# R/helpers_cubic.R In RSA: Response Surface Analysis

#### Documented in caRangeclRange

# helpers_cubic.R
# = functions needed for the interpretation of the cubic models

# ---------------------------------------------------------------------
# Range check function for strict and broad asymmetric congruence models
# ---------------------------------------------------------------------

# function that, for a given point (x,y), computes the one-sided confidence interval of the model-prediction z at (x,y) according to the CA/RRCA model,

ci_pred <- function(obj, x, y, side, n, p, alpha, model){

DV <- obj$DV # compute predicted outcome value at (x,y) z <- predictRSA(obj, x, y, model=model) # vector of predictor values r = c(1, x, y, x^2, x*y, y^2, x^3, x^2*y, x*y^2, y^3) # get covariances of the vector beta of estimated coefficients COV0 = vcov(obj$models[[model]])
covBeta = COV0[c(paste0(DV,"~1"), "b1", "b2", "b3", "b4", "b5", "b6", "b7", "b8", "b9"),][,c(paste0(DV,"~1"), "b1", "b2", "b3", "b4", "b5", "b6", "b7", "b8", "b9")]

# Compute standard error of the predicted value z
var_z = t(r) %*% covBeta %*% r
se_z = sqrt(var_z)

# compute one-sided confidence interval of z.
if(side=="right"){
lower_z = as.vector(-Inf)
upper_z = as.vector(z + qt(1-alpha, df=n-p)*se_z)
}

if(side=="left"){
lower_z = as.vector(z - qt(1-alpha, df=n-p)*se_z)
upper_z = as.vector(Inf)
}

# output
out <- list(z=z, lower_z=lower_z, upper_z=upper_z)
return(out)
}

#' @title Range check for the CA/RRCA model
#'
#' @description
#' Identify data points behind E2 and test how many of them have outcome predictions that significantly differ from predictions for predictor combinations on E2 (that have the same level)
#'
#' @details
#' When testing an asymmetric congruence hypothesis with the CA or RRCA model, the \code{caRange} function helps to determine whether the hypothesis is supported for the whole range of realistic predictor combinations. It computes the position of the second extremum line E2 and tests how many predictor combinations are in the data which lie "behind" this line and, at the same time, have a significantly different outcome prediction than points on E2.
#'
#' When plotting the estimated model (CA or RRCA) with \code{\link{plot}}, you can plot the line E2 and the surface above this line by calling "E2" in the options \code{project} and \code{axes}.
#'
#' @aliases caRange
#'
#' @importFrom stats pnorm
#' @importFrom stats qnorm
#' @importFrom stats qt
#'
#' @export
#' @param object An RSA object
#' @param alpha Alpha level for the one-sided confidence interval of the outcome predictions on E2
#' @param verbose Should extra information be printed?
#' @param model Either "CA" or "RRCA"
#' @param alphacorrection Set "Bonferroni" to adjust the alpha level for multiple testing when testing the outcome predictions of all data points behind E2
#'
#' @references
#' Humberg, S., Schönbrodt, F. D., Back, M. D., Nestler, S. (in preparation). \emph{Cubic response surface analysis: Investigating asymmetric and level-dependent congruence effects with third-order polynomial models.} Manuscript submitted for publication.

caRange <- function(object, alpha=0.05, verbose=TRUE, model="CA", alphacorrection="none"){

rsa = object

# this function only makes sense for the models CA and RRCA
if (!model %in% c("CA","RRCA")) stop("This function only makes sense for the models CA and RRCA.")

# check whether model is contained in the RSA object and has converged
if (!model %in% names(rsa$models[!unlist(lapply(rsa$models, is.null))])) stop("Please provide an RSA object that contains estimates for the asymmetric congruence model (CA or RRCA).")

if (inspect(rsa$models[[model]], "converged") == FALSE) { warning("The model has not converged!") return(NULL) } ## Extract information from RSA object # get variable names of outcome, x, and y DV <- rsa$DV
IV1 <- rsa$IV1 IV2 <- rsa$IV2

# get coefficient estimates of CA model
c0 = as.numeric(coef(rsa$models[[model]])[paste0(DV,"~1")]) c1 = as.numeric(coef(rsa$models[[model]])["b3"])
c2 = as.numeric(coef(rsa$models[[model]])["b6"]) # the present analysis makes no sense if the quadratic or cubic term coefficient is zero, because there is no second extremum line in these cases if (c1 == 0) stop("The quadratic term coefficient c1 is zero.") if (c2 == 0) stop("The cubic term coefficient c2 is zero.") # get raw data that was used to estimate the model data = rsa$data
df = data[data$out==FALSE, ] # create variable with predicted outcome values df$z.pred <- NA
for (k in 1:nrow(df)){df[k,"z.pred"] = predictRSA(rsa, df[k,IV1], df[k,IV2], model=model)}

# ---------------------------------------------------------------------
# How many percent of the data points lie behind the second extremum line?

# define variable e = y - (x + 2c1/3c2) to be used in the next step
df$e <- df[,IV2] - (df[,IV1] + 2*c1/(3*c2)) if(c1*c2 < 0){ # how many data points / what percentage of data points lie behind the second extremum line <-> satisfy y < x + 2c1/3c2 <-> e = y - (x + 2c1/3c2) < 0 df$behind <- df$e < 0 df$behind[is.na(df$behind)] <- FALSE behind = sum(df$behind)
p.behind = 100*(behind/nrow(df))
}

if(c1*c2 > 0){

# how many data points / what percentage of data points lie behind the second extremum line  <->   satisfy y > x + 2c1/3c2  <->  e = y - (x + 2c1/3c2) > 0
df$behind <- df$e > 0
df$behind[is.na(df$behind)] <- FALSE
behind = sum(df$behind) p.behind = 100*(behind/nrow(df)) } # --------------------------------------------------------------------- # How many percent of the data points behind the second extremum line have outcome predictions that differ significantly from the outcome prediction on the second extremum line? # In the following, determine the number of "bad" data points behind the second extremum line, separately for CA versus RRCA model. # Note that in fact, one could apply the procedure defined for the RRCA model (comparing the outcome prediction at each data point behind E2 # to the outcome prediction at the point on E2 with the same level as the data point), as the CA model is a special case of the RR model. # The results would be exactly the same as with the procedure defined for the CA model in the following. # Despite these redundancies, we strictly follow the descriptions in the manuscript here. # if requested, apply a Bonferroni correction to the alpha level (number of tests = number of data points behind E2) if(alphacorrection=="Bonferroni"){alpha <- alpha/behind} if(model == "CA"){ ## compute the confidence interval of the model prediction zr at the intersection point of E2 and the LOIC # Identify x value xr of intersection point of LOIC and second extremum line and compute the confidence interval around the predicted value zr at this point xr = -c1/(3*c2) yr = -xr # compute one-sided confidence interval of z. The direction of one-sidedness depends on whether the surface goes up versus down behind the second extremum line, which depends on the sign of c1. z_ci <- ci_pred(obj=rsa, x = xr, y = yr, side=ifelse(c1<0, "right", "left"), n = nrow(df), p = 10, alpha=alpha, model=model) zr <- z_ci$z

## Count points behind second extremum line whose outcome prediction differs significantly from zr
df$lower_z <- z_ci$lower_z
df$upper_z <- z_ci$upper_z
df$badcase <- ifelse( (df$behind==T & df$z.pred < df$lower_z) | (df$behind==T & df$z.pred > df$upper_z) , TRUE, FALSE) bad = sum(df$badcase)

response <- paste0(p.bad, "% of the data points lie behind the second extremum line AND have a significantly different outcome prediction than the points on the second extremum line.")

# build results object for case of CA model
res <- list(
data.used = df,
behind = behind,
percentage.behind = p.behind,
reversion_point_xy = paste0("(",round(xr,2),", ",round(yr,2), ")"),
reversion_point_z = round(zr,2),
reversion_point_z_lower = round(z_ci$lower_z,2), reversion_point_z_upper = round(z_ci$upper_z,2),
reversion_point_z_ci = paste0("[", round(z_ci$lower_z,2),", ",round(z_ci$upper_z,2),"]"),
which_bad_points = which(df$badcase==TRUE), how_many_bad_points = bad, percentage_bad_points = p.bad, alpha.used = alpha, response = response ) } # end of computation for CA model if(model == "RRCA"){ # for the RRCA model, the confidence interval to which the predicted value of each data point behind E2 must be compared is specific to the respective data point. # for each data point, compute the coordinates (xr,yr) of the point on E2 at the same level as the data point df$xr <- 0.5*(df[,IV1] + df[,IV2]) - (c1/(3*c2))
df$yr <- 0.5*(df[,IV1] + df[,IV2]) + (c1/(3*c2)) # create variables containing the zr value [ = predicted value at (xr,yr) ] and its confidence interval per data point df[,"zr"] <- apply(df, 1, function(data){ci_pred(obj=rsa, x=data["xr"], y=data["yr"], side=ifelse(c1<0, "right", "left"), n = nrow(df), p = 10, alpha=alpha, model=model)$z})

df[,"lower_zr"] <- apply(df, 1, function(data){ci_pred(obj=rsa, x=data["xr"], y=data["yr"],
side=ifelse(c1<0, "right", "left"),
n = nrow(df), p = 10, alpha=alpha,
model=model)$lower_z}) df[,"upper_zr"] <- apply(df, 1, function(data){ci_pred(obj=rsa, x=data["xr"], y=data["yr"], side=ifelse(c1<0, "right", "left"), n = nrow(df), p = 10, alpha=alpha, model=model)$upper_z})

## Count points behind second extremum line whose outcome prediction differs significantly from zr
df$badcase <- ifelse( (df$behind==T & df$z.pred < df$lower_zr) | (df$behind==T & df$z.pred > df$upper_zr) , TRUE, FALSE) bad = sum(df$badcase)

response <- paste0(p.bad, "% of the data points lie behind the second extremum line AND have a significantly different outcome prediction than the same-level point on the second extremum line.")

# build results object for case of CA model
res <- list(
data.used = df,
behind = behind,
percentage.behind = p.behind,
which_bad_points = which(df$badcase==TRUE), how_many_bad_points = bad, percentage_bad_points = p.bad, alpha.used = alpha, response = response ) } # end of computation for RRCA model # print results if(verbose==T){ print(response) } # return results return(res) } # --------------------------------------------------------------------- # Range check function for strict and broad level-dependent congruence models # --------------------------------------------------------------------- # function to compute roots of the quadratic function ax^2 + bx + c quad_roots <- function(a,b,c) { sqrt_term = b^2 - 4*a*c if (sqrt_term > 0) { pos_root <- ((-b) + sqrt(sqrt_term)) / (2*a) neg_root <- ((-b) - sqrt(sqrt_term)) / (2*a) } if (sqrt_term == 0) { pos_root <- ((-b) + sqrt(sqrt_term)) / (2*a) neg_root <- NA } if (sqrt_term < 0) { pos_root = neg_root = NA } # output out <- c(pos_root,neg_root) return(out) } # function to compute the test statistic of the curvature of gk for a fixed value of k pick_a_point <- function(c1,c3,cov,k){ # compute curvature of gk eta = 4*c1 + 8*k*c3 # variance of eta var = 16*cov["b3","b3"] + 64*cov["b3","b6"]*k + 64*cov["b6","b6"]*k^2 # test statistic of eta teststat = eta/sqrt(var) # p value p = 2*pnorm(-abs(teststat)) # output res = c(eta=eta, se=sqrt(var), teststat=teststat, p.value=p) return(res) } #' @title Range check for the CL/RRCL model #' #' @description #' Compute the regions of significance and test their intersection with the data #' #' @details #' When testing a level-dependent congruence hypothesis with the CL or RRCL model, the \code{clRange} function helps to determine whether the hypothesis is supported for the whole range of realistic predictor combinations. It computes the mean predictor levels k1 and k2 at which the curvature of the surface changes its significance status. For each of the resulting intervals, the function informs whether the curvature is significantly negative, nonsignificant, or significantly positive in the respective interval. #' #' When plotting the estimated model (CL or RRCL) with \code{\link{plot}}, you can plot the lines at which the significance status of the curvature changes and the surface above these lines by calling "K1" and "K2" in the options \code{project} and \code{axes}. #' #' @aliases clRange #' #' @importFrom stats pnorm #' @importFrom stats qnorm #' @importFrom stats qt #' #' @export #' @param object An RSA object #' @param alpha Alpha level for the regions of significance of the surface's curvature #' @param verbose Should extra information be printed? #' @param model Either "CL" or "RRCL" #' #' @references #' Humberg, S., Schönbrodt, F. D., Back, M. D., Nestler, S. (in preparation). \emph{Cubic response surface analysis: Investigating asymmetric and level-dependent congruence effects with third-order polynomial models.} Manuscript submitted for publication. clRange <- function(object, alpha=0.05, verbose=TRUE, model="CL"){ rsa = object # this function only makes sense for the models CL and RRCL if (!model %in% c("CL","RRCL")) stop("This function only makes sense for the models CL and RRCL.") # check whether the model is contained in the RSA object and has converged if (!model %in% names(rsa$models[!unlist(lapply(rsa$models, is.null))])) stop("Please provide an RSA object that contains estimates for the level-dependent congruence model (CL or RRCL).") if (inspect(rsa$models[[model]], "converged") == FALSE) {
warning("The model has not converged!")
return(NULL)
}

## Extract information from RSA object

# get variable names of outcome, x, and y
DV <-  rsa$DV IV1 <- rsa$IV1
IV2 <- rsa$IV2 # get coefficient estimates of CL model c0 = as.numeric(coef(rsa$models[[model]])[paste0(DV,"~1")])
c1 = as.numeric(coef(rsa$models[[model]])["b3"]) c3 = as.numeric(coef(rsa$models[[model]])["b6"])

# the present analysis makes no sense if the quadratic or cubic term coefficient is zero
if (c1 == 0) stop("The quadratic term coefficient c1 is zero.")
if (c3 == 0) stop("The coefficient c3 is zero.")

# get covariances of the vector beta of estimated coefficients
COV0 = vcov(rsa$models[[model]]) # get raw data that was used to estimate the model data = rsa$data
df = data[data$out==FALSE, ] # --------------------------------------------------------------------- ## Prepare result table tab = data.frame(region = NA, lower_bound = NA, upper_bound = NA, data_points = NA, percent_data = NA) tab <- tab[-1,] # --------------------------------------------------------------------- ## Determine the regions of negative/non/positive significance # compute k values where the curvature of gk changes from one significance state to another # = roots of the quadratic function ak^2 + bk + c, with a, b, c computed as follows z = qnorm(1-alpha/2) a = 4*c3^2 - 4*COV0["b6","b6"]*z^2 b = 4*c1*c3 - 4*COV0["b3","b6"]*z^2 c = c1^2 - COV0["b3","b3"]*z^2 roots = quad_roots(a,b,c) # if there are two roots, the significance status changes at each of them. Determine the significance status of each interval. if (sum(is.na(roots)) == 0) { # name roots k1 <- lower_root <- min(roots) k2 <- higher_root <- max(roots) # check significance status in the interval ]-Inf, lower_root] row = nrow(tab)+1 tab[row, c("lower_bound","upper_bound")] <- c(-Inf, lower_root) testvalue = lower_root - 10 teststat = pick_a_point(c1, c3, cov=COV0, k=testvalue)["teststat"] if (teststat <= -z){ tab[row, "region"] <- "neg. sign." } else if (teststat >= z) { tab[row, "region"] <- "pos. sign." } else { tab[row, "region"] <- "nonsign."} # check significance status in the interval [lower_root, higher_root] row = nrow(tab)+1 tab[row, c("lower_bound","upper_bound")] <- c(lower_root, higher_root) testvalue = (lower_root+higher_root)/2 teststat = pick_a_point(c1, c3, cov=COV0, k=testvalue)["teststat"] if (teststat <= -z){ tab[row, "region"] <- "neg. sign." } else if (teststat >= z) { tab[row, "region"] <- "pos. sign." } else { tab[row, "region"] <- "nonsign."} # check significance status in the interval [higher_root, Inf[ row = nrow(tab)+1 tab[row, c("lower_bound","upper_bound")] <- c(higher_root, Inf) testvalue = higher_root + 10 teststat = pick_a_point(c1, c3, cov=COV0, k=testvalue)["teststat"] if (teststat <= -z){ tab[row, "region"] <- "neg. sign." } else if (teststat >= z) { tab[row, "region"] <- "pos. sign." } else { tab[row, "region"] <- "nonsign."} } # if there is one root, the significance status changes at it. Determine the significance status of each of the two intervals. if (sum(is.na(roots)) == 1) { # name roots k1 <- single_root <- roots[1] k2 <- NA # check significance status in the interval ]-Inf, single_root] row = nrow(tab)+1 tab[row, c("lower_bound","upper_bound")] <- c(-Inf, single_root) testvalue = single_root - 10 teststat = pick_a_point(c1, c3, cov=COV0, k=testvalue)["teststat"] if (teststat <= -z){ tab[row, "region"] <- "neg. sign." } else if (teststat >= z) { tab[row, "region"] <- "pos. sign." } else { tab[row, "region"] <- "nonsign."} # check significance status in the interval [single_root, Inf[ row = nrow(tab)+1 tab[row, c("lower_bound","upper_bound")] <- c(single_root, Inf) testvalue = single_root + 10 teststat = pick_a_point(c1, c3, cov=COV0, k=testvalue)["teststat"] if (teststat <= -z){ tab[row, "region"] <- "neg. sign." } else if (teststat >= z) { tab[row, "region"] <- "pos. sign." } else { tab[row, "region"] <- "nonsign."} } # if there are no roots, the significance status is the same for all k. Determine which one it is. if (sum(is.na(roots)) == 2) { k1 <- k2 <- NA # check significance status for the whole surface row = nrow(tab)+1 tab[row, c("lower_bound","upper_bound")] <- c(-Inf, Inf) teststat = pick_a_point(c1, c3, cov=COV0, k=0)["teststat"] if (teststat <= -z){ tab[row, "region"] <- "neg. sign." } else if (teststat >= z) { tab[row, "region"] <- "pos. sign." } else { tab[row, "region"] <- "nonsign."} } ## check how many data points lie in each region # for each data point (xi,yi), determine k so that (xi,yi) lies on gk df$k = (df[,IV1] + df[,IV2])/2

# how many data points lie within each region?
for (i in 1:nrow(tab)){
tab[i, "data_points"] <- length(which(df$k > tab[i,"lower_bound"] & df$k < tab[i,"upper_bound"]))
}

tab$percent_data = 100*(tab$data_points/nrow(df))

# ---------------------------------------------------------------------
# build results object
res <- list(
data.used = df,
k1 = k1,
k2 = k2,
regions = tab
)

# return results
return(res)
}


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RSA documentation built on March 24, 2021, 1:10 a.m.