REVERSE HAZARD 7269 period effects and these cannot always safely
be neglected [4].
A fully worked example with three peri-ods and three treatments on 12 cows in one block of six and two blocks of three was given by Lucas [7], though this in fact used unifor-mity trial∗data. A more realistic example of a design with three periods and three treat-ments on six groups of lactating cows was described by Smith [8]. The treatments here were three methods of preparing alfalfa for feeding to the cows, and the records were vitamin A potency per pound of butter fat and milligrams of carotene intake, vitamin A being adjusted by carotene in an analysis of covariance∗.
Reversal designs may have more than three periods, and Brandt also considered four periods. An experiment to compare the responses of two groups of rabbits to injec-tion with different insulin mixtures used a fourperiod reversal design [3]. Blood samples were taken at various times after injection and blood sugar levels determined for each sample.
Reversal designs may be of use in any situ-ation where living subjects receive a sequence of treatments. Changeover designs are com-monly used in clinical trials∗, most often with only two treatments and two periods; however, more than two treatments or peri-ods, or both, can be used. Although reversal designs are available in this context, recent reviews [1,6] do not explicitly mention them. As has been shown above, the theory of the designs has been known for very many years, but practical application appears to have lagged behind theory. Taylor and Arm-strong [9] compared the results from reversal designs and other changeover designs for dairy husbandry trials conducted at various stages during lactation. They concluded that reversal designs with p periods were more efficient than other changeover designs when the yields of milk or fat from successive peri-ods closely conformed to a curve of degree (p− 2), but that otherwise some other form of changeover design was more efficient. Ker-shner and Federer [5] compared three- and four-period reversal designs for two treat-ments with extra-period changeover designs for the same number of periods and treat-ments. Under a model allowing for direct
and residual treatment effects they showed that reversal designs gave higher estimates to the variances of contrasts and so did not recommend them.
REFERENCES
1. Bishop, S. H. and Jones, B. (1984). J. Appl.
Statist., 11, 29–50.
2. Brandt, A. E. (1938). Res. Bull. 234, Iowa State Agricultural Experiment Station. (The paper that introduced these designs, and the leading reference.)
3. Ciminera, J. L. and Wolfe, E. K. (1953).
Biomet-rics, 9, 431–446.
4. Cox, C. P. (1958). Biometrics, 14, 499–512. 5. Kershner, R. P. and Federer, W. T. (1981).
J. Amer. Statist. Ass., 76, 612–619.
6. Koch, G. G., Amara, I. A., Stokes, M. E., and Gillings, D. B. (1980). Int. Statist. Rev., 48, 249–265.
7. Lucas, H. L. (1956). J. Dairy Sci., 39, 146–154. 8. Smith, H. F. (1957). Biometrics, 13, 282–308. 9. Taylor, W. B. and Armstrong, P. J. (1953).
J. Agric. Sci., Camb., 43, 407–412. (Best
comparison of reversal designs with other changeover designs.)
See also CHANGEOVERDESIGNS.
G. H. FREEMAN
REVERSAL TESTS (FOR INDEX
NUM-BERS).
See INDEXNUMBERSREVERSE HAZARD
Although the reverse-hazard rate was implic-itly introduced in the work of Lynden-Bell in 1971 [6] and was later studied in Woodroofe [8], the term first appeared explic-itly in 1988 [5]. It has also been called retro-hazard [1]. As will become clear, it extends the concept of hazard function to a reverse time direction. This function naturally arises in the analysis of right truncated observa-tions, and recent attention to such models is partly due to their applicability to certain data that arise in AIDS research. We first define the reverse-hazard function.
Suppose X is a random variable with distribution function (d.f.) F(x)= P(X x).
7270 REVERSE HAZARD
Without loss of generality assume that X is nonnegative. If X is absolutely continuous with probability density function (pdf) f(x), the well-known hazard function λ(x) of X is defined as λ(x)= f (x)/[1 − F(x)], and the inte-grated hazard ∧(x) =0xλ(u)du is known as the cumulative hazard function. The cumu-lative hazard function plays a key role in the analysis of censored data∗, which is typical in survival analysis∗; see HAZARDRATE AND OTHER CLASSIFICATIONS OF DISTRIBUTIONS. For right truncated data, the reverse hazard function defined below is of similar impor-tance:
1. Suppose X is continuous with pdf f(x).
The reverse-hazard function r(x) of X is defined as r(x)= f (x) F(x) = lim δx→0 P(x− δx < X x|X x) δx
2. Suppose X is a discrete random variable
with probability mass f (xi) at the points
x1< x2<· · ·. Then the reverse hazard is defined as
r(xi)=
f (xi)
F(xi)
.
From the foregoing definitions, it is seen that the reverse-hazard function r(x) describes the probability of an immediate ‘‘past’’ failure, given that the item has already failed at time x, as opposed to the immediate future failure described by the hazard function above. A generalization to mixed discrete–continuous distributions is omitted for reasons of space. A further generalization to bivariate data, where discontinuities are also allowed, has been made [2,3].
Note that the reverse-hazard function also uniquely determines the d.f., as established by the following expressions. Let
R(x)=
x
dF(u) F(u). Then, for the continuous case,
F(x)= exp − ∞ x r(u)du = exp[−R(x)], (1)
and for the discrete case, letting r(x)= −[R(x) − R(x−)], we have
F(x)=
i:xi>x
[1− r(xi)]. (2)
The reverse-hazard function naturally arises with right-truncation models. To see this, suppose that, instead of a random sam-ple, one observes the random pairs (Xi, Ti), i= 1, . . . , n, for which Xi Ti, where T is a con-tinuous, nonnegative random variable inde-pendent of X with unknown d.f. G(t). Then X is said to be right truncated by T. Such data are observed in transfusion-related AIDS cases, where X denotes the incubation period, i.e., the time from infection with HIV to the onset of the disease. If the observation period is terminated at time Te, then only those individuals for whom X T = Te− s are observed, where s is the time of diag-nosis with AIDS. Another application is to modeling the reporting lag distribution in insurance [4]. Let F∗(x) and G∗(t) be the d.f.’s of the observed X and T, so that
F∗(x)= P(X x|X T), G∗(x)= P(T x|X T). Then R(x)= ∞ x dF(u) F(u) = ∞ x dF∗(u) F∗(u)− G∗(u−). (3) A direct nonparametric estimator of the integrated reverse hazard R(x) can now be obtained from the observed truncated data. This also leads to an estimator of F(x) via (2). In particular, for 0 x < ∞ we have
Rn(x)= i:xi>x 1 nCn(Xi) ,
where Cn(x)= Fn∗(u)− G∗n(u−), and Fn∗(x) and
G∗n(x) are the empirical distribution func-tions (see EMPIRICALDISTRIBUTIONFUNCTION (EDF) STATISTICS) of the observed X and T, respectively. Considering the discrete nature of the empirical distribution functions, we
R´EV´ESZ ESTIMATOR OF REGRESSION FUNCTION 7271 obtain Fn(x)= i:xi>x [1− rn(xi)] = i:xi>x 1− 1 nCn(xi) , (4)
where rn(x)= −[Rn(x)− Rn(x−)]. The estima-tor (4) was first derived by Lynden-Bell [6] as the nonparametric maximum likelihood esti-mator of F in the context of an application in astronomy, where the observation of some of the celestial objects was prevented due to selection bias. Further asymptotic properties of (4) have been derived [8]. A smooth non-parametric estimator for r(x) can be obtained via kernel methods [7]. On the other hand, nonparametric estimation of the hazard rate λ(x) for the right-truncation model involves technical complications, due to the difficulty of consistently estimating the distribution function in the tails of the distribution.
REFERENCES
1. Gross, S. T. and Huber-Carol, C. (1992). Regression models for truncated survival data.
Scand. J. Statist., 19, 193–213.
2. G ¨urler, ¨U. (1996). Bivariate estimation with right truncated data. J. Amer. Statist. Ass., 91, 1152–1165.
3. G ¨urler, ¨U. (1997). Bivariate distribution and the hazard functions when a component is randomly truncated. J. Multivariate Anal., 60, 20–47.
4. Kalbfleisch, J. D. and Lawless, J. F. (1991). Regression models for right truncated data with application to AIDS incubation times and reporting lags. Statist. Sinica, 1, 19–32. 5. Lagakos, S. W., Barraj, L. M., and
DeGrut-tola, V. (1988). Nonparametric analysis of trun-cated survival data with application to AIDS.
Biometrika, 75, 515–523.
6. Lynden-Bell, D. (1971). A method of allowing for known observational selection in small sam-ples applied to 3CR Quasars. Monthly Notices
R. Astron. Soc., 155, 95–118.
7. Uzuno ˇgullari, ¨U and Wang, J. -L. (1992). A comparison of the hazard rate estimators for left truncated and right censored data.
Biometrika, 79, 297–310.
8. Woodroofe, M. (1985). Estimating a distribu-tion funcdistribu-tion with truncated data. Ann. Statist.,
13, 163–177.
See also HAZARDRATE ANDOTHERCLASSIFICATIONS OF
DISTRIBUTIONS; LYNDEN-BELLESTIMATOR; and
SURVIVALANALYSIS.
¨
ULK ¨UGURLER¨
REVERSE MARTINGALE.
See M ARTIN-GALESREVERSION, COEFFICIENT OF
This term was coined by Galton∗[1] to describe the ‘‘exceedingly simple law connect-ing parent and offsprconnect-ing seeds.’’ The current name for it is regression coefficient∗.
REFERENCE
1. Galton, F. (1877). Proc. R. Inst., 8, 282–301.
BIBLIOGRAPHY
MacKenzie, D. A. (1981). Statistics in Britain
1865–1930. Edinburgh University Press,
Edin-burgh, Scotland.
See also GALTON, FRANCIS; LINEARREGRESSION; and REGRESSION(Various).
R´EV´ESZ ESTIMATOR OF REGRESSION
FUNCTION
Let (Xi, Yi)(i= 1, 2, . . . , n) be n mutually independent pairs of random variables with ranges of variation 0 Xi 1 and −∞ Yi +∞, respectively. The regression function∗of Y on X is
r(x)= E[Y|X = x].
R´ev´esz [2] suggested the following non-parametric estimator of this function:
Take an arbitrary function r0(x), trans-forming the interval [0, 1] into R (i.e., −∞ to ∞), and define {rn(x)} by the recursive relation rn+1(x)= rn(x)− 1 (n+ 1)an+1 + K x− Xn+1 an+1 (Yn+1− rn(x)),
