Deconvolution of Microfluorometric Histograms with B Splines

Deconvolution of Microfluorometric Histograms with B SplinesAuthor(s): John Mendelsohn and John RiceSource: Journal of the American Statistical Association, Vol. 77, No. 380 (Dec., 1982), pp. 748-753Published by: American Statistical AssociationStable URL: http://www.jstor.org/stable/2287301 .Accessed: 16/06/2014 08:03

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

American Statistical Association is collaborating with JSTOR to digitize, preserve and extend access to Journalof the American Statistical Association.

http://www.jstor.org

This content downloaded from 185.2.32.152 on Mon, 16 Jun 2014 08:03:35 AMAll use subject to JSTOR Terms and Conditions

http://www.jstor.org/action/showPublisher?publisherCode=astatahttp://www.jstor.org/stable/2287301?origin=JSTOR-pdfhttp://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/page/info/about/policies/terms.jsp

Deconvo ution of Microfi uorometric Histog rams

With B Splines JOHN MENDELSOHN and JOHN RICE*

We consider the problem of estimating a probability density from observations from that density which are further contaminated by random errors. We propose a method of estimation using spline functions, discuss the numerical implementation of the method, and prove its consistency. The problem is motivated by the analysis of DNA content obtained by microfluorometry, and an example of such an analysis is included.

KEY WORDS: Deconvolution; Probability density estimation; Regularization; Splines.

1. INTRODUCTION

Suppose that Y1, Y2, . . ., Y,, are observations from a probability distribution with density g that satisfies

g(s) = f w(s, t) f(t) dt, (1.1)

where w(s, t) is a given probability density for each t, and that f(t) is an unknown density. From Y1, Y2,

Yn we construc-t a nonparametric estimate, gn, of g (Tapia and Thompson 1978), and we wish to "invert" (1.1) and construct an estimate of f. This problem arises when one is not able to record observations from f, but observations from f subject to further random error. Our work was motivated by an application arising from the analysis of cell DNA content; various other applications are discussed in Medgyessy (1977), for example.

We write (1.1) symbolically as g = Af and we assume throughout that the solution is identifiable in the sense that there is only one density f satisfying g = Af. For- mally, f = A - lg. The practical difficulty in solving the problem is that typically high frequency variations in f can give rise to only small perturbations of g; thus, small perturbations incurred in measuring g will give rise to large fluctuations in the solution. This is a classical case of an "ill-posed" problem (Tikhonov and Arsenin 1977).

For example, in the convolution case, where w(s, t) = w(s - t), it is easy to see that if one formally divides the empirical characteristic function corresponding to g, gn say, by known characteristic function wi corresponding

* John Mendelsohn is Associate Professor of Medicine and Director of the UCSD Cancer Center, University of California at San Diego. John Rice is Associate Professor, Department of Mathematics, Uni- versity of California at San Diego, La Jolla, CA 92093. Research was partially supported by NSF Grant MCS-7901800 AOI and NIH Grants CA-2666 and CA-1 1971. The manuscript was prepared while the second author was on sabbatical leave at the U.S. National Bureau of Stan- dards. The authors acknowledge the thorough and helpful comments of the referees and an associate editor.

to w and then transforms to estimate f, the variance of the estimate diverges. This is because wi tends to zero rapidly whereas gn does not.

The remainder of this article is organized as follows: a proposed approach to the problem is presented in Sec- tion 2. In Section 3 we discuss the DNA problem and present the results of some computations. Section 4 consists of a discussion in which the method is compared to some other techniques. An appendix contains some results on the consistency of the method.

2. DESCRIPTION OF THE METHOD

We approximate f from a finite dimensional family of densities 9?p of dimension p, say, choosing the approximate t to minimize 11 gn, - Af 11. In our application we use the L2 norm (with respect to Lebesgue measure). The instability of solving (1.1) is controlled by choosing p to be small relative to n. As p increases the bias of the estimate f will decrease but the variance will increase.

The family ?,p should be chosen to have good approximation properties and should be convenient computationally. For these reasons we choose to approximate f by a spline function with fixed knot locations. The B- spline basis for splines with a given knot sequence is especially convenient to work with; we represent 3 as a linear combination of B splines, f(t) = -P 1 jBj(t) and thus (Af)(s) = Pj 131(ABj)(s). The coefficients I3j are determined by a least squares fit of Al to gn. Figure 1 shows the B-spline basis for cubic splines with interior knots 1.2, 1.4, 1.6, 1.8. DeBoor (1978) contains an ex- tensive discussion of B splines and their properties.

In our proof of consistency of the estimate it is important that f be a probability density, that is, that it is nonnegative and integrates to 1. Practically, too, our ex- perience shows that this is important in controlling the fluctuations of f. The practical utility of imposing nonnegativity constraints has also been noted by Wahba (1981). Asymptotic results in Wahba (1973) tend to sug- gest that the imposition of such constraints may limit the rate of convergence. Fortunately, there is a simple and convenient means of enforcing these constraints via the B-spline representation; if Bj is the jth B spline of degree k - 1 corresponding to the knot sequence TI ... , Tp+k f B/Xt) dt = (T+ - Tk)/k and B/,t) > 0 (Fig. 1). Thus the

? Journal of the American Statistical Association December 1982, Volume 77, Number 380

Applications Section

748


http://www.jstor.org/page/info/about/policies/terms.jsp

Mendelsohn and Rice: Deconvolution of Microfluorometric Histograms 749

B(x) B-Spline Basis 1.0

0.8

0.6

0.4

0.2

1.2 1.4 1.6 1.8 x

Figure 1. 8-SPLINE BASIS. For cubic splines with interior knots 1.2, 1.4, 1.6, 1.8.

solution of the constrained problem min 11g - Af 11 subject to 1 >- 0 and cTI3 = 1 (where cj T - Tj)Ik) is a probability density. We note that these constraints are sufficient, but not necessary, to require that f be a probability density, and that they may limit the convergence rate (de Boor and Daniel 1974). From another point of view, we are approximating f by a mixture of B-spline densities.

3. DNA MEASUREMENTS

Flow microfluorometry is a widely used technique for measuring such attributes of cells as DNA content, RNA content, cell size, nuclear size, and protein content. We are concerned with DNA measurements that are obtained in the following way: cellular DNA is labeled with a flu- orescent dye and a batch of labeled cells are placed in suspension. The fluid containing the cells is forced to flow through a thin tube at the rate of 25,000 cells per minute. Each cell in this stream in turn passes through a laser or other highly focused light, which causes the dye to fluoresce, the light thus emitted being proportional to the amount of dye and thus the DNA content. The pulses are picked up by a photomultiplier, sent through an amplifier and then to a pulse height analyzer and are recorded digitally. This results in a histogram of the individual cell measurements. The histogram typically consists of a few hundred channels, or bins, in which measurements of about iO5 cells are recorded.

In principle the DNA distribution consists of a discrete mass at DNA content 1, say, of cells in GI phase, a discrete mass at DNA content 2 consisting of cells in G2 + M phase-cells that have replicated their DNA but have not yet divided-and a density of cells in (1, 2)(S phase) that ar tin thue prtotce f rpatinefgn,0 their DNA

Owin to errorese from vaiable staueinte absopion,lgh scapot-oa terng electrounic noisde, and ohsther souces cothentru dis-

tribution is not the recorded distribution, however. The error appears to be approximately Gaussian (Dean and Anderson 1975) with a constant coefficient of variation usually on the order of 5-10 percent. Frequently the cells have undergone some experimental treatment and there is the additional complication that there are dead cells and other debris present with DNA content less than 1. The resulting DNA histogram must be "deconvolved" to determine the proportion of live cells in GI and G2 + M and the shape of the density in S phase. The method of flow microfluorometry allows a physician to obtain information on cell cycle patterns in an individual tumor. For further discussion of the technique and its applications we refer the reader to Dean and Jett (1974), Fried (1977), Haanen, Hillen, and Wessels (1975), Watson and Taylor (1977), and Watson (1977).

Denoting the proportions of cells in GI and G2 + M by 1PI and 12, the density of the debris (cells with DNA content less than 1) by h, and the density in S phase (cells with DNA content between 1 and 2) by f, the observable density is

g(s) = I3(A8)(s) + 2(AA2)(s) + (Ah)(s) + (Af)(s),

where 81 and 82 are point masses at 1 and 2 and s denotes DNA content. It should be noted that h and f do not integrate to 1. An example from the UCSD Cancer Center Laboratory of the first author is shown in Figure 3. 27,455 human lymphocite cells were measured. The error distribution was modeled as Gaussian (as mentioned earlier) with a constant coefficient of variation of 6.2 percent. The laboratory routinely produces many such histograms every week.

Our analysis approximates the density h, which is as- sumed to have support on an interval [a, 1] by a spline function, and the density f on [1, 2] by a spline function. These functions do not necessarily agree at 1. The left edge of the data is initially tapered smoothly down to zero. Denoting a B-spline basis of degree k - 1 in [a, 1] with a given set of knots by Cl, . . . , Cq, and a B-spline basis in [1, 2] by B1, . . ., Bp, we fit a density of the form

(si) = I3(Abl)(si) + 2(AA2)(sd) p q

+ E PI+2(ABj)(si) + E yj(ACj)(s1) j=l j=l

to the histogram data g(si), i = 1, . . , m. As mentioned, the histogram is the output of the measurement procedure. The fit is carried out by least squares subject to the constraints referred to above.

The integrals (ABj)(s), (ACj)(s) are numerically eval- uated using Simpson's rule and the fact that each B spline has finite support. We perform the computations on a VAX computer at UCSD using spline function routines from deBoor (1978). The constrained least squares minimization is done in double precision using a QR decomposition from LINPACK (Dongarra et al. 1979) and the subroutine LDP from Lawson and Hanson (1974). The



750 Journal of the American Statistical Association, December 1982

results are plotted on a Tektronix 4013- terminal. The interactive program allows the user to choose the degree of the spline and the location of the knots. In the example presented here, the knots are equally spaced. The program is written in Fortran; listings are available on re- quest, although no attempt was made to assure portability.

To obtain a smooth solution, the dimension of the approximation space should be low. Increasing the dimension decreases the bias but increases the variance, man- ifested by increasing oscillation of the solution. Figure 2 shows an approximation by cubic splines with four interior knots in the debris range and four interior knots in S phase. The smooth solid curves show the densities h and f and the bars indicate the mass (divided by 100) at G, and the G2 + M. Figure 3 shows the fit (smooth curve) to the observed distribution. We sometimes find it useful to examine the residuals, and since the counts in the histogram bins are approximately Poisson, the square root transformation stabilizes the variance. A "rootograph" (Tukey 1971) corresponding to Figure 3 is shown in Figure 4. It is seen that the fit is off primarily in the peak regions. We also calculate a goodness-of-fit statistic that is a standardized version of the residual sum of squares. Its value in this case is 26; insofar as statistical significance is at issue, there is no doubt.

The dashed line in Figure 2 shows a fit of cubic splines with four interior knots in the debris region and eight interior knots in S phase. Although this solution oscillates more than in the previous case, there is very little dif- ference in the fitted values or in the rootograph. This is due to the ill-posedness of the problem-the operator A smooths out the oscillations. The goodness-of-fit statistic decreases only slightly, to 22.5. The comparison of these

Estimated Densities f(t)

I I I I I I

A

.01

.005 _

25 50 75 100 125 150 t

Figure 2. ESTIMATED DENSITIES. The curves show the estimated densities h (debris) and f (S phase). The vertical bars show the location of G1 and G2 + M phase and the estimated mass (divided by 100 for scaling) at those phases. The solid curves show densities with 4 interior knots in the debris range and 4 interior knots in S phase. The dashed curves show densities with 4 interior knots in the debris range and 8 interior knots in S phase.

(Af)(s) Fitted Density (Af)(s)

[1,X11 L

.03-

.02-

.01

25 50 75 100 125 150 175

s

Figure 3. HISTOGRAM AND FITTED DENSITY. The solid line shows the DNA histogram. The dashed line shows a fit using 4 interior knots in the debris range and 4 interior knots in S phase.

two solutions illustrates an inherent limitation on the con- clusions that can be drawn from such data and from the solutions of ill-posed problems with random noise in general. There may be no way to rule out mathematically the possibility of a rough or oscillatory solution since when such a solution is fitted to the data after the application of the smoothing operator (A in our case), the result may be virtually indistinguishable from the result produced by a smooth solution. In choosing a smooth solution, one is invoking a prior assumption.

If the underlying model is a very good fit to the real situation, an insignificant value of the goodness-of-fit statistic may be a guide to when to stop adding knots. In our case the statistic rarely reaches the range of statistical insignificance but often does level off at values corre-

r(s) Residual Rootograph

, , , , , I I

.01

0

-.01

-.02-

25 50 75 100 125 150 175

s

Figure 4. RESIDUAL ROOTOGRAPH. Plot of the differences of the square root of the histogram values and the square root of the fitted values from Figure 3.




Table 1. Proportions of Live Cells

Degree of Spline Knots G1 S G2 + M Z

1 0,0 .32 .55 .13 167 2 0,0 .31 .57 .12 109 2 2,0 .21 .66 .13 70 3 2,2 .38 .49 .13 38 3 4,4 .39 .48 .13 26 3 4,6 .40 .47 .13 24.5 3 4,8 .42 .46 .12 22.5

sponding to extremely small significance levels. In practice we choose p by increasing its value until high frequency oscillations occur and then return to a smaller value of p. This is similar to a visual procedure of choosing a bandwidth in estimating a probability density function. The underlying model is probably inaccurate in some respects-the assumption of Gaussian errors, of constant coefficient of variation, the location of GI and G2 + M, and the assumption of independent measurement errors. It should be kept in mind, however, that with a large amount of data any model will likely be rejected. An approximation may still be reasonable and useful even though its misfit can be detected statistically.

Table 1 shows the estimates of the proportions of live cells in Gl, Sl, and G2 + M for various spline schemes. The areas change little as the number of knots in S phase is increased from 2 to 8. In a few other examples the S- phase density peaks increasingly at G, or G2 + M. This can be an indication that the locations of G, and G2 + M are incorrect, but one cannot a priori disallow the possibility that the peak is a real phenomenon. The results are generally sensitive to the location of G, and G2 + M channels; unless the channel locations can be determined precisely through calibration procedures, one must ac- cept the possibility of an error of a few percent in proportions in Gl, S, and G2 + M, since it is quite possible that cells in G1, for example, are being counted as if they were in S. (In one extreme case, shifting the G, location half a channel caused a 10 percent change in the estimated proportion of cells in Gl.) If calibrations are not available, the G, and G2 + M channels must be estimated from the data by locating peaks in the histogram. The results are somewhat less sensitive to the value of the coefficient of variation.

In the process of developing the procedure, we found it useful to run simulations with known, smooth, pdf's f, and sample sizes comparable to those encountered in practice. These simulations convinced us that we could achieve reasonable resolution.

4. DISCUSSION

Regularization is a common procedure for solving (1.1). There is a large literature on this subject, for example Phillips (1962), Tikhonov and Arsenin (1977), and Wahba (1977). This method places a penalty on rough solutions by finding a solution fs that minimizes 11 g - AfA 112 +

AQ(f), where fl(f) is a measure of roughness. A com- monly used Q is fl(f) = f [f"l(t)]2 dt. The smoothing is controlled via the parameter X.

In comparing the methods it is instructive to consider the simpler but closely related problem of estimating the function f given observations Yi = f(iln) + Ei, i = 1, ... , n. The smoothing spline (Reinsch 1967) results from the application of regularization to this problem. Rice and Rosenblatt (1981, 1982) show that the convergence rate of the smoothing spline depends on the boundary behavior of f. Similar phenomena may occur in the general problem. The convergence rate of a spline with a rela- tively small number of knots fit by least squares (the analog of the method presented here) is not affected by the bounaary behavior, however (Agarwal and Studden 1980). If the number of data points is large, as in our case, large matrices (of the order of the number of data points) must be used to implement the regularization procedure, whereas the matrices for our approach are of the size of the dimension of the approximating subspace. It is, of course, possible to use a hybrid of the two approaches (Wahba 1980). Further, in applying the method of regularization, one should insure that the solution is nonnegative, which may be computationally nontrivial since a large number of constraints are involved. With our approach the number of constraints are again of the order of the size of the approximating subspace.

Other means of stabilizing solutions to integral equations of the first kind by restricting the solution to a finite dimensional space have been proposed. Grabar (1967) proposed using Chebyshev polynomials. Truncating the singular value decomposition of A was proposed by Han- son (1971). It may be difficult to enforce the nonnegativity and integral constraints with these approaches.

For any smoothing procedure, the most difficult problem is determining how much to smooth. In the case of regularization one must choose X and in our case one must choose the number of knots. Cross-validation was proposed for the method of regularization (Golub, Heath, and Wahba 1979) and could be applied to our procedure as well.

In our application, the data arrive in the form of a histogram. In general, of course, it would not be necessary to use a histogram or a probability density function estimate of any kind as a "pre-processor," as noted by a referee. It might well be possible to use a likelihood- based approach to determine the coefficients of the approximating mixture, perhaps using the EM algorithm.

The Appendix contains a proof of consistency of our method, but no results on rates of convergence are pres- ently available. The nonlinear constraints complicate the analysis. Thus, it is not possible at this time to prescribe p as a function of n. Based on some results in Rice and Rosenblatt (1982), we conjecture that the rate of convergence depends on the rate of decay of the eigenvalues of A*A the faster the decay, the slower the rate of convergence.

Finally, we remark that if it is not necessary to decon-



752 Journal of the American Statistical Association, December 1982

volve the S-phase density, but it is only desired to estimate the proportions in GI, G2 + M, and S phases, the procedure of Rice (1982) may be used.

APPENDIX

This Appendix gives a proof of consistency, in which it is interesting that the constraints play a crucial role. It would also be desirable to obtain expressions for the local bias and variance and the rate at which the integrated mean squared error decreases, but to date we have not been able to obtain such results.

Lemma. Suppose that the density f belongs to a linear space X with norm 11 lIx and that g belongs to a linear space Y with norm 11II l. Suppose that A is a' bounded linear operator from X to Y and that there is a probability density foEX such that Afo = g. Also assume:

1. There is a sequence of estimates g, of g such that 1g,, - g IIY O as n >w.

2. There is a sequence of closed convex sets SLp(n)CX and for each n, fn is the unique function in -Tp(n) minimizing 11 Affn - gn 11Y

3. The sequence {IEp(n)} has the property that infhEe,(f) 1I fo- h 11 -? 0.

ThenhIAfn-g11Y - >y0-. Proof. 11 Afn - g IlY - 11 Afn - gn IIY + 11 gn - g IIY

The second term tends to 0 by assumption 1. As for the first term,

11 Afn - gn IIY yC 11 Ahn -gn II Y --- Ahn - g IIY + 11 gn - g II Y

where hn is the best approximation to fo from -Tp(n). Fi- nally, 11 Ahn - g 11 = 11 Ahn - Afo IIY C 11 hn - fo lix

- 0.

We now have to show that 11 Afn - g IIY -- 0 implies 11 fo - fn I1 O_ 0. It is important to note that in general this is not true. What makes it true in this case is that fo, fn, and g are probability densities.

Proof. For each n let Fn be the cumulative distribution function corresponding to fn. By Helly's lemma we may choose a subsequent Fnk F weakly. Then

gn,,(s) = f p(s, t) dFn,,(t) | p(s, t) dF(t) = g(s), say.

Since 1I gnk - g IY 0 by the lemma, 11 g - g IIY = 0, and thus by the uniqueness assumption F = Fo.

We now make some comments relating the Lemma and Theorem to the case at hand. First, our Fo is not abso- lutely continuous but has discrete mass at 1 and 2 as well. The statements and proofs can be modified in a straight- forward way at the cost of additional notational com- plexity. We have given them in a simpler form in order that the arguments be more transparent. Second, the lemma and theorem as stated have no stochastic content, merely dealing with convergence in norm. Recent results such as Silverman (1978) on global almost sure properties

of probability density function estimates allow us to as- sert that I1 g, - g IIY - 0 a.s. and to conclude that II fn - fo IIX - 0 a.s. Third, ?4(n) in our case is convex by definition and is closed because it is finite dimensional. That in*fhep(fl) 11 fo - h lIx -o 0 as n -X oc follows from deBoor and Daniel (1974).

[ Received May 1981. Revised August 1982.]

REFERENCES AGARWAL, T., and STUDDEN, J. (1980), "Asymptotic Integrated

Mean Square Error Using Least Squares and Bias Minimizing Splines," Annals of Statistics, 6, 1307-1325.

CRESSMAN, H.A., and TOBEY, R.A. (1974), "Cell-Cycle Analysis in 20 Minutes," Science, 184, 1297-1298.

deBoor, C. (1978), A Practical Guide to Splines, New York: Springer- Verlag.

deBOOR, C., and DANIEL, J. (1974), "Splines With Non-Negative B- Spline Coefficients," Mathematics of Computation, 28, 565-568.

DEAN, P.N., and JETT, J.H. (1974), "Mathematical Analysis of DNA Distributions Derived From Flow Microfluorometry," The Journal of Cell Biology, 60, 523-527.

DONGARRA, J., BUNCH, J., MOLER, C., and STEWARD, C. (1979), LINPACK User's Guide, Philadelphia: SIAM.

FRIED, J. (1977), "Analysis of Deoxyribonucleic Acid Histograms From Flow Cytofluorometry-Estimation of the Distribution of Cells Within S-Phase," The Journal of Histochemistry and Cytochemistry, 25, 942-951.

GOLUB, G., HEATH, M., and WAHBA, G. (1979), "Generalized Cross-Validation as a Method for Choosing a Good Ridge Parame- ter," Technometrics, 21, 215-224.

GRABAR, L. (1967), "An Application of Chebyshev Polynomials Or- thonormalized on a System of Equidistant Points for Solving Integral Equations of the First Kind," Soviet Mathematics, 8, 164-167.

HAANEN, C., HILLEN, H., and WESSELS, J. (ed.) (1975), Pulse Cytophotometry, European Press Medicon.

HANSON, R. (1971), "A Numerical Method for Solving Fredholm Integral Equations of the First Kind Using Singular Values," SIAM Journal of Numerical Analysis, 8, 616-622.

LAWSON, C., and HANSON, R. (1974), Solving Least Squares Prob- lems, New Jersey: Prentice Hall.

MEDGYESSY, P. (1977), Decomposition of Superpositions of Density Functions and Discrete Distributions, New York: John Wiley.

PHILLIPS, D. (1962), "A Technique for the Numerical Solution of Certain Integral Equations of the First Kind," Journal of the ACM, 9, 84-97.

REINSCH, C. (1967), "Smoothing by Spline Functions," Numerische Mathematik, 10, 177-183.

RICE, J. (1982), "An Approach to Peak Area Estimation," Journal of Research of the National Bureau of Standards, 87, 53-65.

RICE, J., and ROSENBLATT, M. (1981), "Integrated Mean Squared Error of a Smoothing Spline," Journal of Approximation Theory, 33, 353-367.

(1982), "Smoothing Splines: Regression, Derivatives, and the Solution of Translation Kernel Integral Equations," Annals of Sta- tistics, to appear.

SILVERMAN, B. (1978), "Weak and Strong Uniform Consistency of the Kernel Estimate of a Density and its Derivatives," Annals of Statistics, 6, 177-184.

TAPIA, R., and THOMPSON, J. (1978), Nonparametric Probability Density Estimation, Baltimore: Johns Hopkins University Press.

TIKHONOV, A., and ARSENIN, V. (1977), Solutions of Ill-Posed Problems, New York: John Wiley.

TUKEY, J.W. (1971), Exploratory Data Analysis (preliminary edition), Boston: Addison-Wesley.

WAHBA, G. (1973), "On the Minimization of a Quadratic Functional Subject to a Continuous Family of Linear Inequality Constraints," SIAM Journal of Control, 11, 1.

(1977), "Practical Approximate Solutions to Linear Operator Equations When the Data are Noisy," SIAM Journal of Numerical Analysis, 14, 651-667.

(1980), "Ill-posed Problems: Numerical and Statistical Methods for Mildly, Moderately, and Severely Ill-posed Problems With Noisy




Data," Technical Report No. 595. University of Wisconsin, Statistics Department.

(1981), "Constrained Regularization for Ill-posed Operator Equations With Applications in Meteorology and Medicine," To appear in Proceedings of the Third Purdue Symposium on Statistical Decision Theory (eds. Gupta and Berger.)

WATSON, J.V. (1977), "The Application of Age Distribution Theory -in the Analysis of Cytofluorometric DNA Histogram Data," Cell Tissue and Kinetics, 10, 157-169.

WATSON, J.V., and TAYLOR, L.W. (1977), "Cell Cycle Analysis in Vitro Using Flow Cytofluorometry After Synchronization," British Journal of Cancer, 36, 281-287.



Article Contentsp. 748p. 749p. 750p. 751p. 752p. 753

Issue Table of ContentsJournal of the American Statistical Association, Vol. 77, No. 380 (Dec., 1982), pp. 707-964Front MatterVolume Information [pp. 958-964]ApplicationsBayesian Optimization of the Estimation of the Age Composition of a Fish Population [pp. 707-713]Round Robin Analysis of Variance Via Maximum Likelihood [pp. 714-724]Estimation of Nonlinear Learning Models [pp. 725-731]The Effects of Asymmetric Filters on Seasonal Factor Revisions [pp. 732-738]On the Design of Seasonal Adjustment Methods Using Linear Programming Techniques [pp. 739-742]Detecting Outliers in Time Series Data [pp. 743-747]Deconvolution of Microfluorometric Histograms with B Splines [pp. 748-753]Playing Safe with Misweighted Means [pp. 754-759]A Fast and Efficient Algorithm for the Estimation of Parameters in Models with the Box-and-Cox Transformation [pp. 760-766]On Graphical Procedures for Multiple Comparisons [pp. 767-772]

Statistical Evidence of Discrimination [pp. 773-783]Statistical Evidence of Discrimination: Comment [pp. 784-787]Statistical Evidence of Discrimination: Comment [pp. 787-788]Statistical Evidence of Discrimination: Comment [pp. 789-790]Statistical Evidence of Discrimination: Rejoinder [pp. 790-792]Theory and MethodsThe Powers and Strengths of Tests for Multinomials and Contingency Tables [pp. 793-802]Some Models for the Analysis of Association in Multiway Cross-Classifications Having Ordered Categories [pp. 803-815]A Time Series Analysis of Binary Data [pp. 816-821]Updating Subjective Probability [pp. 822-830]An Inconsistent Maximum Likelihood Estimate [pp. 831-834]Cluster Inference by Using Transitivity Indices in Empirical Graphs [pp. 835-840]A Hybrid Clustering Method for Identifying High-Density Clusters [pp. 841-847]The Effect of Two-Stage Sampling on Ordinary Least Squares Methods [pp. 848-854]Repeated Significance Testing for a General Class of Statistics Used in Censored Survival Analysis [pp. 855-861]Two-Sample Repeated Significance Tests Based on the Modified Wilcoxon Statistic [pp. 862-868]A Test of Incomplete Additivity in the Multiplicative Interaction Model [pp. 869-877]A Comparison Between Maximum Likelihood and Generalized Least Squares in a Heteroscedastic Linear Model [pp. 878-882]Estimating Latent Variable Systems When Specification is Uncertain: Generalized Component Analysis and the Eliminant Method [pp. 883-889]The Effect of Variable Correlation on the Efficiency of Seemingly Unrelated Regression in a Two-Equation Model [pp. 890-895]On the Effects of Moderate Multivariate Nonnormality on Roy's Largest Root Test [pp. 896-900]Inference Based on Simple Rank Step Score Statistics for the Location Model [pp. 901-907]Prediction and Power Transformations when the Choice of Power is Restricted to a Finite Set [pp. 908-915]Some Robust-Type D-Optimal Designs in Polynomial Regression [pp. 916-921]On Sample-Size Selection and the Evaluation of Discriminability in the Model Choice Problem [pp. 922-928]A Note on Strong Unimodality of Order Statistics [pp. 929-930]Ancillarity Principle and a Statistical Paradox [pp. 931-933]Stein's Paradox is Impossible in the Nonanticipative Context [pp. 934-935]

[List of Book Reviews] [p. 936]Book ReviewsReview: untitled [p. 937]Review: untitled [p. 937]Review: untitled [p. 938]Review: untitled [p. 938]Review: untitled [pp. 938-939]Review: untitled [p. 939]Review: untitled [pp. 939-940]Review: untitled [p. 940]Review: untitled [pp. 940-941]Review: untitled [p. 941]Review: untitled [pp. 941-942]Review: untitled [pp. 942-943]Review: untitled [p. 943]Review: untitled [pp. 943-944]Review: untitled [p. 944]Review: untitled [pp. 944-946]Review: untitled [pp. 946-947]Review: untitled [p. 947]Review: untitled [pp. 947-948]Review: untitled [p. 948]Review: untitled [pp. 948-949]Review: untitled [pp. 949-950]Review: untitled [p. 950]Review: untitled [pp. 950-951]

Publications Received [pp. 952-953]Corrigenda: Analysis of Coarsely Grouped Data from the Lognormal Distribution [p. 954]Corrigenda: Trimmed Least Squares Estimation in the Linear Model [p. 954]Corrigenda: Asymmetric Time Series [p. 954]Corrigenda: Concavity of the Log Likelihood [p. 954]Corrigenda: K-Sample Rank Tests for Umbrella Alternatives [p. 954]Corrigenda: A Seasonal Adjustment Principle and a Seasonal Adjustment Method Derived from This Principle [p. 954]Corrigenda: A Variant of the Acceptance-Rejection Method for Computer Generation of Random Variables [p. 954]Corrigenda: Mean Squared Error Properties of Generalized Ridge Estimators [p. 954]Corrigenda: A Gaussian Approximation to the Distribution of the Sample Variance for Nonnormal Populations [pp. 954-955]Corrigenda: Optimal Selection from a Finite Sequence with Sampling Cost [p. 955]Corrigenda: The Identical Distribution Hypothesis for Stock Market Prices-- Location- and Scale-Shift Alternatives [p. 955]Back Matter [pp. 956-957]

Documents

Deconvolution of Microfluorometric Histograms with B Splines