Theoretical and Applied Genetics

, Volume 110, Issue 7, pp 1236–1243

The distribution of genetic parameter estimates and confidence intervals from small disconnected diallels

Authors

    • Department of Forestry and Environmental ResourcesNorth Carolina State University
  • Dennis D. Boos
    • Department of StatisticsNorth Carolina State University
  • Bailian Li
    • Department of Forestry and Environmental ResourcesNorth Carolina State University
Original Paper

DOI: 10.1007/s00122-005-1957-0

Cite this article as:
Isik, F., Boos, D.D. & Li, B. Theor Appl Genet (2005) 110: 1236. doi:10.1007/s00122-005-1957-0
  • 102 Views

Abstract

The distributions of genetic variance components and their ratios (heritability and type-B genetic correlation) from 105 pairs of six-parent disconnected half-diallels of a breeding population of loblolly pine (Pinus taeda L.) were examined. A series of simulations based on these estimates were carried out to study the coverage accuracy of confidence intervals based on the usual t-method and several other alternative methods. Genetic variance estimates fluctuated greatly from one experiment to another. Both general combining ability variance (σ2g) and specific combining ability variance (σ2s) had a large positive skewness. For σ2g and σ2s, a skewness-adjusted t-method proposed by Boos and Hughes-Oliver (Am Stat 54:121–128, 2000) provided better upper endpoint confidence intervals than t-intervals, whereas they were similar for the lower endpoint. Bootstrap BCa-intervals (Efron and Tibshirani, An introduction to the bootstrap. Chapman & Hall, London 436 p, 1993) and Hall’s transformation methods (Zhou and Gao, Am Stat 54:100–104, 2000) had poor coverages. Coverage accuracy of Fieller’s interval endpoint(J R Stat Soc Ser B 16:175–185, 1954) and t-interval endpoint were similar for both h2 and rB for sample sizes n≤10, but for n=30 the Fieller’s method is much better.

Introduction

Confidence intervals for genetic variance components and their ratios have been determined using a variety of techniques. Öfversten (1993) developed methods for obtaining exact F-tests of variance components for three unbalanced mixed linear models. The methods were extended and generalized by Christensen (1996) and Fayyad et al. (1996) to test the null hypothesis that a variance component is equal to zero. The standard errors of variance components and their ratios were also approximated with chi-square distributions using the expected mean squares (Lu and Graybill 1987). Assuming standard normal distribution properties, Knapp et al. (1987) defined parametric interval estimators for heritability, a ratio of additive genetic variance to phenotypic variance. They used F-distributions to establish exact confidence intervals for the point estimates. Knapp et al. (1989) used Monte Carlo simulation to estimate Jackknife intervals of family-mean heritability and expected selection response. Their methods were based on simulated data for a balanced one-factor random linear model and expected mean squares.

The literature cited here-above mainly focused on the precision of variance components and their functions for specific experiments. In practice, parents of a breeding population are grouped into smaller units, such as diallels, for mating and progeny testing. Diallel mating designs have been widely used in plant breeding programs to estimate genetic variances (Baker 1978). Usually, a small number of parents (4–12) are allocated to a diallel because of logistic difficulties and the cost-associated limitations of breeding and testing (Wilson et al. 1978; Cisar et al. 1982; Boyle 1987; Xiang et al. 2003). Then, each diallel group is field-tested and analyzed separately to obtain variances. Thus, variance components may vary dramatically from one experiment to another because of the “bottleneck” from random sampling a small number of parents (Wright 1985). The question we address: when there are numerous empirical variance components estimated by maximum likelihood methods (ML or REML) and heritability estimates from different experiments (diallels), how can statistical inferences be made about the population?

We utilized a unique empirical data set from the North Carolina State University-Industry Cooperative Tree Improvement Program to make a statistical inference about a breeding population. For the second-cycle breeding of loblolly pine (Pinus taeda L.), 3,800 parents were grouped into six-parent disconnected half diallels. The parents were mated, and 15 crosses were produced for each diallel group (Li et al. 1999). Progeny of two diallels were tested at four locations. Two pairs of diallels were analyzed together to estimate variance components (Li et al. 1996). In this study, we utilized 105 variance components and their ratios from 105 pairs of diallels of the Coastal breeding population of loblolly pine. The data provided an excellent opportunity to examine the distribution properties of variance components from different experiments and to make statistical inference about them.

The objectives of the study reported here were to answer the following questions. (1) How are variance components and their ratios, estimated from individual diallels, distributed? (2) Depending on the distribution properties, what is the true coverage of t-confidence intervals? (3) How can we improve the true coverage of confidence intervals if there are significant departures from the normal distribution?

Materials and methods

Data

The North Carolina State University-Industry Cooperative Tree Improvement Program used a six-parent disconnected-half-diallel mating design (referred to as ‘diallel’ hereafter) to generate progenies for the second-generation breeding program of loblolly pine (Fig. 1). Parent trees of the Coastal breeding population were allocated to diallels for mating according to their flower production. For each diallel, six parents were mated and 15 crosses (full-sib families) were produced. Progenies of 30 crosses from two diallels were tested together using a randomized complete blocks design with six blocks. Each cross was represented by six-tree row plots in each six block. The experiments were replicated on four sites (Xiang et al. 2003). Each parent was represented by approximately 720 progeny across four sites. Height, volume, disease infection and stem straightness of each tree were recorded at age 6 years.
Fig. 1

The disconnected half-diallel mating design used for the second-generation loblolly pine improvement program in the southern USA. Two disconnected half-diallels (parents 1–12) were field-tested and analyzed together to estimate variance components

A pair of diallels tested at the same locations were analyzed together. A linear mixed model was fitted to the data (Xiang et al. 2003) to estimate variance components using individual tree measurements.
$$ Y_{ijoklm} = \mu + T_i + B_{j(i)} + D_o + G_{k(o)} + G_{l(o)} + S_{kl(o)} + {\text{TG}}_{ik(o)} + {\text{TG}}_{il(o)} + {\text{TS}}_{ikl(o)} + P_{ijokl} + E_{ijoklm} $$
(1)
where Yijoklm is the mth observation of the jth block for the klth cross in the oth diallel in the ith test;  μ is the overall mean; Ti is the ith fixed test environment (location) effect; i=1 to t; Bj(I) is the fixed effect of the jth block within the ith test; j=1 to b; Do is the oth fixed diallel effect; o=1; 2, Gk(o) or Gl(o) is the random general combining ability (GCA) effect of the kth female or the lth male within the oth diallel, assumed to be normally and independently distributed (NID) (0, σ2g); k, l=1 to p, and k < l; Skl(o) is the random specific combining ability (SCA) effect of the kth and the lth parents within the oth diallel (kl) ~NID (0, σ2s); TGik(o) or TGil(o) is the random interaction effect of the ith test by the kth female or the lth male GCA ~NID (0, σ2gt); TSikl(o) is the random interaction effect of the ith test by the klth SCA ~NID (0, σ2st); Pijokl is the random plot effect for the klth cross within the oth diallel in the jth block ~NID (0, σ2p); l = 1 to n; Eijoklm is the random within-plot error term ~NID (0, σ2e).

The model was analyzed with the sas mixed procedure by creating dummy variables for GCA effects (Xiang and Li 2001). The variance components of random effects were estimated using the default reml model fitting option. In the data analysis, it was assumed that six parents were randomly assigned into each diallel and that the parent trees in a diallel represent a random sample of the whole breeding population. Diallel effects were negligible and were dropped from the mixed model. Individual-tree narrow-sense heritability (h2) for each diallel was estimated as the ratio of additive genetic to phenotypic variance: h2=4σ2g/(2σ2g + σ2s + 2σ2gt + σ2st + σ2p + σ2e). Type-B genetic correlations (rB) were calculated as rB2g/(σ2g + σ2gt), which is a ratio of family variance components over the sum of the family and genotype-by-environment interaction variance (Yamada 1962). This ratio has been used commonly to quantify the level of genotype-by-environment interaction in progeny trials (Burdon 1977). In this study, we used 105 variance components, heritabilities and type-B genetic correlations estimated from 105 pairs of diallels for 6-year height growth to make inferences about the Coastal breeding population of loblolly pine in the southern USA.

Statistical analysis

Since each diallel mating group was assumed to be a random sample of parents from the breeding population, the estimated variance components from the diallels may be viewed as a random sample derived from the same population. We also assumed that the average of a given variance component from all diallels in the population is a ‘true’ parameter of the population. The descriptive statistics of variance components were produced and appear in Table 1. General combining ability (σ2g) and specific combining ability (σ2s) genetic variances from 105 pairs of diallels had a large positive skewness. In plant breeding experiments, a 2.5% error rate for one-sided t-intervals is considered to be a critical cut-off. In other words, if the lower and upper intervals are appropriate, then two-sided error rates should be close to 5%. A series of simulations were carried out to determine the true error rate for one-sided t-intervals for each variance component. Since genetic variances departed from normality because of skewed distributions, the coverage accuracy of Hall’s transformation method (Zhou and Gao 2000), bootstrap BCa-intervals (Efron and Tibshirani 1993), and the skewness-adjusted t-intervals (Boos and Hughes-Oliver 2000) were also estimated to compare with t-intervals. For simulation, random samples with replacement of n size were drawn from the population of a variance component, where n=2, 3, 4,..., N (N=105). For each n, the mean and 95% intervals based on the mean were calculated. The process was repeated 1,000 times for each sample size.
Table 1

Descriptive statistics of variance components, heritability and type-B genetic correlation for 6-year height for the Coastal breeding population of loblolly pine (N=105)

Variablea

Mean

Standard deviation

Range

Coefficient of variation (%)

Skewness

Kurtosis

σg2

0.23

0.17

0.0–1.11

74

2.09

6.75

σs2

0.09

0.17

0.0–1.41

188

5.88

40.21

σgt2

0.05

0.03

0.0–1.15

65

1.13

1.70

σst2

0.02

0.03

0.0–1.14

155

1.89

3.03

σp2

0.31

0.17

0.0–0.69

53

−0.40

−0.33

σe2

3.69

1.04

1.96–6.49

28

1.10

0.72

hb2

0.19

0.12

0.0–0.62

60

1.34

2.36

rB_b

0.79

0.17

0.0–1.00

21

−1.93

5.56

a σg2 is the general combining ability variance; σs 2, the specific combining ability variance; σgt2, the general combining ability and environment interaction variance; σst2, the specific combining ability and environment interaction variance; σp2, the plot-to-plot variance; σe2, the residual variance; hb2; the biased narrow-sense individual-tree heritability; rB-b, the biased type-B genetic correlation

The intervals from Boos and Hughes-Oliver (2000) (hereafter to be called BH-intervals) may provide better coverage than nominal t-intervals for skewed data and for small sample size. Using the simulated data, we estimated the miss left (ML) and miss right (MR) error rates for one-sided intervals for each sample size n. When the confidence interval falls to the left of the “true” mean, the interval misses the “true” mean on the left and it is called “miss left”. If the confidence interval falls to right of the “true” mean, then it is called a “miss right.” The following two regression equations given by Boos and Hughes-Oliver (2000) were inverted to get skewness adjusted α values to be used in the t-interval formula:
$$ {\text{MR}} = \frac{\alpha} {2} + \left( {- 0.73 + 0.71\,\exp ( - \alpha /2)} \right)\frac{{\sqrt {\beta _1} (X)}} {{\sqrt n}} $$
(2)
$$ {\text{ML}} = \frac{\alpha} {2} + \left( {0.19 + 0.026\,\ln (\alpha /2)} \right)\frac{{\sqrt {\beta _1} (X)}} {{\sqrt n}}, $$
(3)
where α/2 is the acceptable one-sided miss rate (0.025 in our case), \(\sqrt {\beta _{\text{1}}} (X)\) is the estimated Pearson skewness coefficient of the distribution X, and n is the sample size. The proportion of times out of 1,000 that an interval contains the “true” value was calculated. The average length of t-intervals (Lt) and BH-intervals (LBH) for a given sample size were estimated by taking the average over the 1,000 estimated intervals.
$$ L_t = (2t_{n - 1} )\frac{{{{\sum\limits_i s} \mathord{\left/ {\vphantom {{\sum\limits_i s} {1,000}}} \right. \kern-\nulldelimiterspace} {1,000}}}} {{\sqrt n}} $$
(4)
$$ L_{{\text{BH}}} = \frac{1} {{1,000}}\sum\limits_i {(U_i - L_i )} $$
(5)
where, Ui and Li are the upper endpoint and lower endpoint for the ith data set. i.e., the average length over the simulated samples, s is sample standard deviation averaged over 1,000 samples and n is the sample size. The narrower the length the more accurate it is for a given sample size.
The heritability (h2) and the type-B genetic correlation (rB) estimated from a pair of diallels are biased because they are not random samples derived from the same population but are based on ratios of specific variance components. Since h2 and rB are ratios of variance components, the “true” population values in our simulation study are based on the means of the 105 estimated variance components:
$$ h_{- t}^2 = \frac{{4\sigma _{{\text{g}}({\text{m}})}^2}} {{{\text{2}}\sigma _{{\text{g(m)}}}^2 + \sigma _{{\text{s(m)}}}^2 + {\text{2}}\sigma _{{\text{gt(m)}}}^2 + \sigma _{{\text{st(m)}}}^2 + \sigma _{{\text{p(m)}}}^2 + \sigma _{{\text{e(m)}}}^2}} $$
(6)
$$ r_{B - t} = \frac{{\sigma _{{\text{g(m)}}}^2}} {{\sigma _{{\text{g(m)}}}^2 + \sigma _{{\text{gt(m)}}}^2}}, $$
(7)
where the subscript “m” means that we have averaged over the 105 estimates. Because heritability and type-B genetic correlation are ratios of variance components, they may have different distribution properties. To examine the distributions of h2 and rB, 1,000 random samples with replacement were carried out for each n sample size, and approximately unbiased estimates for h2unb and rB-unb were obtained by using the above formulas (Eqs. 6 and 7) using sample averages of variance component estimates. The biases for h2 and rB were estimated as the difference between the “true” values and the values based on n sample means. As the sample size increases, the biases for both h2 and rB are expected to approach zero.

Since the estimates h2−unb and rB−unb are the ratio of the sample means, we used Fieller’s (1954) method to estimate their confidence intervals and compared these to nominal t-intervals. Fieller’s method is a clever way to get a confidence interval for the ratio of two means for random pairs. The idea is to define a new variable Z=Y-yx)X, using the mean μy of the numerator variable Y and the mean μx of the denominator variable X. The Z variable has mean 0 and a variance that is estimated from the sample Z values. Then, the Fieller’s confidence interval finds all values of μyx such that t2 (Z)<t2(0.975,n-1), where t(Z) is the t-statistic on the Z-variable and t2(0.975,n-1) is the usual t-interval cut-off.

For planning purposes, how large should n be for estimating σ2g, σ2s, h2 and rB? The sample size for a desired length of interval can be obtained from the standard formula, n=[tα2,n-1* s/d]2, where α/2 is the error rate (typically 0.025), s is the sample standard deviation (which must be guessed), and d is the margin of error. Recall that the t-interval has length of 2tα2,n-1s/\( {\sqrt n } \) or margin of error tα2,n-1s/\( {\sqrt n } \). Later we give a numerical example for specific standard errors of heritability and type-B genetic correlation.

Results

Descriptive statistics

Variance components and their ratios fluctuated considerably among experiments within the breeding population (Table 1). General combining ability variance (σ2g) ranged from 0.0 to 1.11, whereas narrow-sense heritability (h2) ranged from 0.0 to 0.62. Type-B genetic correlation (rB) fluctuated between the lower (0.0) and upper (1.0) theoretical limits. High variation in variance components among diallels is reflected in high coefficients of variation (CV): rB had the smallest CV (21%), whereas specific combining ability variance (σ2s) had the highest (188%) CV. σ2g and σ2s had a positive skewness of 2.09 and 5.88, respectively, indicating a significant departure from normality. The distribution of σ2g and σ2s estimates also had large kurtosis values. Plot-to-plot (σ2p) and within-plot (σ2e) error variances had the lowest skewness and kurtosis among the variance components.

Interval coverage of genetic variances (simulation results)

The one-sided coverages of t-intervals as a function of sample size for σ2g and σ2s from simulations are presented in Fig. 2, and numeric results of t- and BH-intervals for four arbitrary selected sample size (5, 10, 30, and 100) are given in Table 2. ML and MR error rates are not equal for both genetic variances, with more intervals missing left (ML) than right (MR). For example, at n=30, a nominal 95% t-interval for σ2s would fall 25% of the time to the left of σ2s, whereas it would be about less than 1% times to the right of σ2s (Table 2). Although σ2g had similar MR error rates to σ2s, the ML error rates of σ2g were substantially smaller than the ML error rates of σ2s. At n=30, the t-interval for σ2g would fall 9% of the time to left of σ2g. The ML and the MR error rates are expected to converge to 0.025 as the sample size increases. Although the trend of error rates approached 0.025, we did not see convergence for σ2s genetic variance, even for large sample sizes (Fig. 2). However, we observed convergence for σ2g, particularly for BH-intervals at n=100.
Fig. 2

The one-sided coverage of t-intervals as a function of sample size n for genetic variances σ2g and σ2s. MR and ML proportions are expected to convergence to 0.025 as the sample size further increases. We can not see this convergence for either σ2g or σ2s, even for k>100 because of the skewness in the data

Table 2

Simulation results for σg2 and σs2 genetic variances. MR, ML proportions, total coverage and the length of coverage for nominal 95% t-intervals and BH-intervals for sample sizes n=5, 10, 30, and 100. Entries are based on 1,000 Monte Carlo replications. The standard errors of the miss rates, coverage estimates, and interval lengths are all approximately 0.01

Variable

n

Miss right (MR)

Miss left (ML)

Total coverage

Interval length

BH

t

BH

t

BH

t

BH

t

σg2

5

0.01

0.01

0.11

0.11

0.89

0.88

0.43

0.37

10

0.01

0.01

0.07

0.09

0.92

0.91

0.24

0.22

30

0.01

0.01

0.06

0.08

0.93

0.91

0.13

0.12

100

0.02

0.01

0.03

0.05

0.96

0.95

0.07

0.07

σs2

5

0.00

0.00

0.19

0.22

0.81

0.78

0.31

0.25

10

0.00

0.00

0.17

0.23

0.83

0.78

0.19

0.17

30

0.00

0.00

0.19

0.25

0.80

0.75

0.11

0.11

100

0.01

0.00

0.11

0.15

0.88

0.85

0.07

0.06

Because of high ML error rates, the true coverage of nominal 1-α intervals was much lower than desired. For example, at n=30, the true coverage of 95% t-intervals for σ2s would be only 75%. In another words, σ2s will be not be in the nominal 95% t-interval for about 25% of the time. The total coverage of t-intervals for σ2g was better than the total coverage for σ2s intervals. For the same sample size, the true coverage of nominal 95% interval for σ2g would be about 91%. This total coverage is only 4% smaller than the nominal 95% interval coverage. As the sample size increases, the accuracy of t-intervals for both genetic variances increased. For example at n=100, the true coverage of nominal 1-α intervals was approximately the same as the desired level of 95% interval. The interval lengths of both genetic variances were also considerably smaller (0.07) than the interval lengths at n=30.

The BH-intervals improved the coverage probability for σ2s, whereas the improvement over the t-intervals for σ2g was modest (Table 2). Bootstrap BCa and Hall’s transformed intervals had poorer coverages than BH-intervals and were thus not reported. The true coverage of BH nominal 95% t-intervals for sample size n=30 was around 80% for σ2s and 93% for σ2g. The improvement over the t-intervals was 5% for σ2s and 2% for σ2g for the same sample size. The total coverage of BH-intervals for σ2g is only 2% smaller than 95% interval coverage. For a smaller sample size, the BH-intervals had a greater interval length than the t-intervals. However, as the sample size increases, the interval length difference between t and BH diminished although their improvement of total coverage remained about the same.

Interval coverage of h2 and rB

Nominal 95% t-intervals and Fieller’s intervals, the one-sided error rates and intervals length for heritability and type-B genetic correlation are presented in Table 3. In general, the Fieller’s method had better coverage than t-intervals for h2 and rB. The two-sided coverage of nominal 95% t-intervals for h2 did not change much as the sample size increased. The coverage of t-intervals for h2 was as good as Fiellers’ intervals for all three sample sizes. For example, at n=30, the total coverage of Fieller’s and t-intervals were 0.92 and 0.89, respectively, and the improvement of Fieller’s over the t-interval was only 3%. However, the Fieller’s method had much better coverage of the ‘true’ rB mean than did the t-intervals for large samples, i.e., 0.95 vs. 0.82 for n=30 and 0.94 vs. 0.31 for n=100. These low coverages are due to the fact that the sample mean of the rB values is biased for estimating the true value given in Eq. 7. This bias becomes more apparent as the sample size is increased to n=100.
Table 3

Simulation results for nominal 95% Fieller and t-confidence intervals for heritability (h2) and type-B genetic correlation (rB) based on sample sizes n=5, 10, 30, and 100. Entries are based on 1,000 Monte Carlo replications. The standard errors of the miss rates, coverage estimates, and interval lengths are all about 0.01

Variable

n

Miss right (MR)

Miss left (ML)

Total coverage

Interval length

Fieller

t

Fieller

t

Fieller

t

Fieller

t

h2

5

0.00

0.00

0.09

0.10

0.91

0.90

0.27

0.25

10

0.00

0.00

0.09

0.10

0.91

0.90

0.17

0.16

30

0.01

0.00

0.07

0.11

0.92

0.89

0.09

0.09

100

0.01

0.00

0.04

0.12

0.95

0.88

0.05

0.05

rB

5

0.02

0.03

0.02

0.02

0.96

0.96

0.49

0.35

10

0.02

0.02

0.03

0.04

0.96

0.94

0.20

0.22

30

0.02

0.00

0.04

0.18

0.95

0.82

0.10

0.12

100

0.02

0.00

0.03

0.69

0.94

0.31

0.05

0.07

Discussion

Variation

Partitioning of parents into small groups of diallels for breeding is commonly employed for monoecious woody perennials such as conifers (Yanchuk 1996; Yeh and Heaman 1987; Johnson and King 1998). Despite the several advantages in using diallels, genetic sampling of parents from a large population for a given diallel is subject to random genetic drift. Genetic differences among diallel groups may cause significant variation in estimates of genetic variances (Hill 1985). In our study, over 100 variance components estimates of six-parent disconnected half-diallels were used to make inferences about a breeding population. Thus, the observed large variation of variance components among individual diallels was not surprising. Similar large differences in genetic variance estimates between diallels were also observed in Douglas-fir (Pseudotsuga menziessi var. menziessii) (Yeh and Heaman 1987) and in another population of loblolly pine (Balocchi et al. 1993).

In this study, among the variance components studied, σ2s had a wider range and higher coefficient of variation than any other estimates. This is partly due to the nature of σ2s estimation from the diallels. The estimation of σ2s is based on specific crosses of one parent and is subject to larger standard errors (Yanchuk 1996). On the other hand, σ2g is estimated from many more progenies in a diallel with a smaller variance, as shown by the smaller coefficient of variation. SCA by environment interaction variance (σ2st) followed σ2s in having a large variance. The error variance components (σ2p, σ2e) can be approximated with a normal distribution as they had a smaller skewness and kurtosis than the genetic variances.

In general, the pattern of distribution of h2 followed the distribution pattern of σ2g. This is expected because h2 is mainly the product of σ2g. However, h2 had a smaller kurtosis value than σ2g. Although rB is also the product of σ2g, we observed a negative skewness for rB. This is mainly due to the high frequency of large σ2gt values and sporadic zero σ2gt values. The kurtosis of rB was similar to the σ2g value.

Error rates and the coverage of the intervals

The dependence of coverage probabilities on the skewness have been covered extensively in the literature (Chaffin and Rhiel 1993; Chen 1995; Boos and Hughes-Oliver 2000). In the present study, very high ML error rates of intervals observed for σ2g and σ2s were mainly the effect of a large skewness in the distributions of these two genetic variances. Among the variance components studied, σ2s had a larger skewness and greater standard deviation than any other variance components. Thus, the high error rates of two-sided intervals for σ2s are not surprising. If breeders basically average σ2s variances from different experiments to obtain a mean with a nominal 95% confidence interval, then the ‘true’ σ2s mean would likely be out of that interval at least 25% of the time. We suggest using the BH skewness-adjusted t-intervals for variance components such as σ2s and σ2g. Although the BH intervals were longer than the t-intervals, the improvement can be substantial for smaller samples. The length of coverage of BH-intervals approaches the length of nominal t-intervals as the sample size increases. The drawback of the BH intervals is that the right-side coverages are not perfect. In fact, the t and BH right side error rates are similar, and they converge slowly to 0.025, but even n=100 is not large enough to see the convergence.

Knapp et al. (1987, 1989) found similarities between nominal interval estimates of point heritability and variance components. In their study, confidence interval widths of heritability decreased slightly as sample size increased. Their heritability estimates and precision statistics were based an analysis of variance and expected mean squares but not maximum likelihood estimation procedures. When the error rates were taken into consideration, Fieller’s intervals for h2 and rB were not superior to t-intervals, particularly for small sample size n. However, one needs to consider the length of the interval along with the coverage. Fieller’s intervals were superior to t-intervals as shown by narrower interval lengths for both h2 and rB. The decrease in length of the Fieller’s intervals and t-intervals in our study was clearly pronounced as the sample size increased. We observed a sharp drop in the coverage of the t-interval for rB at n=30 and n=100. This is due to the bias of averaging rB values.

Accuracy of the population statistics and the sample size n

Plant breeders may ask about the sample size (number of estimates) to obtain a reliable sample statistics that is a representative of the population? The answer to the above question varies depending on the accuracy sought. For the Coastal breeding population of loblolly pine, let us say the true population standard deviation of h2 is σ=0.12. If we are satisfied with a d=0.05 margin of error, then for a 95% confidence level we need to have at least n=(t×σ/d)2=(1.96×0.12/0.05)2=22 diallels, where 1.96 is the Z-value for the 95% level. The difference between a Z -value and a t-value is not very large. We used the Z-value 1.96 as an approximation to the t-value because of simplicity. For the same margin of error, the number of diallels required for rB would be double (about 44), because the population standard deviation for rB (0.17) is greater than h2. If d=0.10 is an acceptable margin of error, then the sample size needed for h2 and rB would be 6 and 11 diallels, respectively. For this specific population, assuming that the population standard deviations for h2 and rB are 0.12 and 0.17, respectively, the sample size as a function of a series of error margins is given in Fig. 3. For planning proposes, population standard deviations of h2 and rB can be used to estimate an approximate sample size. Here, we used t-intervals to calculate approximate sample size for h2 and rB because the actual coverage of t-intervals was as good as Fieller’s intervals for small samples.
Fig. 3

Sample size as a function of error margins for heritability (h2) and type-B genetic correlation (rB) for a specific situation. The sample size required for any given error of margins were based on the assumptions that the population standard deviation for h2 and rB are 0.12 and 0.17, respectively. Larger sample sizes are required for rB than for h2 for the same error margins because of a greater population standard deviation of rB

Conclusions

The genetic variance component estimates σ2g and σ2s from disconnected diallels for 6-year height growth had a considerable large positive skewness. The results are based on empirical estimates of variance components from a breeding population of loblolly pine. Because of this skewness, the true coverage of nominal 95% t-intervals for genetic variances would likely be less than the nominal coverage. The BH-intervals adjusted for skewness provided better coverage than regular t-intervals. However, BH-intervals are not optimal. Their miss right error rates were as poor as t-intervals. In contrast to genetic variances, h2 and rB had smaller skewness, but because they are ratios of variance components, it is wise to use Fieller’s intervals over the t-intervals for h2 and rB although regular t-intervals are not much worse in small samples. When statistics (mean, standard deviation, and intervals) are estimated from a sample of genetic variances and their functions, the distribution properties should be taken into account for more reliable inferences.

Copyright information

© Springer-Verlag 2005