library(multinma)
options(mc.cores = parallel::detectCores())
#> For execution on a local, multicore CPU with excess RAM we recommend calling
#> options(mc.cores = parallel::detectCores())
#>
#> Attaching package: 'multinma'
#> The following objects are masked from 'package:stats':
#>
#> dgamma, pgamma, qgamma
This vignette describes the analysis of 10 trials comparing reduced
fat diets to control (non-reduced fat diets) for preventing mortality
(Hooper et al.
2000; Dias et al. 2011). The data are
available in this package as dietary_fat
:
head(dietary_fat)
#> studyn studyc trtn trtc r n E
#> 1 1 DART 1 Control 113 1015 1917.0
#> 2 1 DART 2 Reduced Fat 111 1018 1925.0
#> 3 2 London Corn/Olive 1 Control 1 26 43.6
#> 4 2 London Corn/Olive 2 Reduced Fat 5 28 41.3
#> 5 2 London Corn/Olive 2 Reduced Fat 3 26 38.0
#> 6 3 London Low Fat 1 Control 24 129 393.5
Setting up the network
We begin by setting up the network - here just a pairwise
meta-analysis. We have arm-level rate data giving the number of deaths
(r
) and the person-years at risk (E
) in each
arm, so we use the function set_agd_arm()
. We set “Control”
as the reference treatment.
diet_net <- set_agd_arm(dietary_fat,
study = studyc,
trt = trtc,
r = r,
E = E,
trt_ref = "Control",
sample_size = n)
diet_net
#> A network with 10 AgD studies (arm-based).
#>
#> ------------------------------------------------------- AgD studies (arm-based) ----
#> Study Treatment arms
#> DART 2: Control | Reduced Fat
#> London Corn/Olive 3: Control | Reduced Fat | Reduced Fat
#> London Low Fat 2: Control | Reduced Fat
#> Minnesota Coronary 2: Control | Reduced Fat
#> MRC Soya 2: Control | Reduced Fat
#> Oslo Diet-Heart 2: Control | Reduced Fat
#> STARS 2: Control | Reduced Fat
#> Sydney Diet-Heart 2: Control | Reduced Fat
#> Veterans Administration 2: Control | Reduced Fat
#> Veterans Diet & Skin CA 2: Control | Reduced Fat
#>
#> Outcome type: rate
#> ------------------------------------------------------------------------------------
#> Total number of treatments: 2
#> Total number of studies: 10
#> Reference treatment is: Control
#> Network is connected
We also specify the optional sample_size
argument,
although it is not strictly necessary here. In this case
sample_size
would only be required to produce a network
plot with nodes weighted by sample size, and a network plot is not
particularly informative for a meta-analysis of only two treatments.
(The sample_size
argument is more important when a
regression model is specified, since it also enables automatic centering
of predictors and production of predictions for studies in the network,
see ?set_agd_arm
.)
Meta-analysis models
We fit both fixed effect (FE) and random effects (RE) models.
Fixed effect meta-analysis
First, we fit a fixed effect model using the nma()
function with trt_effects = "fixed"
. We use
prior distributions for the treatment effects
and study-specific intercepts
.
We can examine the range of parameter values implied by these prior
distributions with the summary()
method:
summary(normal(scale = 100))
#> A Normal prior distribution: location = 0, scale = 100.
#> 50% of the prior density lies between -67.45 and 67.45.
#> 95% of the prior density lies between -196 and 196.
The model is fitted using the nma()
function. By
default, this will use a Poisson likelihood with a log link function,
auto-detected from the data.
diet_fit_FE <- nma(diet_net,
trt_effects = "fixed",
prior_intercept = normal(scale = 100),
prior_trt = normal(scale = 100))
Basic parameter summaries are given by the print()
method:
diet_fit_FE
#> A fixed effects NMA with a poisson likelihood (log link).
#> Inference for Stan model: poisson.
#> 4 chains, each with iter=2000; warmup=1000; thin=1;
#> post-warmup draws per chain=1000, total post-warmup draws=4000.
#>
#> mean se_mean sd 2.5% 25% 50% 75% 97.5% n_eff Rhat
#> d[Reduced Fat] -0.01 0.00 0.05 -0.11 -0.04 -0.01 0.03 0.10 3568 1
#> lp__ 5386.16 0.06 2.49 5380.38 5384.77 5386.50 5387.97 5389.91 1586 1
#>
#> Samples were drawn using NUTS(diag_e) at Mon Sep 16 13:35:08 2024.
#> For each parameter, n_eff is a crude measure of effective sample size,
#> and Rhat is the potential scale reduction factor on split chains (at
#> convergence, Rhat=1).
By default, summaries of the study-specific intercepts
are hidden, but could be examined by changing the pars
argument:
The prior and posterior distributions can be compared visually using
the plot_prior_posterior()
function:
plot_prior_posterior(diet_fit_FE)
Random effects meta-analysis
We now fit a random effects model using the nma()
function with trt_effects = "random"
. Again, we use
prior distributions for the treatment effects
and study-specific intercepts
,
and we additionally use a
prior for the heterogeneity standard deviation
.
We can examine the range of parameter values implied by these prior
distributions with the summary()
method:
summary(normal(scale = 100))
#> A Normal prior distribution: location = 0, scale = 100.
#> 50% of the prior density lies between -67.45 and 67.45.
#> 95% of the prior density lies between -196 and 196.
summary(half_normal(scale = 5))
#> A half-Normal prior distribution: location = 0, scale = 5.
#> 50% of the prior density lies between 0 and 3.37.
#> 95% of the prior density lies between 0 and 9.8.
Fitting the RE model
diet_fit_RE <- nma(diet_net,
trt_effects = "random",
prior_intercept = normal(scale = 100),
prior_trt = normal(scale = 100),
prior_het = half_normal(scale = 5))
Basic parameter summaries are given by the print()
method:
diet_fit_RE
#> A random effects NMA with a poisson likelihood (log link).
#> Inference for Stan model: poisson.
#> 4 chains, each with iter=2000; warmup=1000; thin=1;
#> post-warmup draws per chain=1000, total post-warmup draws=4000.
#>
#> mean se_mean sd 2.5% 25% 50% 75% 97.5% n_eff Rhat
#> d[Reduced Fat] -0.02 0.00 0.09 -0.19 -0.06 -0.02 0.03 0.15 1581 1
#> lp__ 5379.23 0.11 3.74 5371.41 5376.76 5379.38 5381.88 5385.92 1217 1
#> tau 0.13 0.00 0.12 0.01 0.05 0.10 0.18 0.45 819 1
#>
#> Samples were drawn using NUTS(diag_e) at Mon Sep 16 13:35:12 2024.
#> For each parameter, n_eff is a crude measure of effective sample size,
#> and Rhat is the potential scale reduction factor on split chains (at
#> convergence, Rhat=1).
By default, summaries of the study-specific intercepts
and study-specific relative effects
are hidden, but could be examined by changing the pars
argument:
The prior and posterior distributions can be compared visually using
the plot_prior_posterior()
function:
plot_prior_posterior(diet_fit_RE, prior = c("trt", "het"))
Model comparison
Model fit can be checked using the dic()
function:
(dic_FE <- dic(diet_fit_FE))
#> Residual deviance: 22.5 (on 21 data points)
#> pD: 11.3
#> DIC: 33.8
(dic_RE <- dic(diet_fit_RE))
#> Residual deviance: 21.4 (on 21 data points)
#> pD: 13.5
#> DIC: 34.9
Both models appear to fit the data well, as the residual deviance is close to the number of data points. The DIC is very similar between models, so the FE model may be preferred for parsimony.
We can also examine the residual deviance contributions with the
corresponding plot()
method.
plot(dic_FE)
plot(dic_RE)
Further results
Dias et al. (2011) produce absolute predictions of the
mortality rates on reduced fat and control diets, assuming a Normal
distribution on the baseline log rate of mortality with mean
and precision
.
We can replicate these results using the predict()
method.
The baseline
argument takes a distr()
distribution object, with which we specify the corresponding Normal
distribution. We set type = "response"
to produce predicted
rates (type = "link"
would produce predicted log
rates).
pred_FE <- predict(diet_fit_FE,
baseline = distr(qnorm, mean = -3, sd = 1.77^-0.5),
type = "response")
pred_FE
#> mean sd 2.5% 25% 50% 75% 97.5% Bulk_ESS Tail_ESS Rhat
#> pred[Control] 0.06 0.05 0.01 0.03 0.05 0.08 0.21 4142 3904 1
#> pred[Reduced Fat] 0.06 0.05 0.01 0.03 0.05 0.08 0.21 4118 3909 1
plot(pred_FE)
pred_RE <- predict(diet_fit_RE,
baseline = distr(qnorm, mean = -3, sd = 1.77^-0.5),
type = "response")
pred_RE
#> mean sd 2.5% 25% 50% 75% 97.5% Bulk_ESS Tail_ESS Rhat
#> pred[Control] 0.07 0.06 0.01 0.03 0.05 0.08 0.22 4379 3576 1
#> pred[Reduced Fat] 0.07 0.06 0.01 0.03 0.05 0.08 0.21 4394 3531 1
plot(pred_RE)
If the baseline
argument is omitted, predicted rates
will be produced for every study in the network based on their estimated
baseline log rate
:
pred_FE_studies <- predict(diet_fit_FE, type = "response")
pred_FE_studies
#> ------------------------------------------------------------------- Study: DART ----
#>
#> mean sd 2.5% 25% 50% 75% 97.5% Bulk_ESS Tail_ESS Rhat
#> pred[DART: Control] 0.06 0 0.05 0.06 0.06 0.06 0.07 6187 3190 1
#> pred[DART: Reduced Fat] 0.06 0 0.05 0.06 0.06 0.06 0.07 7818 3376 1
#>
#> ------------------------------------------------------ Study: London Corn/Olive ----
#>
#> mean sd 2.5% 25% 50% 75% 97.5% Bulk_ESS Tail_ESS
#> pred[London Corn/Olive: Control] 0.07 0.03 0.03 0.06 0.07 0.09 0.13 8422 3035
#> pred[London Corn/Olive: Reduced Fat] 0.07 0.02 0.03 0.06 0.07 0.09 0.13 8512 3110
#> Rhat
#> pred[London Corn/Olive: Control] 1
#> pred[London Corn/Olive: Reduced Fat] 1
#>
#> --------------------------------------------------------- Study: London Low Fat ----
#>
#> mean sd 2.5% 25% 50% 75% 97.5% Bulk_ESS Tail_ESS Rhat
#> pred[London Low Fat: Control] 0.06 0.01 0.04 0.05 0.06 0.06 0.08 7697 2749 1
#> pred[London Low Fat: Reduced Fat] 0.06 0.01 0.04 0.05 0.06 0.06 0.08 8588 2715 1
#>
#> ----------------------------------------------------- Study: Minnesota Coronary ----
#>
#> mean sd 2.5% 25% 50% 75% 97.5% Bulk_ESS Tail_ESS Rhat
#> pred[Minnesota Coronary: Control] 0.05 0 0.05 0.05 0.05 0.06 0.06 5035 3586 1
#> pred[Minnesota Coronary: Reduced Fat] 0.05 0 0.05 0.05 0.05 0.06 0.06 6263 3698 1
#>
#> --------------------------------------------------------------- Study: MRC Soya ----
#>
#> mean sd 2.5% 25% 50% 75% 97.5% Bulk_ESS Tail_ESS Rhat
#> pred[MRC Soya: Control] 0.04 0.01 0.03 0.04 0.04 0.04 0.05 7530 2448 1
#> pred[MRC Soya: Reduced Fat] 0.04 0.01 0.03 0.04 0.04 0.04 0.05 8515 2983 1
#>
#> -------------------------------------------------------- Study: Oslo Diet-Heart ----
#>
#> mean sd 2.5% 25% 50% 75% 97.5% Bulk_ESS Tail_ESS Rhat
#> pred[Oslo Diet-Heart: Control] 0.06 0.01 0.05 0.06 0.06 0.07 0.08 6664 3113 1
#> pred[Oslo Diet-Heart: Reduced Fat] 0.06 0.01 0.05 0.06 0.06 0.07 0.08 8209 3279 1
#>
#> ------------------------------------------------------------------ Study: STARS ----
#>
#> mean sd 2.5% 25% 50% 75% 97.5% Bulk_ESS Tail_ESS Rhat
#> pred[STARS: Control] 0.02 0.01 0.01 0.01 0.02 0.03 0.05 6720 2887 1
#> pred[STARS: Reduced Fat] 0.02 0.01 0.01 0.01 0.02 0.03 0.05 6802 2867 1
#>
#> ------------------------------------------------------ Study: Sydney Diet-Heart ----
#>
#> mean sd 2.5% 25% 50% 75% 97.5% Bulk_ESS Tail_ESS Rhat
#> pred[Sydney Diet-Heart: Control] 0.03 0 0.03 0.03 0.03 0.04 0.04 7106 2707 1
#> pred[Sydney Diet-Heart: Reduced Fat] 0.03 0 0.03 0.03 0.03 0.04 0.04 7635 2816 1
#>
#> ------------------------------------------------ Study: Veterans Administration ----
#>
#> mean sd 2.5% 25% 50% 75% 97.5% Bulk_ESS
#> pred[Veterans Administration: Control] 0.11 0.01 0.1 0.11 0.11 0.12 0.13 5091
#> pred[Veterans Administration: Reduced Fat] 0.11 0.01 0.1 0.11 0.11 0.12 0.12 6910
#> Tail_ESS Rhat
#> pred[Veterans Administration: Control] 3139 1
#> pred[Veterans Administration: Reduced Fat] 3250 1
#>
#> ------------------------------------------------ Study: Veterans Diet & Skin CA ----
#>
#> mean sd 2.5% 25% 50% 75% 97.5% Bulk_ESS
#> pred[Veterans Diet & Skin CA: Control] 0.01 0.01 0 0.01 0.01 0.02 0.03 5371
#> pred[Veterans Diet & Skin CA: Reduced Fat] 0.01 0.01 0 0.01 0.01 0.02 0.03 5342
#> Tail_ESS Rhat
#> pred[Veterans Diet & Skin CA: Control] 2466 1
#> pred[Veterans Diet & Skin CA: Reduced Fat] 2556 1
plot(pred_FE_studies) + ggplot2::facet_grid(Study~., labeller = ggplot2::label_wrap_gen(width = 10))