Posterior covariance matrix from Social Anxiety NMA

Description

The posterior covariance matrix of the variables d (basic treatment effect parameters) and delta (shrunken random effects estimates for each study).

Usage

SocAnx.post.cov

Format

An object of class matrix (inherits from array) with 186 rows and 186 columns.

Source

Generated from WinBUGS output, using the WinBUGS code from Mayo-Wilson et al. (2014). See also vignette("Examples", package = "nmathresh").

References

Mayo-Wilson E, Dias S, Mavranezouli I, Kew K, Clark DM, Ades AE, et al. Psychological and pharmacological interventions for social anxiety disorder in adults: a systematic review and network meta-analysis. Lancet Psychiatry 2014;1:368-76. http://dx.doi.org/10.1016/S2215-0366(14)70329-3

See Also

SocAnx.post.summary

Posterior summary from Social Anxiety NMA

Description

A summary.mcmc object of the type produced by the coda package, containing the requisite posterior summary information on the variables d (basic treatment effect parameters), delta (shrunken random effects estimates for each study), and diff (contrasts of treatment effect parameters).

Usage

SocAnx.post.summary

Format

A summary.mcmc object. The key components for our use are:

statistics

Matrix containing the posterior summary statistics of the variables d, delta, and diff, with columns for Mean, SD, Naive SE, and Time-series SE (also known as the Monte-Carlo standard error)

quantiles

Matrix containing the posterior 2.5%, 25%, 50%, 75%, and 97.5% quantiles of the variables d, delta, and diff

Source

Generated from WinBUGS output, using the WinBUGS code from Mayo-Wilson et al. (2014). See also vignette("Examples", package = "nmathresh").

References

Mayo-Wilson E, Dias S, Mavranezouli I, Kew K, Clark DM, Ades AE, et al. Psychological and pharmacological interventions for social anxiety disorder in adults: a systematic review and network meta-analysis. Lancet Psychiatry 2014;1:368-76. http://dx.doi.org/10.1016/S2215-0366(14)70329-3

See Also

summary.mcmc, SocAnx.post.cov

Posterior covariance matrix from Thrombolytics NMA

Description

The posterior covariance matrix of the basic treatment effect parameters.

Usage

Thrombo.post.cov

Format

An object of class matrix (inherits from array) with 5 rows and 5 columns.

Source

Generated from WinBUGS output, using the WinBUGS code from Caldwell et al. (2005). See also vignette("Examples", package = "nmathresh").

References

Caldwell DM, Ades AE, Higgins JPT. Simultaneous comparison of multiple treatments: combining direct and indirect evidence. Brit Med J 2005;331:897-900. http://dx.doi.org/10.1136/bmj.331.7521.897

See Also

Thrombo.post.summary

Posterior summary from Thrombolytics NMA

Description

A summary.mcmc object of the type produced by the coda package, containing the requisite posterior summary information on the variables dd, the contrasts of the treatment effect parameters.

Usage

Thrombo.post.summary

Format

A summary.mcmc object. The key components for our use are:

statistics

Matrix containing the posterior summary statistics, with columns for Mean, SD, Naive SE, and Time-series SE (also known as the Monte-Carlo standard error)

quantiles

Matrix containing the posterior 2.5%, 25%, 50%, 75%, and 97.5% quantiles

Source

Generated from WinBUGS output, using the WinBUGS code from Caldwell et al. (2005). See also vignette("Examples", package = "nmathresh").

References

Caldwell DM, Ades AE, Higgins JPT. Simultaneous comparison of multiple treatments: combining direct and indirect evidence. Brit Med J 2005;331:897-900. http://dx.doi.org/10.1136/bmj.331.7521.897

See Also

summary.mcmc, Thrombo.post.cov

Convert contrast indexing

Description

Functions for converting between d_{ab} indexing of contrasts (useful notationally) and d[i] indexing used by R.

Usage

d_ab2i(a, b, K)

d_i2ab(i, K)

Arguments

Value

d_ab2i returns a vector of indices i. d_i2ab returns a data frame of indices a and b.

Functions

• d_ab2i: Convert d[i] type indices to d_{ab} type indices.

• d_i2ab: Convert d_{ab} type indices to d[i] type indices.

Note

By convention, 1 \le a < b \le K. If this is not the case, an error will be thrown. For a given number of treatments K, the total number of possible contrasts d_{ab} is K(K-1)/2, and hence i \le K. Again, if this is not the case, an error will be thrown.

Examples

d_ab2i(c(1,1,1, 2,2, 3), c(2,3,4, 3,4, 4), K=4)
d_i2ab(1:6, K=4)

Get thresholds from U matrix

Description

Return the positive and negative thresholds for an observation, given a vector of possible threshold solutions. This function is intended for internal use, and is called by nma_thresh automatically.

Usage

get.int(
  x,
  kstar,
  trt.code,
  contrs,
  mcid = FALSE,
  mean.dk = NULL,
  inflmat = NULL,
  opt.max = NULL
)

Arguments

Value

Data frame of thresholds and new optimal treatments with columns lo, lo.newkstar, hi, and hi.newkstar.

Calculate thresholds and invariant intervals

Description

This function calculates decision-invariant bias-adjustment thresholds and intervals for Bayesian network meta-analysis, as described by Phillippo et al. (2018). Thresholds are derived from the joint posterior, and reflect the amount of change to a data point before the treatment decision changes. Calculation is achieved using fast matrix operations.

Usage

nma_thresh(
  mean.dk,
  lhood,
  post,
  nmatype = "fixed",
  X = NULL,
  mu.design = NULL,
  delta.design = NULL,
  opt.max = TRUE,
  trt.rank = 1,
  trt.code = NULL,
  trt.sub = NULL,
  mcid = 0,
  mcid.type = "decision"
)

Arguments

Details

This function provides bias-adjustment threshold analysis for both fixed and random effects NMA models, as described by Phillippo et al. (2018). Parameters mean.dk, lhood, and post are always required, however there are differences in the specification of post and other required parameters and between the fixed and random effects cases:

FE models

The design matrix X for basic treatment parameters is required. The posterior covariance matrix specified in post should only refer to the basic treatment parameters.

RE models

The design matrix mu.design for additional parameters (e.g. covariates) is required, as is the design matrix delta.design for random effects terms. The posterior covariance matrix specified in post should refer to the basic treatment parameters, RE terms, and additional parameters in that order; i.e. post should be the posterior covariance matrix of the vector (d^T, \delta^T, \mu^T)^T.

Value

An object of class thresh.

Model details

The FE NMA model

The fixed effects NMA model is assumed to be of the form

Prior:

d \sim \mathrm{N} ( d_0, \Sigma_d )

Likelihood:

y|d \sim \mathrm{N} ( \delta, V )

FE model:

\delta = Xd + M\mu

The additional parameters \mu may be given any sensible prior; they do not affect the threshold analysis in any way.

The RE NMA model

The random effects NMA model is assumed to be of the form

Priors:

d \sim \mathrm{N} ( d_0, \Sigma_d ), \quad \mu \sim \mathrm{N} ( \mu_0, \Sigma_\mu )

Likelihood:

y|d,\mu,\tau^2 \sim \mathrm{N} ( L\delta + M\mu, V )

RE model:

\delta \sim \mathrm{N} ( Xd, \tau^2 )

The between-study heterogeneity parameter \tau^2 may be given any sensible prior. In the RE case, the threshold derivations make the approximation that \tau^2 is fixed and known.

Decision rules

The default decision rule is maximal efficacy; the optimal treatment is k^* = \mathrm{argmax}_k \mathrm{E}(d_{k}).

When \epsilon = mcid is greater than zero and mcid.type = 'decision', the decision rule is no longer for a single best treatment, but is based on minimal clinically important difference. A treatment is in the optimal set if \mathrm{E}(d_k) \ge \epsilon and \max_a \mathrm{E}(d_a) - \mathrm{E}(d_k) \le \epsilon.

When mcid.type = 'change', the maximal efficacy rule is used, but thresholds are found for when a new treatment is better than the base-case optimal by at least mcid.

See Also

recon_vcov, thresh_forest, thresh-class.

Examples

# Please see the vignette "Examples" for worked examples including use of
# this function, including more information on the brief code below.

vignette("Examples", package = "nmathresh")

### Contrast level thresholds for Thrombolytic treatments NMA
K <- 6   # Number of treatments

# Contrast design matrix is
X <- matrix(ncol = K-1, byrow = TRUE,
            c(1, 0, 0, 0, 0,
              0, 1, 0, 0, 0,
              0, 0, 1, 0, 0,
              0, 0, 0, 1, 0,
              0, -1, 1, 0, 0,
              0, -1, 0, 1, 0,
              0, -1, 0, 0, 1))

# Reconstruct hypothetical likelihood covariance matrix using NNLS
lik.cov <- recon_vcov(Thrombo.post.cov, prior.prec = .0001, X = X)

# Thresholds are then
thresh <- nma_thresh(mean.dk = Thrombo.post.summary$statistics[1:(K-1), "Mean"],
                     lhood = lik.cov,
                     post = Thrombo.post.cov,
                     nmatype = "fixed",
                     X = X,
                     opt.max = FALSE)

Reconstruct likelihood covariance matrix

Description

Reconstruct the contrast-level likelihood covariance matrix from prior and posterior covariance matrices. The resulting likelihood covariance matrix can then be used to perform a contrast-level threshold analysis with the function nma_thresh.

Usage

recon_vcov(
  post,
  prior.prec = 1e-04,
  prior.vcov = diag(1/prior.prec, dim(post)[1]),
  X = NULL,
  verbose = FALSE
)

Arguments

Details

Full details of the calculation are given by Phillippo et al. (2018). Briefly, the aim is to recover the contrast-level likelihood covariance matrix V that would have led to the posterior covariance matrix \Sigma being obtained from a fixed effects NMA, with design matrix X and prior covariance matrix \Sigma_d for a normal prior on the basic treatment parameters. This is possible in this case via the equation (resulting from conjugacy):

\Sigma^{-1} = X^TV^{-1}X + \Sigma_d^{-1}.

When the treatment network is complete (i.e. fully connected), this equation may be rearranged exactly.

When the treatment network is incomplete (i.e. not all treatments are directly compared), this equation may be solved through the use of non-negative least squares (NNLS).

When NNLS is used, some additional diagnostics are printed (and returned as attributes). Firstly, the residual sum-of-squares (RSS) from the NNLS fit. The RSS is further split into fixed RSS, from structural zeros in the reconstructed posterior according to the design matrix (and hence not fitted) that are non-zero in the true posterior, and fitted RSS, from the other fitted elements. Secondly, the Kullback-Leibler divergence of the reconstructed posterior from the true posterior. Interpreting the KL divergence as a log Bayes factor, values less than 1 indicate negligible differences between the reconstructed posterior from the true posterior, whilst values greater than 3 indicate considerable differences.

Value

A matrix; the reconstructed likelihood covariance matrix. If NNLS is used, the residual sum-of-squares and Kullback-Leibler divergence diagnostics (as printed to the console) are returned as additional attributes rss.total, rss.fixed, rss.free, kl.divergence.

See Also

nma_thresh.

Examples

# Please see the vignette "Examples" for worked examples including use of
# this function, including more information on the brief code below.

vignette("Examples", package = "nmathresh")

### Contrast level thresholds for Thrombolytic treatments NMA
K <- 6   # Number of treatments

# Contrast design matrix is
X <- matrix(ncol = K-1, byrow = TRUE,
            c(1, 0, 0, 0, 0,
              0, 1, 0, 0, 0,
              0, 0, 1, 0, 0,
              0, 0, 0, 1, 0,
              0, -1, 1, 0, 0,
              0, -1, 0, 1, 0,
              0, -1, 0, 0, 1))

# Reconstruct hypothetical likelihood covariance matrix using NNLS
lik.cov <- recon_vcov(Thrombo.post.cov, prior.prec = .0001, X = X)

The thresh class

Description

The function nmathresh returns S3 objects of class thresh.

Details

Objects of class thresh have the following components:

thresholds

A data frame with columns lo and hi for the lower and upper thresholds, and lo.newkstar and hi.newkstar for the new optimal (or rank-trt.rank) treatments at each of the thresholds.

U

The threshold solutions matrix. One column for each data point m, one row for each contrast d_{ab} (in ascending order). The elements U_{ab,m} describe the amount of adjustment to data point y_m required to reverse the relative ranking of treatments a and b. This matrix is particularly useful for deriving thresholds for more complex decisions (e.g. bias-adjustment thresholds for a new treatment entering the top two, for any change in rank of the top three, etc.)

Ukstar

The threshold solutions matrix limited to contrasts involving k^*. In other words, the rows of U corresponding to contrasts of the form d_{ak^*} or d_{k^*a}. Elements U_{ak^*,m} of this matrix describe the amount of adjustment to data point y_m required to make treatment a optimal (or rank-trt.rank) over k^*.

H

The influence matrix of the data on the basic treatment parameters. One column for each data point m, one row for each basic treatment parameter d_k. Elements H_{k,m} describe the influence of data point y_m on parameter d_k. This matrix can be used to derive more complex thresholds (e.g. 2D thresholds for simultaneous adjustments to two data points, or thresholds for common adjustments to a group of data points).

kstar

The base-case optimal (or rank-trt.rank) treatment k^*.

call

A list containing all the arguments defined in the original call to nma_thresh.

See Also

nma_thresh

Producing two-dimensional invariant regions

Description

This function produces two-dimensional threshold lines and invariant regions, as shown by Phillippo et al. (2018).

Usage

thresh_2d(
  thresh,
  idx,
  idy,
  xlab = paste("Adjustment to data point", idx),
  ylab = paste("Adjustment to data point", idy),
  xlim = NULL,
  ylim = NULL,
  breaks = waiver(),
  xbreaks = breaks,
  ybreaks = breaks,
  fill = rgb(0.72, 0.8, 0.93, 0.7),
  lwd = 1,
  fontsize = 12
)

Arguments

Value

A ggplot object containing the 2D threshold plot, which is returned invisibly and plotted (unless assigned).

Examples

# Please see the vignette "Examples" for worked examples including use of
# this function, including more information on the brief code below.

vignette("Examples", package = "nmathresh")

### Contrast level thresholds for Thrombolytic treatments NMA
K <- 6   # Number of treatments

# Contrast design matrix is
X <- matrix(ncol = K-1, byrow = TRUE,
            c(1, 0, 0, 0, 0,
              0, 1, 0, 0, 0,
              0, 0, 1, 0, 0,
              0, 0, 0, 1, 0,
              0, -1, 1, 0, 0,
              0, -1, 0, 1, 0,
              0, -1, 0, 0, 1))

# Reconstruct hypothetical likelihood covariance matrix using NNLS
lik.cov <- recon_vcov(Thrombo.post.cov, prior.prec = .0001, X = X)

# Thresholds are then
thresh <- nma_thresh(mean.dk = Thrombo.post.summary$statistics[1:(K-1), "Mean"],
                     lhood = lik.cov,
                     post = Thrombo.post.cov,
                     nmatype = "fixed",
                     X = X,
                     opt.max = FALSE)

# Produce an invariant region for simultaneous adjustments to both arms of Study 1
thresh_2d(thresh, 1, 2,
          xlab = "Adjustment in Study 1 LOR: 3 vs. 1",
          ylab = "Adjustment in Study 1 LOR: 4 vs. 1",
          xlim = c(-1.5, 0.5), ylim = c(-2, 14),
          ybreaks = seq(-2, 14, 2))

Producing threshold forest plots

Description

This function produces threshold forest plots, overlaying the decision-invariant intervals on the data points and their confidence/credible intervals, as shown by Phillippo et al. (2018).

Usage

thresh_forest(
  thresh,
  y,
  CI.lo,
  CI.hi,
  label,
  orderby = NULL,
  data = NULL,
  CI.title = "95% Confidence Interval",
  label.title = "",
  y.title = "Mean",
  II.title = "Invariant Interval",
  xlab = "",
  xlim = NULL,
  sigfig = 3,
  digits = NULL,
  refline = NULL,
  clinsig = NULL,
  cutoff = NULL,
  II.colw = rgb(0.72, 0.8, 0.93),
  II.cols = rgb(0.93, 0.72, 0.8),
  II.lwd = 8,
  CI.lwd = 1,
  pointsize = 4,
  fontsize = 12,
  xbreaks = NULL,
  add.columns = NULL,
  add.columns.title = NULL,
  add.columns.after = -1,
  add.columns.hjust = 0.5,
  add.columns.uline = TRUE,
  calcdim = display,
  display = TRUE
)

Arguments

Value

Displays the forest plot on the current plot device (if display = TRUE). Also returns invisibly the underlying gtable object, which can be further manipulated.

Examples

# Please see the vignette "Examples" for worked examples including use of
# this function, including more information on the brief code below.

vignette("Examples", package = "nmathresh")

### Contrast level thresholds for Thrombolytic treatments NMA
K <- 6   # Number of treatments

# Contrast design matrix is
X <- matrix(ncol = K-1, byrow = TRUE,
            c(1, 0, 0, 0, 0,
              0, 1, 0, 0, 0,
              0, 0, 1, 0, 0,
              0, 0, 0, 1, 0,
              0, -1, 1, 0, 0,
              0, -1, 0, 1, 0,
              0, -1, 0, 0, 1))

# Reconstruct hypothetical likelihood covariance matrix using NNLS
lik.cov <- recon_vcov(Thrombo.post.cov, prior.prec = .0001, X = X)

# Thresholds are then
thresh <- nma_thresh(mean.dk = Thrombo.post.summary$statistics[1:(K-1), "Mean"],
                     lhood = lik.cov,
                     post = Thrombo.post.cov,
                     nmatype = "fixed",
                     X = X,
                     opt.max = FALSE)

# Get treatment codes for the contrasts with data
d.a <- d.b <- vector(length = nrow(X))
for (i in 1:nrow(X)){
  d.a[i] <- ifelse(any(X[i, ] == -1), which(X[i, ] == -1), 0) + 1
  d.b[i] <- ifelse(any(X[i, ] == 1), which(X[i, ] == 1), 0) + 1
}

# Transform from d_ab style contrast references into d[i] style from the full set of contrasts
# for easy indexing in R
d.i <- d_ab2i(d.a, d.b, K = 6)

# Create plot data
plotdat <- data.frame(lab = paste0(d.b, " vs. ", d.a),
                      contr.mean = Thrombo.post.summary$statistics[d.i, "Mean"],
                      CI2.5 = Thrombo.post.summary$quantiles[d.i, "2.5%"],
                      CI97.5 = Thrombo.post.summary$quantiles[d.i, "97.5%"])

# Plot
thresh_forest(thresh, contr.mean, CI2.5, CI97.5, label = lab, data = plotdat,
              label.title = "Contrast", xlab = "Log Odds Ratio", CI.title = "95% Credible Interval",
              xlim = c(-.3, .3), refline = 0, digits = 2)