I understand how Gibbs sampling works to allow estimations of joint probability distributions in a simple sense. For example, using a simple two dimensional example (A and B are two different tennis players whose results are not independent, and 0=losing, 1 = winning), you might start with the following conditional probabilities:
P(A=0| B=0) = 0.5,
P(A=0| B=1) = 0.125
P(B=0|A=0) = 0.5
P(B=0| A =1) = 0.125
To get things moving, you would then suppose a starting value for A (let it be 0). We can then use that value of A (A=0) for the next iteration for B, where we therefore look at P(B=0|A=0). As shown above, there is a 0.5 probability that B=0 in that case. Let's run a random number (0-1) generator, which happens to yield 0.67. As this is greater than the probability of 0.5, we take B=1 (using the rule that if the random number is lower than the conditional probability, 0 is yielded, and if the random number is greater than the conditional probability, 1 is yielded). This gives us the first pair of joint values: A=0 and B=1 [which can be written as 0,1]. We now run the next iteration for A using that last value of B (B=1). P(A=0|B=1) is 0.125. The random number generator yields 0.28, so we take A =1. We then look at P(B=0|A=1) [as the last value of A yielded was 1], which is 0.125. The random number generator yields 0.34, which means we take B to be 1. So our second pair of values is: A=1, B=1 [or, 1,1]. We can repeat this process for a very large number of iterations, and if we then count the numbers of paired values that are 0,0; 0,1; 1,0; and 1,1, we should be able to estimate the joint probability distribution. I have attached a simple excel program that carries out such a Gibbs sampling. It can be seen that the estimation of the joint probability distribution is very close to the actual joint probability distribution from which the conditional probabilities were calculated (in practice, of course you wouldn't have access to the true joint probabilities as then you'd have no reason to do the Gibbs sampling). I have largely based my example on an excellent YouTube video by Ben Lambert.
However, this is where I need advice and help. I do not understand how the above example relates to network meta-analyses (NMAs). For example, imagine a network meta-analysis of three treatments A, B and C. How do the data from these studies relate to conditional probabilities? For example, if the odds ratio of outcome X is 0.5 for the comparison of A vs B, the odds ratio of outcome X is 0.2 for the comparison of B vs C, and the odds ratio of outcome X is 0.1 for the comparison of A versus C (clearly no incoherence here!), how do we proceed? I have a vague idea that we could use the odds ratios to calculate conditional probabilities, but can't quite grasp exactly what should happen. I have looked at most of the relevant documents (like the DSU document TSD2) but these don't explain exactly what occurs in the Gibbs sampling itself. Can anyone describe, in simple terms, how the sampling would proceed in an NMA, in relation to the model of Gibbs sampling I have given earlier?
Thank you Rana! That was extremely helpful. However I still lack a concrete understanding of what actually happens during the Gibbs sampling. Can we imagine we have a very simple model of treatments A vs B vs C that form a loop, and the data (in the form of risks) are as follows (for the simple binary outcome of patient satisfaction)?
Study 1: A 20/100, B 30/100; RR [A vs B]: 0.66
Study 2: A 25/75, C 50/75; RR [A vs C]: 0.5
Study 3: B 40/60, C 55/60; RR [B vs C]: 0.73
This is clearly a very coherent model (the indirect estimation of B vs C , based on the data from study 1 and 2, would be a RR of 0.75, which is very close to the direct estimate given above). Using these simplified data, would it be possible to show me a limited number of iterations (just one or two should be enough to give me a concrete understanding) through the Gibbs sampler, showing how the sampler uses previous iterations to gradually generate a coherent set of parameter estimates?
NMA is a statistical technique used to compare and rank multiple interventions in a network of studies. Gibbs sampling is one of the methods used to perform NMA. Here's a simplified explanation of how network meta-analyses can be performed using Gibbs sampling:
1. Data collection: Gather relevant studies that compare different interventions of interest. Extract relevant outcome data such as mean values, standard deviations, or odds ratios.
2. Define the network structure: Create a network diagram showing the relationships between the interventions based on the available evidence. Each node in the network represents an intervention, and the lines connecting the nodes represent direct comparisons between interventions.
3. Model specification: Specify a statistical model for the network meta-analysis. This typically involves defining prior distributions for the treatment effects and other parameters of interest. The choice of the model depends on the nature of the outcome data (continuous, binary, or time-to-event), the assumptions made, and the complexity of the network.
4. Markov Chain Monte Carlo (MCMC) simulation: Gibbs sampling is a type of MCMC algorithm used to estimate the parameters of the statistical model. The algorithm iteratively samples values for each parameter from its conditional distribution given the current values of the other parameters. In the context of NMA, Gibbs sampling allows for the estimation of treatment effects and other model parameters.
5. Burn-in period: Discard the initial iterations of the Gibbs sampling algorithm, known as the burn-in period, to allow the Markov chain to converge to the target distribution. The length of the burn-in period depends on the convergence properties of the chain, which can be assessed using diagnostic tools.
6. Sampling iterations: After the burn-in period, continue the Gibbs sampling algorithm to generate a sufficient number of iterations. Each iteration produces a sample of parameter values from the joint posterior distribution.
7. Posterior inference: Once a suitable number of samples have been obtained, conduct posterior inference to summarize the results. This involves calculating summary statistics such as means, medians, or credible intervals for the treatment effects of interest.
8. Model assessment: Assess the goodness of fit of the model to the data and evaluate the assumptions made. Various diagnostic tools and sensitivity analyses can be used to evaluate the robustness and reliability of the results.
9. Reporting and interpretation: Finally, report the results of the network meta-analysis, including the estimated treatment effects, ranking probabilities, and any other relevant findings. Interpret the results in light of the study objectives and the limitations of the analysis.
It is complex and requires expertise in Bayesian statistics and computational methods. Specialized software packages, such as WinBUGS, JAGS, or Stan, are often used to implement the Gibbs sampling algorithm for NMA.
Good luck
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