Economy of Effort in Electron Microscope Morphometry Jonathan Shay, MD, PhD
Statistical techniques known as the analysis of variance make it possible for the electron microscopist to plan work in such a way as to get quantitativ e data with the greatest possible economy of effort. This paper explains how to decide how many measurements to make per micrograph, how many micrographs per tissue block, how- many blocks per experimentally treated organ, and how many organs per experimental treatment group. (Am J Pathol 81:503-512, 1975)
help the electron microscopist in experimental pathology answver the perennial questions: How many measurements must be made per picture? How many pictures per section? How many blocks per animal? How many animals in each experimnental condition?' This article will attempt to answer these questions in a practical wa- for the morphologist by adaptation of statistical techniques developed over the last 20 or so years. I shall not cite the original statistical work, but rather refer the reader to a recent and lucid text, i.e., Snedecor and Cochran's Statistical Methods.' The exigencies of biologic microscopy force upon us a kind of random sampling called nested sampling, cluster sanmpling, or samples within samples. The microscopist interested in quantitative data is rightfully reluctant to limit his measurements to one mitochondrion per liver, one synapse per brain, or the like. He sections a number of blocks per animal, takes micrographs of a number of fields per grid, and may measure a number of cell parts per field. Fortunately, the analysis of such subsampled or nested data is well understood and approachable by the techniques kno-n as the analysis of variance. Application of this statistical technique generates the quantities necessar to decide how many blocks, microscopic fields, measurements, etc., to use. Let us first look at a single group of experimental material -hich has all been treated the same \-ay-. Let us consider a number (n0) organs. These would be the actual undivided tissue of interest; e.g.. the n0 livers of LITTLE
IS
CURREN-TLY
AV AILABLE
to
From the Mlixter Laboratorv for Electron \Microscopy of the Neurosurgical Service of the MlassachuSetts General Hospital. Boston. \Massachusetts Supported by Grant NS-12023-O1 from the National Institutes of Health -ccepted for publication Xugust 1.3. 197,3. Address reprint requests to Dr Jonathan Shay. Mlixter Laboratory. Neurosurgical Service. Mlassachusetts General Hospital. Boston. MIA 022114 503
504
SHAY
American Journal of Pathology
animals treated with such and such a drug or the no isolated retinas incubated under the same conditions, etc. From each of these Tb organs, nb tissue blocks are taken at random for study. For simplicity, let us take only one thin section from each block for microscopy. From each of the n x nb sections we take rip micrographs at random, and within each of the no X nb X np pictures we measure a random sample of flm objects of one particular kind in which we are interested. In this experimental condition there are: nO
organs
n0 X nbi blocks n,XX nbX n, pictures and n. X nb X n, X n,, measurements. This is always the form of sampling in electron microscopy with the variation of more or fewer levels of nesting or clustering. If there is only one measurement per picture (e.g., a ratio of point counts) there would be one level fewer than in the present example. If there are more than one organ per animal or there are anatomic regions within the organ, such as lobes of the liver, the analysis would have more levels. Our laboratory is conducting studies of the effects of oxygen and glucose deprivation on isolated rabbit retina. Table 1 gives an actual analysis of some pilot electron microscope data from this work. Table 1 gives results of analysis of 25 mitochondrial measurements from each of four pictures from each of two blocks from each of four isolated retinas incubated under identical conditions. Starting from desk calculator tapes giving the mean and variance of each of the 32 sets of 25 measurements, this analysis took about 3 hours. In our laboratory, the original caliper measurements are entered directly on the calculator, and the tapes thus become the record of the raw measurements. (It should be noted that neither a costly calculator nor computer time is required for any of the computations described in this paper.) Let iMobpm = a single measurement (the mth) coming from a given picture (the pth) of a given block (the bth) of the oth organ. For the data in Table 1, Mlobpm was the diameter of a single mitochondrion from a single picture from the only tissue section taken from a tissue block of isolated rabbit retina (the oth). These original 800 measurements do not appear in Table 1. The entries in the first column of Table 1 are the sums of squares between measurements within single pictures, i.e.,
SS(M.bpn)
=
E _
-
E a
M0b2, /n.)2
c
-c
cO00
T-
C'I)
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3 3
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0 C_
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6
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0 0 O C
_O 0
c
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10 C') C')
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506
SHAY
American Journal of Pathology
This was easily computed from the calculator tape by multiplying the variance by nn - 1. The total sum of squares for measurements within pictures is TSS(M)
=
E E (M0obp.- 'ia M.b,,/n .)2 E E o b p -
which is to sav, the total of all the individual sums of squares between measurements within individual pictures. We will use the convention that a capital P will stand for the mean, within a given picture, of all the measurements in that picture, a capital B the mean of the picture means in a given block, a capital 0 the mean of the blocks within a given organ. Let us define the mean of measurements within a given picture, i.e., P ob
M.obus11n.
=
This comes right off the calculator tape. The sum of squares between pictures within a single block is thus (Pob, EP:bpb/n,)2 SS(Pobp) = xE p P The total sum of squares between pictures within blocks is then -
TSS(P) = n.
E Eb E (Pobp- E P.bpnp)2 p 0
p
Defining the individual block mean as above Bob = E P0b,//nr Then the sum of squares between blocks within a single organ is SS(B b) = n1.n,
(Bob -E B.b/nb)2 E b b
and the total sum of squares for blocks is TSS(B)
=
n.nup
E o
(Bb- E: B.b/nb)2 E b b
As above 0
=
E
b
B.b/nb
then the total sum of squares between organs within a given experimental condition is
TSS(O)
=
n.nfnub
(0. o
-
E 0.X'n.)2 o
Vol. 81, No. 3 December 1975
EXPERIMENTAL DESIGN IN EM MORPHOMETRY
507
The degrees of freedom for calculating the mean square are DF(o) = nO- 1 DF(b)
nfo(nb- 1)
=
DF(p) = n. X nb(np-1) DF(m) = n0 X nb X n,(n. -1)
The respective mean squares are then MS(o) MS(b) MS(p) MS(m)
= TSS(o)/DF(o) = TSS(b)/DF(b) = TSS(p)/DF(p) = TSS(m)/DF(m)
The sample variances (variance estimates) due to measurements, to
blocks, etc., are computed in the following manner: Sample variance of measurements Sm2= MS(m)
Sample variance of pictures 2
MS(p) -MS(m) n,.
Sample variance of blocks M2 S(b)
-
MS(p)
n,. X nb
Sample variance of organs M2 S(o) - MS(b) n. X nf X nb
Calculation of Most Economical Ratios The above variance estimates for each sampling level can be used to plan further experiments so that one gets the most for one's time and money. For example, if the main source of variation is between organs, there is little to be gained from sectioning many blocks per organ, taking many pictures per block and making dozens of measurements per picture. Subject to certain simple restrictions, one can calculate the most economical distribution of effort.
508
SHAY
American Journal of Pathology
Let
CO = cost per organ Cb = cost per block C, = cost per picture C. = cost per measurement These costs can be expressed in any terms the investigator finds appropriate for his own circumstances. As long as the units are consistent, the "cost" can be purely subjective, or a mixture of weighted subjective and objective inputs. For example, a given investigator who has to do his own microtomy might take the aggravation of thin sectioning into account, or if several investigators must compete for time on a single electron microscope, the time required to make pictures could be weighted accordingly among the costs. Thus the optimum ratios reflect the circumstances of the individual investigator and do not represent eternal verities. The most economical ratios of measurements per picture etc. would be:
C.,s2
optimum n.=
|~~CbSP
*
[C,ASb2 optimuim nb
=
'
CbSo2
Obviouslv, there can be no less than one measurement per picture, one picture per block or one block per organ. From a practical point of view, it would be unwise to have less than two at any level so that continuing quality control and rechecking of the sources of variability can be exercised. Table 2 shows the calculated variance estimates from the data of Table 1 and the estimated costs in person-hours of incubating a single retina, Table 2-Calculated Variance Estimates and Most Efficient Ratios n
Measurements 25 Pictures 4 Blocks 2
TSS
DF
3.764 0.2288 0.2438 1.3140
768 24 3 3
Organs (retinas) 4 * Cost calculated for our laboratory in
MS 0.0049 0.0095 0.0813 0.4380 person-hours.
S2
Cost*
Most efficient ratio
0.0049
0.02 0.1 3.5 3.5
0.95
0.0002 0.0007 0.0018
2.07
EXPERIMENTAL DESIGN IN EM MORPHOMETRY
Vol. 81, No. 3 December 1975
509
sectioning a block, imaging and processing an electron micrograph, and measuring a single mitochondrion. As indicated in the last column of Table 2, the optimum ratio of measurements to pictures, pictures to blocks, and blocks to organs are quite different than those actually used in the pilot study to derive these quantities. This simply means that a good deal of effort would have been wasted had the original ratios been used throughout the experimental study. The standard error of the mean (SEM-the standard deviation of the no, organ means divided by the square root of n0) is the measure of sample dispersion which actually enters into the statistical inference performed in the t test. One usually wants to know whether the experimentally altered organ group differs from the control group. The sensitivity one enjoys in detecting such a difference increases as the SEMs decrease. Equation 1 shows the SEM partitioned into the components which arise from each sampling level.
SE-1(5)
S
) n,
+2
n, X n,
X n X np +SW X nb n,,
nx:n
n
pXfl
(1)
Consideration of the ever increasing denominators of individual terms of Equation 1 suggests the generalization that the variance due to differences between organs is usually going to make the largest contribution to the SEM as a whole. In Table 2, 5,n2 iS larger than 02 but it should be evident that So2 is a much larger component of the SEM from Equation 1. If one wished to reduce the SEM to some particular magnitude, the most efficient way of doing this would usually be to increase To and not any of the other ratios. Having determined the most efficient distribution of effort in studying a single tissue specimen, we are still left with the unanswered question: How many specimens we must have in each experimental category. This question can never be answered apart from the specific interests and objectives of the investigator. The question of how many organs one must have in each treatment group can only be answered if the following information is available: 1. How small a difference between groups does one wish to detect? 2. How great a chance of coming up with the wrong answer is acceptable? What constitutes a wrong answer depends in part on the relative interests of the experimenter. There are two kinds of wrong answers in statistical inferences about the difference between two samples. Either a difference between populations is inferred where none actually exists, or a difference is missed where one of importance actually is present.
510
SHAY
American Journal of Pathology
3. What is the sample variance of organs (S2)? The first two pieces of information can be derived from a priori considerations relating to the needs and interests of the investigator. The third can only come from a pilot study, previous work, or pure guess. If d is the size of the difference one wishes to detect and S2 iS the between organ variance estimate as calculated above, the sample size is a function of the following three quantities: 1. 2S02/d2 2. Pi = size of the desired probability of finding the difference if it really is equal to or greater than d. 1 - P is the probability of missing a real difference equal to or greater than d. 3. a = the size of the acceptable probability (risk) that there is actually no difference when the test of hypothesis indicates there is one. Table 3, which is adapted from Section 4.13 of Snedecor and Cochran, estimates sample size in multiples of 2So2/& for one-tailed t tests and thus based on the assumption of equal variances and a normal distribution. The t test assumption of sampling from normal populations is usually well fulfilled bv nested or cluster samples. Even if the underlying distribution is skewed, the successive subsampling repeatedly brings the normalizing effect of the central limit theorem into play. To give a concrete example: S02 calculated as above from pilot
data for diameters of mitochondria of normal retinal receptor cells (Table 2) is 0.0018 sq g, and suppose we wish to detect swelling of 0.1 s over the mean (0, Table 1) of 0.38 u with a probability of 0.90 and are unwilling to accept more than a 0.05 chance of an apparent 0.1 L difference arising by chance. 2SO /d2
=
0.01
=
0.36
The multiplier from Table 3 is 8.6. Thus, 0.36 times 8.6 or a minimum of three retinas per group would probably suffice to answer these desired conditions. Where the assumptions of normality and approximate equality of variance are not met, the assistance of a statistical consultant will undoubtedlv be required. Table 3-Multipliers for Estimating Minimum Sample Size
=0.01 =
P'=0.80 PI = 0.90 P' = 0.95
10.0 13.0 15.8
a
=
0.05
6.2 8.6 10.8
a =0.10 4.5 6.6 8.6
Vol. 81, No. 3 December 1975
EXPERIMENTAL DESIGN IN EM MORPHOMETRY
511
The methods described above apply not onlyT to data such as caliper measurements of objects within pictures but also to data such as ratios of point counts. For example, the ratio for a given picture of grid points falling over mitochondria to the total points falling over cells would constitute a measurement analyzable by the methods described above. The only difference is that there would be only one such measurement per picture. The analysis would thus have one level less than described above, i.e., n., = 1, DFni = 0. Concluding Comment It is worthwhile to reemphasize that the early application of sound statistical principles to experimental design in morphology can result in sometimes astonishing savings of labor. While this brief article has attempted to instruct the reader on how pilot data can be used to maximize ones efficiency of effort, the soundest course for most investigators would be to consult a professional statistician to aid them in planning meaningful application of statistical principles to their research.
Acknowledgments I am grateful to John P. Gilbert. PhD of the Statistics Department of Harvard University for his criticism during preparation of this manuscript.
Reference 1.
Snedecor GW. Cochran WG: Statistical Methods, Sixth edition. Ames, Iowa, Iowa State Universitv Press, 1967.
512
SHAY
American Journal of Pathology
[End of Article]
Statistical techniques known as the analysis of variance make it possible for the electron microscopist to plan work in such a way as to get quantitativ e data with the greatest possible economy of effort. This paper explains how to decide how many measurements to make per micrograph, how many micrographs per tissue block, how- many blocks per experimentally treated organ, and how many organs per experimental treatment group. (Am J Pathol 81:503-512, 1975)
help the electron microscopist in experimental pathology answver the perennial questions: How many measurements must be made per picture? How many pictures per section? How many blocks per animal? How many animals in each experimnental condition?' This article will attempt to answer these questions in a practical wa- for the morphologist by adaptation of statistical techniques developed over the last 20 or so years. I shall not cite the original statistical work, but rather refer the reader to a recent and lucid text, i.e., Snedecor and Cochran's Statistical Methods.' The exigencies of biologic microscopy force upon us a kind of random sampling called nested sampling, cluster sanmpling, or samples within samples. The microscopist interested in quantitative data is rightfully reluctant to limit his measurements to one mitochondrion per liver, one synapse per brain, or the like. He sections a number of blocks per animal, takes micrographs of a number of fields per grid, and may measure a number of cell parts per field. Fortunately, the analysis of such subsampled or nested data is well understood and approachable by the techniques kno-n as the analysis of variance. Application of this statistical technique generates the quantities necessar to decide how many blocks, microscopic fields, measurements, etc., to use. Let us first look at a single group of experimental material -hich has all been treated the same \-ay-. Let us consider a number (n0) organs. These would be the actual undivided tissue of interest; e.g.. the n0 livers of LITTLE
IS
CURREN-TLY
AV AILABLE
to
From the Mlixter Laboratorv for Electron \Microscopy of the Neurosurgical Service of the MlassachuSetts General Hospital. Boston. \Massachusetts Supported by Grant NS-12023-O1 from the National Institutes of Health -ccepted for publication Xugust 1.3. 197,3. Address reprint requests to Dr Jonathan Shay. Mlixter Laboratory. Neurosurgical Service. Mlassachusetts General Hospital. Boston. MIA 022114 503
504
SHAY
American Journal of Pathology
animals treated with such and such a drug or the no isolated retinas incubated under the same conditions, etc. From each of these Tb organs, nb tissue blocks are taken at random for study. For simplicity, let us take only one thin section from each block for microscopy. From each of the n x nb sections we take rip micrographs at random, and within each of the no X nb X np pictures we measure a random sample of flm objects of one particular kind in which we are interested. In this experimental condition there are: nO
organs
n0 X nbi blocks n,XX nbX n, pictures and n. X nb X n, X n,, measurements. This is always the form of sampling in electron microscopy with the variation of more or fewer levels of nesting or clustering. If there is only one measurement per picture (e.g., a ratio of point counts) there would be one level fewer than in the present example. If there are more than one organ per animal or there are anatomic regions within the organ, such as lobes of the liver, the analysis would have more levels. Our laboratory is conducting studies of the effects of oxygen and glucose deprivation on isolated rabbit retina. Table 1 gives an actual analysis of some pilot electron microscope data from this work. Table 1 gives results of analysis of 25 mitochondrial measurements from each of four pictures from each of two blocks from each of four isolated retinas incubated under identical conditions. Starting from desk calculator tapes giving the mean and variance of each of the 32 sets of 25 measurements, this analysis took about 3 hours. In our laboratory, the original caliper measurements are entered directly on the calculator, and the tapes thus become the record of the raw measurements. (It should be noted that neither a costly calculator nor computer time is required for any of the computations described in this paper.) Let iMobpm = a single measurement (the mth) coming from a given picture (the pth) of a given block (the bth) of the oth organ. For the data in Table 1, Mlobpm was the diameter of a single mitochondrion from a single picture from the only tissue section taken from a tissue block of isolated rabbit retina (the oth). These original 800 measurements do not appear in Table 1. The entries in the first column of Table 1 are the sums of squares between measurements within single pictures, i.e.,
SS(M.bpn)
=
E _
-
E a
M0b2, /n.)2
c
-c
cO00
T-
C'I)
0i
0
E 0 co °
3 3
a:0
Cl ')co)l
g 0 E
' 0e
C
c
0~ 1-
0
~~0
CM
0 0
C')
00
C') C')
C')
0 V
a: E
co 0
0 0
C&
0
00
o 0 0 0 0
0
,
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co'0.
Y .0
CO a,
CO d
0 C_
0
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6
6
C
6
6
(
0 0 O C
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c
Q
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10 C') C')
m E
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03 1010 O (0 CJ 0)C') ~J CJ
0
0
t Z_s cm
(0011LOLO
_ Ote0
19ts
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a:
s
506
SHAY
American Journal of Pathology
This was easily computed from the calculator tape by multiplying the variance by nn - 1. The total sum of squares for measurements within pictures is TSS(M)
=
E E (M0obp.- 'ia M.b,,/n .)2 E E o b p -
which is to sav, the total of all the individual sums of squares between measurements within individual pictures. We will use the convention that a capital P will stand for the mean, within a given picture, of all the measurements in that picture, a capital B the mean of the picture means in a given block, a capital 0 the mean of the blocks within a given organ. Let us define the mean of measurements within a given picture, i.e., P ob
M.obus11n.
=
This comes right off the calculator tape. The sum of squares between pictures within a single block is thus (Pob, EP:bpb/n,)2 SS(Pobp) = xE p P The total sum of squares between pictures within blocks is then -
TSS(P) = n.
E Eb E (Pobp- E P.bpnp)2 p 0
p
Defining the individual block mean as above Bob = E P0b,//nr Then the sum of squares between blocks within a single organ is SS(B b) = n1.n,
(Bob -E B.b/nb)2 E b b
and the total sum of squares for blocks is TSS(B)
=
n.nup
E o
(Bb- E: B.b/nb)2 E b b
As above 0
=
E
b
B.b/nb
then the total sum of squares between organs within a given experimental condition is
TSS(O)
=
n.nfnub
(0. o
-
E 0.X'n.)2 o
Vol. 81, No. 3 December 1975
EXPERIMENTAL DESIGN IN EM MORPHOMETRY
507
The degrees of freedom for calculating the mean square are DF(o) = nO- 1 DF(b)
nfo(nb- 1)
=
DF(p) = n. X nb(np-1) DF(m) = n0 X nb X n,(n. -1)
The respective mean squares are then MS(o) MS(b) MS(p) MS(m)
= TSS(o)/DF(o) = TSS(b)/DF(b) = TSS(p)/DF(p) = TSS(m)/DF(m)
The sample variances (variance estimates) due to measurements, to
blocks, etc., are computed in the following manner: Sample variance of measurements Sm2= MS(m)
Sample variance of pictures 2
MS(p) -MS(m) n,.
Sample variance of blocks M2 S(b)
-
MS(p)
n,. X nb
Sample variance of organs M2 S(o) - MS(b) n. X nf X nb
Calculation of Most Economical Ratios The above variance estimates for each sampling level can be used to plan further experiments so that one gets the most for one's time and money. For example, if the main source of variation is between organs, there is little to be gained from sectioning many blocks per organ, taking many pictures per block and making dozens of measurements per picture. Subject to certain simple restrictions, one can calculate the most economical distribution of effort.
508
SHAY
American Journal of Pathology
Let
CO = cost per organ Cb = cost per block C, = cost per picture C. = cost per measurement These costs can be expressed in any terms the investigator finds appropriate for his own circumstances. As long as the units are consistent, the "cost" can be purely subjective, or a mixture of weighted subjective and objective inputs. For example, a given investigator who has to do his own microtomy might take the aggravation of thin sectioning into account, or if several investigators must compete for time on a single electron microscope, the time required to make pictures could be weighted accordingly among the costs. Thus the optimum ratios reflect the circumstances of the individual investigator and do not represent eternal verities. The most economical ratios of measurements per picture etc. would be:
C.,s2
optimum n.=
|~~CbSP
*
[C,ASb2 optimuim nb
=
'
CbSo2
Obviouslv, there can be no less than one measurement per picture, one picture per block or one block per organ. From a practical point of view, it would be unwise to have less than two at any level so that continuing quality control and rechecking of the sources of variability can be exercised. Table 2 shows the calculated variance estimates from the data of Table 1 and the estimated costs in person-hours of incubating a single retina, Table 2-Calculated Variance Estimates and Most Efficient Ratios n
Measurements 25 Pictures 4 Blocks 2
TSS
DF
3.764 0.2288 0.2438 1.3140
768 24 3 3
Organs (retinas) 4 * Cost calculated for our laboratory in
MS 0.0049 0.0095 0.0813 0.4380 person-hours.
S2
Cost*
Most efficient ratio
0.0049
0.02 0.1 3.5 3.5
0.95
0.0002 0.0007 0.0018
2.07
EXPERIMENTAL DESIGN IN EM MORPHOMETRY
Vol. 81, No. 3 December 1975
509
sectioning a block, imaging and processing an electron micrograph, and measuring a single mitochondrion. As indicated in the last column of Table 2, the optimum ratio of measurements to pictures, pictures to blocks, and blocks to organs are quite different than those actually used in the pilot study to derive these quantities. This simply means that a good deal of effort would have been wasted had the original ratios been used throughout the experimental study. The standard error of the mean (SEM-the standard deviation of the no, organ means divided by the square root of n0) is the measure of sample dispersion which actually enters into the statistical inference performed in the t test. One usually wants to know whether the experimentally altered organ group differs from the control group. The sensitivity one enjoys in detecting such a difference increases as the SEMs decrease. Equation 1 shows the SEM partitioned into the components which arise from each sampling level.
SE-1(5)
S
) n,
+2
n, X n,
X n X np +SW X nb n,,
nx:n
n
pXfl
(1)
Consideration of the ever increasing denominators of individual terms of Equation 1 suggests the generalization that the variance due to differences between organs is usually going to make the largest contribution to the SEM as a whole. In Table 2, 5,n2 iS larger than 02 but it should be evident that So2 is a much larger component of the SEM from Equation 1. If one wished to reduce the SEM to some particular magnitude, the most efficient way of doing this would usually be to increase To and not any of the other ratios. Having determined the most efficient distribution of effort in studying a single tissue specimen, we are still left with the unanswered question: How many specimens we must have in each experimental category. This question can never be answered apart from the specific interests and objectives of the investigator. The question of how many organs one must have in each treatment group can only be answered if the following information is available: 1. How small a difference between groups does one wish to detect? 2. How great a chance of coming up with the wrong answer is acceptable? What constitutes a wrong answer depends in part on the relative interests of the experimenter. There are two kinds of wrong answers in statistical inferences about the difference between two samples. Either a difference between populations is inferred where none actually exists, or a difference is missed where one of importance actually is present.
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American Journal of Pathology
3. What is the sample variance of organs (S2)? The first two pieces of information can be derived from a priori considerations relating to the needs and interests of the investigator. The third can only come from a pilot study, previous work, or pure guess. If d is the size of the difference one wishes to detect and S2 iS the between organ variance estimate as calculated above, the sample size is a function of the following three quantities: 1. 2S02/d2 2. Pi = size of the desired probability of finding the difference if it really is equal to or greater than d. 1 - P is the probability of missing a real difference equal to or greater than d. 3. a = the size of the acceptable probability (risk) that there is actually no difference when the test of hypothesis indicates there is one. Table 3, which is adapted from Section 4.13 of Snedecor and Cochran, estimates sample size in multiples of 2So2/& for one-tailed t tests and thus based on the assumption of equal variances and a normal distribution. The t test assumption of sampling from normal populations is usually well fulfilled bv nested or cluster samples. Even if the underlying distribution is skewed, the successive subsampling repeatedly brings the normalizing effect of the central limit theorem into play. To give a concrete example: S02 calculated as above from pilot
data for diameters of mitochondria of normal retinal receptor cells (Table 2) is 0.0018 sq g, and suppose we wish to detect swelling of 0.1 s over the mean (0, Table 1) of 0.38 u with a probability of 0.90 and are unwilling to accept more than a 0.05 chance of an apparent 0.1 L difference arising by chance. 2SO /d2
=
0.01
=
0.36
The multiplier from Table 3 is 8.6. Thus, 0.36 times 8.6 or a minimum of three retinas per group would probably suffice to answer these desired conditions. Where the assumptions of normality and approximate equality of variance are not met, the assistance of a statistical consultant will undoubtedlv be required. Table 3-Multipliers for Estimating Minimum Sample Size
=0.01 =
P'=0.80 PI = 0.90 P' = 0.95
10.0 13.0 15.8
a
=
0.05
6.2 8.6 10.8
a =0.10 4.5 6.6 8.6
Vol. 81, No. 3 December 1975
EXPERIMENTAL DESIGN IN EM MORPHOMETRY
511
The methods described above apply not onlyT to data such as caliper measurements of objects within pictures but also to data such as ratios of point counts. For example, the ratio for a given picture of grid points falling over mitochondria to the total points falling over cells would constitute a measurement analyzable by the methods described above. The only difference is that there would be only one such measurement per picture. The analysis would thus have one level less than described above, i.e., n., = 1, DFni = 0. Concluding Comment It is worthwhile to reemphasize that the early application of sound statistical principles to experimental design in morphology can result in sometimes astonishing savings of labor. While this brief article has attempted to instruct the reader on how pilot data can be used to maximize ones efficiency of effort, the soundest course for most investigators would be to consult a professional statistician to aid them in planning meaningful application of statistical principles to their research.
Acknowledgments I am grateful to John P. Gilbert. PhD of the Statistics Department of Harvard University for his criticism during preparation of this manuscript.
Reference 1.
Snedecor GW. Cochran WG: Statistical Methods, Sixth edition. Ames, Iowa, Iowa State Universitv Press, 1967.
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American Journal of Pathology
[End of Article]
