vignettes/vignettes_biometry.Rmd
vignettes_biometry.Rmd
The function find_outlier()
may be used to identify possible outliers in a dataframe. It is suggested that before applying any statistical procedures, outliers be checked out.
library(metan) data_out <- data_ge2 data_out[12, 4] = 5 find_outliers(data_out, var = PH, plots = TRUE) # Number of possible outliers: 1 # Line(s): 12 # Proportion: 0.6% # Mean of the outliers: 5 # Maximum of the outliers: 5 | Line 12 # Minimum of the outliers: 5 | Line 12 # With outliers: mean = 2.499 | CV = 15.58% # Without outliers: mean = 2.483 | CV = 13.46%
To check the outliers in different levels of a factor, use the argument by
. As an example, we will find possible outliers for each level of the factor ENV
.
find_outliers(data_ge2, PH, by = ENV) # No possible outlier identified. # # No possible outlier identified. # # Number of possible outliers: 4 # Line(s): 7 11 14 15 # Proportion: 11.4% # Mean of the outliers: 2.438 # Maximum of the outliers: 2.766 | Line 11 # Minimum of the outliers: 1.71 | Line 7 # With outliers: mean = 2.167 | CV = 10.309% # Without outliers: mean = 2.136 | CV = 7.377% # # No possible outlier identified. # ENV outliers # 1 A1 0 # 2 A2 0 # 3 A3 4 # 4 A4 0
To group by more than one variable, use the function group_by()
is used.
data_ge2 %>% group_by(ENV) %>% find_outliers(PH) # No possible outlier identified. # # No possible outlier identified. # # Number of possible outliers: 4 # Line(s): 7 11 14 15 # Proportion: 11.4% # Mean of the outliers: 2.438 # Maximum of the outliers: 2.766 | Line 11 # Minimum of the outliers: 1.71 | Line 7 # With outliers: mean = 2.167 | CV = 10.309% # Without outliers: mean = 2.136 | CV = 7.377% # # No possible outlier identified. # ENV outliers # 1 A1 0 # 2 A2 0 # 3 A3 4 # 4 A4 0
Pearson’s linear correlation does not consider the influence a set of traits on the relationship between two traits. For example, the hypothetical correlation of r = 0.9 between x and y may be due to the influence of a third trait or group of traits acting together. To identify this linear effect between x and y controlling statistically the effect of others traits, the partial correlation is used. From Pearson’s simple correlation matrix, the partial correlation is calculated by the following equation:
\[ {r_{xy.m}} = \frac{{ - {a_{xy}}}}{{\sqrt {{a_{xx}}{a_{yy}}} }} \]
Where \({r_{xy.m}}\) is the partial correlation coefficient between the traits * x * and * y , excluding the effects of the m * remaining traits of the set; \(- {a_{ij}}\) is the inverse element of the correlation matrix corresponding to xy, \({a_{ii}}{a_{jj}}\) are the diagonal elements of the inverse matrix of correlation associated with trait x and y , respectively. The significance of this correlation is also tested by the test * t * according to the following expression:
\[ t_{calc} = r_{xy.m} \sqrt \frac{n-v}{1-r_{xy.m}^2} \]
Where \(t_{calc}\) is the calculated Student * t * statistic; $ r_{xy.m} $ is the partial correlation coefficient for the traits x and y excluding the effect of the other * m * traits; * n * is the number of observations; and * v * is the number of traits. Both the linear and partial correlation coefficients may be obtained using the function lpcor()
.
lpc1 <- data_ge2 %>% select(starts_with("N")) %>% lpcor() print_table(lpc1$results)
# Compute the correlations for each level of the factor ENV lpc2 <- lpcor(data_ge2, by = ENV) print(lpc2) # # A tibble: 420 x 6 # ENV Pairs linear partial t prob # <fct> <chr> <dbl> <dbl> <dbl> <dbl> # 1 A1 PH x EH 0.425 0.981 25.1 0 # 2 A1 PH x EP -0.124 -0.971 -20.1 0 # 3 A1 PH x EL -0.00287 -0.169 -0.841 0.408 # 4 A1 PH x ED 0.0729 0.301 1.55 0.135 # 5 A1 PH x CL -0.158 -0.311 -1.60 0.122 # 6 A1 PH x CD 0.140 0.183 0.911 0.372 # 7 A1 PH x CW -0.0111 0.155 0.771 0.448 # 8 A1 PH x KW 0.0825 0.155 0.770 0.449 # 9 A1 PH x NR -0.0142 0.389 2.07 0.0496 # 10 A1 PH x NKR 0.140 0.271 1.38 0.181 # # ... with 410 more rows
Using the pairs_mantel()
function, it is possible to compute a Mantel’s test (Mantel 1967) for all pairwise correlation matrices of the above example.
lpc2 %>% pairs_mantel(names = paste("H", 1:4, sep = ""))
This same plot may be obtained by passing correlation matrices with the same dimension to an object of class lpcor
and then applying the function pairs_mantel()
.
as.lpcor(cor(data_ge2[1:30, 5:ncol(data_ge2)]), cor(data_ge2[31:60, 5:ncol(data_ge2)]), cor(data_ge2[61:90, 5:ncol(data_ge2)]), cor(data_ge2[91:120, 5:ncol(data_ge2)]), cor(data_ge2[121:150, 5:ncol(data_ge2)])) %>% pairs_mantel(diag = TRUE, pan.spacing = 0, shape.point = 21, col.point = "black", fill.point = "red", size.point = 1.5, alpha.point = 0.6, main = "My own plot", alpha = 0.2)
The function corr_coef()
can be used to compute Pearson producto-moment correlation coefficients with p-values. A correlation heat map can be created with the function plot()
.
We can use a select helper function to select variables. Here, we will select variables that starts with “C” OR ends with “D” using union_var()
.
ccoef2 <- corr_coef(data_ge2, union_var("C", "D")) print(ccoef2, digits = 2) # --------------------------------------------------------------------------- # Pearson's correlation coefficient # --------------------------------------------------------------------------- # CL CD CW CDED ED # CL 1.00 0.300 0.74 0.708 0.70 # CD 0.30 1.000 0.48 0.045 0.39 # CW 0.74 0.484 1.00 0.300 0.74 # CDED 0.71 0.045 0.30 1.000 -0.01 # ED 0.70 0.390 0.74 -0.010 1.00 # --------------------------------------------------------------------------- # p-values for the correlation coefficients # --------------------------------------------------------------------------- # CL CD CW CDED ED # CL 0.0e+00 1.4e-04 3.9e-28 4.9e-25 4.7e-24 # CD 1.4e-04 0.0e+00 1.5e-10 5.7e-01 4.9e-07 # CW 3.9e-28 1.5e-10 0.0e+00 1.4e-04 5.3e-28 # CDED 4.9e-25 5.7e-01 1.4e-04 0.0e+00 9.0e-01 # ED 4.7e-24 4.9e-07 5.3e-28 9.0e-01 0.0e+00
The function corr_plot()
may be used to visualize (both graphically and numerically) a correlation matrix. Pairwise of scatterplots are produced and may be shown in the upper or lower diagonal, which may be seen as a nicer and customizable ggplot2-based version of the pairs()
base R function.
a <- corr_plot(data_ge2, CD, EL, PERK, NKR) b <- corr_plot(data_ge2, CD, EL, PERK, NKR, lower = NULL, upper = "corr") c <- corr_plot(data_ge2, CD, EL, PERK, NKR, shape.point = 19, size.point = 2, alpha.point = 0.5, alpha.diag = 0, pan.spacing = 0, diag.type = "boxplot", col.sign = "gray", alpha.sign = 0.3, axis.labels = TRUE) d <- corr_plot(data_ge2, CD, EL, PERK, NKR, CW, NKE, prob = 0.01, shape.point = 21, col.point = "black", fill.point = "orange", size.point = 2, alpha.point = 0.6, maxsize = 4, minsize = 2, smooth = TRUE, size.smooth = 1, col.smooth = "black", col.sign = "cyan", col.up.panel = "black", col.lw.panel = "black", col.dia.panel = "black", pan.spacing = 0, lab.position = "tl") arrange_ggplot(a, b, c, d, labels = letters[1:4])
It is also possible to use a categorical variable of the data to map the scatterplot by colors.
corr_plot(iris, col.by = Species)
The function corr_coef()
can be used to compute Pearson product-moment correlation coefficients with p-values. A correlation heat map can be created with the function plot()
The function covcor_design()
may be used to compute genetic, phenotypic and residual correlation/(co)variance matrices through Analysis of Variance (ANOVA) method using randomized complete block design (RCBD) or completely randomized design (CRD).
The phenotypic (\(r_p\)), genotypic (\(r_g\)) and residual (\(r_r\)) correlations are computed as follows:
\[ r^p_{xy} = \frac{cov^p_{xy}}{\sqrt{var^p_{x}var^p_{y}}} \\ r^g_{xy} = \frac{cov^g_{xy}}{\sqrt{var^g_{x}var^g_{y}}} \\ r^r_{xy} = \frac{cov^r_{xy}}{\sqrt{var^r_{x}var^r_{y}}} \]
Using Mean Squares (MS) from the ANOVA method, the variances (var) and covariances (cov) are computed as follows:
\[ cov^p_{xy} = [(MST_{x+y} - MST_x - MST_y)/2]/r \\ var^p_x = MST_x / r \\ var^p_y = MST_y / r \\ cov^r_{xy} = (MSR_{x+y} - MSR_x - MSR_y)/2 \\ var^r_x = MSR_x \\ var^r_y = MSR_y \\ cov^g_{xy} = [(cov^p_{xy} \times r) - cov^r_{xy}]/r \\ var^g_x = (MST_x - MSE_x)/r \\ var^g_y = (MST_x - MSE_y)/r \\ \]
where MST is the mean square for treatment, MSR is the mean square for residuals, and r is the number of replications.
The function covcor_design()
returns a list with the matrices of (co)variances and correlations. Specific matrices may be returned using the argument type
, as shown bellow.
# environment A1 data <- subset(data_ge2, ENV == "A1") gcor <- covcor_design(data, gen = GEN, rep = REP, resp = c(PH, EH, NKE, TKW, CL, CD, CW, KW), type = "gcor") %>% as.data.frame() print_table(gcor, rownames = TRUE)
pcor <- covcor_design(data, gen = GEN, rep = REP, resp = c(PH, EH, NKE, TKW, CL, CD, CW, KW), type = "pcor") %>% as.data.frame() print_table(pcor, rownames = TRUE)
rcor <- covcor_design(data, gen = GEN, rep = REP, resp = c(PH, EH, NKE, TKW, CL, CD, CW, KW), type = "rcor") %>% as.data.frame() print_table(rcor, rownames = TRUE)
In this example we will obtain the residual (co)variance for each environment.
cov <- covcor_design(data_ge2, gen = GEN, rep = REP, resp = c(PH, EH, NKE, TKW, CL, CD, CW, KW), type = "rcov")
The residual (co)variance matrix and the means (obtained using type = "means"
) may be used into the function mahala()
to compute the Mahalanobis distance
res <- covcor_design(data, GEN, REP, resp = c(PH, EH, NKE, TKW, CL, CD, CW, KW), type = "rcov") means <- covcor_design(data, GEN, REP, resp = c(PH, EH, NKE, TKW, CL, CD, CW, KW), type = "means") D2 <- mahala(.means = means, covar = res, inverted = FALSE) %>% as.data.frame() print_table(D2, rownames = TRUE)
Recently, a Gaussian-independent estimator for the confidence interval of Pearson’s correlation coefficient was proposed by Olivoto et al. (2018). This estimator is based on the sample size and strength of associations and may be estimated using the function corr_ci()
. It is possible to estimate the confidence interval by declaring the sample size (n) and the correlation coefficient (r), or using a dataframe. The following code computes the confidence interval and make a plot to show the results.
In the following examples, the confidence interval is calculated by declaring the sample size (n) and the correlation coefficient (r). Using the argument by = ENV
the confidence interval can be calculated within each level of the factor ENV
.
# Inform n and r corr_ci(n = 145, r = 0.34) # ------------------------------------------------- # Nonparametric 95% half-width confidence interval # ------------------------------------------------- # Level of significance: 5% # Correlation coefficient: 0.34 # Sample size: 145 # Confidence interval: 0.1422 # True parameter range from: 0.1978 to 0.4822 # ------------------------------------------------- # Compute the confidence for each level of ENV CI2 <- corr_ci(data_ge2, contains("E"), by = ENV) # # A tibble: 21 x 6 # Pair Corr n CI LL UL # <chr> <dbl> <int> <dbl> <dbl> <dbl> # 1 EH x EP 0.839 39 0.185 0.655 1.02 # 2 EH x EL 0.148 39 0.320 -0.172 0.467 # 3 EH x ED -0.0385 39 0.349 -0.387 0.310 # 4 EH x CDED 0.186 39 0.310 -0.125 0.496 # 5 EH x PERK -0.205 39 0.306 -0.510 0.101 # 6 EH x NKE -0.324 39 0.278 -0.602 -0.0458 # 7 EP x EL 0.150 39 0.319 -0.169 0.469 # 8 EP x ED -0.0840 39 0.336 -0.420 0.252 # 9 EP x CDED 0.328 39 0.277 0.0502 0.605 # 10 EP x PERK -0.221 39 0.302 -0.523 0.0801 # # ... with 11 more rows # # A tibble: 21 x 6 # Pair Corr n CI LL UL # <chr> <dbl> <int> <dbl> <dbl> <dbl> # 1 EH x EP 0.899 39 0.176 0.722 1.08 # 2 EH x EL 0.498 39 0.242 0.255 0.740 # 3 EH x ED 0.748 39 0.199 0.549 0.947 # 4 EH x CDED 0.211 39 0.304 -0.0935 0.515 # 5 EH x PERK 0.0187 39 0.354 -0.335 0.373 # 6 EH x NKE 0.269 39 0.290 -0.0217 0.559 # 7 EP x EL 0.365 39 0.269 0.0960 0.634 # 8 EP x ED 0.600 39 0.224 0.376 0.823 # 9 EP x CDED 0.249 39 0.295 -0.0463 0.544 # 10 EP x PERK -0.0505 39 0.345 -0.396 0.295 # # ... with 11 more rows # # A tibble: 21 x 6 # Pair Corr n CI LL UL # <chr> <dbl> <int> <dbl> <dbl> <dbl> # 1 EH x EP 0.844 39 0.184 0.659 1.03 # 2 EH x EL -0.0977 39 0.333 -0.430 0.235 # 3 EH x ED 0.363 39 0.270 0.0935 0.633 # 4 EH x CDED -0.282 39 0.287 -0.569 0.00544 # 5 EH x PERK 0.126 39 0.325 -0.199 0.451 # 6 EH x NKE 0.279 39 0.288 -0.00876 0.567 # 7 EP x EL -0.194 39 0.308 -0.502 0.114 # 8 EP x ED 0.200 39 0.307 -0.107 0.507 # 9 EP x CDED -0.00956 39 0.357 -0.366 0.347 # 10 EP x PERK 0.0365 39 0.349 -0.313 0.386 # # ... with 11 more rows # # A tibble: 21 x 6 # Pair Corr n CI LL UL # <chr> <dbl> <int> <dbl> <dbl> <dbl> # 1 EH x EP 0.781 39 0.194 0.587 0.975 # 2 EH x EL 0.250 39 0.295 -0.0446 0.545 # 3 EH x ED 0.404 39 0.261 0.143 0.665 # 4 EH x CDED 0.143 39 0.321 -0.178 0.464 # 5 EH x PERK 0.250 39 0.295 -0.0447 0.545 # 6 EH x NKE 0.246 39 0.296 -0.0492 0.542 # 7 EP x EL 0.241 39 0.297 -0.0559 0.538 # 8 EP x ED 0.210 39 0.304 -0.0942 0.514 # 9 EP x CDED 0.236 39 0.298 -0.0626 0.534 # 10 EP x PERK 0.156 39 0.318 -0.162 0.473 # # ... with 11 more rows
corr_ss(r = 0.6, CI = 0.1) # ------------------------------------------------- # Sample size planning for correlation coefficient # ------------------------------------------------- # Level of significance: 5% # Correlation coefficient: 0.6 # 95% half-width CI: 0.1 # Required sample size: 194 # -------------------------------------------------
The following codes compute a complete collinearity diagnostic of a correlation matrix of predictor traits. Several indicators, such as Variance Inflation Factor, Condition Number, and Matrix Determinant are considered (T. Olivoto et al. 2017; Olivoto T. et al. 2017) The diagnostic may be performed using: (i) correlation matrices; (ii) dataframes, or (iii) an object of class group_factor
, which split a dataframe into subsets based on one or more grouping factors.
cold2 <- colindiag(data_ge2)
cold3 <- colindiag(data_ge2 , by = ENV)
pcoeff <- path_coeff(data_ge2, resp = KW) # Severe multicollinearity. # Condition Number = 7865.84 # Please, consider using a correction factor, or use 'brutstep = TRUE'.
pcoeff2 <- path_coeff(data_ge2, resp = KW, pred = c(PH, NKE, TKW), verbose = FALSE) print(pcoeff2) # ---------------------------------------------------------------------------------------------- # Correlation matrix between the predictor traits # ---------------------------------------------------------------------------------------------- # # A tibble: 3 x 3 # PH NKE TKW # * <dbl> <dbl> <dbl> # 1 1 0.4584 0.5685 # 2 0.4584 1 -0.06516 # 3 0.5685 -0.06516 1 # ---------------------------------------------------------------------------------------------- # Vector of correlations between dependent and each predictor # ---------------------------------------------------------------------------------------------- # PH NKE TKW # KW 0.7534439 0.6810756 0.6730371 # ---------------------------------------------------------------------------------------------- # Multicollinearity diagnosis and goodness-of-fit # ---------------------------------------------------------------------------------------------- # Condition number: 7.1689 # Determinant: 0.4284438 # R-square: 0.981 # Residual: 0.019 # Response: KW # Predictors: PH NKE TKW # ---------------------------------------------------------------------------------------------- # Variance inflation factors # ---------------------------------------------------------------------------------------------- # # A tibble: 3 x 2 # VAR VIF # <chr> <dbl> # 1 PH 2.324 # 2 NKE 1.580 # 3 TKW 1.844 # ---------------------------------------------------------------------------------------------- # Eigenvalues and eigenvectors # ---------------------------------------------------------------------------------------------- # # A tibble: 3 x 4 # Eigenvalues PH NKE TKW # <dbl> <dbl> <dbl> <dbl> # 1 1.699 0.7222 0.4223 0.5478 # 2 1.064 -0.01921 -0.7794 0.6262 # 3 0.2370 0.6914 -0.4628 -0.5548 # ---------------------------------------------------------------------------------------------- # Variables with the largest weight in the eigenvalue of smallest magnitude # ---------------------------------------------------------------------------------------------- # PH > TKW > NKE # ---------------------------------------------------------------------------------------------- # Direct (diagonal) and indirect (off-diagonal) effects # ---------------------------------------------------------------------------------------------- # PH NKE TKW linear # PH 0.02350989 0.01077646 0.01336628 0.7534439 # NKE 0.32834888 0.71632454 -0.04667561 0.6810756 # TKW 0.40158514 -0.04602544 0.70634648 0.6730371 # ----------------------------------------------------------------------------------------------
pcoeff2 <- path_coeff(data_ge2, resp = KW, pred = c(PH, EH, NKE, TKW), exclude = TRUE, verbose = FALSE)
pcoeff3 <- path_coeff(data_ge2, resp = KW, brutstep = TRUE, maxvif = 5) # -------------------------------------------------------------------------- # The algorithm has selected a set of 8 predictors with largest VIF = 3.346. # Selected predictors: NR PERK EP CDED EL NKR TKW PH # A forward stepwise-based selection procedure will fit 6 models. # -------------------------------------------------------------------------- # Adjusting the model 1 with 7 predictors (16.67% concluded) # Adjusting the model 2 with 6 predictors (33.33% concluded) # Adjusting the model 3 with 5 predictors (50% concluded) # Adjusting the model 4 with 4 predictors (66.67% concluded) # Adjusting the model 5 with 3 predictors (83.33% concluded) # Adjusting the model 6 with 2 predictors (100% concluded) # Done! # -------------------------------------------------------------------------- # Summary of the adjusted models # -------------------------------------------------------------------------- # Model AIC Numpred CN Determinant R2 Residual maxVIF # MODEL_1 1127 7 13.67 0.0841 0.933 0.0669 2.59 # MODEL_2 1125 6 12.26 0.1383 0.933 0.0670 2.46 # MODEL_3 1126 5 12.05 0.1989 0.932 0.0683 2.31 # MODEL_4 1138 4 6.40 0.4595 0.925 0.0747 2.13 # MODEL_5 1148 3 1.34 0.9787 0.919 0.0808 1.02 # MODEL_6 1329 2 2.23 0.8555 0.738 0.2616 1.17 # --------------------------------------------------------------------------
pcoeff4 <- path_coeff(data_ge2, resp = KW, pred = c(PH, EH, NKE, TKW), by = ENV) # Weak multicollinearity. # Condition Number = 5.183 # You will probably have path coefficients close to being unbiased. # Weak multicollinearity. # Condition Number = 81.987 # You will probably have path coefficients close to being unbiased. # Weak multicollinearity. # Condition Number = 24.788 # You will probably have path coefficients close to being unbiased. # Weak multicollinearity. # Condition Number = 10.276 # You will probably have path coefficients close to being unbiased.
First of all, we will rename the plant-related traits PH, EH EP
with the suffix _PLA
to show the usability of the select helper contains()
.
data_cc <- rename(data_ge2, PH_PLA = PH, EH_PLA = EH, EP_PLA = EP) # Type the variable names cc1 <- can_corr(data_cc, FG = c(PH_PLA, EH_PLA, EP_PLA), SG = c(EL, ED, CL, CD, CW, KW)) # --------------------------------------------------------------------------- # Matrix (correlation/covariance) between variables of first group (FG) # --------------------------------------------------------------------------- # PH_PLA EH_PLA EP_PLA # PH_PLA 1.0000000 0.9318282 0.6384123 # EH_PLA 0.9318282 1.0000000 0.8695460 # EP_PLA 0.6384123 0.8695460 1.0000000 # --------------------------------------------------------------------------- # Collinearity within first group # --------------------------------------------------------------------------- # The multicollinearity in the matrix should be investigated. # CN = 977.586 # Largest VIF = 229.164618380199 # Matrix determinant: 0.0025852 # Largest correlation: PH_PLA x EH_PLA = 0.932 # Smallest correlation: PH_PLA x EP_PLA = 0.638 # Number of VIFs > 10: 3 # Number of correlations with r >= |0.8|: 2 # Variables with largest weight in the last eigenvalues: # EH_PLA > PH_PLA > EP_PLA # --------------------------------------------------------------------------- # Matrix (correlation/covariance) between variables of second group (SG) # --------------------------------------------------------------------------- # EL ED CL CD CW KW # EL 1.0000000 0.3851451 0.2554068 0.9118653 0.4581728 0.6685601 # ED 0.3851451 1.0000000 0.6974629 0.3897128 0.7371305 0.8241426 # CL 0.2554068 0.6974629 1.0000000 0.3003636 0.7383379 0.4709310 # CD 0.9118653 0.3897128 0.3003636 1.0000000 0.4840299 0.6259806 # CW 0.4581728 0.7371305 0.7383379 0.4840299 1.0000000 0.7348622 # KW 0.6685601 0.8241426 0.4709310 0.6259806 0.7348622 1.0000000 # --------------------------------------------------------------------------- # Collinearity within second group # --------------------------------------------------------------------------- # Weak multicollinearity in the matrix # CN = 66.084 # Matrix determinant: 0.0028626 # Largest correlation: EL x CD = 0.912 # Smallest correlation: EL x CL = 0.255 # Number of VIFs > 10: 0 # Number of correlations with r >= |0.8|: 2 # Variables with largest weight in the last eigenvalues: # KW > EL > ED > CD > CL > CW # --------------------------------------------------------------------------- # Matrix (correlation/covariance) between FG and SG # --------------------------------------------------------------------------- # EL ED CL CD CW KW # PH_PLA 0.3801960 0.6613148 0.3251648 0.3153910 0.5047388 0.7534439 # EH_PLA 0.3626537 0.6302561 0.3971935 0.2805118 0.5193136 0.7029469 # EP_PLA 0.2634237 0.4580196 0.3908239 0.1750448 0.4248098 0.4974193 # --------------------------------------------------------------------------- # Correlation of the canonical pairs and hypothesis testing # --------------------------------------------------------------------------- # Var Percent Sum Corr Lambda Chisq DF p_val # U1V1 0.630438540 78.617161 78.61716 0.7940016 0.30668 177.29224 18 0.00000 # U2V2 0.163384310 20.374406 98.99157 0.4042083 0.82985 27.97651 10 0.00182 # U3V3 0.008086721 1.008433 100.00000 0.0899262 0.99191 1.21794 4 0.87514 # --------------------------------------------------------------------------- # Canonical coefficients of the first group # --------------------------------------------------------------------------- # U1 U2 U3 # PH_PLA 2.609792 5.490798 7.575090 # EH_PLA -2.559005 -7.646096 -12.812234 # EP_PLA 1.191023 2.428742 6.604968 # --------------------------------------------------------------------------- # Canonical coefficients of the second group # --------------------------------------------------------------------------- # V1 V2 V3 # EL -0.01008726 -1.0481893 0.60553720 # ED 0.14629899 0.7853469 -1.30457763 # CL -0.09112023 -1.2989864 -0.07497186 # CD -0.29105227 1.1513083 -1.50589651 # CW -0.12527616 -0.0361706 0.21180796 # KW 1.16041981 -0.1022916 1.34278026 # --------------------------------------------------------------------------- # Canonical loads of the first group # --------------------------------------------------------------------------- # U1 U2 U3 # PH_PLA 0.9856022 -0.08351129 -0.1470178 # EH_PLA 0.9085216 -0.41771278 -0.0102277 # EP_PLA 0.6319736 -0.71449671 0.3001730 # --------------------------------------------------------------------------- # Canonical loads of the second group # --------------------------------------------------------------------------- # V1 V2 V3 # EL 0.4759982 -0.11260907 -0.2944636 # ED 0.8294407 -0.18663860 -0.4477426 # CL 0.3749015 -0.74801793 -0.4937819 # CD 0.3951578 0.02985218 -0.5415818 # CW 0.6225367 -0.41451273 -0.2698904 # KW 0.9570820 -0.07344796 -0.1498587 # Use select helpers cc2 <- can_corr(data_cc, FG = contains("_PLA"), SG = c(EL, ED, CL, CD, CW, KW)) # --------------------------------------------------------------------------- # Matrix (correlation/covariance) between variables of first group (FG) # --------------------------------------------------------------------------- # PH_PLA EH_PLA EP_PLA # PH_PLA 1.0000000 0.9318282 0.6384123 # EH_PLA 0.9318282 1.0000000 0.8695460 # EP_PLA 0.6384123 0.8695460 1.0000000 # --------------------------------------------------------------------------- # Collinearity within first group # --------------------------------------------------------------------------- # The multicollinearity in the matrix should be investigated. # CN = 977.586 # Largest VIF = 229.164618380199 # Matrix determinant: 0.0025852 # Largest correlation: PH_PLA x EH_PLA = 0.932 # Smallest correlation: PH_PLA x EP_PLA = 0.638 # Number of VIFs > 10: 3 # Number of correlations with r >= |0.8|: 2 # Variables with largest weight in the last eigenvalues: # EH_PLA > PH_PLA > EP_PLA # --------------------------------------------------------------------------- # Matrix (correlation/covariance) between variables of second group (SG) # --------------------------------------------------------------------------- # EL ED CL CD CW KW # EL 1.0000000 0.3851451 0.2554068 0.9118653 0.4581728 0.6685601 # ED 0.3851451 1.0000000 0.6974629 0.3897128 0.7371305 0.8241426 # CL 0.2554068 0.6974629 1.0000000 0.3003636 0.7383379 0.4709310 # CD 0.9118653 0.3897128 0.3003636 1.0000000 0.4840299 0.6259806 # CW 0.4581728 0.7371305 0.7383379 0.4840299 1.0000000 0.7348622 # KW 0.6685601 0.8241426 0.4709310 0.6259806 0.7348622 1.0000000 # --------------------------------------------------------------------------- # Collinearity within second group # --------------------------------------------------------------------------- # Weak multicollinearity in the matrix # CN = 66.084 # Matrix determinant: 0.0028626 # Largest correlation: EL x CD = 0.912 # Smallest correlation: EL x CL = 0.255 # Number of VIFs > 10: 0 # Number of correlations with r >= |0.8|: 2 # Variables with largest weight in the last eigenvalues: # KW > EL > ED > CD > CL > CW # --------------------------------------------------------------------------- # Matrix (correlation/covariance) between FG and SG # --------------------------------------------------------------------------- # EL ED CL CD CW KW # PH_PLA 0.3801960 0.6613148 0.3251648 0.3153910 0.5047388 0.7534439 # EH_PLA 0.3626537 0.6302561 0.3971935 0.2805118 0.5193136 0.7029469 # EP_PLA 0.2634237 0.4580196 0.3908239 0.1750448 0.4248098 0.4974193 # --------------------------------------------------------------------------- # Correlation of the canonical pairs and hypothesis testing # --------------------------------------------------------------------------- # Var Percent Sum Corr Lambda Chisq DF p_val # U1V1 0.630438540 78.617161 78.61716 0.7940016 0.30668 177.29224 18 0.00000 # U2V2 0.163384310 20.374406 98.99157 0.4042083 0.82985 27.97651 10 0.00182 # U3V3 0.008086721 1.008433 100.00000 0.0899262 0.99191 1.21794 4 0.87514 # --------------------------------------------------------------------------- # Canonical coefficients of the first group # --------------------------------------------------------------------------- # U1 U2 U3 # PH_PLA 2.609792 5.490798 7.575090 # EH_PLA -2.559005 -7.646096 -12.812234 # EP_PLA 1.191023 2.428742 6.604968 # --------------------------------------------------------------------------- # Canonical coefficients of the second group # --------------------------------------------------------------------------- # V1 V2 V3 # EL -0.01008726 -1.0481893 0.60553720 # ED 0.14629899 0.7853469 -1.30457763 # CL -0.09112023 -1.2989864 -0.07497186 # CD -0.29105227 1.1513083 -1.50589651 # CW -0.12527616 -0.0361706 0.21180796 # KW 1.16041981 -0.1022916 1.34278026 # --------------------------------------------------------------------------- # Canonical loads of the first group # --------------------------------------------------------------------------- # U1 U2 U3 # PH_PLA 0.9856022 -0.08351129 -0.1470178 # EH_PLA 0.9085216 -0.41771278 -0.0102277 # EP_PLA 0.6319736 -0.71449671 0.3001730 # --------------------------------------------------------------------------- # Canonical loads of the second group # --------------------------------------------------------------------------- # V1 V2 V3 # EL 0.4759982 -0.11260907 -0.2944636 # ED 0.8294407 -0.18663860 -0.4477426 # CL 0.3749015 -0.74801793 -0.4937819 # CD 0.3951578 0.02985218 -0.5415818 # CW 0.6225367 -0.41451273 -0.2698904 # KW 0.9570820 -0.07344796 -0.1498587
d1 <- clustering(data_ge2)
mean_gen <- data_ge2 %>% means_by(GEN) %>% column_to_rownames("GEN") d2 <- clustering(mean_gen)
The S3 generic function plot() may be used to plot the dendrogram generated by the function clustering()
. A dashed line is draw at the cutpoint suggested according to Mojena (1977).
plot(d2)
According to the suggested cutpoint, two clusters are formed. The number of clusters may also be found using intensive computation. I suggest the package pcvlust, an R package for assessing the uncertainty in hierarchical cluster analysis. The implementation may be see below.
library(pvclust) pv_clust <- pvclust(t(d2$data), nboot = 100, method.dist = "euclidean") # Bootstrap (r = 0.47)... Done. # Bootstrap (r = 0.6)... Done. # Bootstrap (r = 0.67)... Done. # Bootstrap (r = 0.8)... Done. # Bootstrap (r = 0.87)... Done. # Bootstrap (r = 1.0)... Done. # Bootstrap (r = 1.07)... Done. # Bootstrap (r = 1.2)... Done. # Bootstrap (r = 1.27)... Done. # Bootstrap (r = 1.4)... Done. plot(pv_clust, hang = -1, cex = 0.5) pvrect(pv_clust, alpha = 0.95)
It is possible to indicate the variables from the data_ge2 to compute the distances. To do that is easy. You should only provide a comma-separated list of unquoted variable names after the .data argument. For example, to compute the distances between the genotypes based on the variables NKR, TKW, and NKE, the following arguments should be used.
d3 <- clustering(mean_gen, NKR, TKW, NKE)
When selvar = TRUE
is used, an algorithm for variable selection is implemented. See ?clustering
for more details.
d4 <- clustering(mean_gen, selvar = TRUE) # Calculating model 1 with 15 variables. EH excluded in this step (7.1%). # Calculating model 2 with 14 variables. EP excluded in this step (14.3%). # Calculating model 3 with 13 variables. CDED excluded in this step (21.4%). # Calculating model 4 with 12 variables. PH excluded in this step (28.6%). # Calculating model 5 with 11 variables. CL excluded in this step (35.7%). # Calculating model 6 with 10 variables. NR excluded in this step (42.9%). # Calculating model 7 with 9 variables. PERK excluded in this step (50%). # Calculating model 8 with 8 variables. EL excluded in this step (57.1%). # Calculating model 9 with 7 variables. CD excluded in this step (64.3%). # Calculating model 10 with 6 variables. ED excluded in this step (71.4%). # Calculating model 11 with 5 variables. KW excluded in this step (78.6%). # Calculating model 12 with 4 variables. CW excluded in this step (85.7%). # Calculating model 13 with 3 variables. NKR excluded in this step (92.9%). # Calculating model 14 with 2 variables. TKW excluded in this step (100%). # Done! # -------------------------------------------------------------------------- # # Summary of the adjusted models # -------------------------------------------------------------------------- # Model excluded cophenetic remaining cormantel pvmantel # Model 1 - 0.8656190 15 1.0000000 0.000999001 # Model 2 EH 0.8656191 14 1.0000000 0.000999001 # Model 3 EP 0.8656191 13 1.0000000 0.000999001 # Model 4 CDED 0.8656191 12 1.0000000 0.000999001 # Model 5 PH 0.8656189 11 1.0000000 0.000999001 # Model 6 CL 0.8655939 10 0.9999996 0.000999001 # Model 7 NR 0.8656719 9 0.9999982 0.000999001 # Model 8 PERK 0.8657259 8 0.9999977 0.000999001 # Model 9 EL 0.8657904 7 0.9999972 0.000999001 # Model 10 CD 0.8658997 6 0.9999964 0.000999001 # Model 11 ED 0.8658274 5 0.9999931 0.000999001 # Model 12 KW 0.8643556 4 0.9929266 0.000999001 # Model 13 CW 0.8640355 3 0.9927593 0.000999001 # Model 14 NKR 0.8648384 2 0.9925396 0.000999001 # -------------------------------------------------------------------------- # Suggested variables to be used in the analysis # -------------------------------------------------------------------------- # The clustering was calculated with the Model 10 # The variables included in this model were... # ED CW KW NKR TKW NKE # --------------------------------------------------------------------------
The distances were computed using the variables ED, CW, KW, NKR, TKW, and NKE. By using these variables the highest cophenetic correlation coefficient (0.8658) was observed. The Mantel’s correlation estimated with the distance matrix of Model 10 (selected variables) with the original distance matrix (estimated with all variables) was near to 1, suggesting that the deletion of the variables to compute the distance don’t affect significantly the computation of the distances.
The package dendextend offers a set of functions for extending ‘dendrogram’ objects in R. A simple example is given bellow.
library(dendextend) d4$hc %>% color_labels(k = 2, col = c("red", "blue")) %>% branches_color(k = 2, col = c("red", "blue")) %>% highlight_branches_lwd() %>% plot(horiz = TRUE, xlab = "Euclidean distance")
d6 <- clustering(data_ge2, by = ENV, nclust = 2)
The function pairs_mantel()
may be used to check the relationships between the distance matrices when the clustering is performed for each level of a grouping factor. In this example, we have four distance matrices corresponding to four environments.
pairs_mantel(d6, names = c("A1", "A2", "A3", "A4"))
The low values of correlation between the distance matrices suggest that the genotype clustering should vary significantly among environments.
# Environment E1 data_E1 <- subset(data_ge2, ENV == "A1") D2_des <- mahala_design(data_E1, gen = GEN, rep = REP, resp = c(PH, EH, NKE, TKW, CL, CD, CW, KW)) %>% as.data.frame() print_table(D2_des, rownames = TRUE)
To compute the Mahalanobis distance for each environment (or any grouping variable) we can use the argument by
. Note that pairs_mantel()
is used to compare compute the Mantel’s test for each combination of distance matrices. Let’s do it.
data_ge2 %>% mahala_design(gen = GEN, rep = REP, resp = c(PH, EH, NKE, TKW, CL, CD, CW, KW), by = ENV) # # A tibble: 4 x 2 # ENV data # <fct> <list> # 1 A1 <df[,13] [13 x 13]> # 2 A2 <df[,13] [13 x 13]> # 3 A3 <df[,13] [13 x 13]> # 4 A4 <df[,13] [13 x 13]>
Lets suppose we want compute the Mahalanobis’ distance for each pairwise genotype comparison based on cob-related traits. Note that the function select(contains("C"))
is used to select the cob-relate traits, after computing the mean for each genotype.
means <- data_ge2 %>% means_by(GEN) %>% column_to_rownames("GEN") %>% select(contains("C"))
The next step is to compute the variance-covariance matrix for the means. The first approach combines R base functions with some functions from metan package to compute the covariance matrix. Of course, the simplest way is by using cov()
.
# Compute the covariance matrix (by hand) cov_mat <- matrix(0, 4, 4) dev_scores <- sweep(means, 2, colMeans(means), FUN = "-") comb_v <- comb_vars(dev_scores, FUN = "*") cov_mat[lower.tri(cov_mat, diag = F)] <- colSums(comb_v/(nrow(means) - 1)) rownames(cov_mat) <- colnames(cov_mat) <- colnames(means) cov_mat <- make_sym(cov_mat, diag = diag(var(means))) # Compute the covariance using cov() covmat2 <- cov(means) # Check if the two matrices are equal all.equal(cov_mat, covmat2) # [1] TRUE
After computing the means and covariance matrices we are able to compute the Mahalanobis distance using the function mahala()
.
D2 <- mahala(means, covar = cov_mat) # Dendrogram D2 %>% as.dist() %>% hclust() %>% as.dendrogram() %>% plot()
Rendering engine
This vignette was built with pkgdown. All tables were produced with the package DT
using the following function.
library(DT) # Used to make the tables # Function to make HTML tables print_table <- function(table, rownames = FALSE, digits = 3, ...){ datatable(table, rownames = rownames, extensions = 'Buttons', options = list(scrollX = TRUE, dom = '<<t>Bp>', buttons = c('copy', 'excel', 'pdf', 'print')), ...) %>% formatSignif(columns = c(as.numeric(which(sapply(table, class) == "numeric"))), digits = digits)}
#References
Mantel, N. 1967. “The detection of disease clustering and a generalized regression approach.” Cancer Research 27 (2): 209–20. http://www.ncbi.nlm.nih.gov/pubmed/6018555.
Mojena, R. 1977. “Hierarchical grouping methods and stopping rules: an evaluation.” The Computer Journal 20 (4): 359–63. https://doi.org/10.1093/comjnl/20.4.359.
Olivoto, T., A. D. C Lúcio, V. Q. Souza, M. Nardino, M. I. Diel, B. G. Sari, D .K. Krysczun, D. Meira, and C. Meier. 2018. “Confidence interval width for Pearson’s correlation coefficient: a Gaussian-independent estimator based on sample size and strength of association.” Agronomy Journal 110 (1): 1–8. https://doi.org/10.2134/agronj2017.09.0566.
Olivoto, T., Nardino M., Carvalho I. R., Follmann D. N., Ferrari M., de Pelegrin A. J., V. J. Szareski, de Oliveira A. C., Caron B. O., and V. Q. Souza. 2017. “Optimal sample size and data arrangement method in estimating correlation matrices with lesser collinearity: A statistical focus in maize breeding.” African Journal of Agricultural Research 12 (2): 93–103. https://doi.org/10.5897/AJAR2016.11799.
Olivoto, T., V. Q. Souza, M. Nardino, I. R. Carvalho, M. Ferrari, A. J. Pelegrin, V. J. Szareski, and D. Schmidt. 2017. “Multicollinearity in path analysis: a simple method to reduce its effects.” Agronomy Journal 109 (1): 131–42. https://doi.org/10.2134/agronj2016.04.0196.