Analysis of an augmented randomized complete block design (Federer, 1956), where a few check varieties are replicated in every block and the new (test) entries appear once. Gives the ANOVA with the treatment sum of squares split into checks, tests and checks vs tests, adjusted means of all entries, the four standard errors of difference, a comparison of every test entry with the checks, and genetic parameters of the test entries.
Usage
anova_augmented(
data,
response,
genotype,
block,
check_names,
verbose = TRUE,
alpha = 0.05,
k = 2.063
)Arguments
- data
Data frame containing the data.
- response
Name of the response variable.
- genotype
Name of the entry (genotype) column.
- block
Name of the block column.
- check_names
Character vector with the names of the check entries.
- verbose
Logical. If TRUE (default), prints the report.
- alpha
Significance level for CD. Default 0.05.
- k
Standardized selection differential for genetic advance. Default 2.063 (5% selection intensity).
Value
An object of class aridagri_augmented with
anova_adjusted (treatments after blocks),
anova_partitioned (blocks after treatments, treatments split),
adjusted_means (all entries with type, raw and adjusted mean,
SE and block), se_diff and cd (the four comparisons),
test_vs_checks (difference of each test entry from the check
mean and the best check, with significance), block_effects,
genetic_parameters, mse, df_error,
anova_checks and model.
Details
Block effects are estimated from the checks and removed from the test
entries: adjusted value = observed value - (mean of checks in its block -
overall check mean). This equals the least-squares mean from the model
response ~ block + genotype, which is used here, so the standard
errors are exact. With \(b\) blocks and \(c\) checks the four
standard errors of difference are: two check means
\(\sqrt{2 MSE / b}\); two test entries in the same block
\(\sqrt{2 MSE}\); two test entries in different blocks
\(\sqrt{2 MSE (1 + 1/c)}\); a test entry and a check mean
\(\sqrt{MSE (1 + 1/b + 1/c - 1/(bc))}\).
Two ANOVA tables are given. In the first, treatments are fitted after blocks and their sum of squares is split into tests vs checks, checks and tests, each adjusted for blocks. In the second, blocks are fitted after treatments and the treatment sum of squares (ignoring blocks) is split into checks, tests and checks vs tests (Federer, 1961).
Genetic parameters of the test entries use the block-adjusted mean square among tests from the first table: phenotypic variance \(\sigma^2_p\) = MS(tests), error variance \(\sigma^2_e\) = MSE, genotypic variance \(\sigma^2_g = \sigma^2_p - \sigma^2_e\), broad-sense heritability \(h^2 = \sigma^2_g / \sigma^2_p\), GCV and PCV in percent of the mean of the adjusted test values, and genetic advance \(GA = k h^2 \sigma_p\).
References
Federer, W.T. (1956). Augmented (or hoonuiaku) designs. Hawaiian Planters' Record, 55, 191-208.
Federer, W.T. (1961). Augmented designs with one-way elimination of heterogeneity. Biometrics, 17(3), 447-473. doi:10.2307/2527837
Examples
data(chickpea_augmented)
res <- anova_augmented(chickpea_augmented, response = "seed_yield",
genotype = "entry", block = "block",
check_names = c("C1", "C2", "C3", "C4"))
#>
#> ======================================================================
#> AUGMENTED RANDOMIZED COMPLETE BLOCK DESIGN
#> ======================================================================
#> Response : seed_yield
#> Blocks : 6
#> Checks : 4
#> Test entries : 48
#>
#> ANOVA (treatments adjusted for blocks)
#> Source Df Sum Sq Mean Sq F value Pr(>F)
#> Blocks (ignoring treatments) 5 283298.94 56659.79 16.914 0.0000 ***
#> Treatments (eliminating blocks) 51 1055984.00 20705.57 6.181 0.0002 ***
#> Tests vs Checks 1 75533.36 75533.36 22.548 0.0003 ***
#> Checks 3 84692.33 28230.78 8.427 0.0016 **
#> Tests 47 895758.31 19058.69 5.689 0.0003 ***
#> Residuals 15 50247.67 3349.84
#> Signif. codes: *** p < 0.001, ** p < 0.01, * p < 0.05, NS not significant
#>
#> ANOVA (blocks adjusted; treatments partitioned)
#> Source Df Sum Sq Mean Sq F value Pr(>F)
#> Blocks (eliminating treatments) 5 53633.33 10726.67 3.202 0.0365 *
#> Treatments (ignoring blocks) 51 1285649.61 25208.82 7.525 0.0001 ***
#> Checks 3 84692.33 28230.78 8.427 0.0016 **
#> Tests 47 1125423.92 23945.19 7.148 0.0001 ***
#> Checks vs Tests 1 75533.36 75533.36 22.548 0.0003 ***
#> Residuals 15 50247.67 3349.84
#> Signif. codes: *** p < 0.001, ** p < 0.01, * p < 0.05, NS not significant
#>
#> Grand mean = 1439.861 CV = 4.02% MSE = 3349.8444 (df = 15)
#>
#> Standard errors of difference and CD (alpha = 0.05)
#> Comparison SEd CD
#> Two check means 33.416 71.224
#> Two test entries (same block) 81.852 174.463
#> Two test entries (different blocks) 91.513 195.055
#> Test entry vs check mean 67.868 144.657
#>
#> Check means
#> Entry Raw_mean SE
#> C2 1537.50 23.629
#> C3 1537.33 23.629
#> C4 1475.17 23.629
#> C1 1392.67 23.629
#>
#> Top 10 test entries (adjusted; best check = C2)
#> Entry Adjusted_mean Pct_over_check_mean Pct_over_best_check Sig_vs_best
#> IC-039 1743.92 17.38 13.43 higher
#> IC-028 1712.42 15.26 11.38 higher
#> IC-011 1637.42 10.21 6.50 ns
#> IC-010 1604.42 7.99 4.35 ns
#> IC-022 1592.17 7.17 3.56 ns
#> IC-036 1560.92 5.07 1.52 ns
#> IC-043 1560.67 5.05 1.51 ns
#> IC-037 1559.92 5.00 1.46 ns
#> IC-003 1550.42 4.36 0.84 ns
#> IC-001 1545.42 4.02 0.51 ns
#>
#> Genetic parameters of test entries
#> Parameter Value
#> Mean of tests 1416.958
#> Range (min) 1151.917
#> Range (max) 1743.917
#> Phenotypic variance 19058.687
#> Genotypic variance 15708.843
#> Error variance 3349.844
#> PCV (%) 9.743
#> GCV (%) 8.845
#> Heritability (broad sense, %) 82.424
#> Genetic advance 234.745
#> GA as % of mean 16.567
head(res$test_vs_checks)
#> Entry Adjusted_mean Diff_vs_check_mean Pct_over_check_mean p_vs_check_mean
#> 1 IC-039 1743.917 258.25 17.382769 0.001180930
#> 2 IC-028 1712.417 226.75 15.262508 0.003196341
#> 3 IC-011 1637.417 151.75 10.214270 0.033192761
#> 4 IC-010 1604.417 118.75 7.993045 0.086393570
#> 5 IC-022 1592.167 106.50 7.168499 0.120587354
#> 6 IC-036 1560.917 75.25 5.065066 0.263040711
#> Diff_vs_best_check Pct_over_best_check p_vs_best_check Sig_vs_best
#> 1 206.41667 13.425474 0.008245957 higher
#> 2 174.91667 11.376694 0.021026564 higher
#> 3 99.91667 6.498645 0.161630291 ns
#> 4 66.91667 4.352304 0.339778318 ns
#> 5 54.66667 3.555556 0.433122138 ns
#> 6 23.41667 1.523035 0.734859621 ns
res$genetic_parameters
#> Parameter Value
#> 1 Mean of tests 1416.958333
#> 2 Range (min) 1151.916667
#> 3 Range (max) 1743.916667
#> 4 Phenotypic variance 19058.687352
#> 5 Genotypic variance 15708.842908
#> 6 Error variance 3349.844444
#> 7 PCV (%) 9.742926
#> 8 GCV (%) 8.845350
#> 9 Heritability (broad sense, %) 82.423530
#> 10 Genetic advance 234.745314
#> 11 GA as % of mean 16.566847