### Summary

- Top of page
- Summary
- 1. Introduction
- 2. Sampling with slices: background and purpose
- 3. Variance prediction for measurement functions with fractional smoothness
- 4. A table of the coefficient α(
*q*, τ) - 5. Empirical performance of the new variance predictors on real data
- 6. Discussion
- Acknowledgements
- References
- Appendices

A general variance predictor is presented for a Cavalieri design with slices of an arbitrary thickness *t* ≥ 0. So far, prediction formulae have been available either for measurement functions with smoothness constant *q* = 0, 1, … , and *t* ≥ 0, or for fractional *q* ∈ [0, 1] with *t* = 0. Because the possibility of using a fractional *q* adds flexibility to the variance prediction, we have extended the latter for any *q* ∈ [0, 1] and *t* ≥ 0*.* Empirical checks with previously published human brain data suggest an improved performance of the new prediction formula with respect to the hitherto available ones.

### 1. Introduction

- Top of page
- Summary
- 1. Introduction
- 2. Sampling with slices: background and purpose
- 3. Variance prediction for measurement functions with fractional smoothness
- 4. A table of the coefficient α(
*q*, τ) - 5. Empirical performance of the new variance predictors on real data
- 6. Discussion
- Acknowledgements
- References
- Appendices

Cavalieri sections are extensively used in stereology to estimate volumes, whereas Cavalieri slices may be used to estimate any other geometric measure, notably cell or particle number. The prediction of the variance of the corresponding estimators is therefore of interest. The problem is nontrivial because the observations are systematic, hence dependent in general. The state of the art has recently been reviewed in Cruz-Orive & García-Fiñana (2005) for Cavalieri sections, and in Cruz-Orive & Geiser (2004) for Cavalieri slices.

Suppose that the target is the volume *Q* of a bounded object. Fix a sampling axis and consider the area *f*(*x*) of the intersection between the object and a plane normal to the sampling axis at a point of abscissa *x*. Then, *Q* is the area under the graph of the function *f* : ℝℝ^{+}. In general, *f* is assumed to be square integrable, and it is called the measurement function. The ‘trend’ of the variance of the Cavalieri estimator *Q̂* of *Q* is well represented by a term of the variance called the extension term, Var_{E}(*Q̂*). It turns out that this term is proportional to *T*^{2q+2}, where *T* is the distance between the sampling planes and *q* is the ‘smoothness constant’ of *f*, namely a non negative real constant that represents the order of the first non-continuous (possibly fractional) derivative of *f*. So far, the following cases have been considered:

Kiêu (1997), Kiêu *et al.* (1999) and Gundersen *et al.* (1999) consider *q* = 0, 1, … , (denoted by ‘*m*’ in these papers) and, although they contemplate the case *t* > 0, their variance predictors do not involve the parameter *t*. As a consequence, their variance predictors do not tend to zero (as they should) when the slice thickness *t* approaches the sampling period *T* (namely the distance between slice midplanes). Earlier predictors (Matheron, 1965, 1971; Gundersen & Jensen, 1987; Cruz-Orive, 1989) involve neither *t* nor *q* (they implicitly assume *t* = 0 and *q* = 0).

The problem of predicting the variance of the estimator of *Q* from Cavalieri slices of thickness *t* > 0 for measurement functions with smoothness constant *q* ∈ [0, 1], has hitherto remained open. The purpose of this paper is to fill this gap.

The new prediction formulae are displayed in Section 3, the relevant coefficient α(*q*, τ) is computed in Section 4, and the performance of the formulae is checked in Section 5 for the brain data used in García-Fiñana & Cruz-Orive (2000) and in McNulty *et al.* (2000). For practical applications the reader may concentrate on the formulae (3.3)–(3.5), resorting to Table 1 to compute the α(*q*, τ). The derivation details are given in the Appendixes.

Table 1. Numerical values of the reciprocal 1/α(*q*, τ) of the coefficient α(*q*, τ), see Eq. (3.5), entering in the right hand side of Eq. (3.3), with up to four significant digits. *q* = smoothness constant of the measurement function; τ = t/*T* = slice thickness/sampling period. τ = q | 0.00 | 0.05 | 0.10 | 0.15 | 0.20 | 0.25 | 0.30 | 0.35 | 0.40 | 0.45 | 0.50 | 0.55 | 0.60 | 0.65 | 0.70 | 0.75 | 0.80 | 0.85 | 0.90 | 0.95 |
---|

0.00 | 12.0 | 13.0 | 14.1 | 15.4 | 16.9 | 18.7 | 20.8 | 23.4 | 26.7 | 30.7 | 36.0 | 43.0 | 52.5 | 66.1 | 86.7 | 120.0 | 180.0 | 306.7 | 660.0 | 2520 |

0.05 | 13.7 | 14.6 | 15.7 | 17.1 | 18.8 | 20.7 | 23.1 | 26.0 | 29.6 | 34.2 | 40.1 | 47.9 | 58.7 | 74.1 | 97.5 | 135.6 | 204.3 | 350.3 | 760.0 | 2934 |

0.10 | 15.5 | 16.4 | 17.7 | 19.2 | 21.0 | 23.1 | 25.8 | 29.0 | 33.0 | 38.1 | 44.8 | 53.6 | 65.8 | 83.3 | 109.9 | 153.4 | 232.3 | 400.6 | 875.7 | 3414 |

0.15 | 17.7 | 18.6 | 19.8 | 21.5 | 23.4 | 25.8 | 28.8 | 32.4 | 36.9 | 42.6 | 50.1 | 60.1 | 73.9 | 93.8 | 124.1 | 173.9 | 264.5 | 458.7 | 1010.0 | 3972 |

0.20 | 20.2 | 21.0 | 22.4 | 24.1 | 26.3 | 28.9 | 32.2 | 36.2 | 41.3 | 47.8 | 56.2 | 67.5 | 83.2 | 105.8 | 140.5 | 197.5 | 301.8 | 526.0 | 1165.0 | 4622 |

0.25 | 23.0 | 23.8 | 25.3 | 27.2 | 29.6 | 32.5 | 36.1 | 40.7 | 46.4 | 53.7 | 63.2 | 76.0 | 93.8 | 119.7 | 159.3 | 224.8 | 344.9 | 604.2 | 1346.0 | 5379 |

0.30 | 26.2 | 27.1 | 28.6 | 30.7 | 33.4 | 36.6 | 40.7 | 45.8 | 52.2 | 60.4 | 71.3 | 85.9 | 106.2 | 135.8 | 181.2 | 256.5 | 395.1 | 695.4 | 1558.0 | 6266 |

0.35 | 30.0 | 30.9 | 32.5 | 34.8 | 37.7 | 41.4 | 46.0 | 51.7 | 59.0 | 68.3 | 80.6 | 97.3 | 120.5 | 154.4 | 206.7 | 293.4 | 453.7 | 802.0 | 1806.0 | 7308 |

0.40 | 34.4 | 35.3 | 37.1 | 39.6 | 42.8 | 46.9 | 52.1 | 58.6 | 66.8 | 77.4 | 91.5 | 110.5 | 137.2 | 176.2 | 236.4 | 336.6 | 522.4 | 927.2 | 2098.0 | 8536 |

0.45 | 39.5 | 40.4 | 42.4 | 45.1 | 48.8 | 53.4 | 59.2 | 66.6 | 75.9 | 88.1 | 104.2 | 126.0 | 156.7 | 201.6 | 271.2 | 387.4 | 603.1 | 1075.0 | 2443.0 | 9990 |

0.50 | 45.5 | 46.5 | 48.6 | 51.7 | 55.7 | 61.0 | 67.6 | 75.9 | 86.7 | 100.6 | 119.1 | 144.2 | 179.6 | 231.5 | 312.2 | 447.2 | 698.6 | 1250.0 | 2853.0 | 11720 |

0.55 | 52.6 | 53.6 | 55.9 | 59.3 | 63.9 | 69.9 | 77.4 | 87.0 | 99.3 | 115.3 | 136.6 | 165.7 | 206.6 | 266.9 | 360.8 | 518.2 | 812.0 | 1458.0 | 3341.0 | 13780 |

0.60 | 61.0 | 62.0 | 64.6 | 68.4 | 73.6 | 80.4 | 89.0 | 100.0 | 114.2 | 132.7 | 157.4 | 191.1 | 238.7 | 309.0 | 418.6 | 602.8 | 947.5 | 1706.0 | 3926.0 | 16260 |

0.65 | 70.9 | 72.0 | 74.9 | 79.3 | 85.2 | 92.9 | 102.9 | 115.6 | 132.0 | 153.5 | 182.2 | 221.5 | 277.1 | 359.3 | 487.8 | 704.3 | 1110.0 | 2006.0 | 4632.0 | 19250 |

0.70 | 82.8 | 84.1 | 87.3 | 92.3 | 99.1 | 108.0 | 119.5 | 134.2 | 153.3 | 178.4 | 211.9 | 257.9 | 323.2 | 419.9 | 571.3 | 826.8 | 1307.0 | 2369.0 | 5488.0 | 22890 |

0.75 | 97.3 | 98.6 | 102.3 | 108.0 | 115.8 | 126.2 | 139.5 | 156.8 | 179.1 | 208.5 | 247.9 | 302.1 | 379.1 | 493.4 | 672.6 | 975.9 | 1547.0 | 2812.0 | 6535.0 | 27340 |

0.80 | 114.8 | 116.4 | 120.5 | 127.1 | 136.2 | 148.3 | 163.9 | 184.2 | 210.5 | 245.2 | 291.8 | 356.0 | 447.4 | 583.2 | 796.8 | 1159.0 | 1841.0 | 3357.0 | 7825.0 | 32840 |

0.85 | 136.4 | 138.1 | 142.9 | 150.6 | 161.2 | 175.4 | 193.9 | 217.9 | 249.1 | 290.3 | 345.8 | 422.4 | 531.6 | 694.2 | 950.3 | 1385.0 | 2207.0 | 4034.0 | 9431.0 | 39690 |

0.90 | 163.2 | 165.2 | 170.8 | 179.7 | 192.4 | 209.2 | 231.2 | 259.8 | 297.1 | 346.5 | 413.0 | 505.0 | 636.5 | 832.7 | 1142.0 | 1668.0 | 2664.0 | 4884.0 | 11450.0 | 48320 |

0.95 | 197.0 | 199.3 | 205.8 | 216.4 | 231.5 | 251.6 | 278.0 | 312.4 | 357.5 | 417.1 | 497.7 | 609.3 | 769.0 | 1008.0 | 1385.0 | 2027.0 | 3245.0 | 5964.0 | 14020.0 | 59310 |

1.00 | 240.0 | 242.7 | 250.5 | 263.3 | 281.4 | 305.8 | 337.9 | 379.7 | 434.6 | 507.5 | 606.0 | 742.7 | 938.7 | 1232.0 | 1696.0 | 2488.0 | 3993.0 | 7356.0 | 17330.0 | 73530 |

### 4. A table of the coefficient α(*q*, τ)

- Top of page
- Summary
- 1. Introduction
- 2. Sampling with slices: background and purpose
- 3. Variance prediction for measurement functions with fractional smoothness
- 4. A table of the coefficient α(
*q*, τ) - 5. Empirical performance of the new variance predictors on real data
- 6. Discussion
- Acknowledgements
- References
- Appendices

The coefficient α(*q*, τ) cannot be evaluated directly from Eq. (3.5) for all *q* ∈ [0, 1]*.* In fact, for *q* = 0, 1 we should use Eq. (3.7); for *q* = 1/2 and τ > 0 we should use Eq. (3.10); for *q* ≠ 1/2 and τ = 0 we should use Eq. (3.8), whereas α(1/2, 0) is given by Eq. (3.11). For the remaining values *q* ∈ (0, 1)∖{1/2} and τ ∈ (0, 1), we can use the expression (3.5).

The summation in the right hand sides of Eq. (3.5), (3.10) may be evaluated quickly with the aid of the software packages maple® or mathematica® using the polylogarithm function,

- ((4.1))

In particular Li_{a}(1) =ζ(*a*) is Riemann's Zeta function. Thus,

- ((4.2))

where Re() denotes the real part of the expression within parentheses.

Table 1 and Fig. 1 have been obtained bearing the preceding recommendations in mind.

The graph of 1/α(*q*, τ) plotted in Fig. 1 is intended to give some idea of its pattern of variation; for instance, for τ beyond 0.7 (namely when the slice thickness approaches the sampling period), the value of the coefficient α(*q*, τ) rapidly approaches zero for any *q* ∈ [0, 1]. Numerical values of 1/α(*q*, τ) are better obtained from Table 1. The graph extends the particular case τ = 0 considered in García-Fiñana & Cruz-Orive (2000, 2004).

### 5. Empirical performance of the new variance predictors on real data

- Top of page
- Summary
- 1. Introduction
- 2. Sampling with slices: background and purpose
- 3. Variance prediction for measurement functions with fractional smoothness
- 4. A table of the coefficient α(
*q*, τ) - 5. Empirical performance of the new variance predictors on real data
- 6. Discussion
- Acknowledgements
- References
- Appendices

As a test example we consider the grey matter data of a human brain (including the cerebellum), as used by McNulty *et al*. (2000). Details on the data source are given in the latter paper. The basic data set was obtained from *N*_{0} = 183 consecutive coronal slices of *t* = 1 mm thickness, obtained by noninvasive magnetic resonance imaging techniques with the aid of the Analyze software (Mayo Foundation, MA, U.S.A.). The corresponding data vector can be found in appendix 4 from García-Fiñana & Cruz-Orive (2000). The volume of each basic slice, see Eq. (2.2), was measured with a negligible error by automatic pixel counting (pixel side = 1 mm). Following McNulty *et al*. (2000) we consider four slice thicknesses, namely *t* = 1, 3, 9, 27 mm. For *t* > 1 mm, slice volumes were computed by pooling the pertinent consecutive slices of *t* = 1 mm thickness, see McNulty *et al*. (2000), eq. (13). Thus, in this application local errors were discarded, and consequently we set ν_{n} = 0 in the right hand side of Eq. (3.3). The slice data were analysed without previous smoothing.

For a given slice thickness, *t*, the maximum number of non empty slices is

- ((5.1))

where ‘⌈⌉’ denotes ‘ceiling’ (e.g. ⌈3.1⌉ = 4, ⌈3.0⌉ = 3). For each *t* we considered an initial integer sequence of mean slice numbers *n*_{0} ranging from 1 to *n*_{max}. For each member *n*_{0} of this sequence the adopted sampling period *T*, namely the distance between consecutive slice midplanes, was forced to be an integer, namely

- ((5.2))

where ‘⌊⌋’ denotes ‘floor’, (e.g. ⌊3.9⌋ = 3, ⌊3.0⌋ = 3). Therefore the total number of available Cavalieri slice samples of period *T* was precisely *T.* Moreover, with the adjustment described in *Remark* 1 below, the actual number of slices of thickness *t* in each of the *T* samples of period *T* was also forced to be an integer and it was computed as

- ((5.3))

The number of nonempty slices in a sample could be either *n*, or *n* − 1*.* For each of the *T* samples the corresponding estimator of the grey matter volume was computed via Eq. (2.9), and then the variance between the *T* available volume estimates was also computed. This empirical variance was adopted to represent Var(*Q̂*) for the pertinent pair (*t*, *T*). The full thinner curves plotted in Fig. 2 represent the corresponding coefficients of error, namely the empirical version of

- ((5.4))

as a function of *n*, see Eq. (5.3). Further, Var(*Q̂*) was computed via Eq. (3.3) with ν_{n} = 0 for each of the *T* samples and for *q* = 0, 0.42, 1, respectively. For each *q* the corresponding *T* estimates were averaged. The corresponding predictors of CE(*Q̂*) are also plotted in Fig. 2. The estimate *q̂*= 0.42 was the average of the two estimates (namely 0.44 and 0.40) computed, respectively, from the 92 odd– and the 91 even–numbered slices (that is with *T* = 2 mm), from the basic set of 183 slices of 1 mm thickness. The estimator used was Kiêu–Souchet's,

- ((5.5))

The results are commented in the Discussion.

*Remark 1*. As explained above, for each pair (*t*, *T*) the total number of Cavalieri samples of size *n *[see Eq. (5.3)], is precisely *T*. Because *nT* ≥ *N*_{0}, we adjusted *t* zeros at the beginning and *nT* − *N*_{0} − *t* zeros at the end of the initial vector of *N*_{0} primary slice volumes. In this way we proceeded with a suitable ‘extended’ vector of size *nT* (McNulty *et al*., 2000). Here it helps to realize that, in order to subsample a fixed segment [0, *H*] with a uniform random test segment [*x*, *x* + *t*), the range of *x* is [– *t*, *H*].

### 6. Discussion

- Top of page
- Summary
- 1. Introduction
- 2. Sampling with slices: background and purpose
- 3. Variance prediction for measurement functions with fractional smoothness
- 4. A table of the coefficient α(
*q*, τ) - 5. Empirical performance of the new variance predictors on real data
- 6. Discussion
- Acknowledgements
- References
- Appendices

The new variance predictor (3.3), with the general coefficient (3.5), covers the long standing need for a free choice of the smoothness constant *q* in the interval [0, 1], combined with an arbitrary slab thickness *t* > 0.

The performance of the new variance predictor has been tested for the data set described in Section 5. The results displayed in Fig. 2 look fairly satisfactory in the sense that, for each of the four slab thicknesses considered, the variance prediction (thicker continuous curve) seems to conform to the concept of ‘extension term’ or ‘trend’ of the variance. This impression should, however, be confirmed with further experiments on different objects.

The main snag in the prediction procedure lies, however, in the estimation of *q*. Thus:

- (a)
As explained in the last paragraph of

Cruz-Orive (1999), the estimation of

*q* is precluded in practice for

*t* > 0

*.* Thus, to estimate

*q* by Eq.

(5.5) it is necessary to use a Cavalieri sample of

*n* ≥ 5 sections, namely slices of ‘zero thickness’ or ‘as thin as possible’.

- (b)
As shown in

García-Fiñana & Cruz-Orive (2004), fig. 4(c), the estimator

(5.5) of

*q* is rather unstable for low or moderate sample sizes, and it may vary as a function of the sample size.

The best strategy to circumvent the preceding shortcomings is still open.

For the ‘splitting’ Cavalieri design described in Cruz-Orive (2004), the first factor in the right hand side of Eq. (2.27) – which holds for *q* = 0 – could be updated in the light of the new variance model (3.2). The task, however, would be unlikely to be worthwhile in view of the comments made in the last paragraph of the Discussion of that paper.