# Example: Dietary fat

    library(multinma)
    options(mc.cores = parallel::detectCores())

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 \(\mathrm{N}(0, 100^2)\) prior distributions for the treatment effects \(d_k\) and study-specific intercepts \(\mu_j\). 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  3321    1
    #> lp__           5386.20    0.06 2.36 5380.63 5384.86 5386.53 5387.89 5389.89  1707    1
    #>
    #> Samples were drawn using NUTS(diag_e) at Mon Apr 29 16:38:34 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 \(\mu_j\) are hidden, but could be examined by changing the `pars` argument:

    # Not run
    print(diet_fit_FE, pars = c("d", "mu"))

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 \(\mathrm{N}(0, 100^2)\) prior distributions for the treatment effects \(d_k\) and study-specific intercepts \(\mu_j\), and we additionally use a \(\textrm{half-N}(5^2)\) prior for the heterogeneity standard deviation \(\tau\). 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 0.08   -0.19   -0.06   -0.01    0.03    0.16  2266 1.00
    #> lp__           5379.04    0.12 3.95 5370.59 5376.56 5379.32 5381.81 5386.02  1136 1.00
    #> tau               0.13    0.00 0.11    0.00    0.04    0.10    0.17    0.40   941 1.01
    #>
    #> Samples were drawn using NUTS(diag_e) at Mon Apr 29 16:38:45 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 \(\mu_j\) and study-specific relative effects \(\delta_{jk}\) are hidden, but could be examined by changing the `pars` argument:

    # Not run
    print(diet_fit_RE, pars = c("d", "mu", "delta"))

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: 22.4 (on 21 data points)
    #>                pD: 11.2
    #>               DIC: 33.6

    (dic_RE <- dic(diet_fit_RE))
    #> Residual deviance: 21.4 (on 21 data points)
    #>                pD: 13.6
    #>               DIC: 35

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 \(-3\) and precision \(1.77\). 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]     0.07 0.06 0.01 0.03 0.05 0.09  0.23     3908     4103    1
    #> pred[Reduced Fat] 0.07 0.06 0.01 0.03 0.05 0.09  0.23     3902     4057    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     4015     3994    1
    #> pred[Reduced Fat] 0.07 0.06 0.01 0.03 0.05 0.08  0.22     4075     3951    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 \(\mu_j\):

    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     5693     3013    1
    #> pred[DART: Reduced Fat] 0.06  0 0.05 0.06 0.06 0.06  0.07     7988     3522    1
    #>
    #> ------------------------------------------------------ Study: London Corn/Olive ----
    #>
    #>                                      mean   sd 2.5%  25%  50%  75% 97.5% Bulk_ESS Tail_ESS Rhat
    #> pred[London Corn/Olive: Control]     0.07 0.02 0.03 0.06 0.07 0.09  0.13     6515     2769    1
    #> pred[London Corn/Olive: Reduced Fat] 0.07 0.02 0.03 0.06 0.07 0.09  0.13     6512     2579    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     6725     3040    1
    #> pred[London Low Fat: Reduced Fat] 0.06 0.01 0.04 0.05 0.06 0.06  0.08     7136     3050    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     5351     3247    1
    #> pred[Minnesota Coronary: Reduced Fat] 0.05  0 0.05 0.05 0.05 0.06  0.06     6656     3408    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     7366     2745    1
    #> pred[MRC Soya: Reduced Fat] 0.04 0.01 0.03 0.04 0.04 0.04  0.05     7857     2971    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     6532     3558    1
    #> pred[Oslo Diet-Heart: Reduced Fat] 0.06 0.01 0.05 0.06 0.06 0.07  0.08     7753     3581    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     5497     2633    1
    #> pred[STARS: Reduced Fat] 0.02 0.01 0.01 0.01 0.02 0.03  0.05     5608     2441    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     6777     3053    1
    #> pred[Sydney Diet-Heart: Reduced Fat] 0.03  0 0.03 0.03 0.03 0.04  0.04     7580     3048    1
    #>
    #> ------------------------------------------------ Study: Veterans Administration ----
    #>
    #>                                            mean   sd 2.5%  25%  50%  75% 97.5% Bulk_ESS Tail_ESS
    #> pred[Veterans Administration: Control]     0.11 0.01  0.1 0.11 0.11 0.12  0.13     5754     3391
    #> pred[Veterans Administration: Reduced Fat] 0.11 0.01  0.1 0.11 0.11 0.12  0.13     7228     3335
    #>                                            Rhat
    #> pred[Veterans Administration: Control]        1
    #> pred[Veterans Administration: Reduced Fat]    1
    #>
    #> ------------------------------------------------ Study: Veterans Diet & Skin CA ----
    #>
    #>                                            mean   sd 2.5%  25%  50%  75% 97.5% Bulk_ESS Tail_ESS
    #> pred[Veterans Diet & Skin CA: Control]     0.01 0.01    0 0.01 0.01 0.02  0.03     5450     2838
    #> pred[Veterans Diet & Skin CA: Reduced Fat] 0.01 0.01    0 0.01 0.01 0.02  0.03     5564     3002
    #>                                            Rhat
    #> pred[Veterans Diet & Skin CA: Control]        1
    #> pred[Veterans Diet & Skin CA: Reduced Fat]    1
    plot(pred_FE_studies) + ggplot2::facet_grid(Study~., labeller = ggplot2::label_wrap_gen(width = 10))

## References

Dias, S., N. J. Welton, A. J. Sutton, and A. E. Ades. 2011. “NICE DSU Technical Support Document 2: A Generalised Linear Modelling Framework for Pair-Wise and Network Meta-Analysis of Randomised Controlled Trials.” National Institute for Health and Care Excellence. https://www.sheffield.ac.uk/nice-dsu.
Hooper, L., C. D. Summerbell, J. P. T. Higgins, R. L. Thompson, G. Clements, N. Capps, G. Davey Smith, R. Riemersma, and S. Ebrahim. 2000. “Reduced or Modified Dietary Fat for Preventing Cardiovascular Disease.” Cochrane Database of Systematic Reviews, no. 2. https://doi.org/10.1002/14651858.CD002137.