Baixe sobrevivência celular e outras Manuais, Projetos, Pesquisas em PDF para Biotecnologia, somente na Docsity!
MtCatton Research, 111 (1983) 79-96 79 Elsevier
MTR
Statistical methods for in vitro cell survival assays
a Robert E. Tarone, b Dominic A. S c u d i e r o a n d c J a y H. R o b b i n s
" Btometry Branch, Nattonal Cancer Instttute, Bethesda, MD 20205, h Chemwal Carcmogenests Program, Frederwk Cancer Research Facthty, Frederwk, Maryland 21701, and c Dermatology Branch, Natwnal Cancer lnstttute, Bethesda, M D 20205 (U S A ) (Recewed 8 November 1982) (Revision recewed 4 April 1983) (Accepted 26 April 1983)
Summary
Statistical methods are presented for estimating and comparing survival curves
obtained from experiments in which cells are exposed in vitro to increasing doses of
a DNA-damaging agent. These methods, which are applicable m a variety of cell
survival assays, are illustrated in the evaluation of two sets of experiments in which
the colony-forming ability of fibroblast cell hnes from 9 muscular dystrophy patients
and 17 normal individuals were studied after exposure to N-methyl-N'-nitro-N-
nitrosoguanidine (MNNG).
Over the last three decades, survival studies of cultured mammalian cells have
received considerable attention (Elkind and Whitmore, 1967). Early work con-
centrated on identifying and investigating mechanisms of cell killing. Recently there
has been a rapidly expanding effort to use in vitro cell survival assays to identify
human diseases associated with hypersensiuvlty to DNA-damagmg agents. The first
d~seases shown to have inherited defects in DNA repair which lead to enhanced cell
kdllng m vitro were xeroderma pigmentosum and ataxia telangiectasia (Frledberg et
al., 1979). With ataxia telangiectasia patients, who are abnormally sensitive to
lomzing radiation and radiomimetic chemicals, and most xeroderma pigmentosum
patients, who are abnormally sensitive to UV radiation and UV-mimetic chemicals,
the degree of hypersensitivity is so great that the choice of statistical methods used to
quantify survival differences is not crucial. More recently, however, invesUgatlons
have included the study of cancer-prone disorders and neurological degenerations
for which the degree of hypersensitivity is much less. In the course of such
Abbrevtatton" MNNG, N-methyl-N'-nltro-N-mtrosoguamdme
0027-5107/83/$03 00 © 1983 Elsevier Science Pubhshers B V
invesngations we have found it necessary to develop appropriate and efficient methods for e s n m a n o n and significance testing, with particular reference to prob- lems inherent to m vitro cell survival assays. In this paper we present statistical methods which are applicable in a variety of m vitro cell survival assays, as well as other biological systems. More specifically, we shall discuss the stansncal analysis of dose-response curves which summarize the m vitro survival or D N A - r e p a i r capablhties of various cell lines at increasing dose levels of a D N A - d a m a g i n g agent. We present methods for fitting a survival curve for a single cell line using data from a single experiment, for e s n m a t m g the survival curve for a single cell line using data from several replicate experiments, for summarizing the survival curves of a particular group of cell hnes (e.g.. hnes from patients with a c o m m o n disease status), and for comparing the surwval capabilities of two groups of cell lines (e.g., lines from pauents with a particular disease c o m p a r e d to hnes from a group of disease-free individuals). Most of the e s n m a n o n methods presented m this paper represent an application m the linear regression setting of principles and methods presented by Cochran (1954). and are closely related to methods used by H y d e (1980) in another biological a p p h c a n o n. The basic data obtained m a single survival assay for a parncular cell hne consist of m + 1 pairs (d,, X,), where m denotes the number of non-zero dose levels of the D N A - d a m a g i n g agent, d, denotes the tth dose level of the agent, 0 = d 0 < d~ < .. < d m, and X, > 0 denotes a measurement of a r a n d o m variable representing some aspect of survival or repair capability. Depending on the type of assay. X, could represent, for example, the colony-forrhing efficiency or the vmble cell count measured at some time after the cells under study are exposed to a dose, d,, of the D N A - d a m a g i n g agent, or X 1 could represent the plaque-forming efficiency after the
cells under study are infected with virus which has been exposed to a dose, d,, of the
D N A - d a m a g i n g agent. For reasons apparently unrelated to D N A - r e p m r capabilities, different cell lines can vary considerably with respect to their baseline measurements 0.e., the values obtained for the response variable at dose 0) and the same cell line can give variable base-line measurements m different experiments. Thus, ~t often Is not possible to obtain valid comparisons of survival capabilities using the absolute measurements. X,. In most cases it is necessary to consider measurements of the response variable m a particular experiment relative to their base-line measurement m the same experi- ment. Thus, for the discussion of statistical methods m this paper, we shall consider
the ratio, P, = X , / X o, to be the dependent variable of interest. The goal m fitting a
survival curve is to summarize the relationship between PI and d,, for t = 1, 2.. , m. F o r such a survival curve, the point at which the curve takes a parncular value, say 0.1, represents the point at which the measured response variable (denoted by X) is reduced to 10% of the basehne response. The methods presented in this paper are applicable for general funcnonal forms of the d o s e - r e s p o n s e relationship between P, and d,. For our exphcit derivations, however, we will assume that the observed pairs (d,, P,) are on the exponential portion of the survival curve; that is, we assume that on average, the logarithm of P,
is linearly related to dose in the interval [d l, dm]. Thus, the expected value (i.e., the
a n d that there is a value o > 0 such that V a r ( Z , ) = o 2 f o r t = 0 , 1..... m.
U n d e r these a s s u m p t i o n s , u n b i a s e d estimators of the intercept, a. the slope, b,
a n d the error variance, o 2, can be o b t a i n e d using the m e t h o d of least squares The
e s t i m a t o r for the slope is
m
/)= E d,(YI -- Y ) / S d
where m m
Y = Y'~ Y,/m, d = d , / m and Sd = E ( d l - J )2.
I=l x=l i--
The e s t i m a t o r for the intercept is
gl= Y - b d
a n d the e s t i m a t o r for the error variance is
S 2= ~ ( y - , ~ - [ ~ d , ) 2 / ( m - 2)
i=l
T h e q u a n t i t y s 2 will be referred to as the e s t i m a t e d error variance in subsequent discussions. The variance o f / ; m a y be e s t i m a t e d by "
V(/)) = s2/Sd
a n d the variance of ~ / m a y be e s t i m a t e d by
V ( ~ ) = s 2 1 + dz,/Sd
T h e slope estimator, /;, does not d e p e n d on Z 0, and hence the variance of /; is identical to the variance o b t a i n e d for linear regression with i n d e p e n d e n t response variables; however, fi is a function of Z0, a n d thus V(fi) differs from the s t a n d a r d f o r m u l a o b t a i n e d for linear regression with i n d e p e n d e n t response variables. The e x t r a p o l a t i o n number, n, the D o and the Dp are functions of the intercept and the slope, a n d hence estimators of n, D O and Dp can be o b t a i n e d from fi a n d b. The e x t r a p o l a t i o n n u m b e r can be e s t i m a t e d by
r~ = e x p ( f i )
a n d the large s a m p l e variance of ~ can be e s t i m a t e d by
V ( h ) =,~ 2 V(,~) The D o can be e s t i m a t e d by b o = - 1//; a n d the l a r g e - s a m p l e variance o f / ) o can be e s t i m a t e d by
v(A,) = bJv( ;)
Similarly, Dp can be estimated by
Dp = ( l o g ( p ) - &)//~
and the large-sample variance of/)p can be estimated by
V(Z)P)=(1)°s)2(l+m-'~(d'-~)p)2fSd),=l
Making inferences about the survival parameters of individual cell hnes or groups
of cell lines using data from a single experiment requires additional assumptions
about the distribution of the outcome responses, Z l. If, for example, it is assumed
that the Z, have a normal (i.e., Gausslan) distribution, then confidence intervals for
the survival parameters of a given cell line can be formed using the estimators and
standard errors derived above, based on the t-distribution with m - 2 degrees of
freedom. If we further assume that the error variance, o 2, ~s the same for all cell hnes
(i.e., if experiments on all cell lines are of equal preosion), then comparisons of
survival parameters for different cell lines or for groups of cell lines can be
performed m a standard fashion using t-tests. In the following sections we present
methods which are apphcable under less restrictive assumptions. In deriving these
methods, which require that multiple experiments be performed on each cell hne, we
assume neither that experiments on &fferent cell lines are of equal preos~on, nor
that the multiple experiments on the same cell line are of equal precision.
Illustrative example
Before proceeding with the presentation of estimation methods for multiple
experiments, it will be informative to consider an example which illustrates the need
for proper statistical analysis of in vitro cell survwal data. The data m Table 1 are
contrived to give a perfect linear fit on the logarithmic scale m each experiment.
Although the data are artificial, estimation problems with actual data s~milar to
those in Table 1 led to the derivation of the methods presented in the ensuing
sections. The data in Table 1 summarize the results of three experiments with four
dose levels used in each experiment. One observation at the 600 dose level is missing,
perhaps because of contamination or perhaps because no cells survwed the highest
dose in the third experiment.
TABLE 1
SUMMARY DATA (P,) FROM 3 CELL SURVIVAL EXPERIMENTS Expt Dose level 50 100 200 600 1 0.37 0 22 0 082 0 0015 2 0 45 0.27 0.100 0 0018 3 0 37 0.18 0.
To test for variability among experiments, define
k
Q = E v [ ' ( 0 j - ~v) 2
j = l
where ev = Ek~ ivj- lej/Ek= lVj 1. If there is no rater-experiment variability, Q will be
approximately distributed as a chi-square random variable with k - 1 degrees of
freedom (Cochran, 1954). If Q is too large, then inter-experiment variabdlty is
indicated. For our purposes, it is recommended that significance at the 10%
significance level (i.e., ct = 0.10) be taken as evidence of heterogeneity.
If s~gnificant heterogeneity is indicated, then a summary estimate may be ob-
tained using a modification of the semi-weighted mean (Cochran, 1954) for regres-
stun data. To calculate the semi-weighted mean, first compute
k
j = l
where w. = ~k= 1WJ and ew = Ek= lwje//w-, and
k B= Z s (1-w/w ) J ~ l
Then the contribution to the variance of the estimators, 0j, due to inter-experiment
variability is estimated by
sv = (.4 - B ) /^ ( ' w.-^ wj2/w. /
J = l
Defining vj = (S v + vj)- 1, we find the semi-weighted mean is
Q w = J,j0, / ~ 1,~
j ~ l j = l
and the variance of gsw is estimated by
The semi-weighted mean gives greater weight to estimates with smaller estimated
variances. If the inter-experiment variability is extreme (i.e., if Sv is large relative to
the oj), then the weights, vj, will be approximately equal, and the semi-weighted
mean will differ little from the unweighted mean.
If one or more of the replicate experiments have a particularly large error
variance, it is possible for there to be significant heterogeneity, but for B to be larger
than A. In this situation, S~ is negative and the semi-weighted mean should not be
used. For such a s~tuation, a summary estimate can be obtained using the un-
weighted mean,
k
0 = Y'. 0 / / ,
J ~ l
and the variance of ~ can be estimated by k = , j/k j = l If Q is not significant, indicating homogeneity of the individual estimates. ~, then a summary estimate may be obtained using the partially weighted mean (Yates and Cochran, 1938; Cochran, 1954). Although the weighted mean ~v defined above is a possible estimator in this case, the weighted mean can be dominated by a single experiment with a fortuitously small estimated error variance. The partially weighted mean maintains the property of the weighted mean of giving small weight to experiments with large error variances, while protecting against giving unduly high weight to experiments with fortuitously low estimated error variances In order to describe the calculation of the partially weighted mean, it will be convenient to assume that the experiments are ordered according to the magnitude of their estimated error variances, so that s~ <s22 < <s~. Let mj denote the number of dose levels in the exponential portion of the j th survival curve In
addition, if k is even, let h = k / 2 and if k is odd, let h = (k + 1)/2 Then define
h h , ~ 2 Z ( m , - 2)Sj2/ E ( m , - 2) J--1 J = l Note that ~z is the weighted average of the h smallest estimated error variances For
j = l , 2..... h, let % = w i g 2 and for l = h + 1, h + 2.... k let w j = w j / ~ = t ~ t
Thus for the ( k - h) experiments with the largest estimated error variances, ~0 is simply the inverse of the estimated variance of ~j. Then the partially weighted mean is given by 0 p v ~ = (~k_lcOjOj)/¢O. , where w. = ~=lc0j. The variance of 0p,, can be estimated by (see Cochran, 1954)
v ( ) = + x 1 ('~1 j = h + l
TABLE 2 VALUES OF MULT1PLICATIVE FACTOR, k, FOR C O M P U T I N G V A R I A N C E OF T H E PAR- T I A L L Y W E I G H T E D MEAN
2 3 4 5 6 7 8 9 l 3 3 0 6 3 101 152 203 272 357 4 5 4 546 4 2 0 2 9 3 9 51 61 7 0 7.9 9 3 106 5 17 2 2 2 8 3 4 3 9 4 4 4.8 5 4 6 0 6 15 18 2 2 2 5 2 7 3 0 3 2 35 37 7 14 17 19 21 2 3 2 4 2 5 2 6 2 6 8 13 15 17 18 19 2 0 2.0 21 21 9 12 15 16 17 18 19 19 2 0 2 0 10 12 15 15 16 16 17 17 18 18
an estimated standard error of 0.081. The latter slope estimate is lower because the least squares analysis of the pooled data gives the same weight to each of the five experiments, whereas the partially weighted mean gives less weight to the two experiments with the highest estimated error variances, and these two experiments produced the lowest slope estimates The standard error estimated from the least squares analysis of the pooled data is positively biased (even In those special cases where the slope estimator from such an analysis is a valid estimator) due to the within-experiment correlation of the response variables. Testing for equality of the individual /}01's, one finds that Q = 18.22. indicating significant heterogeneity ( P = 0.001) Thus the semi-weighted mean is the suggested estimator for the D0~. The estimated contribution of Inter-experiment variability to the variance of the D0~ estimates as S, = 0.2455, and the weights for the individual D0~ s are calculated to be v~' = 1 82, v2 = (^) 1.03, v3 = (^) 0.84, v4 =2.42 a n d v 5 = 2 9 8 Thus the semi-weighted mean is found to be 11.0 with an estimated standard error of 0. By way of comparison, the least squares analysis of the pooled data would give a D estimate of 11.1 with an estimated standard error of 0.73. Once again, this latter standard error estimate is positively biased. The estimators presented in this section were derived for the situation in which the experiments on a given cell line have unequal precision. These methods will be appropriate if, in fact, the individual experiments have equal precision, however, they will not be optimal. Hyde (1980) presented analogous methods for the estima- tion of a slope assuming all experiments are of equal precision These methods, which are regression analogs of the methods given by Cochran (1954) for experi- ments with different sizes but the same precision, can be extended easily for the estimation of any of the survival parameters considered above If, in fact, experi- ments of equal precision are anticipated, the estimators suggested by Hyde can be expected to have smaller variance than those presented in this paper
Summary estimation for a group of cell lines
Suppose that several cell lines from a group of individuals have been assayed. The group rmght represent patients with a certain disease Each cell line has been tested in multiple experiments, and estimates of a survival parameter, e, have been obtained using the methods of the last section. Let R denote the n u m b e r of cell lines tested, and assume that the R individuals from w h o m the cell lines were obtained are unrelated. Let Er denote the estimate of e for the r th cell hne and let Vr denote the estimated variance of ~;r, for r = 1, 2..... R. Since the In vitro responses of different cell lines may be expected to be heterogeneous due to genetic differences a m o n g the individuals from whom they were obtained, we will assume heterogeneity in deriving a summary estimator for the group. Let R
P= v,/R
r = l
denote the "average experimental variance" and let R
rE= Z l)
r = l
where E = Y~r=IEr/R. R " Then the intrinsic variance among cell lines (sometimes
referred to as the genetic variance) can be estimated by VG = VE - I7. Letting Ur = (V6 + Vr ) - l, the summary estimator for the group may be obtained using the semi-weighted mean R
Esw = E Uri r/U. r = l
where U. = ~ = 1Ur. The variance of/Tsw may be estimated by
v(Fsw) = v. - '
and approximate (1 - a) × 100% confidence intervals for the group mean are ob- tained as
E~w+[V(F'sw)] ' p t R - i°
where t~_ 1 is the (1 - a / 2 ) × 100 percentile of the t-distribution with R - l degrees of freedom. If V~ is much larger than V, then the semi-weighted mean will be very httle different from the unweighted mean, and the unweighted mean can have greater
precision (Cochran, 1954). Cochran proposed as a rule of thumb that if F - - V E / V is
greater than 4.0, then the group mean could be estimated by the unweighted mean. As noted by Cochran, however, if there is a large variation among the experimental variances, Vr, then the unweighted mean should not be used, even if F > 4.0. Our experience suggests that a reasonable modification of Cochran's rule is that if
Fr = VE/V~ > 3.0 for all r, then the group mean should be estimated by f f with
variance estimator V ( / 7 ) = VE/R. Confidence intervals for the group mean can be
obtained exactly as described above, with /~ substituted for E~w and V(/T) sub- stituted for V(/Tsw). If a particular assay system gives extremely variable results for one or more of the cell hnes, it is possible that V could be larger than VE, leading to a negative estimate for the genetic variance, Vc. In such a case, the semi-weighted mean should not be used. A summary estimate can be obtained using the unweighted mean,/7, but the
variance o f / T should be estimated in thas case as V(/~)= V/R.
Table 5 presents the estimated Doj'S and their estimated standard errors for fibroblast cell lines from apparently normal individuals in two distract series of experiments performed to study colony-forming ability after exposure to M N N G. For the seven normal lines from Series IV using the methods described above, we find V = 1.5711, V E ~--- 11.1129 and VG = 9.5418. Thus the semi-weighted mean gives a value of 17.3 with a standard error estimate of 1.24. Although the standard error of
the first/)01 is somewhat large ( F I = VE/V 1 = 1,7), indicating that the semi-weighted
mean is the estimator of choice, the magnitude of the genetic variance suggests that
Let/T~ denote the summary mean for the first group of R I cell lines and let Vt
denote the estimated variance of Et. Let /T2 denote the summary mean for the
second group of R 2 cell lines and let V2 denote the estimated variance o f / T2. Then, if
there is no difference between the two groups with respect to the survival parameter,
e, the statistic
T= ( - E 2 ) / [ v , + v2l ''
will have an approximate t-distribution. The number of degrees of freedom for the
approximate distribution of T should be calculated using Welch's approximation.
Letting c = V~/(V~ + V2), the approximate degrees of freedom for T is (Welch, 1937)
/ = [c~/( R, - 1 ) + (l - c ) 2 / ( 1 ~ - 1 ) ] - '
To test for survival differences between two groups of cell lines, the computed value
of T should be compared to the percentiles of the t-distribution with f degrees of
freedom.
Table 6 presents the estimated D0~'s and their estimated standard errors for
fibroblast cell lines from patients with muscular dystrophy studied in the same two
series of M N N G colony-forming experiments which produced the normal values
given in Table 5.
The summary D01 for the five muscular dystrophy lines m series IV, calculated from
either the serm-weighted mean or the unweighted mean ( F r > 5.0 for r = 1, 2..... 5)
is found to be 12.8 with a standard error estimate of 0.89. Thus, using the Series IV
data, the test statistic for comparing the normal/)0~'s with the muscular dystrophy
D01~S IS
T = (17.3 - 12.8)/[(1.24) 2 + (0.89)2]1/2 = 2.
Computing the number of degrees of freedom for T, Welch's approximation gives
f = 9.9. Comparing the computed T to the percentiles for the t-distribution with 10
T A B L E 6 E S T I M A T E D Dol'S A N D S T A N D A R D ERRORS (SE) F O R M U S C U L A R D Y S T R O P H Y FIBRO- BLAST LINES A F T E R EXPOSURE TO M N N G IN SERIES IV A N D SERIES V OF ROBBINS et al (1983) Series IV Series V
Cell hne a /)01 SE Cell hne a /)ol SE RB4608 14 2 0 50 RB4369 9 5 0 27 GM3755 14 4 0 88 RB4569 12.7 0 35 RB4364 13 7 0.45 RB4619 11 3 0 47 RB5213 12 3 0 40 GM3783 12 3 0 53 AG4035 9.6 0. a All muscular dystrophy lines were obtained from the Institute for Medical Research, Camden, New Jersey
degrees of freedom we find that P = 0.015, indicating that the muscular dystroph~ hnes were significantly more sensitive to M N N G than were the normal hnes m the Series IV experiments. If the genetic varianon within each group ~s large compared to the experimental variation, it m a y sometimes be possible to ignore differences a m o n g mdlwduals with respect to experimental variation, and perform a somewhat s~mpler analys~s In such a situation, the variation of the esnmate for each cell line m a group wdl be approximately equal to that group's genetic variance, and an approximate s~gmfi- cance test m a y be based on the usual t-test for comparing independent means. If the genetic variances of the two groups differ considerably, it may be advisable to transform the individual esnmates before performing the t-test. For the Series IV data, the values of F = Vr flV defined m the last section are 7. for the muscular dystrophy (^) D0~ s and~ ' 14.1 for the normal /)0t's, indicating that the experimental contribution to the variance of the D0~'s ~s considerably smaller than the genetic contribution. The usual t-test for equality of the muscular dystrophy and normal mean D0~'s gives t = 2.86 with an assocmted P-value of 0.017 In series IV the estimated genetic variance for the muscular (^) ^dystrophy , /)01's ~s 3 7, while the estimated genetic variance for the higher normal D0~ s ~s 9.5. Although the variance ratio of 2.6 is not excessive, a logarithmic transformation might be suggested The t-test after a logarithmic transformation of the Series IV/}0~'s gives t = 2 98 w~th an associated P-value of 0. It ~s sometimes the case, particularly for a.very rare disease, that ~t ~s of interest to c o m p a r e the survival parameter of a cell line from a single patient w~th the surwval parameters of cell lines from several normal mdxwduals. It is difficult to see how to perform such a comparison under the general assumpnons of this paper, except m the situation m which the geneuc contribution to the variance of the parameter estimates in the control 0.e., normal) group is large relative to the experimental contribution (measured by V). In this case, a comparison is possible using an analog of the usual t-test for comparing two means. Even when such a comparison )s possible, drawing conclusions based on tests of a single disease cell hne is a questionable practice, as an example will indicate. Let /~N~, EN2.... ENR be the parameter estimates for R cell hnes from normal individuals, and suppose that the genetic variance is large relative to the experimen- tal variance (e.g., suppose F = VEflV lS greater than 4.0). Let /~D be the parameter estimate for the single disease cell line. Define
R E r = l where E N = y~R J/~N JR. Then, calculate
TD = (/~D-- EN )/[SN(R + 1 ) / R ] 1 / 2
T o obtain an approximate test of whether ~:D is significantly outside the range of the normal values, compare T D to the percentiles of a t-distribution with R - I degrees of freedom. This test should not be used if the standard error of the parameter
TI=D//[V(D/)] I/2 and determine the associated two-sided P-value, P [ , Then
define L X, = - 2 Y~ log(PVl) / - 1
A summary P-value comparing group 1 and group 2 may be obtained by comparing the computed value of X, to the percentiles of the chl-square distribution with 21, degrees of freedom (Fisher, 1958). This method should only be used if the effect of exposure is m the same direction m each series of experiments (e g., if the mean for disease lines is less than the mean for normal lines in each series). Furthermore, ) ( should not be used to combine results from experiments of markedly different quahty (i.e., markedly different statistical power). In such a situation, for example, a highly significant result from a poorly designed experiment using a small number of normal and disease cell lines may combine with a non-significant result from a well designed experiment with a large number of normal and disease cell hnes to incorrectly indicate a significant overall effect of exposure In a set of experiments (labeled Series V m Tables 5 and 6) performed subsequent to the Series IV experiments, 12 normal cell lines were studied concurrently with cell lines from 4 muscular dystrophy patients unrelated to the 5 patients studied in the earher set of experiments. The summary/)01 for the 12 normal cell lines is 15.3 w~th an estimated standard error of 0.63. The summary/~0~ for the 4 muscular dystrophy cell hnes is 11.4 with an estimated standard.error of 0.72. Welch's approximation for the second set of experiments gives 8.1 as the estimated degrees of freedom The computed T for the second set experiments is 4.08 with an associated P-value of 0.0035. Thus, using the P-values of 0 015 from the first set of experiments and 0.0035 from the second set, we find X, = 19.71 Comparing this computed X, to the percentiles of a chl-square distribution with 4 degrees of freedom, we obtain a summary P-value of 0.0006, providing strong evidence of hypersensitivity to M N N G m the muscular dystrophy lines studies xn Series IV and Series V In all analyses discussed above two-sided P-values have been reported. One might argue that increased survival should not be anticipated, and thus that one-sided P-values should be reported m experiments performed to detect hypersensmve l,nes. A recent account of apparent rad~oresistance m a cancer-prone family (Bech-Hansen et al., 1981), however, suggests that the use of two-sided P-values is appropriate As with the estimation methods presented earlier, the methods gwen above for comparing two groups will perform better for symmetrxc distributions. Thus, even if /)o's and extrapolation numbers are to be reported, it is recommended that associ- ated P-values be computed based on the comparison of slopes (for D,,'s) and intercepts (for extrapolation numbers)
Discussion
Most of the procedures, such as the scoring of colonies, involved in determining
the in vitro survival curves with mammalian cells are not automated. It may be
advisable, therefore, to perform cell survival assays on coded hnes m order to
eliminate any possibility of bias (Comings, 1981). In the two sets of experiments
analyzed above, all of the muscular dystrophy hnes in both sets, 5 of the 7 normal
lines in the first set, and 8 of the 12 normal lines in the second set were coded so that
the personnel performing and evaluating the experiments did not know the identity
of the lines. An analysis using the methods described above, but based only on the
13 coded normal lines, is presented elsewhere (Robbins et al., 1983) and also shows
that the muscular dystrophy cell hnes are significantly hypersensitive to M N N G.
An important issue in the method of analysis presented m this paper is how to
select the dose range which constitutes the exponential portion of a curve. An in
vitro cell survwal curve typically has an initial curvilinear portion at lowest dose
levels, followed by the exponential portion at higher doses. Because of the short-term
nature of the in vitro survival assays, preliminary experiments may be performed to
identify, at least approximately, the exponential region for each cell line. The
problem of determining the exponential region is, however, still likely to arise in the
final data analysis. The use of multiple replicate experiments for each cell line allows
one to check for consistent, systematic departures from hnearity. If there is a
question as to whether a dose is on the initial curvilinear portion, a straight line can
be fit for each experiment through the points corresponding to all higher doses. If
the points at the dose level in question lie consistently above or below their
corresponding lines, then the dose should be omitted.
The above approach does not guarantee that the data for each experiment with a
given cell line will be linear m the selected dose range. The data from experiments 2
and 3 m Table 3 are not fit well by straight lines. The lack of fit m these
experiments, however, contributes to their larger corresponding estimated error
variances, and thus to the estimates from these experiments receiving less weight in
our estimation procedure than the estimates from experiments for winch the re-
sponses are more linear. Our approach has been to identify a dose range w~thin
which most of the experiments with a cell line have a linear response, and then to use
all data from this dose range, even if ~t appears that better linear fits for some
experiments could be achieved by selectively ehminating certain points. If our
method of estimation gave equal weight to all experiments, then some such selective
elimination of certain data would be necessary. It has also been our approach to
base the analysis of ekpenments from a gwen series, as nearly as possible, on data
from the same dose range for all cell lines. Thus, for example, the analyses of Series
IV and Series V reported in this paper used data from all dose levels of 6 ~tg/ml or
higher for all cell lines. In a later series of experiments, the 6-/~g/ml dose level
appeared to be on the curvilinear portion of some, but not all, cell lines. Thus, m this
later series, analysis was based on data from all dose levels of 8 ~tg/ml or higher for
all cell lines. Whatever method is used to determine the exponential region, it is
imperative that normal and disease cell lines be treated alike. Accordingly, our
estimation of survival curves is performed on coded data.
