gva.augmentedRCBD {augmentedRCBD}

Perform Genetic Variability Analysis on augmentedRCBD Output

Description

gva.augmentedRCBD performs genetic variability analysis on an object of class augmentedRCBD.

Usage

gva.augmentedRCBD(aug, k = 2.063)

Arguments

aug

An object of class augmentedRCBD.

k

The standardized selection differential or selection intensity. Default is 2.063 for 5% selection proportion (see Details).

Details

gva.augmentedRCBD performs genetic variability analysis from the ANOVA results in an object of class augmentedRCBD and computes several variability estimates.

The phenotypic, genotypic and environmental variance (\(\sigma^{2}_{p}\), \(\sigma^{2}_{g}\) and \(\sigma^{2}_{e}\) ) are obtained from the ANOVA tables according to the expected value of mean square described by Federer and Searle (1976) as follows:

\[\sigma^{2}_{g} = \sigma^{2}_{p} - \sigma^{2}_{e}\]

Phenotypic and genotypic coefficients of variation (\(PCV\) and \(GCV\)) are estimated according to Burton (1951, 1952) as follows:

\[GCV = \frac{\sigma^{2}_{g}}{\sqrt{\overline{x}}} \times 100\]

Where \(\overline{x}\) is the mean.

The estimates of \(PCV\) and \(GCV\) are categorised according to Sivasubramanian and Madhavamenon (1978) as follows:

CV (%)	Category
x \( < \) 10	Low
10 \(\le\) x \( < \) 20	Medium
\(\ge\) 20	High

The broad-sense heritability (\(H^{2}\)) is calculated according to method of Lush (1940) as follows:

\[H^{2} = \frac{\sigma^{2}_{g}}{\sigma^{2}_{p}}\]

The estimates of broad-sense heritability (\(H^{2}\)) are categorised according to Robinson (1966) as follows:

\(H^{2}\)	Category
x \( < \) 30	Low
30 \(\le\) x \( < \) 60	Medium
\(\ge\) 60	High

Genetic advance (\(GA\)) is estimated and categorised according to Johnson et al., (1955) as follows:

\[GA = k \times \sigma_{g} \times \frac{H^{2}}{100}\]

Where the constant \(k\) is the standardized selection differential or selection intensity. The value of \(k\) at 5% proportion selected is 2.063. Values of \(k\) at other selected proportions are available in Appendix Table A of Falconer and Mackay (1996).

Selection intensity (\(k\)) can also be computed in R as below:

If p is the proportion of selected individuals, then deviation of truncation point from mean (x) and selection intensity (k) are as follows:

x = qnorm(1-p)

k = dnorm(qnorm(1 - p))/p

Using the same the Appendix Table A of Falconer and Mackay (1996) can be recreated as follows.

TableA <- data.frame(p = c(seq(0.01, 0.10, 0.01), NA,
                           seq(0.10, 0.50, 0.02), NA,
                           seq(1, 5, 0.2), NA,
                           seq(5, 10, 0.5), NA,
                           seq(10, 50, 1)))
TableA$x <- qnorm(1-(TableA$p/100))
TableA$i <- dnorm(qnorm(1 - (TableA$p/100)))/(TableA$p/100)

Appendix Table A (Falconer and Mackay, 1996)

p%	x	i
0.01	3.71901649	3.9584797
0.02	3.54008380	3.7892117
0.03	3.43161440	3.6869547
0.04	3.35279478	3.6128288
0.05	3.29052673	3.5543807
0.06	3.23888012	3.5059803
0.07	3.19465105	3.4645890
0.08	3.15590676	3.4283756
0.09	3.12138915	3.3961490
0.10	3.09023231	3.3670901
<>	<>	<>
0.10	3.09023231	3.3670901
0.12	3.03567237	3.3162739
0.14	2.98888227	3.2727673
0.16	2.94784255	3.2346647
0.18	2.91123773	3.2007256
0.20	2.87816174	3.1700966
0.22	2.84796329	3.1421647
0.24	2.82015806	3.1164741
0.26	2.79437587	3.0926770
0.28	2.77032723	3.0705013
0.30	2.74778139	3.0497304
0.32	2.72655132	3.0301887
0.34	2.70648331	3.0117321
0.36	2.68744945	2.9942406
0.38	2.66934209	2.9776133
0.40	2.65206981	2.9617646
0.42	2.63555424	2.9466212
0.44	2.61972771	2.9321196
0.46	2.60453136	2.9182048
0.48	2.58991368	2.9048286
0.50	2.57582930	2.8919486
<>	<>	<>
1.00	2.32634787	2.6652142
1.20	2.25712924	2.6028159
1.40	2.19728638	2.5490627
1.60	2.14441062	2.5017227
1.80	2.09692743	2.4593391
2.00	2.05374891	2.4209068
2.20	2.01409081	2.3857019
2.40	1.97736843	2.3531856
2.60	1.94313375	2.3229451
2.80	1.91103565	2.2946575
3.00	1.88079361	2.2680650
3.20	1.85217986	2.2429584
3.40	1.82500682	2.2191656
3.60	1.79911811	2.1965431
3.80	1.77438191	2.1749703
4.00	1.75068607	2.1543444
4.20	1.72793432	2.1345772
4.40	1.70604340	2.1155928
4.60	1.68494077	2.0973249
4.80	1.66456286	2.0797152
5.00	1.64485363	2.0627128
<>	<>	<>
5.00	1.64485363	2.0627128
5.50	1.59819314	2.0225779
6.00	1.55477359	1.9853828
6.50	1.51410189	1.9506784
7.00	1.47579103	1.9181131
7.50	1.43953147	1.8874056
8.00	1.40507156	1.8583278
8.50	1.37220381	1.8306916
9.00	1.34075503	1.8043403
9.50	1.31057911	1.7791417
10.00	1.28155157	1.7549833
<>	<>	<>
10.00	1.28155157	1.7549833
11.00	1.22652812	1.7094142
12.00	1.17498679	1.6670040
13.00	1.12639113	1.6272701
14.00	1.08031934	1.5898336
15.00	1.03643339	1.5543918
16.00	0.99445788	1.5206984
17.00	0.95416525	1.4885502
18.00	0.91536509	1.4577779
19.00	0.87789630	1.4282383
20.00	0.84162123	1.3998096
21.00	0.80642125	1.3723871
22.00	0.77219321	1.3458799
23.00	0.73884685	1.3202091
24.00	0.70630256	1.2953050
25.00	0.67448975	1.2711063
26.00	0.64334541	1.2475585
27.00	0.61281299	1.2246130
28.00	0.58284151	1.2022262
29.00	0.55338472	1.1803588
30.00	0.52440051	1.1589754
31.00	0.49585035	1.1380436
32.00	0.46769880	1.1175342
33.00	0.43991317	1.0974204
34.00	0.41246313	1.0776774
35.00	0.38532047	1.0582829
36.00	0.35845879	1.0392158
37.00	0.33185335	1.0204568
38.00	0.30548079	1.0019882
39.00	0.27931903	0.9837932
40.00	0.25334710	0.9658563
41.00	0.22754498	0.9481631
42.00	0.20189348	0.9306998
43.00	0.17637416	0.9134539
44.00	0.15096922	0.8964132
45.00	0.12566135	0.8795664
46.00	0.10043372	0.8629028
47.00	0.07526986	0.8464123
48.00	0.05015358	0.8300851
49.00	0.02506891	0.8139121
50.00	0.00000000	0.7978846

Where p% is the selected percentage of individuals from a population, x is the deviation of the point of truncation of selected individuals from population mean and i is the selection intensity.

Genetic advance as per cent of mean (\(GAM\)) are estimated and categorised according to Johnson et al., (1955) as follows:

\[GAM = \frac{GA}{\overline{x}} \times 100\]

GAM	Category
x \( < \) 10	Low
10 \(\le\) x \( < \) 20	Medium
\(\ge\) 20	High

Value

A list with the following descriptive statistics:

Mean

The mean value.

PV

Phenotyic variance.

GV

Genotyipc variance.

EV

Environmental variance.

GCV

Genotypic coefficient of variation

GCV category

The \(GCV\) category according to Sivasubramaniam and Madhavamenon (1973).

PCV

Phenotypic coefficient of variation

PCV category

The \(PCV\) category according to Sivasubramaniam and Madhavamenon (1973).

ECV

Environmental coefficient of variation

hBS

The broad-sense heritability (\(H^{2}\)) (Lush 1940).

hBS category

The \(H^{2}\) category according to Robinson (1966).

GA

Genetic advance (Johnson et al. 1955).

GAM

Genetic advance as per cent of mean (Johnson et al. 1955).

GAM category

The \(GAM\) category according to Johnson et al. (1955).

Note

Genetic variability analysis needs to be performed only if the sum of squares of "Treatment: Test" are significant.

Negative estimates of variance components if computed are not abnormal. For information on how to deal with these, refer Robinson (1955) and Dudley and Moll (1969).

References

Lush JL (1940). “Intra-sire correlations or regressions of offspring on dam as a method of estimating heritability of characteristics.” Proceedings of the American Society of Animal Nutrition, 1940(1), 293–301.

Burton GW (1951). “Quantitative inheritance in pearl millet (Pennisetum glaucum).” Agronomy Journal, 43(9), 409–417.

Burton GW (1952). “Qualitative inheritance in grasses. Vol. 1.” In Proceedings of the 6th International Grassland Congress, Pennsylvania State College, 17–23.

Johnson HW, Robinson HF, Comstock RE (1955). “Estimates of genetic and environmental variability in soybeans.” Agronomy journal, 47(7), 314–318.

Robinson HF, Comstock RE, Harvey PH (1955). “Genetic variances in open pollinated varieties of corn.” Genetics, 40(1), 45–60.

Robinson HF (1966). “Quantitative genetics in relation to breeding on centennial of Mendelism.” Indian Journal of Genetics and Plant Breeding, 171.

Dudley JW, Moll RH (1969). “Interpretation and use of estimates of heritability and genetic variances in plant breeding.” Crop Science, 9(3), 257–262.

Sivasubramaniam S, Madhavamenon P (1973). “Genotypic and phenotypic variability in rice.” The Madras Agricultural Journal, 60(9-13), 1093–1096.

Federer WT, Searle SR (1976). “Model Considerations and Variance Component Estimation in Augmented Completely Randomized and Randomized Complete Blocks Designs-Preliminary Version.” Technical Report BU-592-M, Cornell University, New York.

Falconer DS, Mackay TFC (1996). Introduction to Quantitative Genetics. Pearson/Prenctice Hall, New York, NY.

See Also

augmentedRCBD

Examples

# Example data
blk <- c(rep(1,7),rep(2,6),rep(3,7))
trt <- c(1, 2, 3, 4, 7, 11, 12, 1, 2, 3, 4, 5, 9, 1, 2, 3, 4, 8, 6, 10)
y1 <- c(92, 79, 87, 81, 96, 89, 82, 79, 81, 81, 91, 79, 78, 83, 77, 78, 78,
        70, 75, 74)
y2 <- c(258, 224, 238, 278, 347, 300, 289, 260, 220, 237, 227, 281, 311, 250,
        240, 268, 287, 226, 395, 450)
data <- data.frame(blk, trt, y1, y2)
# Convert block and treatment to factors
data$blk <- as.factor(data$blk)
data$trt <- as.factor(data$trt)
# Results for variable y1
out1 <- augmentedRCBD(data$blk, data$trt, data$y1, method.comp = "lsd",
                      alpha = 0.05, group = TRUE, console = TRUE)
# Results for variable y2
out2 <- augmentedRCBD(data$blk, data$trt, data$y2, method.comp = "lsd",
                     alpha = 0.05, group = TRUE, console = TRUE)

# Genetic variability analysis
gva.augmentedRCBD(out1)
gva.augmentedRCBD(out2)