Finding the best model for the Case variable

We have two types of variables under consideration:

For a binary variable, I will consider two types of phylogenetic model, the continuous time Markov Chain (CTMC), and models with a continuous variable plus a Bernoulli draw with the logistic transformation of the latent variable as probability.

Additionally, I will consider a non-phylogenetic model for each variable.

For all models, Bayesian inference will be used, implemented in Stan.

preparation

Im my experience, R is the best environment for working with Stan.

I start with loading relevant packages.

library(repr)
options(repr.plot.width = 12, repr.plot.height = 10)
library(tidyverse)
library(gridExtra)
library(patchwork)
library(stringr)
library(reshape2)
library(rstan)
library(bridgesampling)
library(loo)
library(ape)
library(bayesplot)
library(shinystan)
library(coda)
library(ggtree)

options(mc.cores = parallel::detectCores())
rstan_options("auto_write" = TRUE)
karminrot <- rgb(206/255, 143/255, 137/255)
gold <- rgb(217/255, 205/255, 177/255)
anthrazit <- rgb(143/255, 143/255, 149/255)

Next, the data are loaded into a tibble called d.

d <- read_csv("../data/data_with_counts.csv")
d$Glottocode[d$Glottocode == "kose1239"] <- "awiy1238"
taxa <- d$Glottocode
head(d)
Rows: 827 Columns: 13
── Column specification ────────────────────────────────────────────────────────
Delimiter: ","
chr (5): ISO, Glottocode, Language, Family, Area
dbl (8): N_Speakers, Case, AP_Entropy, VerbFinal, VerbMiddle, v1, vm, vf

ℹ Use `spec()` to retrieve the full column specification for this data.
ℹ Specify the column types or set `show_col_types = FALSE` to quiet this message.
A tibble: 6 × 13
ISO Glottocode Language Family Area N_Speakers Case AP_Entropy VerbFinal VerbMiddle v1 vm vf
<chr> <chr> <chr> <chr> <chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
aai arif1239 Arifama-Miniafia Austronesian Oceania 3470 0 0.7510324 0.8172043 0.15053763 3 14 76
aak anka1246 Ankave Angan Oceania 1500 NA 0.9549905 0.6121212 0.30303030 14 50 101
aau abau1245 Abau Sepik Oceania 7500 1 0.6152539 0.9565217 0.02173913 1 1 44
abt ambu1247 Ambulas Ndu Oceania 33000 1 0.8478617 0.6666667 0.29411765 2 15 34
aby anem1248 Aneme Wake Yareban Oceania 650 1 0.8426579 0.6562500 0.22916667 11 22 63
acd giky1238 Gikyode Atlantic-Congo Africa 10400 0 0.5916728 0.1904762 0.73809524 3 31 8

In the next step, the maximum clade credibility trees are loaded as phylo objects.

mcc_trees <- list.files("../data/mcc_trees", pattern = ".tre", full.names = TRUE)

trees <- purrr::map(mcc_trees, read.tree)

# get a vector of tip labels of all trees in trees

tips <- purrr::map(trees, function(x) x$tip.label) %>% unlist %>% unique

A brief sanity check to ensure that all tips of the trees correspond to a Glottocode in d.

if (length(setdiff(tips, taxa)) == 0 ) {
    print("All tips in trees are in taxa")
} else {
    print("Not all tips in trees are in taxa")
}
[1] "All tips in trees are in taxa"

All Glottocodes in d that do not occur as a tip are isolates (for the purpose of this study).

For each isolate I create a list object with the fields "tip.lable" and "Nnode", to make them compatible with the phylo objects for language families. These list objects are added to the list trees.

isolates <- setdiff(taxa, tips)

# convert isolates to one-node trees and add to trees
for (i in isolates) {
    tr <- list()
    tr[["tip.label"]] <- i
    tr[["Nnode"]] <- 0
    trees[[length(trees) + 1]] <- tr
}

Just to make sure, I test whether the tips of this forest are exactly the Glottocodes in the data.

tips <- c()
for (tr in trees) {
    tips <- c(tips, tr$tip.label)
}

if (all(sort(tips) == sort(taxa))) {
    print("taxa and tips are identical")
} else {
    print("taxa and tips are not identical")
}
[1] "taxa and tips are identical"

Now I re-order the rows in d to make sure they are in the same order as tips.

d <- d[match(tips, d$Glottocode), ]

The trees in trees form a forest, i.e., a disconnected directed acyclic graph. I want to construct a representation of this graph which I can pass to Stan, consisting of

  • the number of nodes N,
  • a Nx2-matrix edges, and
  • a length-N vector of edge lengths, edge_lengths.
  • a vector of root nodes
  • a vector of tip nodes

First I create an Nx4 matrix where each row represents one node. The columns are

  • node number
  • tree number
  • tree-internal node number
  • label

For internal nodes, the label is the empty string.

node_matrix <- c()
node_number <- 1
for (tn in 1:length(trees)) {
    tr <- trees[[tn]]
    tr_nodes <- length(tr$tip.label) + tr$Nnode
    for (i in 1:tr_nodes) {
        tl <- ""
        if (i <= length(tr$tip.label)) {
            tl <- tr$tip.label[i]
        }
        node_matrix <- rbind(node_matrix, c(node_number, tn, i, tl))
        node_number <- node_number+1
    }
}

I convert the matrix to a tibble.

nodes <- tibble(
    node_number = as.numeric(node_matrix[,1]),
    tree_number = as.numeric(node_matrix[,2]),
    internal_number = as.numeric(node_matrix[,3]),
    label = node_matrix[,4]
)
head(nodes)
tail(nodes)
A tibble: 6 × 4
node_number tree_number internal_number label
<dbl> <dbl> <dbl> <chr>
1 1 1 aman1265
2 1 2 wari1266
3 1 3
4 2 1 bora1263
5 2 2 muin1242
6 2 3
A tibble: 6 × 4
node_number tree_number internal_number label
<dbl> <dbl> <dbl> <chr>
1549 95 1 uduk1239
1550 96 1 wiru1244
1551 97 1 kamu1260
1552 98 1 yagu1244
1553 99 1 kark1258
1554 100 1 nucl1454

Next I define a function that maps a tree number and an internal node number to the global node number.

get_global_id <- function(tree_number, local_id) which((nodes$tree_number == tree_number) & (nodes$internal_number == local_id))

Now I create a matrix of edges and, concomittantly, a vector of edge lengths.

edges <- c()
edge_lengths <- c()
for (tn in 1:length(trees)) {
    tr <- trees[[tn]]
    if (length(tr$edge) > 0 ) {
        for (en in 1:nrow(tr$edge)) {
            mother <- tr$edge[en, 1]
            daughter <- tr$edge[en, 2]
            mother_id <- get_global_id(tn, mother)
            daughter_id <- get_global_id(tn, daughter)
            el <- tr$edge.length[en]
            edges <- rbind(edges, c(mother_id, daughter_id))
            edge_lengths <- c(edge_lengths, el)
        }
    }
}

Finally I identify the root nodes and tip nodes.

roots <- c(sort(setdiff(edges[,1], edges[,2])), match(isolates, nodes$label))
tips <- which(nodes$label != "")

Next I define a function computing the post-order.

get_postorder <- function(roots, edges) {
    input <- roots
    output <- c()
    while (length(input) > 0) {
        pivot <- input[1]
        daughters <- setdiff(edges[edges[,1] == pivot, 2], output)
        if (length(daughters) == 0) {
            input <- input[-1]
            output <- c(output, pivot)
        } else {
            input <- c(daughters, input)
        }
    }
    return(output)
}

Case

Non-phylogenetic model

The non-phylogenetic model for a binary variable is straightforward:

\[ \begin{aligned} y_i &\sim \mathrm{Bernoulli}(p)\\ p &\sim \mathrm{Beta}(1,1) \end{aligned} \]

Implementing it in Stan is also straightforward.

stan_code <- "
data {
    int<lower=1> Ndata;
    int<lower=0, upper=1> Y[Ndata];
}

parameters {
    real p;
}

model {
    target += beta_lpdf(p | 1, 1);
    for (i in 1:Ndata) {
        target += bernoulli_lpmf(Y[i] | p);
    }
}

generated quantities {
    // Posterior predictive checks
    int Y_posterior[Ndata];
    real log_lik[Ndata];

    // Prior predictive checks
    int Y_prior[Ndata];

    for (i in 1:Ndata) {
        // Generating new data based on the posterior distribution of p
        Y_posterior[i] = bernoulli_rng(p);

        // Generating new data based on the prior distribution of p
        Y_prior[i] = bernoulli_rng(beta_rng(1, 1));
        log_lik[i] = bernoulli_lpmf(Y[i] | p);
    }
}
"
base_data <- list()
base_data$Ndata <- sum(!is.na(d$Case))
base_data$Y <- filter(d, !is.na(Case))$Case
model_logistic <- stan_model(model_code = stan_code)
fit_logistic_case <- rstan::sampling(
    model_logistic,
    data = base_data,
    chains = 4,
    iter = 10000,
    thin = 10
)
print(fit_logistic_case, pars = c("p", "lp__"))
Inference for Stan model: anon_model.
4 chains, each with iter=10000; warmup=5000; thin=10; 
post-warmup draws per chain=500, total post-warmup draws=2000.

        mean se_mean   sd    2.5%     25%     50%    75%   97.5% n_eff Rhat
p       0.39    0.00 0.02    0.35    0.37    0.39    0.4    0.42  1753    1
lp__ -462.45    0.02 0.71 -464.74 -462.62 -462.17 -462.0 -461.95  1944    1

Samples were drawn using NUTS(diag_e) at Mon Mar 18 11:29:03 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).
(marginal_logistic_case <- bridge_sampler(fit_logistic_case))
Warning message:
“effective sample size cannot be calculated, has been replaced by number of samples.”
Iteration: 1
Iteration: 2
Iteration: 3
Iteration: 4
Bridge sampling estimate of the log marginal likelihood: -465.0188
Estimate obtained in 4 iteration(s) via method "normal".
log_lik <- extract_log_lik(fit_logistic_case, parameter_name = "log_lik", merge_chains = FALSE)
reff <- relative_eff(exp(log_lik))
(loo_logistic <- loo(log_lik, r_eff=reff))

Computed from 2000 by 693 log-likelihood matrix

         Estimate   SE
elpd_loo   -463.0  6.0
p_loo         1.0  0.0
looic       925.9 12.0
------
Monte Carlo SE of elpd_loo is 0.0.

All Pareto k estimates are good (k < 0.5).
See help('pareto-k-diagnostic') for details.

Predictive checks

Following Jäger & Wahle (2021), I define three statistics to be used in prior and posterior predictive checks:

  • total variance (tv)
  • mean lineage-wise variance (lv)
  • cross-family variance (cv)

Here are the corresponding R functions.

total_variance <- function(Y, d=d) var(Y)
lineage_wise_variance <- function(Y)
    d %>% 
        filter(!is.na(Case)) %>%
        mutate(Y = Y) %>%
        group_by(Family) %>%
        summarize(variance = var(Y, na.rm = TRUE)) %>%
        summarize(mean_variance = mean(variance, na.rm = TRUE)) %>%
        pull
crossfamily_variance <- function(Y)
    d %>% 
        filter(!is.na(Case)) %>%
        mutate(Y = Y) %>%
        group_by(Family) %>%
        summarize(centroids = mean(Y, na.rm = TRUE)) %>%
        pull(centroids) %>%
        var

First we get the empirical values.

Y <- base_data$Y
empirical <- tibble(
    tv = total_variance(Y),
    lv = lineage_wise_variance(Y),
    cv = crossfamily_variance(Y)
)

Thanks to the generated quantities block in the Stan programm, fitting the model simultaneously produced prior and posterior predictive samples.

# Extract the generated quantities
generated_quantities <- extract(fit_logistic_case)
prior_pc <- tibble(
    tv = apply(generated_quantities$Y_prior, 1, total_variance),
    lv = apply(generated_quantities$Y_prior, 1, lineage_wise_variance),
    cv = apply(generated_quantities$Y_prior, 1, crossfamily_variance)
)
predictive_plot <- function(prior_pc) {
    p1 <- prior_pc %>%
      mutate(
          x_jittered = 1 + rnorm(n(), mean = 0, sd = 0.01),
          y_jittered = tv + rnorm(n(), mean = 0, sd = sd(prior_pc$tv)/10)
      ) %>%
      ggplot(aes(x = x_jittered, y = tv)) +
      geom_boxplot(aes(x = factor(1), y = tv), width = 0.5, fill=karminrot) +
      geom_point(alpha = 0.5, size = 0.1, position = position_identity()) +
      theme_grey(base_size = 16) +
      theme(
        axis.text.x = element_blank(),
        axis.ticks.x = element_blank(),
        axis.title.x = element_blank(),
        axis.title.y = element_text(size = 18),
        plot.title = element_text(size = 20, hjust = 0.5)
      ) +
      labs(title = "total variance", y = "") +
      geom_point(aes(x=1, y=empirical$tv, color="empirical"), size=5, fill='red') +
      guides(color = "none")

    p2 <- prior_pc %>%
      mutate(
          x_jittered = 1 + rnorm(n(), mean = 0, sd = 0.01),
          y_jittered = lv + rnorm(n(), mean = 0, sd = sd(prior_pc$lv)/10)
      ) %>%
      ggplot(aes(x = x_jittered, y = lv)) +
      geom_boxplot(aes(x = factor(1), y = lv), width = 0.5, fill=gold) +
      geom_point(alpha = 0.5, size = 0.1, position = position_identity()) +
      theme_grey(base_size = 16) +
      theme(
        axis.text.x = element_blank(),
        axis.ticks.x = element_blank(),
        axis.title.x = element_blank(),
        axis.title.y = element_text(size = 18),
        plot.title = element_text(size = 20, hjust = 0.5)
      ) +
      labs(title = "lineage-wise variance", y = "") +
      geom_point(aes(x=1, y=empirical$lv, color="empirical"), size=5, fill='red') +
      guides(color = "none")

    p3 <- prior_pc %>%
      mutate(
          x_jittered = 1 + rnorm(n(), mean = 0, sd = 0.01),
          y_jittered = cv + rnorm(n(), mean = 0, sd = sd(prior_pc$cv)/10)
      ) %>%
      ggplot(aes(x = x_jittered, y = cv)) +
      geom_boxplot(aes(x = factor(1), y = cv), width = 0.5, fill=anthrazit) +
      geom_point(alpha = 0.5, size = 0.1, position = position_identity()) +
      theme_grey(base_size = 16) +
      theme(
        axis.text.x = element_blank(),
        axis.ticks.x = element_blank(),
        axis.title.x = element_blank(),
        axis.title.y = element_text(size = 18),
        plot.title = element_text(size = 20, hjust = 0.5)
      ) +
      labs(title = "cross-lineage variance", y = "") +
      geom_point(aes(x=1, y=empirical$cv, color="empirical"), size=5, fill='red') +
      labs(color = "") +
      guides(color = "none")

    return(grid.arrange(p1, p2, p3, ncol=3))
}
predictive_plot(prior_pc)
Warning message in geom_point(aes(x = 1, y = empirical$tv, color = "empirical"), :
“All aesthetics have length 1, but the data has 2000 rows.
ℹ Did you mean to use `annotate()`?”
Warning message in geom_point(aes(x = 1, y = empirical$lv, color = "empirical"), :
“All aesthetics have length 1, but the data has 2000 rows.
ℹ Did you mean to use `annotate()`?”
Warning message in geom_point(aes(x = 1, y = empirical$cv, color = "empirical"), :
“All aesthetics have length 1, but the data has 2000 rows.
ℹ Did you mean to use `annotate()`?”

We see that the prior does not really cover the empirical values. Here is what happens after the model is fitted:

posterior_pc <- tibble(
    tv = apply(generated_quantities$Y_posterior, 1, total_variance),
    lv = apply(generated_quantities$Y_posterior, 1, lineage_wise_variance),
    cv = apply(generated_quantities$Y_posterior, 1, crossfamily_variance)
)


predictive_plot(posterior_pc)
Warning message in geom_point(aes(x = 1, y = empirical$tv, color = "empirical"), :
“All aesthetics have length 1, but the data has 2000 rows.
ℹ Did you mean to use `annotate()`?”
Warning message in geom_point(aes(x = 1, y = empirical$lv, color = "empirical"), :
“All aesthetics have length 1, but the data has 2000 rows.
ℹ Did you mean to use `annotate()`?”
Warning message in geom_point(aes(x = 1, y = empirical$cv, color = "empirical"), :
“All aesthetics have length 1, but the data has 2000 rows.
ℹ Did you mean to use `annotate()`?”

The fitted model still over-estimates the lineage-wise variance and under-estimates the cross-lineage variance.

CTMC

In the CTMC model, we have the parameters

  • \(\pi = \langle \pi_0, \pi_1\rangle\): equilibrium probabilities of states \(0\) and \(1\), and
  • \(r\): mutation rate.

The state of the root node is drawn from \(\pi\).

The \(Q\)-matrix of the CTMC is given as

\[ Q = r\begin{pmatrix} -\pi_1 & \pi_1 \\ \pi_0 & -\pi_0 \end{pmatrix} \]

The transition probabilites after time \(t\) are then given by the \(P\)-matrix

\[ P(t) = e^{tQ} = \begin{pmatrix} \pi_0 + \pi_1 e^{-rt} & \pi_1 - \pi_1 e^{-rt}\\ \pi_0 - \pi_0 e^{-rt} & \pi_1 + \pi_0 e^{-rt} \end{pmatrix} \]

In a phylogeny, the state of a daughter node is drawn from the distribution \(p(t)_i\), where \(t\) is the branch length between mother node and daughter node, and \(i\) is the state of the mother node.

The likelihood of a vector of tip states, given the parameters and the phylogeny, is then the sum of the likelihoods of all histories generated by the process and consistent with the observations.

The likelihood of a forest is the product of the likelihoods of the trees in the forest.

It is not possible to implement this directly in Stan because the states at the internal nodes are drawn from discrete distributions, and Stan can only deal with continuous latent variables. However, it is possible to marginalize out the internal-node states. This is what Felsenstein’s Pruning Algorithm does.

We have a matrix \(N\times 2\) matrix \(ll\) (for log-likelihood), where \(N\) is the number of nodes in the tree. Let \(n_i (1\leq i\leq N)\) be a post-order of the nodes. Then \(ll\) is recursively defined as follows:

  • If \(i\) is a tip with state \(s\), then \[ \begin{aligned} ll_{i,s} &= 0\\ ll_{i,1-s}&= -\infty \end{aligned} \]

  • If \(i\) is an internal node, let \(j\), \(k\) be its daughters, with \(t_j, t_k\) the corresponding branch lengths.

\[ \begin{aligned} ll_{i, s} &= \sum_{\gamma\in\{j,k\}}\log(\sum_{\alpha\in \{0,1\}} e^{ll_{j,0}} p(t_\gamma)_{s,\alpha} ) \end{aligned} \]

The total likelihood of a tree is then given by

\[ \mathcal L = \pi_0 e^{ll_{1, 0}} + \pi_1 e^{ll_{1, 1}} \]

The likelihood of a forest is the product of the tree likelihoods.

Let us implement this in Stan.

First I implement a function taking to integers \(i, n\) and returns the one-hot vector of length \(n\) with \(1\) in the \(i\)th position.

row_vector onehot(int i, int n) {
    row_vector[n] oh = to_row_vector(rep_vector(0, n));  // initialize a zero vector of length n
    oh[i] = 1;  // set the i-th element to 1
    return oh;
}

Now I initialize the \(ll\) matrix and a vector log_likelihoods holding the log-likelihoods of the trees in the forest.

matrix[Nnodes, 2] ll = rep_matrix(0, Nnodes, 2);
vector[Ntrees] log_likelihood = rep_vector(0, Ntrees);

for (i in 1:Ntips) {
    ll[tips[i]] = log(onehot(Y[i]+1, 2));
}

The vector Y contains the observations. As these are from \(\{0, 1\}\), \(1\) has to be added when constructing the one-hot vector.

Also, I precompute the \(P\)-matrices for each edge. This is part of the transformed parameter part.

matrix[2, 2] Q;
matrix[2, 2] P[Nedges];
Q[1, 2] = total_rate * pi[2];
Q[2, 1] = total_rate * pi[1];
for (i in 1:2) {
    Q[i, i] = 0;
    Q[i, i] = -sum(Q[i]);
};
for (i in 1:Nedges) {
    P[i] = matrix_exp(Q * edge_lengths[i]);
}

Now to the recursive part of the pruning algorithm. Here I deviate somewhat from the mathematical definition given above. I first order the edges according to the post-order of the daughter nodes in R and pass this matrix to Stan. Then I loop through the re-ordered edges.

for (i in 1:Nedges) {
    int mother = edges[i, 1];
    int child = edges[i, 2];
    vector[2] child_log_prob = rep_vector(0, 2);
    for (k in 1:2) {
        child_log_prob = to_vector(log(P[i][k])) + to_vector(logL[child]);
        ll[mother, k] += log_sum_exp(child_log_prob);
    }
}

Then the log-likelihood of each tree is computed.

for (t in 1:Ntrees) {
    log_likelihood[t] = log_sum_exp(log(pi) + to_vector(ll[roots[t]]));
}
  

Finally, we need to set priors for \(\pi\) and \(r\). For \(\pi\), a uniform prior – dirichlet(1,1) – is an obvious choice. For \(r\) I chose cauchy(0, 2.5), which allows for outliers. When defining the parameter \(r\) (called total_rate in the script), it has to be specified that it must be \(> 0\).

pi ~ dirichlet(1,1);
total_rate ~ cauchy(0, 2.5);

To perform predictive checks, we have to add a generated quantities block to the Stan code. Here is a start.

generated quantities {
    matrix[2, 2] Q_prior_predictive;
    matrix[2, 2] P_prior_predictive[Nedges];
    real<lower=0> total_rate_prior_predictive;
    simplex[2] pi_prior_predictive;
    real<lower=0> total_rate_posterior_predictive;
    simplex[2] pi_posterior_predictive;
    matrix[2, 2] Q_posterior_predictive;
    matrix[2, 2] P_posterior_predictive[Nedges];
    real<lower=0> edge_lengths_posterior_predictive[Nedges];
    
    // Prior Predictive Checks

    total_rate_prior_predictive = cauchy_rng(0, 2.5);
    pi_prior_predictive = dirichlet_rng(rep_vector(1, 2));
    
    Q_prior_predictive[1, 2] = total_rate_prior_predictive * pi_prior_predictive[2];
    Q_prior_predictive[2, 1] = total_rate_prior_predictive * pi_prior_predictive[1];
    for (i in 1:2) {
        Q_prior_predictive[i, i] = -Q_prior_predictive[i, 3-i];
    }
    
    for (i in 1:Nedges) {
        P_prior_predictive[i] = matrix_exp(Q_prior_predictive * edge_lengths[i]);
    }

    // Posterior Predictive Checks

    total_rate_posterior_predictive = total_rate;
    pi_posterior_predictive = pi;

    Q_posterior_predictive[1, 2] = total_rate_posterior_predictive * pi_posterior_predictive[2];
    Q_posterior_predictive[2, 1] = total_rate_posterior_predictive * pi_posterior_predictive[1];
    for (i in 1:2) {
        Q_posterior_predictive[i, i] = -sum(Q_posterior_predictive[i]);
    }

    for (i in 1:Nedges) {
        edge_lengths_posterior_predictive[i] = exponential_rng(edge_lengths[i]);
        P_posterior_predictive[i] = matrix_exp(Q_posterior_predictive * edge_lengths_posterior_predictive[i]);
    }
}

This will sample the parameters, but not the observations. To achieve that, we have to implement the generative model directly. Here is code for prior predictive sampling.

int<lower=0, upper=2> z_prior[Nnodes] = rep_array(0, Nnodes); 
int<lower=0, upper=1> Y_prior[Ntips];

for (i in 1:Ntrees) {
    z_prior[roots[i]] = categorical_rng(pi_prior_predictive);
}
for (i in Nedges:1:-1) {
    int mother_node = edges[i, 1];
    int child_node = edges[i, 2];
    vector[2] child_probs = to_vector(P_posterior_predictive[i][z_prior[mother_node]]);
    z_prior[child_node] = categorical_rng(child_probs);
}

for (i in 1:Ntips) {
    Y_prior[i] = z_prior[tips[i]] - 1;
}

Okay, let’s try it out.

First we need to organize the input data into a list.

tips <- match(filter(d, !is.na(Case))$Glottocode, nodes$label)
undefined <- match(filter(d, is.na(Case))$Glottocode, nodes$label)
po <- c()
for (i in get_postorder(roots, edges)) {
    if ((i %in% edges[,2]) && !(i %in% undefined)) {
        po <- c(po, i)
    }
}
edge_po <- match(po, edges[,2])
ctmc_data = list()
ctmc_data$Ntrees <- length(trees)
ctmc_data$Nnodes <- nrow(nodes)
ctmc_data$Y <- filter(d, !is.na(Case))$Case
ctmc_data$Nobservations <- length(ctmc_data$Y)
ctmc_data$Nedges <- length(edge_po)
ctmc_data$observations <- match(filter(d, !is.na(Case))$Glottocode, nodes$label)
ctmc_data$roots <- roots
ctmc_data$edges <- edges[edge_po,]
ctmc_data$edge_lengths <- edge_lengths[edge_po]
functions_code <- "
functions {
    row_vector onehot(int i, int n) {
        row_vector[n] oh = to_row_vector(rep_vector(0, n));  // initialize a zero vector of length n
        oh[i] = 1;  // set the i-th element to 1
        return oh;
    }
}
"
data_code <- "data {
    int<lower=0> Nnodes;                                    // number of nodes in the forest
    int<lower=0> Ntrees;                                    // number of trees in the forest
    int<lower=0> Nedges;                                    // number of edges in the forest
    int<lower=0, upper=Nnodes> Nobservations;               // number of observations
    int<lower=1, upper=Nnodes> edges[Nedges, 2];            // edges
    int<lower=1, upper=Nnodes> observations[Nobservations]; // indices of the tip nodes with defined values
    int<lower=1, upper=Nnodes> roots[Ntrees];               // indices of the root nodes
    int<lower=0, upper=1> Y[Nobservations];                 // observed values 
    real<lower=0> edge_lengths[Nedges];                     // edge lengths
}
"
parameter_code <- "
parameters {
    real<lower=0> total_rate; // total rate in the GTR model
    simplex[2] pi;            // stationary distribution of the GTR model
}

transformed parameters {
    matrix[Nnodes, 2] ll = rep_matrix(0, Nnodes, 2);
    matrix[2, 2] Q;
    matrix[2, 2] P[Nedges];
    Q[1, 2] = total_rate * pi[2];
    Q[2, 1] = total_rate * pi[1];
    for (i in 1:2) {
        Q[i, i] = 0;
        Q[i, i] = -Q[i, 3-i];
    };
    for (i in 1:Nedges) {
        //P[i] = matrix_exp(Q * edge_lengths[i]);
        P[i][1,1] = pi[1] + pi[2]*exp(-total_rate*edge_lengths[i]);
        P[i][1,2] = pi[2] - pi[2]*exp(-total_rate*edge_lengths[i]);
        P[i][2,1] = pi[1] - pi[1]*exp(-total_rate*edge_lengths[i]);
        P[i][2,2] = pi[2] + pi[1]*exp(-total_rate*edge_lengths[i]);
    } 

    for (i in 1:Nobservations) {
        ll[observations[i]] = log(onehot(Y[i]+1, 2));
    }

    for (i in 1:Nedges) {
        int mother = edges[i, 1];
        int child = edges[i, 2];
        vector[2] child_log_prob = rep_vector(0, 2);
        for (k in 1:2) {
            child_log_prob = to_vector(log(P[i][k])) + to_vector(ll[child]);
            ll[mother, k] += log_sum_exp(child_log_prob);
        }
    }
}
"
model_code <- "
model {
    vector[Ntrees] log_likelihood = rep_vector(0, Ntrees);

    target += dirichlet_lpdf(pi | rep_vector(1, 2));
    target += cauchy_lpdf(total_rate | 0, 2.5);
    

    for (t in 1:Ntrees) {
        log_likelihood[t] = log_sum_exp(log(pi) + to_vector(ll[roots[t]]));
    }

    target += sum(log_likelihood);
}
"
generated_quantitites_code <-  "
generated quantities {
    vector[Nobservations] log_lik = rep_vector(not_a_number(), Nobservations);
    int<lower=0, upper=2> z_prior[Nnodes] = rep_array(0, Nnodes);
    int<lower=0, upper=1> Y_prior[Nobservations] = rep_array(0, Nobservations);
    matrix[2, 2] Q_prior_predictive;
    matrix[2, 2] P_prior_predictive[Nedges];
    real<lower=0> total_rate_prior_predictive;
    simplex[2] pi_prior_predictive;

    int<lower=0, upper=2> z_posterior[Nnodes] = rep_array(0, Nnodes); 
    int<lower=0, upper=1> Y_posterior[Nobservations] = rep_array(0, Nobservations);
    
    // Prior Predictive Checks

    total_rate_prior_predictive = fabs(cauchy_rng(0, 2.5));
    pi_prior_predictive = dirichlet_rng(rep_vector(1, 2));
    

    Q_prior_predictive[1, 2] = total_rate_prior_predictive * pi_prior_predictive[2];
    Q_prior_predictive[2, 1] = total_rate_prior_predictive * pi_prior_predictive[1];
    for (i in 1:2) {
        Q_prior_predictive[i, i] = -Q_prior_predictive[i,3-i];
    }
    
    for (i in 1:Nedges) {
        //P[i] = matrix_exp(Q * edge_lengths[i]);
        P_prior_predictive[i][1,1] = pi[1] + pi[2]*exp(-total_rate_prior_predictive*edge_lengths[i]);
        P_prior_predictive[i][1,2] = pi[2] - pi[2]*exp(-total_rate_prior_predictive*edge_lengths[i]);
        P_prior_predictive[i][2,1] = pi[1] - pi[1]*exp(-total_rate_prior_predictive*edge_lengths[i]);
        P_prior_predictive[i][2,2] = pi[2] + pi[1]*exp(-total_rate_prior_predictive*edge_lengths[i]);
    } 

    for (i in 1:Ntrees) {
        z_prior[roots[i]] = categorical_rng(pi_prior_predictive);
    }

    for (i in 1:Nedges) {
        int j = Nedges - i + 1;
        int mother_node = edges[j, 1];
        int child_node = edges[j, 2];
        vector[2] child_probs = to_vector(P_prior_predictive[j][z_prior[mother_node]]);
        z_prior[child_node] = categorical_rng(child_probs);
    }
    

    for (i in 1:Nobservations) {
        Y_prior[i] = z_prior[observations[i]] - 1;
    }
    
    // Posterior Predictive Checks
    
    for (i in 1:Ntrees) {
        vector[2] root_ll = log(pi) + to_vector(ll[roots[i]]);
        root_ll = root_ll - log_sum_exp(root_ll);
        z_posterior[roots[i]] = categorical_rng(exp(root_ll));
        for (n in 1:Nobservations) {
            if (roots[i] == observations[n]) {
                log_lik[n] = log(pi[Y[n]+1]);
                break;
            }
        }
    }

    for (i in 1:Nedges) {
        int j = Nedges - i + 1;
        int mother_node = edges[j, 1];
        int child_node = edges[j, 2];
        vector[2] child_probs = to_vector(P[j][z_posterior[mother_node]]);
        child_probs = child_probs + to_vector(exp(ll[child_node]));
        child_probs = child_probs / sum(child_probs);
        z_posterior[child_node] = categorical_rng(child_probs);
        for (n in 1:Nobservations) {
            if (child_node == observations[n]) {
                log_lik[n] = log(child_probs[Y[n]+1]);
                break;
            }
        }
    }

    for (i in 1:Nobservations) {
        Y_posterior[i] = z_posterior[observations[i]] - 1;
    }
}
"
stan_code <- paste0(
    functions_code,
    data_code,
    parameter_code,
    model_code,
    generated_quantitites_code
)
model_pruning <- stan_model(model_code = stan_code)
#fit_pruning_case <- rstan::sampling(
#    model_pruning,
#    data = ctmc_data,
#    chains = 4,
#    iter=10000,
#    thin=1,
#)
#saveRDS(fit_pruning_case, file = "models/pruning_case.rds")
fit_pruning_case <- readRDS("models/pruning_case.rds")
print(fit_pruning_case, pars = c("pi", "total_rate", "lp__"))
Inference for Stan model: anon_model.
4 chains, each with iter=10000; warmup=5000; thin=1; 
post-warmup draws per chain=5000, total post-warmup draws=20000.

              mean se_mean   sd    2.5%     25%     50%     75%   97.5% n_eff
pi[1]         0.44    0.00 0.04    0.35    0.41    0.44    0.47    0.52 13877
pi[2]         0.56    0.00 0.04    0.48    0.53    0.56    0.59    0.65 13877
total_rate    1.34    0.00 0.21    0.96    1.18    1.32    1.47    1.80 13458
lp__       -276.75    0.01 1.00 -279.43 -277.14 -276.44 -276.04 -275.78  8693
           Rhat
pi[1]         1
pi[2]         1
total_rate    1
lp__          1

Samples were drawn using NUTS(diag_e) at Mon Feb  5 08:06:54 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).
generated_quantities <- extract(fit_pruning_case)

So far so good. Now I can do predictive checks.

prior predictive check

spl <- sample(1:dim(generated_quantities$Y_prior)[1], 1000)

prior_pc <- tibble(
    tv = apply(generated_quantities$Y_prior[spl, ], 1, total_variance),
    lv = apply(generated_quantities$Y_prior[spl, ], 1, lineage_wise_variance),
    cv = apply(generated_quantities$Y_prior[spl, ], 1, crossfamily_variance)
) %>% filter(complete.cases(.))


predictive_plot(prior_pc)
Warning message in geom_point(aes(x = 1, y = empirical$tv, color = "empirical"), :
“All aesthetics have length 1, but the data has 1000 rows.
ℹ Did you mean to use `annotate()`?”
Warning message in geom_point(aes(x = 1, y = empirical$lv, color = "empirical"), :
“All aesthetics have length 1, but the data has 1000 rows.
ℹ Did you mean to use `annotate()`?”
Warning message in geom_point(aes(x = 1, y = empirical$cv, color = "empirical"), :
“All aesthetics have length 1, but the data has 1000 rows.
ℹ Did you mean to use `annotate()`?”

posterior predictive check

spl <- sample(1:dim(generated_quantities$Y_posterior)[1], 1000)

posterior_pc <- tibble(
    tv = apply(generated_quantities$Y_posterior[spl, ], 1, total_variance),
    lv = apply(generated_quantities$Y_posterior[spl, ], 1, lineage_wise_variance),
    cv = apply(generated_quantities$Y_posterior[spl, ], 1, crossfamily_variance)
) %>% filter(complete.cases(.))

predictive_plot(posterior_pc)
Warning message in geom_point(aes(x = 1, y = empirical$tv, color = "empirical"), :
“All aesthetics have length 1, but the data has 1000 rows.
ℹ Did you mean to use `annotate()`?”
Warning message in geom_point(aes(x = 1, y = empirical$lv, color = "empirical"), :
“All aesthetics have length 1, but the data has 1000 rows.
ℹ Did you mean to use `annotate()`?”
Warning message in geom_point(aes(x = 1, y = empirical$cv, color = "empirical"), :
“All aesthetics have length 1, but the data has 1000 rows.
ℹ Did you mean to use `annotate()`?”

Both prior and posterior predictive checks confirm that the model is consistent with the data.

Next I do a direct model comparison via Bayes Factor.

new_mod <- stan(model_code = stan_code, data = ctmc_data, chains = 0)
(marginal_pruning_case <- bridge_sampler(
    fit_pruning_case,
    new_mod
))
the number of chains is less than 1; sampling not done
Iteration: 1
Iteration: 2
Iteration: 3
Iteration: 4
Bridge sampling estimate of the log marginal likelihood: -277.5376
Estimate obtained in 4 iteration(s) via method "normal".

We see that the log-Bayes factor in favor of the pruning model is about 190, so the evidence is overwhelming.

log_lik <- extract_log_lik(
    fit_pruning_case,
    parameter_name = "log_lik",
    merge_chains = FALSE
)
reff <- relative_eff(exp(log_lik))
(loo_pruning_case <- loo(log_lik, r_eff=reff))
Warning message:
“Some Pareto k diagnostic values are too high. See help('pareto-k-diagnostic') for details.
”

Computed from 20000 by 693 log-likelihood matrix

         Estimate  SE
elpd_loo   -100.8 4.9
p_loo        14.0 0.5
looic       201.5 9.7
------
Monte Carlo SE of elpd_loo is NA.

Pareto k diagnostic values:
                         Count Pct.    Min. n_eff
(-Inf, 0.5]   (good)     610   88.0%   11529     
 (0.5, 0.7]   (ok)         3    0.4%   19510     
   (0.7, 1]   (bad)        4    0.6%   19749     
   (1, Inf)   (very bad)  76   11.0%   17235     
See help('pareto-k-diagnostic') for details.

Brownian motion

Next I will consider the phylogenetic Brownian motion model, combined with a logistic component. Here the basic assumptions are

  • there is a continuous latent variable \(z\) which has a value at each node
  • at the root nodes, \(z\) is drawn from a very uninformative distribution like \(\mathcal N(0, 10)\)
  • once the value of \(z\) at a mother node is fixed, the values at the daughter nodes is drawn from \(\mathcal N(z_i, \sigma t_j)\), where \(z_i\) is the value at the root and \(t_j\) is the branch length to the daughter
  • the observed values \(y_i\) at tip \(i\) are distributed as

\[ y_i \sim \mathrm{bernoulli}(\frac{1}{1+e^{-z_i}}) \]

The parameter \(\sigma\) can be interpreted as the rate of evolution.

It is possible to marginalize out the states of \(z\) at the internal nodes.

Let \(i,j\) be two tips in the same tree and \(s_{i,j}\) the length of the path from the root to the latest common ancestor of \(i\) and \(j\). Then, under this model, the covariance of \(z_i\) and \(z_j\) is \(\sigma^2 s_{i,j}\). The expected value of \(z_i\) for all tips \(i\) is identical; it is the value of \(z\) at the root.

This entails that the values of \(z\) at the tips of a tree are drawn from a multivariate normal distribution, with mean and covariance as just explained.

To implement this in Stan I follow an indirect approach because Stan does not like containers of varying sizes. Therefore I create one big matrix containing all covariance matrices for the individual trees.

brown_data = list()
brown_data$Ntrees <- length(trees)
brown_data$Nnodes <- nrow(nodes)
brown_data$Y <- filter(d, !is.na(Case))$Case
brown_data$Nobservations <- length(brown_data$Y)
brown_data$Nedges <- length(edge_po)
brown_data$observations <- match(filter(d, !is.na(Case))$Glottocode, nodes$label)
brown_data$roots <- roots
brown_data$edges <- edges[edge_po,]
brown_data$edge_lengths <- edge_lengths[edge_po]
brown_code <- "data {
    int<lower=1> Ntrees;
    int<lower=0> Nnodes;
    int<lower=1> Nobservations;
    int<lower=1> Nedges;
    int<lower=1,upper=Nnodes> observations[Nobservations]; // indices of the observed tips
    int<lower=0, upper=1> Y[Nobservations]; // values at the observed tips
    int<lower=1, upper=Nnodes> roots[Ntrees];
    int<lower=1, upper=Nnodes> edges[Nedges, 2];
    vector[Nedges] edge_lengths;
}

parameters {
    vector[Nnodes] z_normal;
    real mu;
    real<lower=0> sigma;
}

model {
    vector[Nnodes] z = rep_vector(0, Nnodes);
    target += normal_lpdf(mu | 0, 10);
    target += exponential_lpdf(sigma | 1);
    target += normal_lpdf(z_normal | 0, 1);

    
    for (i in 1:Ntrees) {
        int idx = roots[i];
        z[idx] = mu + z_normal[idx] * sigma;
    }
    
    for (i in 1:Nedges) {
        int j = Nedges - i + 1;
        int mother_node = edges[j, 1];
        int daughter_node = edges[j, 2];
        real local_mu = z[mother_node];
        real local_sigma = sigma * sqrt(edge_lengths[j]);
        z[daughter_node] = local_mu + z_normal[daughter_node] * local_sigma;
    }

    for (i in 1:Nobservations) {
        target += bernoulli_logit_lpmf(Y[i] | z[observations[i]]);
    }
}

generated quantities {
    int Y_prior[Nobservations]; // Prior predictive samples
    int Y_posterior[Nobservations]; // Posterior predictive samples
    vector[Nnodes] z_prior = rep_vector(0, Nnodes);
    vector[Nnodes] z_posterior = rep_vector(0, Nnodes);
    real mu_prior;
    real<lower=0> sigma_prior;
    real log_lik[Nobservations];

    mu_prior = normal_rng(0, 10);
    sigma_prior = exponential_rng(1);

    for (i in 1:Ntrees) {
        z_prior[roots[i]] = normal_rng(mu_prior, sigma_prior);
    }

    for (i in 1:Ntrees) {
        int idx = roots[i];
        z_posterior[idx] = mu + z_normal[idx] * sigma;
    }
    
    for (i in 1:Nedges) {
        int j = Nedges - i + 1;
        int mother_node = edges[j, 1];
        int daughter_node = edges[j, 2];
        real local_mu = z_posterior[mother_node];
        real local_sigma = sigma * sqrt(edge_lengths[j]);
        z_prior[daughter_node] = normal_rng(
            z_prior[mother_node],
            sigma_prior * sqrt(edge_lengths[j])
        );

        z_posterior[daughter_node] = local_mu + z_normal[daughter_node] * local_sigma;
    }
    
    for (i in 1:Nobservations) {
        Y_prior[i] = bernoulli_logit_rng(z_prior[observations[i]]);
        Y_posterior[i] = bernoulli_logit_rng(z_posterior[observations[i]]);
    }
    for (i in 1:Nobservations) {
        log_lik[i] = bernoulli_logit_lpmf(Y[i] | z_posterior[observations[i]]);
    }
}
"
model_brownian_logistic <- stan_model(model_code = brown_code)
# fit_brownian_case <- rstan::sampling(
#    model_brownian_logistic,
#    data = brown_data,
#    chains = 4,
#    iter=100000,
#    thin=10
# )
# saveRDS(fit_brownian_case, file = "models/case_logistic_brownian.rds")
fit_brownian_case <- readRDS("models/case_logistic_brownian.rds")
print(fit_brownian_case, pars = c("mu", "sigma", "lp__"))
Inference for Stan model: anon_model.
4 chains, each with iter=1e+05; warmup=50000; thin=10; 
post-warmup draws per chain=5000, total post-warmup draws=20000.

          mean se_mean    sd     2.5%      25%      50%      75%    97.5% n_eff
mu        1.92    0.01  0.97     0.24     1.25     1.83     2.49     4.06 19386
sigma     6.33    0.01  1.49     4.00     5.28     6.12     7.17     9.84 18961
lp__  -2319.12    0.23 32.13 -2382.34 -2340.85 -2318.76 -2297.12 -2257.08 19591
      Rhat
mu       1
sigma    1
lp__     1

Samples were drawn using NUTS(diag_e) at Mon Feb  5 09:00:18 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).
new_mod <- stan(model_code = brown_code, data = brown_data, chains = 0)
(marginal_brownian_case <- bridge_sampler(
    fit_brownian_case,
    new_mod,
    maxiter = 100000,
    cores = 4
))
the number of chains is less than 1; sampling not done

Warning message:
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Estimate might be more variable than usual.”
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