THE ANATOMICAL RECORD 211:126-132 (1985) Statistical Significance Tests for Autoradiographic Data BETTY J. SKIPPER AND LINDA J. McGUFFEE Department of Family, Community, and Emergency Medicine (B.J.S.) and Department of Pharmacology (L. J. M.), Uniuersity of New Mexico Albuquerque, N M 87131 ABSTRACT The purpose of this paper is to develop statistical methods that take radiation spread into account in analyzing data from different autoradiographic experiments. The method uses the probability circle analysis of Salpeter and McHenry (1973) to obtain the probable source of each radioactive emission and the circle and point counting method of Williams (1969) to estimate the relative area occupied by each cellular site. Two levels of analysis are presented. The first level of analysis is concerned with estimating relative activities and standard errors for cellular items that are larger than the probability circle. The second level of analysis involves estimating relative activities and standard errors for cellular sites that are smaller than the probability circle and are therefore observed in circles containing another item such as cytoplasmic matrix. Two different tests of hypotheses are discussed. The first null hypothesis is that the radioactivity is randomly distributed among the cellular sites. The second null hypothesis is that there is no difference between two different treatments in the relative activities for a given site. Autoradiography a t the level of the electron microscope can be used to determine the relative distribution of a substance among the various components of a cell. The methodology for carrying out this type of experiment is well documented (Kent and Williams, 1974; Salpeter and McHenry, 1973; Salpeter et al., 1978; Williams, 1977). Briefly, the method consists of exposing a tissue to a radioactive beta-emitting isotope, fixing the tissue, and slicing it into thin sections. These sections are then overlayed with a photographic emulsion containing silver halide crystals in a gelatin matrix. The isotope will emit beta radiation in all directions. Some of the beta particles will enter the emulsion layer and hit silver halide crystals. When a beta particle hits a sufficient cross-sectional area of a crystal, a latent image, a speck of reduced silver, appears somewhere in the crystal. When this crystal is exposed to photographic developer, it is subject to faster reduction than are crystals that are not hit by beta particles. After development, the silver grain is visible at the light microscopic level as a black dot or at the electron microscopic level as a darkened area of filamentous silver. An illustrative electron micrograph is shown in Figure 4 of the paper by McGuffee et al. (1985). Because the radioactive particles are emitted in all directions, many particles will not travel in a path that is perpendicular to the thin section and may not appear over the site of origin. Depending on the energy of the beta particle, most of the grains originating from small sources will fall outside of their sources owing to this radiation spread. In smooth muscle, mitochondria and SR can be considered small sources. The “Probability Circle” analysis as described by Salpeter and McHenry (1973) takes the radiation spread into consideration. The method involves experimentally determining the half 0 1985 ALAN R. LISS. INC. distance (HD) for the isotope, i.e., the distance from a line source within which 50%of the grains lie. When a circle with a radius equal to 1.7 HD is drawn around a grain, there is a 50%probability that the circle contains the point source. Salpeter and McHenry (1973) have pointed out the fact that one cannot use a circle that is too large because as the circle size increases toward loo%, the selective association of grains with structures is lost. In the initial step of the analysis, the center of the developed grain, i.e., the midpoint between the extremities, is chosen for each grain (Salpeter and McHenry, 1973) and the circle of radius equal to 1.7 HD is placed over the grain. All structures that lie within this circle, e.g., mitochondria, SR, plasma membrane, or nucleus in smooth muscle, are considered to be possible sources of the radioactive decay. If no organelle occurs within the circle, the cytoplasmic matrix is scored as the probable source. Because different sites occupy different proportions of the area of the cell, the information that can be obtained from the grain count distribution is limited unless one takes into account the relative sizes of the various components of the cell. The stereological methodology used to determine the area distribution for this study is the point and circle analysis of Williams (1969). For this analysis, one uses a plastic overlay on which a n array of circles has been drawn. The radius of each circle is the same as the size of the circle used for obtaining the grain counts and the center of the circle is denoted by a point. The overlay is placed over each autoradiogram. Received May 30, 1984; accepted August 8, 1984. STATISTICAL METHODS FOR AUTORADIOGRAPHY Cov(G,, GJ = -GglgJ 127 Relative Activities and Standard Errors of Individual Constituents of Compound Items If the sample size is relatively large, then the Gi’s are Some cellular sites are smaller than the probability approximately normally distributed. circle so their relative activites cannot be estimated For the proportion of grains in category i, the esti- using the above method. The purpose of this section is mated mean is g, and the estimated variance is to obtain estimated relative activities and standard erV(gJ = g, (1 - gJG. (la) rors for these individual constituents of compound items. Let aij denote the relative activity for a compound item The estimated standard error is containing sites i and j. Assume that site i is large enough to be observed by itself, but site j is too small to SE(gJ = = Jg, (1 - gJG. (lb) be observed by itself. For example, site i could be cytoplasmic matrix while site j could be mitochondria. Let The estimated covariance between proportions g, and gj ai denote the relative activity for the observed item is containing site i by itself. Let C- denote the number of circles containing items i and j. f!et Pi be the number of (IC) Cov(gi, gj) = -gi gjIG. points over site i and let Pj be the number of points over site j. Then pi = Pi/Cij is the proportion of the area of Since not all items occupy the same proportion of the the compound item occupied by site i and pj = P;/Cij is area of the cell, the grain count for each compound item the proportion of the area occupied by site j. Let s; denote must be adjusted for the relative amount of the area the relative activity of site j, the component that is too that is occupied by that item. As described previously, small to be observed by itself. The equation given by the point and circle counting method is used to deter- Williams (1969) and Kent and Williams (1974) for estimine these relative areas. This procedure is carried out mating s; is using the same set of micrographs and the same set of compound items, but is done without regard to the positions of the grains. (5) Let C be the total number of circles counted. Let Ci be the number of circles observed to occur over category i. Since the variance of the difference between two ranLet ci = Ci/C denote the proportion of circles associated with category i. The equation for the expected number dom variables X and Y is of grains in category i is given by Kent and Williams v:v - Y ) yx) t VEf -9 Cnv<X,V! (1974)and Williams (1977)as expected number of grains = G x (Ci/C). The relative activity is the variance of the relative activity for site j can be obtained as ~ a - Gi C, GIC This expression may be rewritten as That is, the relative activity is the proportion of grains associated with observed item i divided by the proportion of the area associated with that item. Since gi must be between 0 and 1, the relative activity, ai, will be between 0 and Uci. It has been previously suggested (Williams, 1969)that if one counts a large number of circles, then the standard errors for the circle count will be so small that they can be ignored. Since the ci’s are constant under this assumption, the estimated variance of the relative activity can be obtained as The estimated standard error is (7a) (7b) The covariance is obtained as follows: 128 B.J. SKIPPER AND L.J. MCGUFFEE Some circles will include a single cellular site, whereas other circles will fall over portions of more than one cellular site. One then records the frequency of occurrence of single items, such as cytoplasmic matrix, and compound items, such as mitochondria and cytoplasmic matrix or SR and cytoplasmic matrix. To obtain results for the individual constituents of these compound items, a n estimate of the portion of the circle that is occupied by each constituent item must be obtained. This is done by recording the frequency with which points (i.e., centers of the circles) fall over each of the individual items. For example, assume that in a series of autoradiograms 100 circles fall over areas that contain portions of mitochondria and cytoplasmic matrix. For those 100 circles, 25 have their centers over mitochondria and 75 have their centers over matrix. One would then estimate that the mitochondria occupy 25% of the area in circles containing mitochondria and matrix while the matrix occupies the remaining 75% of the area. Williams (1977) has used the relative areas obtained from the point and circle analysis to calculate the expected number of grains for a given item assuming random incorporation of the radioisotope. He then divides the observed grain count for each item by the expected grain count to obtain the quantity that he called the “relative crude specific activity.” In this paper, this quantity will be called the “relative activity for observed compound items.” After the relative activities have been obtained for each compound item, one needs to be able to estimate relative activities for the cellular sites that are smaller than the probability circle and therefore are only observed in association with another site, such as cytoplasmic matrix. Williams (1977) and Kent and Williams (1974) have published a n equation for estimating these relative activities, but no estimates of standard errors are included in their papers. In the present paper, we will present equations for the standard errors of the relative activities of compound items, as well as standard errors for the relative activities of the smaller individual constituents of compound items. These standard errors then will be used to test whether the radioactivity associated with individual sites is significantly different from what one would expect from a random distribution of radioactivity throughout the cell. A test of the null hypothesis that two different treatments result in the same relative activity for a given cellular site will also be presented. STATISTICAL METHODS Theorems Used in Deriving Standard Errors Three theorems from mathematical statistics (Mendenhall and Scheaffer, 1973)will be used in deriving the standard errors. Theorem 3: Let Cov (X,Y) be the covariance between random variables and let the letter a denote a constant. Then Cov(uX,Y) = a Cov(X,Y). Assumptions and Definitions Assuming that radioactive decay is a random process, one would expect the number of grains associated with each cellular item to follow a Poisson distribution (Williams, 1977). The total number of grains would also follow a Poisson distribution since it is the sum of independently distributed Poisson variables. However, for analyzing the results of a particular experiment, we need the distribution of the number of grains associated with a given cellular item conditional on the total number of grains that are observed. It can be shown that this conditional distribution is the multinomial distribution (Meyer, 1975). Let G be the total number of grains counted for the probability circle analysis. These grains are classified into N categories based on the cellular sites which are included in the circles around the grains. Some of the N categories will include a single site such as cytoplasmic matrix, while others will include two sites such as mitochondria and cytoplasmic matrix or three sites such as plasma membrane, cytoplasmic matrix, and extracellular space. The items that include two or more sites will be called “observed compound items” in contrast to the individual cellular sites which may be smaller constituents of these items and therefore cannot be directly counted as single items occupying the entire area of the probability circle. The statistical methods for analyzing data from compound items will be developed first and then this analysis will be expanded to give relative activity estimates for the smaller individual constituents of these compound items. Single subscripts will be used for the analysis of grain counts for the observed compound items. This notation will be expanded to include double or triple subscripts when we are estimating the relative activities for the individual constituents of the observed compound items. Relative Activities and Standard Errors for Observed Compound items Let Gi be the number of grains in the ith category. n c 1=1 G, = G. Let gi = Gi/G denote the proportion of grains that are category i. As discussed above, the number of grains in category i will follow a multinomial distribution. For the multinomial distribution (Mendenhall and ScheafTheorem 1: Let V(X) be the variance of a random vari- fer, 1973), the estimated mean is Gi and the estimated able, X, and let the letter a denote a con- variance is V(Gi) = G(gi)(l-gJ. The estimated standard stant. Then V(aX) = a2V(X). error is Theorem 2: Let V(X) and V(Y) denote the variances of random variables X and Y, respectively. Let Cov (X,Y) denote the covariance of X and Y. Then the variance of the difference be- The estimated covariance between Gi, the number of tween X and Y can be obtained as grains in category i, and Gj, the number of grains in category j, is V(X-Y) = V(X) + V(Y) - 2 Cov(X,Y). STATISTICAL METHODS FOR AUTORADIOGRAPHY 129 Testing for Randomness of Radioactivity The derived relative activities, a;, for individual constituents of compound items are obtained by linear transformations of the observed grain counts, Gi. Since the Gi's are approximately normally distributed, the derived relative activities also will be approximately normally distributed. If the radioactivity is randomly distributed throughout the cell, the proportion of the total grains associated with each site will be equal to the proportion of total area occupied by that site. That is, the relative activity will be equal to 1.One can test the null hypothesis that the relative activity is equal to one by calculating the It should be noted that the denominator of the standard error (equation 10)includes pJ and &. The standard error becomes smaller as one increases the number of grains that are counted. Since it is a square root relationship, one would need to quadruple the number of grains counted if one wanted to halve the standard error. The pj's will be small for small sites, such as mitochondria and SR. Division by these small quantities means that standard errors for these sites will tend to be large. The only way to counteract this effect is to count a large number of grains and thus reduce the standard error by increasing the value of G in the denominator. Because of sampling variation, there may be occasions where sJwill be negative. Such results should be interpreted to mean that the particular experiment does not show any radiation coming from source j. An observed compound item may contain three sites, such as plasma membrane, cytoplasmic matrix, and extracellular space. For such items, assume that sites i and j are large items, such as cytoplasmic matrix and extracellular space. Assume that site k is a small site, such as plasma membrane. The estimated relative activity for site k is aijk - (Pi a, sk = + PJ aj) Since this quantity will be approximately normally distributed with a mean of 0 and a standard deviation of 1, one can use standard tables of the normal distribution to determine the statistical significance level (P value) for Z. Comparison of Results for Two Different Experiments The purpose of some autoradiographic studies is to determine if two different treatments result in the same relative distribution of a given substance within the cell. Therefore, one will be testing the null hypothesis that the relative activity for site j in experiment 1is equal to the relative activity for site j in experiment 2. Let syl) be the relative activity for site j for treatment 1and jet s;(2) be the relative activity site j for treatment 2. As mentioned previously, these relative activities will be approximately normally distributed. The difference between the two relative activities, (sj(l1-sj(2)), will also be approximately normally distributed with variance V(s.ci)) + V(S~(~)). Under the assumption of the null hypotLesis that there is no difference between the two treatments, the quantity (11) Pk where pi, p;, and Pk are the relative areas of sites i, j, and k within circles containing these sites. The estimated variance is given by will be approximately normally distributed with mean 0 and standard deviation 1. Tables of the standard normal distribution can be used to determine the statistical significance level for Z. To verify the assumption of normality, we carried out simulation studies based on data from the experiment discussed in the following applications section. Figure la-c shows the histograms of the results of 2,000 simulations comparing relative activities of plasma membrane, mitochondria, and SR for two different experiments. These distributions are approximately normal, as expected. RESULTS OF SPECIFIC APPLICATION The numbers of grains observed in association with each compound item in an experiment using 45Ca in control and K + -contracted smooth muscle from rabbit vas deferens are shown in column 1 of Table 1. Details of the experimental methodology are described in McGuffee et al. (1985).The distribution of the circles that were counted is shown in column 2 of Table 1 and the distribution of points is shown in columns 3-8. 130 B.J. SKIPPER AND L.J. MCGUFFEE 500 t YW - - -- - ; 300- e For illustrative purposes, the calculations for the relative activity in the mitochondridcytoplasmic matrix item are shown below. Extra digits are carried throughout the calculations to minimize rounding error. (0) 7 200- g3 = 100I , a LL 56 1.180 = 0.0475. From equation lb, the standard error is , 4 W ~ These results are shown in column 1 of Table 2. The relative proportion of circles in association with this item is =tdd€uL- 5 - 4 - 3 Q - I 0 I 2 3 175 c3=--4,578 - 0.0382. From equation 2, the relative activity is obtained as 4 5611,180 - 1.2415. 17514.578 a3=-(C) Using equation 4,the standard error is These results are shown in column 3 of Table 2. DIFFERENCE BETWEEN RELATIVE ACTIVITIES FOR CONTRACTU) AND CONTROL TISSUES Fig. 1. Distribution of the results of 2,000 simulationsof the difference between the relative activities of contracted and control tissues. a) Plasma membrane; b) mitochondria; c) SR. TABLE 1. Raw data for 45Ca in control and K-contracted tissue rabbit vas deferens Control Tissue Cyto. matrix P.M./matrix/Ec.S. Mito./matrix S.R./matrix Nuchatrix Ec.S. Total K-contracted tissue Cyto. matrix P.M./matrix/Ec.S. MitoJmatrix S.R./matrix Nuc./matrix Ec.S. Total P.M. points Mito. points S.R. points Nuc. points (2) Cyto. points (3) (4) (5) (6) (7) 2,255 456 142 132 52 0 3,037 0 144 107 1,180 2,255 988 175 163 395 602 4,578 0 0 33 0 0 0 33 0 0 0 31 0 0 31 0 0 0 0 343 0 343 563 136 52 47 49 31 878 3,031 753 263 208 314 263 4,832 3,031 363 227 145 39 0 0 36 0 0 0 36 0 0 0 0 0 0 0 Grains Circles (1) 613 277 56 52 75 0 0 0 0 144 0 153 0 0 0 0 0 3,805 153 63 0 0 63 275 0 275 Ec.S. points (8) 0 388 0 0 0 602 990 0 237 0 0 0 263 500 Cyto., cytoplasmic matrix; P.M., plasma membrane; EcS., Extra-cellular space; Mito., mitochondria; S.R., sarcoplasmic reticulum; Nuc., nucleus. 131 STATISTICAL METHODS FOR AUTORADIOGRAPHY Since the mitochondria are small relative to the size of the circle, they will only OCCLU as a constituent of the compound item including mitochondria and cytoplasmic matrix. The relative activity and the standard error can be estimated from equations 5 and 10, respectively. 1.2415 -(142/175)(1.0546) = 2.0457 33/175 1 (0.0475)(0.9525) SE ( ~ g = ) (33/175)J1,180 (0.0382)2 (142/175)2(0.5195)(0.4805) s3 = + (0.4926)2 1 +(2)(142/175)(0.0475)(0.5195) (0.0382)(0.4926) % o.8977 ~ These results are shown in column 1of Table 3. For the mitochondria in the control preparation, one can determine if the relative activity is significantly different from 1by calculating -1 z=-2.05 - 1.17 0.90 From tables for the standard normal distribution, one finds that for a two-tail test a t the 5% level of significance the critical values are Z = -1.96 and Z = +1.96. Since 1.17 is less than 1.96, the relative activity for mitochondria in this preparation is not significantly different from 1. To determine if there is a significant difference between the relative activities in mitochondria in control and contracted tissues, one can calculate Z= 2.05 -1.50 = 0.38. d(0.90)2 + (1.13)2 This value is shown in column 3 of Table 3. Since 0.38 is less than 1.96, there is no significant difference between the relative activities in mitochondria in the two preparations. DISCUSSION In analyzing the results of autoradiographic experiments, one may want to test the null hypothesis that the radioactivity associated with a given substance is randomly distributed throughout the cell or one may want to compare the amounts of radioactivity associated with a given cellular site under different experimental conditions. To be able to carry out such analyses, one needs to take into account the effects of radiation spread. One method for doing this uses the probability circle of Salpeter and McHenry (1973) for determining the most likely sources of the radioactive emissions. The circle and point counting method described by Williams (1969) is used for obtaining the area distribution of the cellular items. Williams (1977) and Kent and Williams (1974) have published equations for estimating the relative activities for observed compound items, as well as sites that are smaller constituents of the observed compound items. The standard errors presented in this paper may be used in estimating the variability associated with the estimates of relative activity. As can be seen from equation 4 and equation 10, the estimated standard errors for the relative activities are inversely proportional to the square root of G, the total number of grains counted. Therefore, one can reduce the standard errors by counting more grains. However, if one is already counting a reasonably large number of grains, one may have to increase substantially the sample size to obtain the desired results. For example, to halve a n estimated standard error based on counting 1,000 grains, one would have to count 3,000 more grains. Some sites are small and can only be counted in conjunction with another item, such as cytoplasmic matrix. The estimated standard error of the relative activity of such a site is inversely proportional to pj, the portion of the area of the circle that is occupied by that site. Since pj will be small for sites that occupy relatively little area, the standard errors for these sites will be large TABLE 2. Relative activities of compound items in control and K-contracted tissues Relative activity (1) Relative proportion of circles (2) 0.5195 f 0.0145 0.2347 k 0.0123 0.0475 f 0.0062 0.0441 f 0.0060 0.0636 f 0.0071 0.0907 2 0.0084 0.4926 0.2158 0.0382 0.0356 0.0863 0.1315 1.055 0.030 1.088 f 0.057 1.241 k 0.162 1.238 & 0.168 0.737 f 0.082 0.690 i 0.064 0.6412 0.1549 0.0592 0.0535 0.0558 0.0353 0.6273 0.1558 0.0544 0.0430 0.0650 0.0544 1.022 f 0.026 0.994 5 0.078 1.088 k 0.146 1.244 0.176 0.859 k 0.119 0.649 f 0.114 Relative proportion of grains f SE Cellular item Control Tissue Cyto. matrix P.M./matrix/Ec.S. MitoJmatrix S.R./matrix Nuclmatrix Ec.S. K-contracted tissue Cyto. matrix P.M./matrixlEc.S. MitoJmatrix S.R. matrix NucJmatrix Ec.S. Abbreviations as in Table 1. f 0.0162 f 0.0122 f 0.0080 5 0.0076 k 0.0077 f 0.0062 k SE (3) 132 B.J. SKIPPER AND L.J. MCGUFFEE TABLE 3. Relative activities of individual cellular sites in control and K-contracted rabbit vas deferens K-contracted tissue relative activity f SE Cellular site Control tissue relative activity k SE (1) (2) 2 Cyto. matrix P.M. Mito S.R. Nuc. Ec.S. 1.05 0.03 2.27 + 0.50 2.05 0.90 2.02 k 0.92 0.69 f 0.10 0.69 f 0.06 1.02 f 0.03 1.46 + 0.46 1.50 1.13 1.75 f 0.60 0.84 f 0.14 0.65 0.11 0.71 1.19 0.38 0.25 -0.87 0.32 * * Abbreviations as in Table 1. unless one counts a large number of grains to compensate for this effect. As one would expect, it is more difficult to estimate precisely the relative activities for small cellular sites than it is to estimate relative activities for large sites. Two different tests of hypotheses have been presented. The first one tests the null hypothesis that the relative activity for a given site is equal to the relative activity expected from a random distribution. The second test allows one to compare the relative activities of a given site under two different experimental conditions. Both of the tests are relatively easy to calculate once one has calculated the appropriate relative activities and standard errors. Standard normal tables can be used to determine the statistical significance levels. The use of these equations for standard errors and hypothesis testing procedures will permit more rigorous evaluation of the results of autoradiographic experiments. ACKNOWLEDGMENTS This work was partially supported by National Institutes of Health grant 1-R01-GM30003. L.J.M.'s work was done during the tenure of a n Established Investigatorship from the American Heart Association and with funds contributed in part by the New Mexico Affiliate. LITERATURE CITED Kent, C., and M.A. Williams (1974) The nature of hypothalamo-neurohypophyseal neurosecretion in the rat. J. Cell Biol., 60:554. McGuffee, L.J., E.S. Wheeler-Clark, B.J. Skipper, and S.A. Little (1984) The cellular distribution of calcium in freeze-dried rabbit vas deferens using EM autoradiography. Anat. Res. 211t117-124. Mendenhall, W., and R.L. Scheaffer (1973) Mathematical Statistics With Applications. D u b u r y Press, North Scituate, MA., pp. 185193. Meyer, S.L. (1975) Data Analysis for Scientists and Engineers. John Wiley and Sons, Inc., New York, pp. 335-356. Salpeter, M.M., and F.A. McHenry (1973) Electron microscope autoradiography. In: Advanced Techniques in Biological Electron Microscopy. J.K. Koehler, ed. Springer-Verlag, Berlin, pp. 113-152. Salpeter, M.M., F.A. McHenry, and E.E. Salpeter (1978) Resolution in electron microscope autoradiography. J. Cell Biol., 76127. Williams, M.A. (1969) The assessment of electron microscopic autoradiographs. In: Advances in Optical and Electron Microscopy, Val. 3. R. Barer and V.E. Cosslet, eds. Academic Press, New York, pp. 219-277. Williams, M.A. (1977) Quantitative methods in biology. 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