BayesianEvSyn: Bayesian evidence synthesis

In rebeccakuiper/GoricEvSyn: GORIC(A) evidence synthesis (GoricEvSyn) aggregates the evidence for theory-based hypotheses from multiple studies that may use diverse designs to investigate the same central theory.

Bayesian evidence synthesis

Description

Bayesian evidence synthesis (BayesianEvSyn) aggregates the evidence for theory-based hypotheses from multiple studies that may use diverse designs to investigate the same central theory. There is also an interactive web application on my website to perform BayesianEvSyn: https://www.uu.nl/staff/RMKuiper/Websites%20%2F%20Shiny%20apps.

Usage

BayesianEvSyn(
  TypeEv,
  S,
  Param_studies,
  CovMx_studies,
  N,
  SameHypo,
  NrHypos,
  Hypo_studies,
  Safeguard = "unconstrained",
  Name_studies = 1:S,
  PrintPlot = T
)

Arguments

TypeEv	The type of evidence-synthesis approach: Equal-evidence approach (0) or Added-evidence approach (1). In case of an equal-evidence approach, aggregating evidence from, say, 5 studies with n=100 observations is the same as obtaining evidence from 1 study (as if it was possible) with n=500 observations (like meta-analysis does). In the added-evidence approach, the aggregated evidence from, says, 5 studies is stronger than as if the data were combined (as if that was possible).
S	The number of (primary) studies. That is, the results (evidence) of S studies will be aggregated.
Param_studies	List of S 'named' vectors with the k_s (standardized) parameter estimates of interest of Study s. Thus, there are S items in the list and each item is a 'named' vector with k_s elements: the k_s number of parameter estimates relevant for that study. In case each study has the same number of parameters (k) which denote the same (in terms of hypothesis specification), Param_studies can be an S x k 'named' matrix. Note: The names of the vectors (or the column names of the S x 'k' matrix) with estimates should be used in the hypothesis specification.
CovMx_studies	List of the S covariance matrices of the (standardized) parameter estimates of interest (of size k_s x k_s). In case number of parameters are the same, it can also be a S*k_s x k_s matrix. Note: The columns (and rows) do not need to be named.
N	Vector of size S, with the sample size for each study.
SameHypo	Indicator whether the same hypotheses are used (1) or not (0) in all S studies. If SameHypo = 1, then the same estimates in Param_studies should have the same name.
NrHypos	The number of theory-based hypotheses that will be evaluated within each study (is a scalar with an integer value).
Hypo_studies	A vector of strings containing the NrHypos theory-based hypotheses. If SameHypo = 0, then there should be S specifications of the NrHypos theory-based hypotheses, that is S times NrHypos strings.
Safeguard	Indicator of which safeguard-hypothesis should be used: "unconstrained" (default; i.e., all possible theories including the one specified), "none" (only advised when set of hyptheses cover all theories), or (only when 'NrHypos = 1') "complement" (i.e., the remaining theories).
Name_studies	Optional. Vector of S numbers or S characters to be printed at the x-axis of the plot with posterior model probabilities (PMPs). Default: Name_studies = 1:S.
PrintPlot	Optional. Indicator whether plot of posterior model probabilities (PMPs) should be printed (TRUE; default) or not (FALSE). The posterior model probabilities per study are plotted and the cumulative posterior model probabilities (where those for the last study are the final ones).

Value

The output comprises, among other things, the cumulative and final evidence for the theory-based hypotheses.

Examples

# TO DO voeg in voorbeelden hieronder N (vector of length S) toe!


#### Note ####
#BayesianEvSyn uses the R package 'bain'; make sure that this is installed.


### Example 1: 2 ANOVA studies (Monin and Holubar) ###
### We will use the estimates of the ANOVA models

S <- 2
est_1 <- c(1.88, 2.54, 0.02)
names(est_1) <- c("group1", "group2", "group3")
vcov_est_1 <- diag(c(0.2149074, 0.2149074, 0.1408014))
est_2 <- c(0.98, 0.02, 0.27)
names(est_2) <- c("gr1", "gr2", "gr3") # Use different names to show use of different set of hypotheses
vcov_est_2 <- diag(c(0.1382856, 0.1024337, 0.0987754))

# If number of parameters differ per study (but can also be used when they are the same): make lists
#
# beta values from the analyses
Param_studies <- list(est_1, est_2)
#
# standard error of the beta's (from the S primary studies)
CovMx_studies <- list(vcov_est_1, vcov_est_2)

# Set of hypotheses for each study
# Note: in this case we could make the names of the estimates in est_1 and est_2 the same.
# In general, when not same number of parameters they names will not be the same and/or the set of hypotheses, so we use that here.
SameHypo <- 0
NrHypos <- 2
# names(est_1) # Specify restrictions using those names
H11 <- 'group1 = group2 & group2 > group3'
H12 <- 'group2 > group1 & group1 > group3'
# names(est_2) # Specify restrictions using those names
H21 <- 'gr1 = gr2 & gr2 > gr3'  # Note: cannot use gr1 == gr2 > gr3
H22 <- 'gr2 > gr1 & gr1 > gr3'   # Note: cannot use gr2 > gr1 > gr3
Hypo_studies <- c(H11, H12, H21, H22)
#
# Evidence synthesis
TypeEv <- 1 # Added-evidence approach
BayesianEvSyn(TypeEv, S, Param_studies, CovMx_studies, SameHypo, NrHypos, Hypo_studies)



### Example 2: 4 trust studies discussed in Kuiper et al. (2013) ###
### Different statistical models are used: linear regression, probit regression, and three-level logististic regression

S <- 4 # Number of primary studies for which the evidence for a central theory should be determined and synthesized.

# Example 1a #
# If same number of parameters per study
#
NrParam <- 1 # In this example, each study has one parameter of interest (i.e., hypotheses below address only this parameter).
# beta values from the analyses
Param_studies <- matrix(c(0.09, 0.14, 1.09, 1.781), nrow = S, ncol = NrParam)
colnames(Param_studies) <- "beta1" # Should have names and should have the same as in the hypotheses below
#
# standard error of the beta's (from the S primary studies)
CovMx_studies <- matrix(c(0.029^2, 0.054^2, 0.093^2, 0.179^2), nrow = S, ncol = NrParam) # Note: no names needed
#

# OR

# Example 1b #
# If number of parameters differ per study (but can also be used when they are the same)
#
# beta values from the analyses
est_1 <- matrix(c(0.09), nrow = 1)
colnames(est_1) <- "beta1"
est_2 <- matrix(c(0.14), nrow = 1)
colnames(est_2) <- "beta1"
est_3 <- matrix(c(1.09), nrow = 1)
colnames(est_3) <- "beta1"
est_4 <- matrix(c(1.781), nrow = 1)
colnames(est_4) <- "beta1"
Param_studies <- list(est_1, est_2, est_3, est_4)
#
# standard error of the beta's (from the S primary studies)
vcov_est_1 <- matrix(c(0.029^2), nrow = 1)
vcov_est_2 <- matrix(c(0.054^2), nrow = 1)
vcov_est_3 <- matrix(c(0.093^2), nrow = 1)
vcov_est_4 <- matrix(c(0.179^2), nrow = 1)
CovMx_studies <- list(vcov_est_1, vcov_est_2, vcov_est_3, vcov_est_4)

# Set of hypotheses for each study
# Note: in this case the same for each study
SameHypo <- 1
NrHypos <- 3
H0 <- "beta1 = 0"
Hpos <- "beta1 > 0"
Hneg <- "beta1 < 0"
Hypo_studies <- c(H0, Hpos, Hneg)
# Since this covers the whole space / covers all theories, we do not need a safeguard-hypothesis
Safeguard <- "none"
#
# Evidence synthesis
TypeEv <- 1 # Added-evidence approach
BayesianEvSyn(TypeEv, S, Param_studies, CovMx_studies, SameHypo, NrHypos, Hypo_studies, Safeguard)



### Example 3: 3 'fictional' studies; for ease, all 3 use linear regression ###
### where continuous predictor variable need to be standardized and
### where we will the complement of our theory as competing hypothesis ###

S <- 3
ratio <- c(1,1.1,1.2)
n <- c(30, 50, 100)

# Generate data1
n1 <- n[1]
x11 <- rnorm(n1)
x12 <- rnorm(n1)
x13 <- rnorm(n1)
data <- cbind(x11, x12, x13)
# Standardize data - since parameters for continuous variables will be compared
data1 <- as.data.frame(scale(data))
y1 <- ratio[1]*data1$x11 + ratio[2]*data1$x12 + ratio[3]*data1$x13 + rnorm(n1)
# Note: since there is one outcome, the outcome does not need to be standardized
# Fit regression model
fit.lm1 <- lm(y1 ~ 1 + x11 + x12 + x13, data = data1)

# Generate data2
n2 <- n[2]
x21 <- rnorm(n2)
x22 <- rnorm(n2)
x23 <- rnorm(n2)
data <- cbind(x21, x22, x23)
# Standardize data - since parameters for continuous variables will be compared
data2 <- as.data.frame(scale(data))
y2 <- ratio[1]*data2$x21 + ratio[2]*data2$x22 + ratio[3]*data2$x23 + rnorm(n2)
# Note: since there is one outcome, the outcome does not need to be standardized
# Fit regression model
fit.lm2 <- lm(y2 ~ 1 + x21 + x22 + x23, data = data2)

# Generate data3
n3 <- n[3]
x31 <- rnorm(n3)
x32 <- rnorm(n3)
x33 <- rnorm(n3)
data <- cbind(x31, x32, x33)
# Standardize data - since parameters for continuous variables will be compared
data3 <- as.data.frame(scale(data))
y3 <- ratio[1]*data3$x31 + ratio[2]*data3$x32 + ratio[3]*data3$x33 + rnorm(n3)
# Note: since there is one outcome, the outcome does not need to be standardized
# Fit regression model
fit.lm3 <- lm(y3 ~ 1 + x31 + x32 + x33, data = data3)

# Extract estimates and their covariance matrix (per study)
est_1 <- coef(fit.lm1)
est_2 <- coef(fit.lm2)
est_3 <- coef(fit.lm3)
vcov_est_1 <- vcov(fit.lm1)
vcov_est_2 <- vcov(fit.lm2)
vcov_est_3 <- vcov(fit.lm3)
#
## If same number and type of parameters per study (since they will obtain the same name)
NrParam <- length(est_1)
# Parameter estimate values from the S primary studies
Param_studies <- matrix(c(est_1, est_2, est_3), byrow = T, nrow = S, ncol = NrParam)
colnames(Param_studies) <- c("intercept", "x1", "x2", "x3")
# standard error of the beta's (from the S primary studies)
CovMx_studies <- matrix(c(vcov_est_1, vcov_est_2, vcov_est_3), byrow = T, nrow = S*NrParam, ncol = NrParam) # Note: no names needed

# Set of hypotheses for each study
# Note: in this case the same for each study
SameHypo <- 1
# colnames(est) # Specify restrictions using those names
H1 <- 'x1 < x2 & x2 < x3'   # Note: cannot use x1 < x2 < x3
# Since estimates of continuous variables are compared in our theory, we standardized the data before to obtain comparable estimates.
# Since no restrictions on intercept, you can leave it out in 'Param_studies' and 'CovMx_studies'; this does not impact the posterior model probabilities.
NrHypos <- 1
# Since we have only one theory-based hypothesis, we will use the (more powerful) complement of the hypothesis (Vanbrabant, Van Loey, Kuiper, 2019).
# The complement represents the remaining 11 theories, while the unconstrained reflects all 12 possible theories including H1.
Safeguard <- "complement"
#
# Evidence synthesis
TypeEv <- 1 # Added-evidence approach
BayesianEvSyn(TypeEv, S, Param_studies, CovMx_studies, SameHypo, NrHypos, Hypo_studies = H1, Safeguard)

# Change labels on x-axis in PMPs plot #
# For example, let us say that the studies come from the years 2016, 2017, 2019.
# Because of unequal spacing, you may want to use numbers instead of characters:
Name_studies <- c(2016, 2017, 2019)
BayesianEvSyn(TypeEv, S, Param_studies, CovMx_studies, SameHypo, NrHypos, Hypo_studies = H1, Safeguard, Name_studies)

rebeccakuiper/GoricEvSyn documentation built on July 3, 2023, 6:41 a.m.