• path_coeff() computes a path analysis using a data frame as input data.

• path_coeff_seq() computes a sequential path analysis using primary and secondary traits.

• path_coeff_mat() computes a path analysis using correlation matrices as input data.

## Usage

path_coeff(
.data,
resp,
pred = everything(),
by = NULL,
exclude = FALSE,
correction = NULL,
knumber = 50,
brutstep = FALSE,
maxvif = 10,
missingval = "pairwise.complete.obs",
plot_res = FALSE,
verbose = TRUE,
...
)

path_coeff_mat(cor_mat, resp, correction = NULL, knumber = 50, verbose = TRUE)

path_coeff_seq(.data, resp, chain_1, chain_2, by = NULL, verbose = TRUE, ...)

## Arguments

.data

The data. Must be a data frame or a grouped data passed from dplyr::group_by()

resp

<tidy-select> The dependent trait.

pred

<tidy-select> The predictor traits. set to everything(), i.e., the predictor traits are all the numeric traits in the data except that in resp. To select multiple traits, use a comma-separated vector of names, (e.g., pred = c(V1, V2, V2)), an interval of trait names, (e.g., pred = c(V1:V3)), or even a select helper (e.g., pred = starts_with("V")).

by

One variable (factor) to compute the function by. It is a shortcut to dplyr::group_by(). To compute the statistics by more than one grouping variable use that function.

exclude

Logical argument, set to false. If exclude = TRUE, then the traits in pred are deleted from the data, and the analysis will use as predictor those that remained, except that in resp.

correction

Set to NULL. A correction value (k) that will be added into the diagonal elements of the X'X matrix aiming at reducing the harmful problems of the multicollinearity in path analysis (Olivoto et al., 2017)

knumber

When correction = NULL, a plot showing the values of direct effects in a set of different k values (0-1) is produced. knumber is the number of k values used in the range of 0 to 1.

brutstep

Logical argument, set to FALSE. If true, then an algorithm will select a subset of variables with minimal multicollinearity and fit a set of possible models. See the Details section for more information.

maxvif

The maximum value for the Variance Inflation Factor (cut point) that will be accepted. See the Details section for more information.

missingval

How to deal with missing values. For more information, please see stats::cor().

plot_res

If TRUE, create a scatter plot of residual against predicted value and a normal Q-Q plot.

verbose

If verbose = TRUE then some results are shown in the console.

...

Depends on the function used:

• For path_coeff() additional arguments passed on to stats::plot.lm().

• For path_coeff_seq() additional arguments passed on to path_coeff.

cor_mat

Matrix of correlations containing both dependent and independent traits.

chain_1, chain_2

<tidy-select> The traits used in the first (primary) and second (secondary) chain.

## Value

Depends on the function used:

• path_coeff(), returns a list with the following items:

• Corr.x A correlation matrix between the predictor variables.

• Corr.y A vector of correlations between each predictor variable with the dependent variable.

• Coefficients The path coefficients. Direct effects are the diagonal elements, and the indirect effects those in the off-diagonal elements (lines).

• Eigen Eigenvectors and eigenvalues of the Corr.x.

• VIF The Variance Inflation Factors.

• plot A ggplot2-based graphic showing the direct effects in 21 different k values.

• Predictors The predictor variables used in the model.

• CN The Condition Number, i.e., the ratio between the highest and lowest eigenvalue.

• Det The matrix determinant of the Corr.x..

• R2 The coefficient of determination of the model.

• Residual The residual effect of the model.

• Response The response variable.

• weightvar The order of the predictor variables with the highest weight (highest eigenvector) in the lowest eigenvalue.

• path_coeff_seq() returns a list with the following objects

• resp_fc an object of class path_coeff with the results for the analysis with dependent trait and first chain predictors.

• resp_sc an object of class path_coeff with the results for the analysis with dependent trait and second chain predictors.

• resp_sc2 The path coefficients of second chain predictors and the dependent trait through the first chain predictors

• fc_sc_list A list of objects with the path analysis using each trait in the first chain as dependent and second chain as predictors.

• fc_sc_coef The coefficients between first- and second-chain traits.

• cor_mat A correlation matrix between the analyzed traits. If .data is a grouped data passed from dplyr::group_by() then the results will be returned into a list-column of data frames.

## Details

In path_coeff(), when brutstep = TRUE, an algorithm to select a set of predictors with minimal multicollinearity and high explanatory power is implemented. first, the algorithm will select a set of predictors with minimal multicollinearity. The selection is based on the variance inflation factor (VIF). An iterative process is performed until the maximum VIF observed is less than maxvif. The variables selected in this iterative process are then used in a series of stepwise-based regressions. The first model is fitted and p-1 predictor variables are retained (p is the number of variables selected in the iterative process. The second model adjusts a regression considering p-2 selected variables, and so on until the last model, which considers only two variables. Three objects are created. Summary, with the process summary, Models, containing the aforementioned values for all the adjusted models; and Selectedpred, a vector with the name of the selected variables in the iterative process.

## References

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. Agron. J. 109:131-142. doi:10.2134/agronj2016.04.0196

Olivoto, T., M. Nardino, I.R. Carvalho, D.N. Follmann, M. Ferrari, et al. 2017. REML/BLUP and sequential path analysis in estimating genotypic values and interrelationships among simple maize grain yield-related traits. Genet. Mol. Res. 16(1): gmr16019525. doi:10.4238/gmr16019525

## Author

Tiago Olivoto tiagoolivoto@gmail.com

## Examples

library(metan)

# Using KW as the response variable and all other ones as predictors
pcoeff <- path_coeff(data_ge2, resp = KW)
#> Severe multicollinearity.
#> Condition Number: 7865.84
#> Consider using a correction factor with 'correction' argument.
#> Consider identifying collinear traits with non_collinear_vars()

# The same as above, but using the correlation matrix
cor_mat <- cor(data_ge2 %>% select_numeric_cols())
pcoeff2 <- path_coeff_mat(cor_mat, resp = KW)
#> Severe multicollinearity.
#> Condition Number = 7865.84
#> Please, consider using a correction factor, or use 'brutstep = TRUE'.

# Declaring the predictors
# Create a residual plot with 'plot_res = TRUE'
pcoeff3<- path_coeff(data_ge2,
resp = KW,
pred = c(PH, EH, NKE, TKW),
plot_res = TRUE)

#> Weak multicollinearity.
#> Condition Number: 40.232
#> You will probably have path coefficients close to being unbiased.

# Selecting a set of predictors with minimal multicollinearity
# Maximum variance Inflation factor of 5
pcoeff4 <- 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    0.259   2.59
#>  MODEL_2 1125       6 12.26      0.1383 0.933    0.259   2.46
#>  MODEL_3 1126       5 12.05      0.1989 0.932    0.261   2.31
#>  MODEL_4 1251       4  6.66      0.4016 0.846    0.393   1.98
#>  MODEL_5 1308       3  3.05      0.7438 0.774    0.475   1.34
#>  MODEL_6 1329       2  2.23      0.8555 0.738    0.512   1.17
#> --------------------------------------------------------------------------
#>

# When one analysis should be carried out for each environment
# Using the forward-pipe operator %>%
pcoeff5 <- path_coeff(data_ge2, resp = KW, by = ENV)
#> Severe multicollinearity.
#> Condition Number: 13958.438
#> Consider using a correction factor with 'correction' argument.
#> Consider identifying collinear traits with non_collinear_vars()
#> Severe multicollinearity.
#> Condition Number: 8139.667
#> Consider using a correction factor with 'correction' argument.
#> Consider identifying collinear traits with non_collinear_vars()
#> Severe multicollinearity.
#> Condition Number: 11334.047
#> Consider using a correction factor with 'correction' argument.
#> Consider identifying collinear traits with non_collinear_vars()
#> Severe multicollinearity.
#> Condition Number: 12981.917
#> Consider using a correction factor with 'correction' argument.
#> Consider identifying collinear traits with non_collinear_vars()

# sequential path analysis
# KW as dependent trait
# NKE and TKW as primary predictors
# PH, EH, EP, and EL as secondary traits
pcoeff6 <-
path_coeff_seq(data_ge2,
resp = KW,
chain_1 = c(NKE, TKW),
chain_2 = c(PH, EH, EP, EL))
#> ========================================================
#> Collinearity diagnosis of first chain predictors
#> ========================================================
#> Weak multicollinearity.
#> Condition Number: 1.139
#> You will probably have path coefficients close to being unbiased.
#> ========================================================
#> Collinearity diagnosis of second chain predictors
#> ========================================================
#> Severe multicollinearity.
#> Condition Number: 1047.993
#> Consider using a correction factor with 'correction' argument.
#> Consider identifying collinear traits with non_collinear_vars()
pcoeff6$resp_sc$Coefficients
#>           PH         EH        EP        EL    linear
#> PH 1.4800514 -1.3237892 0.4282428 0.1689388 0.7534439
#> EH 1.3791537 -1.4206365 0.5832858 0.1611440 0.7029469
#> EP 0.9448831 -1.2353087 0.6707935 0.1170515 0.4974193
#> EL 0.5627096 -0.5151991 0.1767029 0.4443467 0.6685601
pcoeff6\$resp_sc2
#>    trait      effect         NKE         TKW     residual      total
#> 1     PH      direct  1.45585270 -0.07780424 -0.102002953  1.4800514
#> 2     PH indirect_EH -1.70571124  0.54287910  0.160957033 -1.3237892
#> 3     PH indirect_EP  0.48951552 -0.12747469 -0.066202002  0.4282428
#> 4     PH indirect_EL  0.09405361  0.07201754 -0.002867693  0.1689388
#> 5     PH       total  0.33371058  0.40961771 -0.010115614  0.7534439
#> 6     EH      direct -1.83049968  0.58259569  0.172732518 -1.4206365
#> 7     EH indirect_PH  1.35660461 -0.07250018 -0.095049229  1.3791537
#> 8     EH indirect_EP  0.66674189 -0.17362620 -0.090170068  0.5832858
#> 9     EH indirect_EL  0.08971396  0.06869464 -0.002735378  0.1611440
#> 10    EH       total  0.28256079  0.40516395 -0.015222157  0.7029469
#> 11    EP      direct  0.76677015 -0.19967455 -0.103697874  0.6707935
#> 12    EP indirect_PH  0.92943432 -0.04967118 -0.065119943  0.9448831
#> 13    EP indirect_EH -1.59170362  0.50659374  0.150198865 -1.2353087
#> 14    EP indirect_EL  0.06516626  0.04989828 -0.001986918  0.1170515
#> 15    EP       total  0.16966711  0.30714629 -0.020605871  0.4974193
#> 16    EL      direct  0.24738189  0.18942212 -0.007542670  0.4443467
#> 17    EL indirect_PH  0.55350939 -0.02958086 -0.038781116  0.5627096
#> 18    EL indirect_EH -0.66383754  0.21128050  0.062642093 -0.5151991
#> 19    EL indirect_EP  0.20198546 -0.05259902 -0.027316482  0.1767029
#> 20    EL       total  0.33903919  0.31852275 -0.010998175  0.6685601