illh'lIl.dl 1',·- ments or factors. Sometimes it . known that a
'iulls ;It (',1cll comh in"t ion or I,'wi'-; of II ,Inri C. response variable is subject to several random
sources of variation, and the object of expe-
From this table, we see that at the low level
of C, changing the B factor from low level to rimentation is then to estimate the magnitude
high increases the fabric length from 7.75 to of the cuntribution of each source to the to-
28.75, an increase of magnitude 21.00. On the tal variation. Since variation is usually mea-
other hand, ClliJngingB f"om low level to high, sured by variances, this amounts to estimating
"lith C at 'its high level, yields an increase the variince of each individual source.
in length frol1l15.75 to 21.25, i.e., of magni-
tude 5.50. The difference in size of these in- E,amp l~g:-:i
creases, 21.00 and 5.50, is responsible for
the large BC sum of squares and its near-sig- flhen del :veries of yarn are received from a
nificance. Table 12.17 also shov/s the mean spinner, it is often prudent to carry out
values of all observations at the low and high tests to jetermine whether or not the average
levels of B. The difference between these,
25.00 - 11. 75 = 13.25, is equal to the B 'Fac- count is correct. Each delivery consists of
tor effect' in row 3 of Table 12.15, Similar- several cases of yarn, each containing a num-
ly, the difference in the C means shown in ber of cones. To perform a single count test,
Table 12.17 is the C 'Factor effect' in row 5 one therefore has to choose a cone at random
of Table 12.15. *** from a case, which is itself chosen at randoiJ
from the jelivery; ind the delivrry is one 0,'
One ,disadvantage of 2n,factorial designs is
that, as n becomes larger, the number of ~ series of similar deliveries sent out by the
factor combinations grows rapidly. However, it spinner. The value obtained in a count test is
is often possible to reduce the amount of ex- therefore subject to variation
perimentation needed by choosing a selection
of these combinations in a way that confounds, (i) between deli~eries,
or mixes up, the effects of high-order inter- ( ii) bet\·/eencases with in a de 1ivery, and
actions such as ABC, ABCD, etc., which are not (iii) between cones within a case,
of great interest, while still enabling the
experimenter to estimate the main effects' and and, to determine the error associated with
two-factor interactions. Such designs are
called fractional replications of the complete testing a delivery, it is necessary to know
2n factor1aI. Deta11s of such designs may be
found in, for example, 'Design and Analysis of the variance attributable to each of these
lndust~ial Ex~e~iments', edited by O.L. Davies
(01 iver & Boyd, London)" sources of variation.
An experiment was therefore carried out in
which three deliveries were chosen at random,
four cases were randomly selected from each
delivery, and five random cones were selected
from each case. A standard linear-density test
was then made on each cone. The results of the
experiment are given in Table' 12.18. The data
are to be used eventually to design a routine-
testi ng pl'ocedure.' ***
Table 12.17 The BC Interaction B B C means
+
C- 7.75 28.75 1&.25
C+ 15.75 21.25 1&.50
i B means 11. 75 25.00 18.375
T'bl'
11.18 Linear-density Tests on Three
,1 Yarn Del1venes
Delivery
·,i Case 12 4 3
Cone; 1 2 3 41 1 2 3 1234
i 22
1 08 2 5 8 16 1 5 20 12 20 8
2 9 5 5 16 1 12 5 8 12 1 8 0
3 1 12 12 8 8 5 16 20 8 12 8
4 g 8 1 12 0 16 0 24 28 16 4 12
S 5 155 5 23 5 5 12 5 12 1
Case
total s 24 34 25 46 22 "2 27 64 92 42 ,. 56 29
Delivery 129 185 219
totals
533
Grand
total
t: --_--~~ ____ --'l~('-- -
j
·_---:-:::---1;;2-;"':-A:n:a-:liy~s,i.s._:_o:f:;:_Va"ri'ii"a:n::c"e::::::--'--
Th, final column of this table is differe1t
The data of Tab~e 12.18 are all example of il from that In prevIous ANOVAs. Since we are'
ter2sted in estimating the variances due to1n-
<jeneral eXJl,~nllit~nl in \'I!I'ich d delivl·ries Cll' eddl source of volriolLion, rillher than testin(
hl)lchl'~ lire lIlVO!VPO, C Ld~;l'~ tlfe '>d1llpll'Ll fn;1Il 1'01 slgrnllc"nce, Lhe last column noVi shovls ~
cacll uelivel'y, ancln C011es frUIII edch Cd,e dr'e th,' cOlllbl~atlons at these variances that euch
sele:tecl for teeting. J,t first sight, the data mean square IS expected to estImate. The
appear to be In the form of a complete thl"ee- notation used is as follows;
factor factona1 desIgn, the f'actors beIng
deliveries, cases, and cones. Out a little 0' ; variance due to random variation
between cones ;
thought vii 11 shovi that this is not so, for
ca:e no. 1 in the first de-livery has no con-
nexion at all with case no. 1 in delivery 2,
because cases are selected at random from the variance due to random variation
betvleen cases ;
deliveries. The same is tl"ueo~cones,
since they are selected at random from the
cases. The experif'lental plan is, in fact, an
examnle of a hierarchical cl assification, variance due to random variation
between deliveries
which nlay be llTtIstrated as in FIg. 12.4.
Such data can be analysed by the ANOVAtech-
nique, and the following sums of squares are
needed.
51 ; sum of squares of in !ividual observations
sum of squares of case totals
n
sum of squares of delivery totals
cn
5 _ (grand total)2
, - dcn
The ANOVAis then as in Table 12.19.
Table 12.19 ANOVATable for Hierarchical
C lasslflcatlon WIth Ihree Sources
of Vari at ion
Source of Sums of Degrees of Mean Expected va 11'~S
variation of mean square.
squa res freedom squares
Between 02+no~+cnod'
deliveries 5,-5, d-1 I~d
02+no2
Between cases 52-5, d(c-l ) t~c c
within deliveries 51-52 cd(n-l ) M
02
Between cones 0
within cases
51-5, cdn-1
Tota Is
and therefore s' 33.6
o~ is estimated by (Hc-Ho)/n
Similarly, we rind tho t (12.13) "s'c = 99.2-33.6 = 13.1
(12.14 ) 103.3-99.2
0d is estimated by (~ld-Hc)/clI s'd 20 0.205
*,~*
~ample 12.5 (continued)
The necessary totals are ~hown in Table 12.18,
and we find the followin9 sums of squares
(noting that in this exarrple n = 5, c = 4,
d = 3).
24'+34'+ ... +56'+29' 12.6.2 Standard Error of a Delivery Hean
5
\,hen a de],very is received, the question
129'+185'+219' 98827 _ arises a~ to how many tests should be made to
4x 5 ~~ - 4941.35 . provide a satisfactory estimate of the mean of
the delivery. The precision of the estimate is
533' 284089 = 4734 82 measured by its standard error, and this will
S, = 3 x 4 x 5 60 . be determined by two sources of variation.
1\ To see why this is so, ima9ine a sin9le test
being carr'ed out. First, a case has to be
I chosen at random from the delivery, and then a
cone is selected from the chosen case. Suppose
Md 103.3 the cases ~re numbered, and that case i is
chosen. Imagine al so that the cones within
i Hen:e, by uSlng s2, s6, and s2d to denote each case are numbered and that the cone la-
1 sample estimates of02, 06 and 02d, respecti- belled ij (i.e., the jth cone in the ith casel
ively, the results (12.121, (12.131, and is selected for test. The test result can then
be represented by the model
1'".14) gi"
where ~d is the true mean of the delivery, ci
is the deviation of the ith case from this
mean, and eij is the deviation of "he jth cone
from the mean of the ith case. Then the devia-
2tions ci and eij are random variables, with
variances 0 and 02,respectively.,
1 12.20 p -,
ANOVA T.bl.,-fo_rE_x_am_ _l1_2e_._5
1 "" Sums of Degrees of Mean Expected
!I Source of squares freedom squares values
variation
9 103.3
i Between 0'+50'c+200' d
99.2
l deliveries
~
r Between cases 892.85 0'+50'c
within deliveries
Between cones 1612.80 48 33.6 0'
2712.18 59
~hi" "''' "
Tota 1s
,'---- -.J~c= _
12. Analysis of Variance
Now, suppose that c cases are chosen and n ~c· (0 131 + 0.3n36).
cones selected from each case, giving a total
of cn tests. The~c can be averaged to provide In thc eiperimcnt, fIve tests were carrIed OUt
an unbiJssed estimdLe of thc mea/} of the de- on each of four cases selected at random from
a ci"livery. Thus c ~ 4, n ~ 5, and Equation
livcr y. 1his meJI' \ViII be (12.17) gives
\Vhere the dots indicate that averaging has var-{x..)" ~ (0.131 + 0.~36) 0.0496.
taken p Iace over the val ucs of Lhe suffi x they
replace, as in T,lble 12.2, fOI" example. The Helice the standard error of the over-all mean
precision of this estimate is its standard er- of the 20 tests is
ror, i.e., the square root of its variance.
Since Equation (12.15) is a linear combination se(x ..) ~ Vvarlx ..J
of random 'ariables, the result of (I; 6.3, By using this value, 95% confidence limits
Equation (6.91)sho\Vs that CGJld be fcund for the mean of the delivery
anj would be approximately
var(x .. J ~ var(~.) + var(~ .. )
It is instluctive to consider other combina-
However, E. is the mean of c case deviations tions of nand c. For example, suppose it was
de:ided to do a total 'of 12 tests on a deli-
and therefore has vari ance 02/c, whi 1e e .. is very. There are then several combinations of n
and c such that nc ~ 12, and these are shown
the mean of cn cone deviatio~s having variance in Table 12.21 together with the corresponding
o 2/ cn. standard error of the mean of twelve tests,
calculated from Equation (12.17). It can be
Hence \'Iehave seen how the standard error decreases as the
number of cases sampled increases.
fa,o~ + ~ ~ ~ + ~) In fact, when c ~ 6, n = 2, the standard error
i~ the same as that obtained previously when
C cn c \ c n c'= 4 and n = 5, i.e., the same precis' on has
been obtained for less testing.
The values of c and n can thus be chosen, in
any appl ication, to make var(x .. ) as smal 1 as 12.6.3 The Economics of Routine Testing
vie please. It is worth noting, hovlever, that l1e conclusion that the precision of an esti-
increasing n alone reduces only the second mate of the mean of a del ivery depends on hal'l
term of Equation(12.16); however large we make r.any tests are performed and on how the test
n, var (x .. ) wi 11 never reduce be low 0 2/c. On specimens are chosen raises-the question of
the other hand, increasing c will reduce both hhether there is a most economical way of SE-
terms of Equation (12.16) and will eventuaTTY lecting the test specimens in order to give a
make var (x.. J vel"y small indeed. Thus, from r'equired precision. In general, there are two
the point of view of increasing the precision CJsts involved, namely, the cost of choosing a
of the estimate, it is best to select a few ~ample (selecting and opening cases) and the
cones frum a large number of cases, rather cost of actually carrying out the tests.
than many cones from a few cases. ~uppose S is the cost of,.selecting and opening
ij case and T is the 'cost of choosi ng a cnne
In this example, we have found that estimates from the case and testing it. The total cost
of 02 and 02 are s2 = 33.6 and s2 ~ 13.1. It ~f c cases are chosen and n cones selected dnd
tested from each is
must be rem~mbered, however, thar these are
variances of transformed variables, the trans-
formation used being that shown in Table
12.18. This indicates that all results were
multiplied by 10 before the calculations were
carried out. Hence the variance estimates must
be di vided by 102 = 100 to get them back to
the original tex units. Thus, in original
units, s~ ~ 0.131 and s2 = 0.336, so that
Equation (.2.161 is
Table 12.21 Standard Errors of Means of 12
Tests
c n se(x .. 1
1 12 0.399
2 6 0.306
3 4 0.268
4 3 0.246
6 2 0.223
13. linear Regression
One of the 1II0stinteresting tilsks in scientific Tallle 13.? sh0\4s the rpsults of area-shrinka'1c
<llldlechlloILl1lic,ij1'e',I',It'dclhltldeveloplllelili:,
the investigation of relations between two or lests 011 pieces of the salliefabric that had been
more variables, Consider the folloWing examples, treated with varying strengths of a sllrink re-
si st finish. Tllis time, the de , show that the
In the manufacture of tyre cord, the processing shrinkage decreases as the strength of the
tension has an effect on the properties of the treatment goes up, as 'lOuld be expected, but the
cord, In one investigation, cords were produced graph of Fig. 13.2(a) shows that the trend is not
at a number of different tensions, and their linear. The rate of decrease of shrinkage tends
modulus of elasticity was measured, with the
resul ts shown in Table 13,1. It is immediately to fall as the strength increases. However, if the
apparent from a brief examination of the data logarithm of the shrinkage is plotted against
that the modulus tends to increase as the tension the strength of treatment, as in Fig. 13.2(b),
increases, and this tendency is even more obvious the result is a linear trend downwards. ***
when the data are plotted on a graph as in Fig.
13.1. In particular, it is seen that, if a A feature of the two experiments described above
single smooth curve were to be used to repre-
sent the relation between the variables, a was that in each case the variables involved
straight line would be most appropriate. Such a
relatlonship is called linear. *** were of two types. One variable was under the
control of the experimenter; the values of pro-
cessing tension in Example 13.1 and of strength
of shrink-resist finish in Example 13.2 were
deliberately chosen by the experimenters, Such
controlled va~s are called independent
variables. The other variable in each exaniPle is
Table 13.1 Processing Tension and Modulus of
Tyre Cords
Processing
tension
,J t )~[ _
j'
j
J re~p""s,~ lu llie vdllle 01 'lie illllcpendenl VJri- 't IS of considel'"ble 'illterest to find the E '
J~rC '",id 'j, 1Ilerelun' cdlle,1 i1 dependent v.lri- 11011 of the curvl; or llne tllJt best fits theqUe
iJ~le. rlius tlie eliJstic 1II0duIJsi'n'rXai'lpTe 13.1
uepenJs on the pr()cess'"~ l,,"sion used. vlhi le "Jtd so tlwt HI luture thc lIIean value of the
tlie slil'inkd~e 111 [Xdlllple 13.2 depends on Vtt,Jt
slrcngLiI of f,,"sli \'IJS Jppllcd to the rJbric. oependcnt vanJble can be I!.~:edlcted I·then the
IStrength of v due of the lndepende~t vilrlub"'TeTs knO'tn. if
I S-R finish IS the depend'cnt vJna"le and the indeper.den~ y
, V<Jr'lab Ie' s X. thl s cune 1s ca 11cd the regre',
% Area shrinka,e
sIan of y on x. ~~,
% area
~hrinkagp.
In (% area S, -"ngth of
shrinkage) tredtment
+ Strength of
treatment
In ~eneral, the dependent variable is subject to .
random errors, and the method of fitting a curve
to the data must take account of this. The meth- These deviations measure the 'closeness' of the
od usually adopted is called the met.hod of ](~ast line to ':hc ;",,,ts and in the method of least
squares. It.is i1PP""('It,. fl\liltl.h~ CW'(;"i~'Xdiil'ljTes-'--
'giv'cnabove thilt anilllpO(t.ililcllass of rei a- squares a an,' !l are chosen so that the sum of
tionships is the linear one, vlhich has an equa-
tion of the type squares of ",..ei v"lues is a minimum, ~we
iiii riTiJiiZ c
dnd we therefore begin by applying the method
of least squares to the problem of fitting such n n
equations to experimental data.
S = L e' L (yi-a-bxi)'·
13.2 Fitting a Straight Line
i= 1 1 i= 1
The coefficients a and b in Equation (13.1) can
be thought of as the 'population' parameters of This s~m is a function of a and b and is mini-
the 'true' re~ression of y on x. The task of mized by putting
fitting a line to sample data involves making
estimates of a and B; these estimates we shall a"Sa=as<IS
denote by a and b, respectively. Thus the equa-
tion of the line actually fitted is Differentiating S partially with respect to a,
vie fi nd :h at
Suppose the data are in the form of n pairs of
,bservations (Xl, Yl), (X2, Y2); ... , (Xi, Yi) <IS = L 2 (y. - a - bx . ) ( -1 )
... , (xn' Y n), which can be plotted on a graph "a
as in Fig. 13.3. The fitted straight line is oil 1
also drawn on the graph, and, because of the
random elements in the Y values, it wi 11 not Putting this equal to zero and cancelling the
pass through all the plotted points. Obviously, common factor -2 gives
the closer the points lie to the line, the bet-
ter is the line's fit to the data and the more Ey - na - :'EX = D
reliable it is likely to be for prediction pur-
poses. We must therefore choose a and b so that S = ~ {(y i - y) - b(xi - x)}' •
the line passes as close to the points as pos- 1
sible. Now, the predicted mean value of y
corresponding to any X value, say, X = xi, is L2{(y.-Y)-b(x.-x)}(-(X.-x)} = O.
given by the equation of the line, and 1S i1
1 1
Y i = a + bx i .
wh ich 1 eads to
The difference between this and'the value of
y(= Yi) actually observed when X = xi is b = E(X-X)(y-y)
E(X-X)'
e i = Y i - Y i = Y i - (a + bx i) .
B
~
t C(Xi. Yi)
• t ei
----e E(xi. Yi)
I
I
I
I
I
.r- L
1
Equations (13.4) and (13.5) are the least- of the regression line is to provide an estimate
squares estimates of a and B. Substituting for of •.he mean y value at a given value of x. The
a in the equation of the line, Equation (13.2), mean of the above distribution is thus given by
,;e get the equation of the regression line when x • Xi,
1.('., 1t 1S at
whicn shows that the fitted line passes through
the mean point 0, whose co-ordinates are (x, 2). y = a + BX i .
For the coo7utation of b, it is best to make use
of the formul ae ,Ie ;hall also suppose that the distribution of y
valJes lS normal, "l1th thlS mean and variance 0'
The calculation of a and b for this example is whlse value is the.same for all values of x. The'
shown in Table 13.3. The steps in the calcula- ass~med situation lS as shown in Fig. 13.4.
tion should be self-explanatory and yield
a = 27.18, b = 0.5982. On rounding these off, Tab le 13.3 Calculation of a and b for
the Equation of the regression line is estimated Example 13.1
as
, ,
Suppose that several measurements are made on
the dependent variable y at the same value of x y x xy Y
x (= xi, say). It has already been pointed out
that the y values are subject to random fluctu- 5 30.1 25 150.5 906.01
ations. Consequently, the repeated y measure-
~2nts will' vary and will therefore have a dis- 7 32.1 49 224.7 1030.4'\
tribution. It wi 11 be recalled that the purpose
9 32.2 81 289.8 1036.84
10 32.9 100 329.0 1082.41
11 32.9 121 361. 9 1082.4;
12 34.7 144 416.4 1204.0'1
13 35.5 169 461.5 1260.2:;
Tot] 1s 67 230.4 689 2233.8 7602.4('
x Lx/n 67/7 = 9.571
y Lyin
230.4/7 = 32.91
7602.42-(230.4)2/7 18.9686
~(x-x)(y-y) rxy-(rx)(ry)/n
2233.8-(67)(230.4)/7 28.i429
b r(x-x)(y-y)/r(x-x)2
28.5429/47.7143 = 0.5982
"'distribution of
y Wh2n x = Xi
The variance 0' measures the'variability of the dence limi:s for a and B, just as confidence li-
measured data about the regression line and mits for the mean and other population parame-
hence is a measure of how well the Ilne fits the ters were found in Chapter 7 of Part I.
data. The smaller 0' is, the less variation
there is about the line ~nd thereforc the closcr { }a~t ;' '
is its fit to ti,e c1~tJ. It is LllUS illlportilnto s I x'
hJve an estimate of the vJlue of 0'. It can be
shown that this is provided by -+---
k,y n [(x-x)'
where Smin is the minimum value of the sum of 1
squ~res of deviJtions, defined in Equation
113.3J. This minimum is achieved \'Ihena and b b ~ tk s/l[(x-x)'j' ,
~re calculated by using Equations (13.4) and ,y
(13.5) and is therefore
where k = ~-2 and s is the standard deviation
about the regression line.
fl, S· = 2.: (y . - a - bx. )'
mln ill
2.:{(y. - y)-b(x. - x)}' To find 95% confidence limits, we must put
ill 0'= 0.025 ln the above formulae, and, with
k = n-2 = 5, we find that t5,O.025= 2.57, from
2.:(y.-y)'-2b2.: (x.-x)(y.-y)+b'2:{x-x)' . Table A4. Hence, since
.i 1 ill i1
we find that 95% confidence limits for d.are, by
Substituting for b from Equation (13.5), we find using Equation (13.12),
:hat
1 9.571'
(2.i:ll(x.-x)(y-Y)) "-
-+---
2.: (x-x)'
7 47.7143
i1
All the sums of squares and products on the 0.5982 ~ 2.57 x 0.6151 J47.7143
right-hand side of this equation have already
been found while computing a and b in Table
13.3, so tha~ the calculation of s2 is.a simple
matter.
I.
]' Example 13.1 (continued)
.1.} In Table 13.3, we found that
It will be remembered that a principal use for
the regression equation is to predict future
values of y when a value of x is known. These
predicted vclues of yare subject to error be-
cause of che uncertainties in the estimates of a
and B, and confidence limits can also be found
for them.
. Suppose the known value of x is x = xf, then the
regression Equation provides an estimate of the
15ml.n mean y value at this value of x. equal to
s = JO.3788 = 0.615 y = a + bXf .
The values of a and b calculated from Equations The 100(1-22)% confidence limits for the 'true'
(13.4) and (13.5) are sample estimates of the mean value of yare then given by
I population values a and B and are therefore
, likely to vary from one sample to another. It is 1 -X)2}'(Xf 1
thus necessary to be able to calculate confi- {a + bXf ± tk ,y s -n + -l(: x-x_)'
•
In many applications, it may be necessary to
calculate an interval within which future indi-
vidual values of y may I,e expected to lie.~
can be shown that a future y value will lie
within the l~mits
(x -XF}; ,.
a + bX ~t '; 1 + l + _f__
f k ,) { n [(x-x)'
~ )~L _
'13. Linear Regression
~!_,!:I~.~_.~L.l(,C"-"-lll-"~,d) onc .,lcLhod of doing lilis is by meiJns of lhe
He shill I c.clcul<JLe t:sL11l1JLes of the modulus Vlhen an<Jlysis of vuri<1l1cc. rhe technique is besed on
rtile pI'oces~ing Lellsion 15 o. Putting xf ; 8 and a rL ,ult thot can be foullo by first considering
; 0.025 in EquJlion (13,14) gives the follovling the idenlity ,
95'.:, confidcnc~ lill1il~; for' lile lliean modulus of
tyre cord pl'oducecl Jt. lIlJt t.ensi'Oil: •
y-~ = (y ·-9) + (9,-Y)
l' 1 1
{.!.27.2 + 0.6 x 8' 2.57 x 0.615 x + (8-9.571£.}; where Yi ; a + bXi is the value of y calculated
from the regression line >!hen x ; xi· By squar·
7 47.7143 ing and adding such expressions for all values
of i, it can be shOVin that
2: (y,-9·)2+ 2:(9·-Y)2
i '1 i1
Again, Equation (13.15) gives an interval \vithin
Vlhich individual modulus tests can be expected after noting that the first term on the right-
to lie. We find, \'Ihen Xf; 8,
{ }'1 (8-9.571 )2 2 hand side is the SUIliof squared deviations that
1+7'+ 47.7143
,;as minimizec in calculating a and b. This is
27.2+0.6.x8~2.57xO.615x often referred to as the sum of squares about
the regression. The second term on the rTgllf='
nan'f slde of Equation (13.16) is the sum of
squares of the vertical deviations of the points
on the line (Xi, Yi) from their mean (x, y).
Th,s term is called the sum of squares due to
~ession or ~lained by regression. Tfiel€ft- I·
hand5'i<leof EquatlOn (13.16) is the sum of ~
squares of the deviations of the y values from t
Thus, provided that other processing conditions their mean, usually called the total sum of . jt. j
do not change, future tyre cords made by uSlng a
processing tension of 8 can be expected to have sql03res.
a mean modulus lying betVleen 31.3 and 32.7,
vlhile individual test pieces will have moduli total sum} {sum of squares j
={ of squares
ranging between 30.3 and 33.7. *** about regression }J} {sum of squares
+ due to regress ion
13.5 Analysis of Variance of Regression
and this forms the basis of the ANOVAshown
8efore a regression line, calculated as de- . in Table 13.4. This table shows that the mean
scribed above, is used to predict y values, lt square about regression, Mo; Smin/(n - 2), is
must be verified that a real regression exists, the estimate s20f the variance 02 about the
i.e., th"t the y values reall,) do depend on x. recression, Equation (13.10). It can also be shO~ln
We have seen that the regression line can be
written in the form of Equation (13.:), i.e., thet the mean square due to regression, Mr, is
an estimate of
y = ~ Y + B(x-~ x )
If the null hypothesis is true, i.e., 6; 0,
and y wi 11 be dependent on x only if 6 f O. Thus this expression reduces to 02, and then Mo and f'lr
we must devise a test 0,' the null nypothesis: arEo both estimates of 0.2 and should not be sig-
ni~icantly diffe-ent.
On the other hand, if 6"0, then t~r would be
expected to be significantly greater than Mo'.
and this can be tested by calculating the ratlo
Source of Sums 'Jf Oegrees of Mean F F0 ; M/Mo
va ri a t i on squa r·2S
freedom squa res and referring the result to --able AS wi:h 1 and
Oue to (n - 2) degrees of freedom. A test of t le null
regression 2: (9i-y)2 1 t~r M/Mo hl Jothesis is thus made available.
1 n-2 M
About n-1 The calculation of the ANoVAtable is simplified
regression 2: (Yi-9i)2 = Smin 0 by comparing Equations (13.11) and (13.16). From
this comparison, vie see th'Dt
Total 1
2: (Yi-y)2 ;( 2: (x-x)(Y'-Y~'/2: (x.-x)'
ill i1
1
The total sum of ~qual'es, L (Yi-Y)', is al- is the proportion of the total variation ex-
pl <lineelby the regression anel is therefore a
I mCdsure or how well Lhe line fits the elata. The
r,ltio, dC''loledby "2, is cal led the coefficient
reaely known; hcnce Smin C;\ll be round by dif- of detemination <lnd its definition sh-owsL1i5f'
ference. iT,;" 'v',il'it,ji'ue''sYTiebel.wec·nzero and one. If this
value is 'era, this implies that
2: (y·-Y)' = 0
i1
"'hich ca,. only be true if Yi = y for all values
of i, i.e., the slope of the regression line
must be zero. On the other hand, if r2 = 1, then
Smin=O,i.e.,
and this Equation can be satisfied only if
Yi = Yi hr all i, i .e, if the data pairs
(Xi, Yi) ;ie exactly on the regression line.
If we substitute for L CYi _y)2 from Equation
(13.17) iq the Equation for r2, we find that
=r' (E(X-X)(Y-Y))'
E(X-x)'E(y-y)' /
which is .]convenient r6rmfor calculation,
since all the quantities on the right-hand side
have already been found in estimating a and b.
The value of F, when compared "lith Table A5, is Although ~he coefficient of determination is, in
seen to be significant at the 1% level. The null
hypothesis is therefore rejected, and we con- many ways. the most satisfactory measure of the
clude that the y values really do depend on x. linear aS30ciation between x and Y, the most
commonly used measure is its square root, which
*** is called the correlation coefficient:
E(X-X)(Y-Y)
It is often useful to be able to measure the VE(X-I.)'E(Y-Y)'
strength of the linear cssociation between two
variables. Such a measure can be derived by con- 13.6. 1 T~e Interpretation of r
sidering Equa~ion (13.15), which may be written
in the form It must be emphasized that the value of r is a
measure of the linear association between two
total } _ {variation } {variation due} variables. If tneoata pairs (xi, Yi) fall
{ variation - about regression + to regression . exactly G1 a straight line, the correlation is
SaTcftO be perfect. In this case, r2 = 1, i.e.,
If the data pairs (x i, Yi) fall close to a r = ±1, and the situation is as illustrated in
straight line, i.e., if the linear association Fig. 13.5 (a) and (bJ,
is good, the variation about regression (= Smin) If the data points dl,viate from the straight
wlll be small, and consequently the variation line, then r will have a value lying between +1
due to regression will provide the major contri- and -1. The sign of r depends on whether Y tends
bution to the total variation. The ratio to increase or to decrease as x increases, and
Fi g. 13.5 (c) shows a set of data for which
variation due to regression L:: (y;-y)' r.+0.7.
total variation
i. Fig. 13.5 (d) shows a situation in "Ihich there
is obviously no dependence of Y on x, and in
L:: (Yi-Y)' such caseo r will be found to have a value near
to zero. However, it is very dangerous to con-
1 clude that there is no association between x and
Y just because r is close to zero. It must again
Table 13.5 ANOVA for Example 13.1 be stres9!d that the correlation coefficient is
of value only when the relation between x anG Y
-+- Source of Sums of Degrees of ~lean F
vari dtion squares freedom squares 45. 1 is linear. Flg. 13.5 (e) shows an lnstance In-
I
Due to 17.0745 17.0745 WhlCh there is an obvious relation between x and
1 y, but it is non-linear, and the absolute value
r'egression of r for these data would be founu to be very
small.
About 1.8941 0.3788
18.9686 Another pjtfall in interpreting correlation
II regression 6 coefficients lies in concluding that, because a
correlation is significant, this necessarily
I Total lmplles a cause-and-effect relati&n between the
two variavles involved. In tx'ample 13.1, this
danger did not arise, since it is known from
.L ]~C __- _
oth(~r (~Vil[.>rl<-l' LlldL Chdll~ ill(J I.h-, ~rocessill~ Lell-
sian IIlusL ineviLalJly couse 0 chdn~e in the phy-
sico! properties of the resultillg tyrL cord. It L occasionolly desirilble to carry ouL 0 Lest
of s1gnificance of the correlotion coefficient
!lut sOliletirlll" Lhe cor'reldtioll C.l1I be spurious, Till: Ilull hypothes-is is that there is really no'
alld Lhis Cdll IH' ~adiclilurly UlIl' when the daLa corn-lotion bdwc"rr the t./O voriobles, i.e.,
hdve beell coll"cLcd over- d pel'liJd vI' Lillie. l-ul' ~hot Lhe popultltioll correldtion coefficient
exalllple, the sLacks of ,',--1 cotton vary with p ~ O. The alternative hypothesis is that p: O.
tillie, as does the production of cotton under- f!hen the null hypothesis is true, it can be
.Iear. There may l'/ell be a significant correla- sho.m that
tion between stocks of cotton and the production
of underwear, lJut this does not mean that run-
ning down stocks of raw cotton will automati-
cally increase (or decrease) the production of
ullderwear. fJhat is happening, in fact, is that
buth variables are reacting to outside influen- has a t distribution with (n - 2) degrees a·
ces and are varying with a third variable, name- freerlJm, and may therefore be tested for
ly, time. sigl1lficance by comparing to with the vollies in
Tabll: A4. This test is equivalent to that in the
ANDI". of Section ,3.5.
r = +1
y x
•• • • • (c) •0
• • •
•x •• •• 0
••••••• 0 • • e0
• (e) 0
• •
• •0 0
• r = 0.7 • •Q III
r=O
y
• ••
•c
•0
• r=0
0
•. x
~e 13.1 (continued) Confidence limits are given hv Equations analo-
gous to EquJtions (13.13), ..J.14), and (13.15).
By usi ng the values of sums of squares a,ld pro- For G, the 100(1 23")% confidence limits are
ducts found in Table 13.3, the coerficient of
determination is, from Equation (1~.)8), b~ tk,ys/fD',
28.5429' ~or the mean value of y when x = xf, the confi-
47.7143x 18.9686 dence 1imi t s are
and hence a proportion 0.9, or 90%, of the total and, for a future individual y value when x xi,
variation of the y values is explained by the they are
regression of y on x. The correlation coeffi-
cient is !
This is obviously very high, but a formal test bXf ± tk,yS(1+x}/[x2j'
of significance is made by calculating to from
Eqcation (13.20), i.e., It has long been argued that a relationship of
the type
0.95 x {5 C = k/t
1" Vl-0.95' exists between the loop length ~ and the courses
per unit length of plain-knitted fabrics when
, ,-,ith5 degrees of freedom. Reference to Table A4 the fabric is in a relaxed state. This equation
shO\-JSthat the significance level is 2 x 0.005 = is of the fom
0.01, the doubling being necessary since a two- y = Bx
if "Ie put y C, x = 1h, and B = kc. The data
'1l~. tilil test is appropriate. The null hypothesis is of Table 13.6 are the results of an experiment
therefore rejected, and we conclude that the as- to investigate this relationship for wet-relaxed
sociation between processing tension and cord fabrics. A graph of the data is shown in Fig.
modulus is real. This agrees with the conclusion ~3.6, and suggests a linear trend.
of Section 13.5. ***
Table 13.6 Measurements on Wet-relaxed Plain-
Occasionally, it is kno\-Inbefore an experiment knitted Fabri cs (All Lengths in em)
I 1S earned out, perhaps from physical considera-
1.72 1. 55 1.50 1.38 1.36 1.23 1. 18 1. 13 1. 07 1.04
i tions, t'lat the regression line should pass
8.61 7.3B 7.00 6.92 6.71 5.86 5.92 5.51 5.09 5.20
1 through the origin. The regression Equation is
1 then of the fom Fig.13.6
{ y=C
and there is only one parameter to estimate. If
., b is the least-squares estimate of B, its value
I is found by minimizing the sum of squares of de-
viations from
S = L (y._y.)2 = L (y.-bX.)2
ill ill
'l 0,
'j i.e., when
2 L (y.-bx.)(-x.)
ill 1
and an estimate of the variance about the
regression line is
s' = Smi/(n-l ) (13.24) "
4 I II I II Itiii X = !
1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8
13. Linear Regression tio~ship is of a certain kind, the methods of
b ~ 86.6974 ~ 4.8779 the last few sections can be extended to deal
17.7736
wit~ this case, WhlCh is called multiple linear
reg,_~!~~. ------
A w0rd of explanation is necessary about what
tyP('s of relationship can be dealt with by these
metl;ods. The gene,-al linea,- Equation is
"Ihere y is the dependent; -iable and Xl, x2,
... , xkare the independent variables. This
Equation is a linear function of the parameters
0, 81,82' ._., ak' and, pro v ided th at it is pro-
pos:d to fit an equation that can be written in
this form, the methods of multiple linear re-
gression can be used. Thus, if there are two in-
derendent variables, the following Equations are
al; linear in the parameters and can be fitted
to suitable data:
Th.t these are of the form of Equatior (13.28)
can be seen by taking (b) as an example. All we
have to do is write x3 ~ xlx2 and then (b)
from Equation (13.23), and then Equation (13.24) bE":om2s '
gives
wh ich is of the requi red form. An important caSE
Since k n - 1 ~ 9, and 3' ~ 0.025 for 95% con- is when a polynomial in-one independent variable
fidence limits, Table A4 gives t9,0.025 ~ 2.26, is requirEd. If vie put
and the confidence limits for Bare
then Equation (13.28) becomes
Y ~ 0+ B1X + BzX z + •.• + BkX k ,
after using Equation (13.25). i~ not a linear function of the parameters and
If a fabric is knitted with a stitch length therefore cannot be fitted by the methods of
& ~ 0.77 :m, then xf ~ 1/& ~ 1.30, and the con- multiple linear regression.
fidence limi ts for the mean number of courses
per cm for this fabric are, from Equation T,le case vlhen there are two independent vari-
( 13.26), ables is of frequent occurrence and we shall
4.8779x 1.30 ± 2.26 x 0.1832 x 1.30/ V17 .7736 describe it in some detail, the following
e~ample being used to illustrate the steps in
So far, we have been concerned with the rela- the calculation. The method can be extended
tionship betvleen two variables. It often hap- fairly easily to the case when more th.an two
pens, however, thar-the dependent variable is independent variables are involved.
determined by the values of more than one inde-
pendent variable and, provided that the rela- 11e saw in Example 13.2 that the area shrinkage
of a knitted fabric is affected by the 'strength'
uf the shrink-resist finish applied to it. How-
2ver, the inherent shrinkage of such fabrics
also seems to be influenced by the tighcness of
· ·tting, IS lI\('ilsured by the loop length. Conse- J3.8.2 The Nor~_01uat~
~n~l1tlY, ail experiment was carried out ·in which Suppose an equation of the form of Equation
(13.29) is to be fitted to n data sets
qU e loaP 1ength and treatment strength we:e var-
t~d Systematically and the re5ultlng fabncs
~sted for shrinkage. The results are given in
fable 13.7.
saw in discussing Example 13.2 that the rela-
~ n between strength of finish and shrinkage \ole shall use a, bland b2 to denote the sample
tl~ non-linear but that it could be converted to estimates of ", 6 1'and 62 respectively. Their
byvalues are fOUld
~a'ear form by taking the lOjarithm of the the method of leilst S~Uilres,
",:,~inkage. T,lble 13. 7 thu~; shows the lQ9iJrillllllS i.e., by minimizing the sum of squilres of devia-
r tile shrinkilge villues, ilno these hilve heen tions of the measured y values from the corre-
ysponding
;Iotted against x 1 (th~ strength of the SR values given by the regression equa-
',eatmentl in Fig. 13.1(a). It can be seen that tion. This sum of squares,
;~ each,level of x2 the trend is linear. Simi-
l:'rlY, in Fig. 13.7(b), vie see that wY vanes :L: Y :L:S = (Yi- i)2 = (Yi-,I-b,x1i-b2x2i)2 ,
':nearly with x 2 at each lev!'l of Xl. Further- 11
':ore, the trends ln each graph are roughly pa- is a mi nimum \~\len
.,11el, and these observations lead us to sug-
_ ;est that a mode 1 of the type adS ~ = ~ =
da db, db2
',b le 13.7 Data for Example 13.4
Xl x2 Yy xl = 1st rength I of shrink-resist finish
x 2 = loop length (cm)
a 0.3 15.6 2.75 Y percentage area shri nkage
0.3 10.6 2.36 y wY.
0.5 0.3 4.~ 1. 50
1.0 0.3
1.5 0.3 1.9 0.64
:, 2.0 1.5 0.41
0.5
a 0.5 25.5 3.24
0.5 15.0 2.71
0.5 0.5
1.0 0.5 9.5 2.25
1.5 4.2 1.44
2.0 0.69
2.0
0 0.7 62.8 4.14
0.5 0.7
1.0 0.7 24.8 3.21
1.5 0.7 17.5 2.86
7.7 2.04
2.0 0.7 3.5 1. 25
19.13.7 Graphs of data for Example 13.4
0.5 ~,,_
0.3
t========)~(
~tting, IS mcasured by the loop length. Conse- 13.8.2 The Normi\l_-',(luat~
~lntlY, a~ experiment was carried out ,in which Suppose an equation of the form of Equation
(13.29) is to be fitted to n data sets
~~ lOOP length and treatment strength we:e var-
. t~d Systematically and the re.iult1llg fabncs
::sted for shrinkage. The results are given in
Tab 1e 13.7.
saw in discussing Example 13.2 that the rela-
ft n between strength of finish and shrinkage We shall use a, b1 and b2 to denote the sample
t1~ non-linear but that it could be converted to estimates of n, B l'and 62 respectively. Their
byvalues are [ou'ld
,a'ear form by taking the 10jarithm of the the method of least squares,
I;,~inkagc. T,lble 13. 7 thu~; shows the 1"9i1r'ilh",s "i.e., by minirllizing the sum of squares of dev'ia-
If the shrinkage values, and these have bcen tions of the measured y values from the corre-
~Iotted againstx 1 (~h~ strellgth of the SR 7sponding
:eatment) In FIg. lo./(a). It can be seen that values given by the regression equa-
:; each level of x2 the trend is linear. Simi-
tion. This sum of squares,
]:'rly, in Fig. 13.7(b), Vie see that tnY vanes S = L: (Yi-Yi)2 = L: (Yi-·I-blxli-b2x2i)2 ,
l,lnearly with x 2 at eaeh level of Xl. Further- 11
~re, the trends in each graph are roughly pa- is a mi nimum when
,llel, and these observations lead us to sug-
;est that a model of the type
Xl x2 y y Xl = ' strength' of shrink-resist finish
x 2 = loop 1ength (em) y
a 0.3 15.6 2.75
0.3 10.6 2.36 Y percentage area shri nkage
0.5 0.3 4.::> 1. 50 y ~nY.
11.0 0.3 0.64
" 1. 5 0.3 1.9 0.41
:- 2.0 1.5
0.5 3.24
0 0.5 25.5 2.71
0.5 0.5 15.0 2.25
1.0 0.5 9.5 1. 44
1.5 0.5 4.2 0.69
2.0
2.0 0.7 4.14
0.7 62.8 3.21
0 0.7 24.8 2.86
0.5 0.7 17.5 2.04
1.0 0.7 1. 25
1.5 7.7
3.5
2.0
ig.13.7 Graphs of data for Example 13.4
Ii (a)
4
[-------
The first of these conditions gives
J 0 y-b1x1-L,,,x;>, Ih~ Ldlcul"tion uf d, 01. "fld o~ is shO\,n ifl
f, i I in T<lble 13:8 "nd, provided that lile COIllUU-
L"tlUns ore carrlcdollt ~.ysL<:IJIJLlc,llly. they'
·.I'ould pn,~.eflt flU ulff'lculLy.
f,' this sLage, it is lVortll rCIJI<lf'king a I Lhe
design of this experiment, since it ~Ia; chosen
\'Iith a vie\'1 to simplifying the calculations. Tho
vaiable xl \'Ias investigated at five levels e
(0, 0.5, 1.0, 1.5, 2.0), \'Ihich are seel to be
equally spaced. Similarly, x2 \'Ias vari:d at
three levels (0.3, 0.5, 0.7l, and these are alsc
spaced at equal intervals. When the le~els are
r~osen in this lVay, it turns out that the sum of
produc ts
512 = E{x1-x1)(xZ-X2)
hJS <I zero value, and this makes the solution of
the normal equations particularly simple. In
fact, lVe find that
S1/s11
S2y = E{XZ-XZ){y-y) = EX2Y-{EX2){Ey)/n . bo s2/sZ2
Equations (13.30) and (13.31) are called the
normal equations, and they are sufficient to
determlne <I, 51, <lnd b2'
They can be generalized quite easily. For the It can be sholVn that the total sum of sauares
general linear regression lVith k independent Syy = ~{y_y)2 can be split into two components
variables, the normal equations are
(i) due to regression, and
(ii) about regression,
sllbl+X1Zb2+···+s1kbk Sly as for the case of two variables, Section 13.5,
s12bl + s22b2 + ... + s2kbk S2y
and this provides the basis of an analysis of
variance. The sum of squares due to regression
is now given by
Table 13.8 Calculation of a, bl,and b2 for
Example 13.4
n 15 .
EX1 15.0; Xl EX";n = 15.0/15 1.0
EX2 7.5; x 2 Ex2/n 7.5/15 0.5
'iy 31. 49; y oy!n 31.4~/15 = 2.0993
1EX, 22.50; 511 E: -{Ex1)2/n 22.50-15.02/15 7.50 ~
4.15-7.5'/15
EXZ 4.15; s22 EXZ-(EX2}2/n = 0.40
EX1x2 7.50; 512 EX1X2-{EX1}{Ex2)/n 7.50-15x 7.5/15 0.00 ..
EX1Y 21.630; SlY EX1Y-{EX1}(Ey}/n 21.630-15 X 31.49/15 = -9.86
EX2Y 16.913; S2y = Ex2y-(EX2)([y)/n = 16.913-7.5x 31.49/15 = 1.168
Ey2 82.7611; Syy = Ey2_{Ey}2/n = 82.7611-31.492/15 = 16.6531
~ 7.50 b1 + 0.b2 = -9.86 Using the sums of squares and products in Table
13.8, .Ie find that Equation (13.32) gives:
.lnd
T~e ANOVA table is therefore as shown in Table
b, -1.3147 13.10. The C ratio for thft regression of y on .,
and x2 is 351, with 2 and 12 degrees of freedom.
b2 2.92. This is significant beyond the 1% level when
compared .Iith TabIe A5, and we therefore
Equation (13.30) then gives = 1.954 . conclude that the regression is real.
t a = 2.0993-(-1.3147)(1.0)-(2.92)(0.5) However, there is a possibility that it is not
really necessary to include both xl and x2 in
The estimated regression Equation is therefore the regre ss;on, and the fo 11owing procedure
can be adop~ed to test this. Imagine that the
y = 1.95-1.31x1+2.92x2 ' experiment had been carried out with x2 ignored.
Then I.e should have determined the regression of
after rounding off the estimates, and the ANOVA y on x 1 alene. Using Equation (13.17), we should
table is as shown in Table 13.9. have found
A test. of the significance of the overall re-
gression is provided by the ratio flr/f-o,wh'ch
~ust be compared with the F values in Table A5.
Table 13.9 ANOVA for Multiple Regre~sion with Similarly, if the experiment had been performed
Iwo Indepenoent varlables
with Xl ignured, we should have determined the
Sources of Sums of Degrees of Mean regression on x2 alone. The appropriate sum of
variation squares .freedom - squares
F squares would be .
Due to b1 Sly + 2 Mr M/Mo
regress ion b2S2y
About by
regression difference n-3 M
0
Total Syy n-l Hence we see that the contribution of Xl to the
total sum of squares Syy = 16.6531 is the
Table 13.10 ANOVA for Example 13.4 greater, and Xl can therefore be regarded as the
major independent variable. Its contribution has
.'1Jources of Sums of Degrees of Mean F thus been entered in t~e second row of Table
-Bariation squares freedom squares 351 13.10. The third row is the additional sum of
556 squares due to regression brought about by
Jue to 16.3735 2 8.1868 146 adding x2 to the regression of y on Xl' This sum
~1!greSSiOn on 12.9626 of squares -isfound from the difference of the
,'\ and x2 3.4109 12 12.9626 entries in l'OWS 1 and 2, i.e., 3.4109 = 16.3735-
14 3.4109 12.9626.
f 0.0233
The two degrees of freedom for the regression on
'Due to xl Xl and x2 are divided, and each degree of free-
_~ alone dom is ascribed to a component sum of squares.
The mean squares are found in the usual way, and
---'-Vn. Addn. of their ratios with the mean square about regres-
sion providE tests of significance for the terms
• x2 in Xl and XI in the regression Equation. In this
example, both ratios are highly significant, and
:tout we therefor~ conclude that it is worthwhile re-
:egression taining both Xl and x2 in the regr ,sion equation.
***
Iota1 ~~,'" 6~ ~
.\~. ),
'.
~
lmmmc _
As for the case of two variables, it is pos- For 95% confidence limits, \;e must put}' = 0.025,
sible, and desirable, to calculate confidence and Slnce k = 12, Table A4 glVes t12,0.025= 2.18
limi ts for the regression param~_ers, Cl, 8.1,and The cunfldence llmlts are therefore as fol10;s.
02' and for iI fulul'c predicled v,lIue of y. Lel (i) For a
I • OoIJ,'J L. IU' X 0 • OUu"I' x {J.'O15O + 10• " x (J..'1 + 05• ' x '1.5,0
where Mo is the 'about-regression' mean square
in the ANOVA of Table 13 9. Al so suppose that (iiil For 82
k = n-3 is the number of degrees of freedom asso- 2.9'-± 2. 18 x 0.0881 x {7.50
ciaterl with this mean square. The following ex-
pressions then give 100(1-2'0)1 confidence li-
nn t s.
a±tk,y(s/D){D2/n+xls22+X2S11
_ _ 1.
-2x1x2s12}2
(ii) For 81
b1±tk,y(siDl/Sz;
(iv) For the mean value of y when xl ±2.13x 0.0881 x {3i~0 + (1.3 - 1.0)2 x 0.4
= x1f,x2 = x2" +(0.4 - 0.5)2 x 7.50
±tk,y(S/D){D2/n+(x1f-xl)2s22 Since thnse are 95% confidence limits for 9
+(x2f-x2)2s11
= inY, the 95% confidence limits for the actual
_1 shri nkage Yare
-2(xlf-xl)(x2f-x2)S12}2
Example 13.4 (continued)
Using the results of Table 13.8, we find that
An experimcni. l'las performed to determi ne the 5. The strength of a joint in a pari'r:",te web-
bing (y) depelds on the length (l 'verlap in
con'elution bctl'/ecil Lhl' ,'csults obtuined on " the joint (XI) and on the stitclllllg density
louo.-aLo,-y carpel-Vie",' Leste,' alld the life of (x2)· [n an experiment, thc following data
tile carpet in actual I·/ear. Six carpets \·,ere wve obtained.
tested on the tester and in everyday use,
It is sUJgested that a m0del of the type
rlith the follOI·/ing results. y = ~ + 81xl + 8?x2 fits the data. Estimate
I ~, 81, and 82' Cdlculate 95% confidence
limits for the mean joint strength when
I
Xl = 7 and x2 = 5.
..,er of rub-
·ocvrleson y = 8,x1 + 82x2
j~e('" to form
'ole (' 000) is to be fitted to n data sets (Xli, x2i, Yi),
i = 1, 2, ..... , n. Show that the least-squares
I estimates, bl and b2, of 81 and 82 are the solu-
tions of the equations
iMr of pas-
:eS over carpet x 0.1 0.2 0.5 1.0 5.0 10.0
'fo.-m a ho Ie Y 50 25 12 10 15 30
A certain theory :uggests that the relation
:00) between X and y L of the form
y = G1X +Bzlx
~culate the correlation coefficient and test Find estlmates of 81 and 82'
\ significance.
The following data show the relation between
the relative viscosity of a dye liquor and
the ,jye uptake on a certain fabric. (The
~its have been transformed.)
.Iati ve 0 6 8 11
6 8 12
,\seasi ty
!1.-'l
"! uptake
_culate an equation for estimating the mean
! uptake when the relat've viscosity is known.
~ are the 95% confidence limits for the mean
.uptake when the relative viscosity = 4?
Th,! fo]1OI.ing data were obtained in an ex-
.peI'iment to investigate the relation between
jth~ tenacity of a sliver and the processing
,sp~ed .
r. ,city (mN/tex) 0.76 0.70 0.65 0.59 0.57
4culate the Equation of the regression of
~acity on speed dnd find 95% confidence limits
• the mean tenacity when the speed is 10 m/min
The data below relate the thickness loss
during calendering of a viscose needle-
punched fabric and the load on the calender
bowl.
l' ~ 0.5 1.0 1.5 2.0 2.5 3.0
, 13 14 20 24 33
ils) (x) :
'ckne ss
is (i: (y) : 4
: an equati on of the form y = 8 X to these
:,. Calculate 95% confidence limits for r. and
',he thicknESS loss when the load = 2.3 tons.
)~[--------
13. Linear Regression
5.29 ~Iith k = 3. [> 0.05; hence no WJrning lililits= 5 ~ 4.3 = (0.7, 9.3).
significant differences among Ac tion 1im I ts = 5 ~ 6.8 = (0, 11.8).
machines. All points in control except for days G,
17. Omitting these gives:
6.30 with k = 2. 0.025 < (1< 0.05; Revised wal'ning limits = 4 ± 3.9 =
hence proportions differ signifi- (0.1, 7.9).
cantly from nominal. Revised action limits = 4 ~ 6.1 =
(0, 10.1).
15.07 with k = 6. a < 0.025; hence
there are differe~ces among t~2 \'Iarninglimits = 9.625 ~ 6.076 = (3.5, 15.7)
treatments. Action limits = 9.625 ± 9.579 = (0, 19.2).
15.49 ~Iith k = 8. a ~ 0.05, 'dhich 3. Average Chart
suggests there may be differences in
machine behaviour in the different Warning limits = 57 ± 1.0 = (56.0, 58.0).
filctories. l\ction limits = 57 ± 1.6 = (55.4, 58.6).
1.67 ~Iith k = 1. (1 > 0.05; not enough War nin g Iim its = (0.61, 4.03).
evidence to say there was a signifi- Action limits = (0.21, 5.37).
cant difference between males and
females. Sourc e Sums of Degrees of Mean F -. I
squares freedom squares i
0.905, significant at 1% level. Hence I
good agreement between judges.
Betwe en I10.5 **
0.327, to = 1.38 with k = 1Ii. a> 0.05, fabri cs 81.58 3 27.19
which suggests that preliminary test ,
is not a good predictor of employee Within
potential. ,fabrics 20.67 8 2. :,84 I
0.75, which is just significant at 5% Total 102.25 11 !:
level, suggesting there may be an
association between flexural rigidity ** = significant at the 1% le'I~~:..___.
and handle.
Least significant difference between fabrics
,0. 72 5 , F 0 = 7, 38, k 1 = 3. 5, k 2 = 10 . 5 . 1 and 2 or between 3 and 4 = 3,03.
a < 0.01; hence good agreement among
judges. Least significant difference between fibre-type
means = 2.14.
0.3, Fo = 1~25, k, = 4.5, k2 = 13.5.
Source Sums of Degrees of Mean I
a> 0.05, which suggests poor agree- squares freedom squares
ment among groups, Technologists seem Between F
to be 'odd men out'. Neglecting tech- locations
nologists, we find Within I
W = 0.721, Fo = 4.78, k, = 4.3, locations
k2 = 8.7. a < 0.05, which suggests 16 .. 1, ,. 3 5.367 2.85
agreement among sales directors, Total
buyers, and the public. 67.8 36 1.883 I
83.9 39
0.806, Fo = 16.4, k, = 7.6, k2' 30.4. J
a < 0,01; hence good agreement among The F ratio is just significant at the 5% level.
observers. By using Tukey's procedure, the least signifi-
Chapter 10 cant difference between any two means is about
1.6. Thus the difference between locations I and
1. n = 238, c 8; n = 175, c = 5. II is just significant.
2. n 310, c 12; n = 200, c = 8; n 345,
c 13.
3. n 26, XL = 6.04 kg; n = 33, XL = 6.02 kg.
4. n = 50, x must lie between 455 ± 21 for the
delivery to be acceptable; n = 41,
x must lie between 455 ± 2.33.
Chapter 13
Source Sums of Deg!'ees of ~lean F ~. r ~ 0.967. t ~ 7.58 with 4 degrees of
10.7** freedom, si gni fi cant at the 1% 1eve 1.
Between squares f,'eedolll squares
proces ses
---------- ----
2. Y 0.31 t 1.03x; (3.29, 5.57)
1442 2 721
3. Y 0.801 - 0.00817x; (0.694, 0.745)
Between
fabrics 498 3 166 (ns) 4. Y 1O.33x; (9.29,11.37); (21.4, 26.1)
Residual 406 6 67.66 5. y - 33.~; + 4.67x1 + 7.75x2,; (34.6,41.3)
Total ------ -- 6. y 2.93x + 4.9B/x.
2346 11 --
,-
not si gnifi cant at 1% level
nearly significant
Source' Sums of Degrees of I~ean F
squares freedom squares
Between 687.150 4 171.8 41.6**
techno-
logists
Between 3 397.1 96.3**
machines 1191.275 12 6.904 1. 67( ns)
Tech no' 82.850
logists x
machi nes
Resi dual 82.500 20 4.125
Total 2043.775 39
significant at 1% level
'not significant
Lease significant di"ference between any two
machines ~ 2.54. -
To test between laboratories, we use the
contrast
I Hence the difference between laboratories is
si gnificant.
_II
.J
I Factor Factor Factor
effect SSqs
~J Mean, 15.5 -
- 21. 55 928.8
A
B - 4.1 33.6
AB 5.35 57.3
2.80 15.7
C 3.1
AC - 1. 25 14.6
9.3
BC 2.7
ABC - 2.15
I Strictly, no factor is significant, with the ABC
.: 85. _1_ SSqs used as a measure of experimental error.
However, A, B, and AB seem worth further inves-
tigation.
)~C= _
Table Al: Areas in the Tail of the Standard Normal Distribution
The entries in this table are values of
CL = Pr (u ~ U)
for the given values of U
U CL U CL U CL U CL U CL
0.00 0.5000 0.60 0.2743 1.20 0.1151 1.80 0.0359 2.40 0.0082
0.02 0.4920 0.62 0.2676 1.22 0.1112 1.82 0.0344 2.42 0.0078
0.04 0.4840 0.64 0.2611 1.24 0.1075 1.84 0.0329 2.44 0.0073
0.06 0.4761 0.66 0.2546 1.26 0.1038 1.8.6 0.0314 2.46 0.0070
0.08 0.4681 0.68 0.2483 1.28 0.1003 1.88 0.0301 2.48 0.0066
0.10 0.4602 0.70 0.2420 1.3C 0.0968 1.90 0.0287 2.50 0.0062
0.12 0.4522 0.72 0.2358 1.32 0.0934 1.92 0.0274
0.14 0.4443 0.74 0.2296 1.34 0.0901 1.94 0.0262 2.60 0.0047
0.16 0.4364 0.76 0.2236 1.36 0.0869 1.96 0.0250
0.18 0.4286 0.78 0.2177 1.38 0.0833 1.98 . 0.0239 I
0.20 0.4207 1.40 0.0808
0.22 0.4129 0.80 0.2119 1.42 0.0778 2.00 0.0228 I
0.24 0.4052 0.82 0.2061
0.26 0.3974 1.44 0.0749 2.02 0.0217 2.70 0.0034
0.28 0.3897 0.84 0.2005 1.46 0.0721
0.30 0.3821 1.48 0.069~ 2.04 0.0207 2.80 ...0...0.026 ..
0.32 0.3745 C.86 0.1949 1.50 0.0668
0.34 0.3669 0.88 0.1894 1.52 0.0643 2.06 0.0197 2.90 0.0019
0.36 0.3594 0.90 0.1841 2.08 0.0188
0.38 U.3520 0.92 0.1788 1.54 0.0613 2.10 0.0179 I
0.40 0.3446 0.94 0.1736 1.56 0.0594 2.12 0.0170
0.42 0.3372 0.96 0.1685 2.14 0.0162 3.00 0.0014
0.44 0.3300 0.98 0.1635 1.58 0.0571 2.16 0.0154
0.46 0.3228 1.00 O.1587 1.60 0.0548 2.18 0.0146
0.48 0.3156 1.02 0.1539 1.62 0.0526 2.20 0.0139
0.50 0.3085 1.04 0.1492 2.22 0.0132
0.52 0.3015 1.06 O.1446 e 2.24 0.0126
0.54 0.2946 ·1.03 0.1401 2.26 0.0119
0.56 0.2877 1.10 0.1357 1.64 0.0505
1.12 0.1314 1.66 0.0485 2.28 0.0113
~2810 0.2743 1.14 O.1271 2.30 0.0107
1.16 0.1230 ·1.68 0.0465 2.32 0.0102
0.60 1.18 0.1190 1.70 0.0446
1.20 0.1151 1.72 0.0427 2.34 0.0096
1.74 0.04l9
1.76 0.0392 2.36 0.0091
1.78 0.0375 .2.38 0.0086
1.80 0.03S9 2.40 0.0082
The tables in these Appendices have been extracted from 'Biometrika
Tables for Statisticians'. and are reproduced by permission of the
Biometrika Trustees.
Table A2: Probability Points of the Standard Nor"mal
Distribution
The entries in this table are values
of U corresponding to the given values
of ex
ex U ex U ex U
00500 0.0000 0.034 1.8250 0.015 201701
0.450 0.1257 0.032 1 .8522 0.014 2.1973
0.400 0.2533 0.030 1.8808 0.013 2.2262
0.350 0.3853 0.029 1 .8957 0.012 2.2571
0.300 0.5244 0.028 1.9110 0.011 2.2904
0.250 0.674S 0.027 1 .9268 0.010 2.3263
0.200 0.8416 0.026 1.9431 0.009 2.3656
0.150 1 .0364 0.025 1.9600 0.008 2.4089
0.100 1 .2816 0.024 1.9774 0.007 2.4573
0.050 1.6449 0.023 1 .9954 0.006 2.5121
0.048 1 6646 0.022 2.0141 0.005 2.57:8
0.046 1.6849 0.021 2.0335 0.004 2.6521
0.044 1.7060 0.020 2.0537 0.003 2.7478
0.042 1.7279 0.019 2.0749 0.002 2.8782
0.040 1. / 507 0.018 2.0969 0.001 3.0902
0.038 1.7744 0.017 2.1201 0.0005 3.2905
0.036 1 .7991 0.016 2.1444
: ]~c= __
Table A3: Probability Points of the X2 Distribution
LThe entri:s in this table are
the values of X~,a
corresponding to the
given values of a
a
k 0.995 0.975 0.950 0.050 0.025 0.005
1 0.000 0.001 0.004 3.841 5.024 7.879
2 0.010 0.051 0.103 5.991 7.378 10.60
3 0.072 0.216 0.352 7.815 9.348 12.84
4 0.207 0.484 0.711 9.4b8 11.14 14.86
5 0.412 0.831 1.145 11.07 12.83 . 16.75
6 0.676 1.237 1.635 12.59 14.45 18.55
7 0.989 1.690 2.167 14.07 16.0' 20.28
8 1.344 2.180 2.733 15.51 17.53 21.96
9 1.735 2.700 3.325 16.9? 19.02 23.59
10 2.156 3.247 3.940 18.31 20.48 25.19
12 3.074 4.404 5.226 21.03 23.34 28.30
14 4.07;3 5.629 6.571 23.68 26.12 31.32
16 5.142 6.908 7.962 26.30 28.85 34.27
18 6.265 8.231 9.390 28.87 31.53 37.16
20 7.434 9.591 10.85 31.41 34.17 40.0LJ
25 10.52 13. 12 14.61 37.65 40.65 46.93
30 13.79 16.79 18.49 43.77 46.98 53.67
40 20.71 24.43 26.51 55.76 59.34 66.77
50 27.99 32.36 34.76 67.50 71.42 79.49
The entries in this table
are values of tk,a
corresponding to the
?iven values of a
t k,a
a
k 0.05 0.025 0.01 0.005
'~- 1 6.31 12.70 31.80 63.70
..~,.. 2 2.92 4.30 6.97 9.93
.,., 3 2.35 3.18 . 4.54 5.84
""'I 4 2. 13 2.78 3.75 4.60
..y' 5 2.02 2.57 3.37 4.03
6 1. 94 2.45 3.14 3.71
.•.
7 1. 90 2.37 3.00 3.50
-I) 8 1.86 2.31 2.90 3.36
9 1.83 2.26 2.82 3.25
"'" 10 1. 81 2.23 2.76 3.17
12 1. 78 2.18 2.68 3.06
..., 14 1. 76 2.15 2. 6:~ 2.98
16 1. 75 2.12 2.58 2.92
""' 18 1. 73 2.10 2.55 2.88
20 1. 73 2.09 2.53 2.85
22 1.72 2.07 . 2.51 2.82
24 1. 71 2.06 2.~9 2.80
26 1. 71 2.06 2.48 2.78
28 1. 70 2.05 2.47 2.76
30 1.70 2.04 2.46 2.75
Table A5: Probability Points of the F Distribution
The entries in this table are vaJues of Fk1k2a corres
to the given values of a
a = 0.05
k2 1 2 3 4 5 6
1 161 200 216 225 230 234
2 18.5 ~9.0 19.2 19.3 19.3 19.3
3 10. 1 9.6 9.3 9.1 9.0 8.9
4 7.7 6.9 6.6 6.4 6.3 6.2
5 6.6 5.8 5.4 5.2 5. 1 5.0
6 6.0 5. 1 4.8 4.5 4.4 4.3
7 5.6 4.7 4.4 4.1 4.0 3.9
8 5.3 4.5 4.1 3.8. 3.7 3.6
9 5. 1 4.3 3.9 3.6 3.5 3.4
10 5.0 4. 1 3.7 3.5 3.3 3.2
15 4.5 3.7 3.3 3.1 2.9 2.8
20 4.4 3.5 3.1 2.9 2.7 2.6
;
00 3.8 3.0 2.6 2.4 2.2 2.1
sponding
F k1k2C(
k1 8 9 10 20 00
7
239 241 242 248 254
4 237 19.4 19.4 19.4 19.5 19.5
19.4 8.9 8.8 8.8 8.7 8.5
6.0 6.0 6.0
9 8.9 4.8 4.8 4.7 5.8 5.6
2 6.1 4.6 4.1\
0 4.9 4.2 4.1 4.1
3.7 3.7 3.6 3.9 3.7
3 4.2 3.4 3.4 3.4 3.4 3.2
9 3.8 3.2 3.2 3. "j 3.2 2.9
6 3.5 3. 1 3.0 3.0 ;:.9
4 3.3 2.8 C~. ./ 7
2 3.1 2.6 2.6 2.5
2.5
8 2.7 2.5 2.4 2.4
2.3 2.1
6 2.5 1.9 1 .(1 1.8
2.1
1 2.0
1.6 1.0
~
Table A5 (continued)
Ci = 0.025
k2 1 2 3 4
1 648 800 864 900
2 38.5 39.0 39.2 39.3
3 17.4 16.0 15.4 15.1
4 12.2 10.7 10.0 9.6
5 10.0 8.4 7.8 7.4
6 8.8 7.3 6.6 6.2
7 8.1 6.5 5.9 5.5
8 7.6 6.1 5.4 5.1
9 7.2 5.7 5. 1 4.7
10 6.9 5.5 4.8 4.5
15 6.2 4.8 4.2 3.8
20 5.9 4.5 3.9 3.5
co 5.0 3.7 3.1 2.8
k1
5 6 7 8 9 10 20 co
0 922 937 948 957 963 9b9 993 1018
3 39.3 39.3 39.4 39.4 39.4 39.4 39.5 39.5
1 14.9 14.7 14.6 14.5 14.5 14.4 14.2 13.9
6 9.4 9.2 9.1 9.0 8.9 8.8 8.6 8.3
4 7.2 7.0 6.9 6.8 6.7 6.6 6.3 6.0
2 6.0 5.8 5.7 5.6 5.5 5.5 5.2 4.9
5 5.3 5. 1 5.0 4.9 4.8 4.8 4.5 4. 1
1 4.8 4.7 4.5 4.4 4.4 4.3 4.0 3.7
7 4~ 4.3 4,2 4.1 4.0 4,0 3.7 3.3
5 4.2 4.1 4.0 3.9 3.8 3.7 3.4 3. 1
8 3.6 3.4 3.3 3.2 3.1 3.1 2.8 2.4
5 3.3 3.1 3.0 2.9 2.8 2.8 2.5 2. 1
8 2.6 2.4 2.3 2.2 2. 1 2.1 1.7 1.0
k2 1 2 3 4 5
1 4052 5000 5403 5625 5764
2 98.5 99.0 99.2 99.3 99.3
3 34.1 30.8 29.5 28.7 28.2
4 21. 2 18.0 16.7 16.0 15.5
5 16.3 13.3 12. 1 11. 4 11.0
5 13.8 10.9 9.8 9.2 8.8
7 12.3 9.6 8.5 7.9 7.5
8 11.3 8.7 7.6 7.0 6.6
9 10.6 8.0 7.0 6.4 6.1
10 10.0 7.6 6.6 6.0 5.6
15 8.7 6.4 5.4 4.9 4.6
/'
20 .8.1 5.9 4.9 4.4 4. 1
co 6.6 4.6 3.8 3.3 3.0
-:
hI
I6 7 8 9 10 20 OJ
I
5859 5928 5982 6022 6056 6209 6366 I
I99.3 99.4 99.4 99.4 99.4 99.5 99.5
I27.9 27.7 27.5 27 .4 27.2 26.7 26.1
15.2 15.0 14.8 14.7 14.6 14.0 13.5
10.7 10.5 10.3 10.2 10. 1 9.6 9.0 I
I
8.5 8.3 8.1 8.0 7.9 7.4 6.9
I7.2 7.0 6.8 6.7 6.6 6.2 5.7
6.4 6.2 6.0 5.9 5.8 5.4 4.9
5.8 5.6 5.5 5.4 5.3 4.·8 4.3
5.4 5.2 5.1 4.9 4.9 4.4 3.9
4.3 4. 1 4.0 3.9 3.8 3.4 2.9
3.9 3.7 3.6 3.5 3.4 2.9 2.4
2.8 2.6 2.5 2.4 2.3 1.9 1.0
I
Table A6 The SLudentized Range qc,k
(for calculating the least significant difference at the 5% level)
;~4 5 6 7 g 9 10
5 5.22 5.67 6.03 6.33 6.58 6.80 6.99
6 4.90 5.30 5.63 5.90 6.12 6.32 6.49
7 4.68 5.06 5.36 5.61 5.82 6.00 6. '16
8 4.53 4.89 5. 17 5.40 5.60 5.77 5.92
9 4.41 4.76 5.02 5.24 5.43 5.59 5.74
10 4.33 4.65 4.91 5.12 5.30 5.46 5.60
12 4.20 4.51 4.75 4.95 5. 12 5.27 5.39
14 4. 11 4.41 4.64 4.83 4.99 5. 13 5.25
16 4.05 4.33 4.56 4.74 4.90 5.03 5. 15
18 4.00 4.28 4.49 4.67 4.82 4.96 5.07
20 . 3.96 4.23 4.45 4.62 4.77 4.90' 5.01
30 ' 3.85 4.10 4.30 4.46 4.60 4.82
40 3.79 4.04 4.23 4.39 4.52 4.72 4.73
4.63
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