plotting with confidence: graphical comparisons of two populations

BiometrUca (1976), 63, 3, pp. 421-34 4 2 1With 2 text-figures

Printed in Great Britain

Plotting with confidence: Graphical comparisonsof two populations

B Y KJELL A. DOKSUM

Department of Statistics, University of California, Berkeley

AND GERALD L. SIEVERS

Department of Mathematics, Western Michigan University, Kaiamazoo

SUMMARY

Statistical methods that give detailed descriptions of how populations differ are con-sidered. These descriptions are in terms of a response function A(x) with the property thatX + A(X) has the same distribution as Y. The methods are based on simultaneous confidencebands for the response function computed from independent samples from the two popula-tions. Both general and parametric models are considered and comparisons between thevarious methods are made.

Some key words: Behrens-Fiaher model; Confidence bands; Empirical probability plot; Nonlinearmodel; Q-Q plot; Response function; Shift function; Two-sample problem.

1. INTRODUCTION

We consider the problem of comparing two populations with distribution functions Fand 0 on the basis of two independent random samples Xlt..., Xm and Tlt ...,Yn, respectively.Instead of the usual shift model where F(x) = O(x + 6) for all x, we treat the general case.where F(x) = 0{x + A(a;)} for some function A(x). If the X'B are control responses and theY'B are treatment responses, A(a;) can under certain conditions be regarded as the amountthe treatment adds to a potential control response x (Doksum, 1974). Since it gives theeffect of the treatment as a function of the response variable, we call it the response function.Under general conditions it is the only function of x that satisfies X + A(X) ~ Y, where~ denotes distributed as. Thus A(.) is the amount of 'shift' needed to bring the X'B up tothe Y'B in distribution and it is also referred to as the shift function.

Assume that F and 0 are continuous. Let F-1^) = inf {x: F(x) > u} be the left inverseof F. Then we can write

If in fact a shift model holds, that is, F(x) = O(x + 6) for some constant 6, then A(x) = 6.A natural estimate of Q-x{F(x)} is G^F^x)}, where Fm and On denote the empirical

distribution functions based on the X and Y samples. The Q-Q plot considered by Wilk &Gnanadesikan (1968) is essentially O^1 Fm evaluated at the X order statistics. Doksum(1974) referred to it as the empirical probability plot and derived the asymptotic distribu-tion of A » = G?{Fm(x)} -x.

Suppose that a beneficial treatment leads to large responses. Then certain naturalquestions arise: (i) Is the treatment beneficial for all the members of the population, i.e. isA(a;) > 0 for all a;? (ii) If not, for which part of the population is the treatment beneficial,i.e. what is {x: A(x) > 0}?

This kind of model in effect also yields information about how the treatment works and

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422 K J E L L A. DOKSUM AND GERALD L. SIEVBBS

about which statistical analysis is appropriate. Thus the following questions are of interest:(iii) Does a shift model hold, i.e. is A(x) = 6, for some 6 and all x ? (iv) If not, does a shift-scalemodel hold, i.e. is A(x) = a+fix, for some a and /? and for all x ?

These questions can be answered by giving a confidence band [A*(x), A*(x)] for A(x),simultaneous for all x. Thus (i) is answered in the affirmative if A*(x) > 0 for all x, (ii) hassolution {x: A*(x) > 0}, (iii) is rejected if no horizontal line fits in the confidence band, and(iv) has a negative response if no straight line fits in the band. Note that the first twoanswers only required a lower confidence boundary A* (x).

Such confidence bands have been considered by Doksum (1974), Switzer (1976), andG. L. Sievers in an unpublished report. Switzer and Sievers derive a band based on thetwo-sample Kolmogorov-Smimov statistic and thereby obtain the exact confidence co-efficient of the band of Doksum (1974). Similar bands have been considered by Steck,Zimmer & Williams (1974) in connection with the 'acceleration function' OF'1. Here weconsider bands based on statistics of the form

sup \Fm(x)-On(x)\li/r{HN(x)},am<x<bm

where N = m + n, A = m/N and HN{x) =± AFm(x) + (1 - A) (?n(x).Efficiency comparisons are made between such bands in terms of squared ratios of

widths, and it is found that the choice of (u) = {u( 1 — u)}i leads to an efficient band.If a location-scale model can be assumed, a band which improves on the above general

bands is constructed from order statistics and its asymptotic efficiencies with respect tothe general bands are given.

For the normal Behrens-Fisher model, the likelihood ratio band is derived when n = m,and it is shown to be much more efficient than the general bands in the normal model. Aband based on the maximum likelihood estimate of A(x) is also considered for this model.

2 . NONPARAMETBIO STMTTLTANEOTTS CONFIDENCE BANDS FOB THE RESPONSE FUNCTION

In the nonparametrij case, it is natural to construct confidence bands for A(x) usingpivots based on the empirical distribution functions Fm and On. The key to finding suchpivots is that O^n, denned by

ia distributed as the empirical distribution of a sample of size n from F. Note that

<?n{A(y) + 2/} = [no. of 7, ^ G^F^n = [no. of i 1 - 1 ^ , ) } < y]/»

and that J'-1{G1( T)} has distribution F.Now if 4>{Fm, On) is a distribution-free level a test function for 11$: F = 0, then

{A( . ) : ^ m , ^ i n ) = 0} (1)

is a distribution-free level (1 — a) confidence region for the response function A(.).These regions will reduce to simple bands if we consider distribution-free test statistics

T(Fm, On) with the property that the inequality T(Fm, On) < K is equivalent to

K{FJx)} < Gn(x) < h*{Fm(x)} (2)

for all x, for some functions h+ and h*. Typically these functions are nondecreasing. Forinstance, let N = m + n, M = mn/N and suppose

T(Fm,On) = DN = MiBapa\Fm(x)-Gn(x)\,the Kolmogorov-Smirnov statistic. Then A,(x) = x—K\M\ and h*(x) = x + K/Ml.



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Plotting with confidence 423

From (2), we derive confidence bands as follows. Let G^}(u) and (?nJ(u) = sup {a;: On{x)u} be the left and right inverses of On, and suppose that K is chosen so that

Q*)< #}=!-<*• (3)Then, simultaneously for all x,

1 -a = pr,_0{r(J'm,<?n) < Z) = p r , , ^ * ^ G ± n ) < £}

±n(x) < h*{Fm(x)}]

)}]-* < A(«) < G?[h*{Fm(x)}]-x).

Thus we have the following theorem.

THEOREM 1. If (2) and (3) AoW, tten

[G^[h.{Fm(x)}]-x, G?\h*{Fn(x)}]-x) (4)

cw x raTigesfrom -co toco gives a level (1 — a) simultaneous, distribution-free confidence bandfor the response function A(a;).

Now suppose that K8a has been chosen from the Kolmogorov-Smimov tables (Pearson& Hartley, 1972, Table 55) so that prJ _o(X>JV < KSa) = 1 -a.

Remark 1. A level (I — a) simultaneous distribution-free confidence band for A(a;)( — oo < x < oo) is given by

- x, G~i{FJx) + K8JMi} - x).

This band, which we call the 8 band and denote by [8^(x), 8*(x)), was obtained bySwitzer (1976) and by G.L. Sievers and should replace a similar band given by Doksum(1974).

Let [f] denote the greatest integer less than or equal to t; let <<) be the least integer greaterthan or equal to t; let X(l) < ... < X(m) and F(l) < ... < T(n) denote the order statisticsof the X and T samples, and define Y(j) = — oo (j < 0) and Y(j) = oo (j > n+1). Then theband (4) can be expressed by

[A,(*),A*(aO) = [ r ^ ^ - x . F ^ . ^ j + l)-*) (5)

for xe[X(i),X(i + l)) (i = 0, l,...,m) with X(0) = -oo and X(m+l) = oo. This rep-resentation was used to produce Fig. 1 where the 8 band,

is given for X and Y samples from #(0, 1) and N(l, 4) distributions, respectively. In thisfigure, m = n = 100 and a = 0-05.

The general method can also be applied to a weighted sup norm statistic

N N ( m n ) sup l ^ - y l , (6)

where HN(x) = AFm(x) + (1 - A) Gn(x), A = m/N and 0 < o < b < 1. However if we choose= [t(l-t)}i, then we give approximately equal weight to each x in the sense that

Mi{Fm(x)-Gn(x)W{HN{x)}



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424 KJELL A. DOKSTJM AND , GERALD L. SIEVEBS

has asymptotic variance independent of x. If we consider one-sided test statistics,without the absolute value, in the class (6) with 0 < a < b < 1, this choice of IP asymp-totically maximizes the Tninimnm power when testing HQ : F = 0 against Hx: F(x) — G(x) > 8for some x (S > 0) (Borokov & Sycheva, 1968).

To apply Theorem 1, we need to solve the inequality |TJjv(.Fm, On)\ < K for On. WhenW(t) = {f (1 — t)}i this inequality becomes

{On(x)-Fm(x)Y ^ K*{AFm(x) + (l-\)Gn(x)}[l-{AFm(x) + (l-A.)Gn(x)}]/M

for x such that a < Fn(x) < b.Let c = K?\M, u = Fm(x) and v = OJx); then the inequahty can be written as d(v) < 0,

whered(v) =

Since the coefficient of v% is positive, d(») < 0 if and only if v is between the two real rootsof the equation d(v) = 0. These roots are

I t follows that with probability (1 — a), On(x) is in the band

h~{FJx)} < 0n(x) < h+{FJx)}for all x G {x: a < Fm(x) < 6}.

Applying Theorem 1, we have shown the following.

Remark 2. Let ~pTF_o(WN < K) = 1— a; then the level (1 -a ) simultaneous confidenceband for A(x) based on Ws with ^(f) = {f(l - «)}* is

[C-itA-^^)}] - x, G?\h+{FJx)}] -x), xe{x:a< FJx) < 6}.

We refer to this hand as the W band and write it \W+ (x), W*(x)). As with the S band, it iscomputed from the order statistics by using (5). Monte Carlo values of K are given byCanner (1976) when a = 1 — 6 = 0. Figure 1 gives this band for X-samples and F-samplesfrom N(0,1) and N(l,4) distributions, respectively. Here m — n= 100, a = 0-05, andK = 3-02 is obtained from Canner.

For a > 0, b < 1, asymptotic critical values K are given by Borokov & Sycheva (1968).A third band can be obtained by considering the Renyi statistic

This statistic is reasonable when one wants to give more weight to smaller X'B. If the X'aand Y'B are lifetimes, small X'B correspond to high risk members of the population. If onewants a band which is accurate for these x's, the band based on BN could be considered.The inversion of this statistic is straightforward.

Let r denote the level a critical value for BN and define

then the R band [R^{x), R*(x)) is obtained by substituting h% for h^ and h* in (5); x isrequired to be in {x: Fm(x) > c}.

Asymptotic critical values r can be obtained from the tables of Renyi (1953).



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-3 _ 2 - l 0 1

Fig. 1. Synthetio data; the level 0-96 3 band, solid line, and W band, dashes.

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426 KJKLL A. DOKSTJM AND GERALD L. SIEVEBS

3 . COMPABISON OF THE NONPABAMETKIO BAITDS

We compare the bands in terms of their widths and their limiting widths

wuJx)=MiGz*[h*{Fm{x)}]-MiW\h,{FJz)}l wa(x) = lim wMia(x),

where the limit here and below is in probability. When computing this limit, it is convenientto introduce the notation u = Fm(x) and e = M~t. Moreover, to emphasize the dependenceon e, we write H+(e, u) for h+(u) and H*(e, u) for h*(u). Now

(e, a)} -l{G-I(u)-G-1(u)}

s

i-l

say.The limit of wMa(u) is infinite unless G is strictly increasing at C~1(«). We make this

assumption; thus i6 = 0.Suppose furthermore that G has a continuous, nonzero derivative g. Using the arguments

of Doksum (1974, pp. 272-3) and a random change of time argument (BiUingsley, 1968,pp. 144-6), we find that 73 + 74 converges in law to

Ai[U{F(x)}lg{&(x) + x}- U{F(x)}lg{A(x)+x}]

on every interval [F-^S), F-^l — d)] (0 < d < J), where U denotes the Brownian Bridgeon D[0,1], and A = lim (m/N) (0 < A < 1). It follows that I3 + /4 converges in probability tozero. We add the assumption that Z7*(e, u) and H*(e, u) converge to u as e->- 0 +, and thatH+(e, u) and H*(e, u) have right-hand derivatives A#(0, u) and A*(0, u) with respect to e ate = 0 that are continuous in u.

Then we can compute the limits of Ix and I% as follows:

lim I lim g-J{g*(*.tt)}-g-J{g*(0,tt)} h*{0,F(x)} K{0,F(x)}JS^'iSi < " g[G->{F(x)}Y !^h

To summarize, we have Theorem 2.

THEOREM 2. Under the above stated conditions, the. asymptotic toidths of the bands (4) aregiven by

h*{0,F(x)}-K{0,F(x)}a

Let the asymptotic relative efiiciency of two bands be the square of the ratio of thereciprocals of their asymptotic widths. Thus for two bands based on functions {A* («), A*(u)}and {k+ (u), k *(«)}, we have that the efficiency is

Tk*{o,F{x)}-K{o.nx)}yeh-k{X}~[h*{0,F(x)}-hm{0,F{x)}\-

If we think of this efficiency as a function of the quantiles xq = F^q) oiF, we see that it isindependent of the form of F and G.

In the case of the >Sf band, H^,(e,u) = u—KSaesJidH*(e,u) = u+Kg, ae; thus its asymp-totic width is

or



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Plotting ivith confidence 427

In the case of the X band, we replace c by j£ae* in (7) and differentiate with respect to eto obtain At(0, u) and A*(0, u). This yields

as the asymptotic width of the W band. Thus the asymptotic efficiency of the 8 band to theW band is

*? 2(1-3) (a < g < 6),

where ifs>a and -ff^,, denote the asymptotic critical values of the DN and WN statistics,respectively.

Using the results of Borokov & Sycheva, we obtain the approximation to the asymptoticKWa given in Table 1. Combining this with the asymptotic tables for DN, we obtain theefficiencies given in Table 2. The values for q > 0-5 are the same as those for 1 — q. Differentvalues of a yielded similar results. We find that the 8 band is better in the centre of the Xdistribution at the expense of being worse in the tails up to + xa. Technically, the asymptotioefficiency is infinite for x outside ± xa. However, the 8 band is only valid for Fm(x) in(K8JJM, 1-KgJjM). When m = n and a = 0-01, K8JJM > 0-1 forn < 53i, while fora = 0-1, K8iJ^M > 0-1 for n < 297.

Table 1. Asymptotic critical values of WN; b = I — a,\a 0-2 0-1 004 002

0-26 2-109 2-482 2-879 31380-1 2-482 2-789 3-138 3-371

Table 2. The asymptotic efficiency esw^cg) of the 8 bandto the W band when a = 0-1

a \0 1

a \0-1

0-25

0-78

0 1

0-47

(a) a =

0-30

0-87

(6) a =0-2

0-83

= 1 - 6

0-35

0-94

= 1 - 60

1-

. 0-

= 0•3

09

25

0-40

100

•10-4

1-25

0

1-

45

02

0

1-

•5

30

0

1-

•50

04

Table 3. Approximate values of the asymptotic efficiency, a = 1 — b = 0\g . 0:01 005 01 0-2 ' 0-3 0-4 0-5o\

0050-01

0-060-06

0-260-24

0-500-46

0-890-82

1-17108

1-341-24

1-391-29

Canner (1975) gives Monte Carlo critical values of WN with o = 1 — 6 = 0 whenn = m = 2000 (a = 0-06) and n = m = 1000 (a = 0-01). Using these in place of asymptoticcritical values, we obtain the approximations in Table 3. These values are close to thevalues of Table 2 with a = 0-1.

We also computed the efficiencies for finite sample sizes, i.e. reciprocal ratios of widthsfor actual samples, using Canner's critical values and computer generated samples. Someresults are given in Table 4 for X samples from a N(0,1) distribution and T samples froma N(l, o~l) distribution. These results are qualitatively close to the asymptotic results but



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428 KJELL A. DOKSTJM AND GERALD L. SIEVEES

favour the W band more. They vary little from sample to sample or from distribution todistribution.

Taken together, the tables show that in terms of width the W band is preferable to the 8band. The 8 band is better in the centre of the X distribution at the expense of being muchworse in the tails. This is also clear from Fig. 1. I t is interesting to note that in this figure,the W band leads correctly to the rejection of a shift model while the 8 band does not; see(iii), § 1. The 8 band has the advantage of being simpler and its critical values are moreextensively tabulated. I t is preferable if the central part of the X distribution is of moreinterest than the tails.

Table 4. Finite sample size efficiency of the 8 band to the W band; a = 0-05

0-620

<r,\0-520

0-1

oo/oooo/oo

0-2

00

(a) m = n = 50

0-3 04 0-5

00

0-610-61

1-011-18

0-6

1-001-00

0-05

oo/oooo/oo

0-1

00

0-2

00

(6) m = n = 100

0-3 0-4 0-5

0-870-80

0-97090

1-181-14

0-6

1-181-19

0-7

0-801-09

0-7

0-760-66

0-8

0-650-66

0-8

oo/oooo/oo

0-9

oo/oooo/oo

For the R band, Theorem 2 yields the following asymptotic width

(c < q < 1),wR,x\

where K^a is the asymptotio critical value of the Renyi statistic RN. Using the results ofRenyi (1953), we obtain the asymptotic K^a of Table 5. Combining this with the corre-sponding entries of Table 1, we compute the efficiencies of Table 6. The W band is preferableto the R band except when only small xg are of interest.

Table 5. Asymptotic critical values of RN

0-2 0 1 0-05\ c<z\

010-05

3-9214-484

6-8816-726

8-5469-772

Table 6. Asymptotic efficiency of the R band to the W band;a = 0-l,o = 1-6

005 0 1 0-2 0-25 0-3 0-4 0-5

0-25 0-2 oo oo oo 1-20

0-6 0-7

a0-1 0 0 5

0-100

00

0-962-02

0-430-90

0-320-67

0-250-52

0-160-34

0110-22

0070-15

0-050-10

0-93 0-60 0-40 0-27 0-17

4. THE LO0ATI0N-80AIiB MODEL

In this section, we consider the model where



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for some continuous distribution function H. Then O~x{u) = fi2 + cr^'^u) and

A{x) =/ii + <ri (x - II^)\<TX - x.

In this model it is common to treat fiz—/i^ as the parameter of interest. However, asseen in § 1, A(a;) will yield additional information about how the populations differ.

Since A(x) is linear in x, a simultaneous band can be constructed by specifying intervalsof values for A(a;) at just two points x. Thus if xx < xt and

pr{?; < Ate) < T*, 8* < Ate) < 8*} = l - « ,

then for x e [a , x^], the upper boundary would consist of the line connecting the pointste,T*) and (x^S*), while the lower boundary would be the line connecting (a ,T+) andte, 8+). For x > x%, the upper boundary would be the line through the points (a , T+) andte>#*)> ftnd so on. Here xx and x% may be random; in the following, they are orderstatistics.

Remark 3. Suppose that a location-scale model holds. Let rk, ik and sk (k =1,2) beintegers such that

< X(ik) < Y(sk) {k = 1,2)} = 1 - a. (8)

Then a simultaneous (1 — a) confidence band for A(a;) is determined by

Y(rk) - X(ik) < A{X(it)} < T(sk) - X(ik) (jfc = 1,2). (9)

The choice of integers rk, ik and ak satisfying (8) could be made using the bivariate hyper-geometric distribution, although convenient tables do not seem to be available.

Alternatively, Switzer (1976) has suggested a conservative procedure using the Bon-ferroni inequality and the hypergeometric distribution. A third possibility, when samplesizes are large, is to use a bivariate normal approximation. To do this, let Zx be the numberof Yj < X(%i), and Zt the number of Yj > X(ia). Then the probability in (8) equals

< Z2 «s n-rt),

and can be approximated using Lemma 1.

LEMMA 1. For 0 < & < y?2 < 1, let ik = [mfid + 1 (k = 1,2), let gx = filt g% = l - / ? 2 ,and let

If F = O, then (Vv Vt) has a standard bivariate normal limiting distribution with correlation

The lemma is proved by noting that the conditional distribution of (Zu Z2) given X(ij),X(»a) is trinomial and after standardization, asymptotically bivariate normal. SinceE{Zk\X(ik) (i = 1,2)} is asymptotically normal, an application of Theorem 2 of Sethuraman(1961) or a bivariate version of Hajek & Sidak (1967, problem 6, p. 195) gives the result.

The following remark gives a particular application of the lemma.

Remark 4. Let tx = [m/3] +1 and »2 = [m(l— /?)] + l for some/? e(0, £). Then an asymptotic(1 — a) simultaneous confidence band for A(x) is determined by (9) with

for r2 = n+1-8 X and «2 = n + 1 —f, where ca satisfies pr {|P | < ca (k = 1,2)} = I—a for(Vv V2) standard bivariate normal with correlation p = — /?/(! — fi).



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430 K J E L L A. DOKSUM AND GERALD L. SEEVEBS

If LM(x) denotes the width of this band at x, and if H has a density h, then

= L(x),

say, for x = <rx zp +fa with fi < p < 1 — /? and zp the pth quantile of H. The convergence isin probability.

If the density h(x) is symmetric about 0, then

L(x) = 2ca{/?(l -

We call the band determined by (9) and (10) the 0 band, the order statistics band, andcompare it with the W band of §2. In the location-scale model, the asymptotiowidth of the W band is 2KW a{p(l -p)}^a-Jh(zp). When H is normal, we obtain the asymp-totic efficiencies given in Table 7. These efficiencies will not be much different for otherreasonable' bell-shaped' h. Thus we conclude that if a location-scale model can be assumed,a considerable gain in efficiency is possible by using the 0 band provided only that h(zfi)and h(z1_^) are not too close to zero.

Table 7. The asymptotic efficiency of the 0 band to the W bandwith a = 1 — b = ft in the normal model 0; a = 0-1

0-10 0-20 0-26 0-30 0-40 0-60

1-301-64

1-221-63

11

•13•42

1101-38

010 205 1-430-26 — —

5. THE NOEMAL MODEL

If we let H in the location-scale model be ^(0, 1), then we have the Behrens-Fishermodel. In this case we can write A(x) = ax+b — x, where o = crij(Tx and 6 = /t,—/tj aajo'1.

In the case that m = n, we will construct the likelihood ratio test for the hypothesisHQ : a = o0, b = 60 for fixed o0 e (0, oo) and b0 e (— oo, oo). The collection of all (a,,, 60) that isaccepted by this test for a given set of data provides a confidence region for (a, b) that is anellipsoid. This ellipsoid will be translated into a likelihood ratio simultaneous confidenceband for A(x).

If L denotes the likelihood function, then

The maximum for unrestricted fa, /ij, <rlt <Jt is well known. Substituting a\ = o§ erf andfi2 = b0 + a0fa, the maximum of L under H^ is found using standard methods. Let (x, y, S\, S\)denote the usual unrestricted maximum likelihood estimate of (fa, fit, a\, <rj), then thelikelihood ratio is

where A =fn/iV.The space of (fa, fax, crv <rt) with cra = a0 ax and fa = bo + aofa is linear with 2 dimensions.

Thus, under H^, by classical results, — 2 log KN has a limiting chi-squared distribution withtwo degrees of freedom.

Let xa(l - a) denote the (1 — a)th quantile of this distribution and let



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then the acceptance region of the test when m = n and A = $ is

Write 6Q = bo + aoz; then the collection of (a,,, 6J) for which fij, is accepted is the inside ofan ellipse, namely

S Z S ) « £« »l). (11)

Let ( denote this ellipse arid let

then [#"(£), ^(a;)] is the level (1 — a) likelihood ratio confidence band for A(i). Theorem 3specifies S±.

THEOBEM 3. When n = m, the level (1 — a) likelihood ratio confidence band for A(x) innormal models is given by

Proof. We use the Cauohy-Schwara inequality in the form

ISa^ i l < (Sa?Eu>$)*, (12)

with equality if and only if wt is proportional to a^. Simply note that (11) is equivalent tow{ + w\ ^d2, where wx =

If we apply (12) with a^ = — {x — x)l(^2S-^ and ctj = 1, the result follows.

Remark 5. The above proof is very similar to the derivation of Scheffe"'s simultaneousconfidence intervals for contrasts. The interpretation is also similar: if the likelihood ratiotest of HQ : A(x) = 0 for all x rejects, that is o = 1, 6 = 0, then for some x the band doesnot contain 0, and vice versa.

When n + m, we can use the maximum likelihood estimate

Mx) = y + ^(x-x)-x

of A(x) to obtain a confidence band for A(a;). Let TN{x) = Mi{A(x) — A(x)}. By using aTaylor expansion of 2^(3;) in terms of X, V, Sx and Slt we find that TN(x) converges in law toT(x), say, equal to (T^^ + tVJ^), where t = (x-/h)l(Ti a n ( i \ an(* 2 a r e independentstandard normal variables. Write

T1 = var

Then we can use'supx|?^r(a;)[/f as a pivot to obtain the following maximum likelihood band.

THEOEBM 4. An asymptotic (1 — a) simultaneous confidence band for A(x) is given by

for ail x, where x*(l — a) is the (1 — a)th quantile of the x\ distribution.

Proof. Now TN{x)jf converges in law to the process (^ + <Ty^/2)/(l + ^2) i . Hence8Mpx\TN(x)/f converges in law to

say. By considering the equation l'(t) = 0, we find that for almost all {Vlt Vt), the maximum



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432 KJELL A. DOKSUM AND GERALD L. SEEVEBS

of |/(<)| is attained at i = ^iVJVlt a Cauohy random variable. Thus the maximum is (Ff + F})i,which is the square root of a #5 variable. The result follows.

By standard asymptotic theory, the likelihood ratio and maximum likelihood bandsshould be asymptotically equivalent. This can be established directly when n = m by usingthe first two terms in the Taylor expansion of e* about z = 0 in the expression

Ka = exp{s»(l -a)IN}.

The likelihood ratio band is preferable since it is based on a more accurate approximation.The above expansion also yields the following.

Remark 6. The likelihood ratio and maximum likelihood bands both have asymptoticwidth 2o-2a;(l-a)(l + #2)*, where* = {x-fij)/^.

I t is interesting to compare this asymptotic width with the asymptotic widths of thegeneral methods of the previous sections to find out how much these general methods lqseif in fact the correct model is normal.

We see that the asymptotic relative efficiency of the S band to the likelihood ratio bandin normal models is

(width L band)'eg>L(xa) - ^ (width,S band)'

_ <f>*(t) s»(l - a) (1 + m (1 + #*) s»(l - a)

where <f>(t) denotes the standard normal density. Similar expressions hold for the W and Obands. Some numerical results are given in Table 8. The efficiency of the S band is surprisinglylow, much smaller than the familiar 2/TT = 0-64.

Table 8. Asymptotic efficiencies of the bands W, 8 and 0 with respectto the likelihood ratio band in normal models, t = ^^(p), a = 0-10, a = 1 — b

Band p = 0-1 p = 0-2 p = 0-3 p = 0-4 p = 0-5

W, a = 01 0-37 0-39 0-39 0-38 0-38W, a = 0-25 0 0 0-49 0-48 0-483 017 0-33 0-43 0-48 0-49O, /? = 01 0-75 0-56 0-47 043 0-41O, P = 0-25 — — 0-75 0-68 0-66

Table 9. Finite sample size efficiency of the W band with a = 1 - b === 0relative to the likelihood ratio band, a — 0-05 and m — n

\ p 01 0-2 0-3 0-4 0-5 0-6 0-7 0-8 0-9n <r,\

50 0-6 0 0-31 0-34 0-46 0-63 0-62 0-44 0 02-0 0 0-35 0-27 0-48 0-53 0-57 0-66 0 0

100 0-5 0-31 0-33 0-28 0-46 0-73 0-51 0-60 0-31 02 0 0-33 0-36 0-27 033 0-66 0-52 0-57 0-34 0

We also give in Table 9 some finite sample size 'efficiencies' computed for the sameN(0, 1) and N(l, a\) samples as in § 3. By multiplying the entries of this table with theentries of Table 4, we get the corresponding table for the S band. The asymptotic efficienciesevidently give a good indication of the finite sample size performance of the bands.

The results show that the< general bands are quite inefficient if the correct model isnormal. On the other hand, the bands designed for the normal model are quite sensitive tothe normality assumption in the sense that skewness or high kurtosis in the F and O dis-



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tributions will alter the level of the band. Also note that the 0 and likelihood ratio bandscannot be used to test whether a location-scale model holds. Finally, the general methodsW, 8 and 0 have the advantage that they can be applied to censored data.

6. AN ILLTTSTBATION

Doksum (1974) gives an example involving experimental data where a linear responsefunction and thus a location-scale model is indicated, and where the response functionshows that high risk members of a guinea pig population are affeoted entirely differentlyby a tubercle bacillus dose than low risk members.

Here we give an illustration involving data from an experiment designed to studyundesirable effects of ozone, one of the components of California smog. One group of 22seventy-day-old rats were kept in an ozone environment for 7 days and their weight gainsyt noted. Another group of 23 similar rate of the same age were kept in an ozone free environ-ment for 7 days and their weight gains x noted. The results, provided by Brian Tarkington,California Primate Research Center, University of California, Davis, are given in Table 10.

Table 10. Weight gains oftvx) groups of rats in grams

ControlOzone

ControlOzone

41010-1

21-8140

38-46-1

15-46-6

24-420-4

27-4121

25-7-

19-16-

93

27

21-914-3

22-439-9

18-315-6

17-7-16-9

131- 9 - 9

26-054-6

27-36-8

29-4-14-7

28-528-2

21-444-1

-16-917-9

26-6- 9 0

26-- 9 -

22-

00

7

17-- 1 2 -

49

20 -

-10 10 20 30 40

Fig. 2. The estimate A(«) and the level 0-90 S band for theresponse function in the ozone experiment.

Figure 2 gives the estimate A(x) = O^F^x)} — x and the #-band with exact level 0-90.I t is fairly obvious that x = — 16-9 is an outlier. If it is removed, only slight changes occurin the graphs beyond the removal of the long straight lines on the left of A and 8*. Then Asuggests that, even though ozone reduces average weight gain, large weight gains are madeeven larger. Also 8* shows that weight gain is reduced significantly for x below the controlweight gain 22-5.

From the 8 band, we cannot reject a shift model assumption, even though A stronglyindicates that it does not hold. This may be because the sample sizes are too small leavingthe band too wide. The W band would be preferable, but we do not have the critical valuesfor the sample sizes of this experiment.

Now A with the outlier left out indicates that the response could well be linear and thus



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434 K J K L L A. DOKSUM AND GERALD L. S I E V E B S

the narrower 0 band could be used. For sample sizes m = n = 22, a good choice of y? is0-296 and then the 0 band is determined by

-37-3 ^ A(21-4) *S -7-4, -19-3 < A(26-6) sj 28-0.

Thus, for this experiment, the 8 and 0 bands are remarkably close and the shift modelassumption can still not be rejected.

We are indebted to P. Nanopoulos for programming the plots and the finite sample sizeefficiencies. The paper was prepared with the partial support of the National ScienceFoundation and the Norwegian Research Council. Doksum's work was partly donewhile visiting the Department of Mathematics, University of Oslo, and Sievers's workwhile visiting the Department of Statistics, University of California, Berkeley.

REFERENCES

BnimGBUDY, P. (1968). Convergence of Probability Measures. New York: Wiley.BOBOKOV, A. A. <fc SYOKBJVA, N. M. (1968). On asymptotically optimal non-parametrio criteria.

Theory Prob. Applie. 13, 369-93.CANKBB, P. L. (1976). A simulation study of one- and two-sample Kolmogorov-Smirnov statistics

with a particular weight function. J. Am. Statist. Assoc. 70, 209-11.DOKSUM, K. A. (1974). Empirical probability plots and statistical inference for nonlinear models in

the two sample case. Ann. Statist. 2, 267-77.HAJKK, J. <fc Smix, Z. (1967). Theory of Rank Tests. New York: Academio Press.PBABSOK, E. S. & TTA-RTT.THV, H. O. (1972). Biometrika Tables for Statisticians, Vol. 2. Cambridge

University Press.RBKYI, A. (1953). On the theory of order statistics. Ada Math. Acad. Sci., Hung. 4, 191-231.SBTHUWAMATf, J. (1961). Some limit theorems for joint distributions. SankhyA A 23, 379-86.SarEOK, G. P., ^T rwint, W. J. & WILLIAMS, R. E. (1974). Estimation of parameters in acceleration'

models. In Proc. Annual Reliability and Maintainability Symposium. New York: IJE.E JE.SWTTZEH, P. (1976). Confidence procedures for two sample problems. Biometrika 53, 13-26.WXLK, M. B. & GNANADBSTKAK, R. (1968). Probability plotting methods for the analysis of data.

Biometrika 55, 1-17.

[Received February 1976. Revised May 1976]



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plotting with confidence: graphical comparisons of two populations

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