# Analyzing FBD Parameters

#### 2022-05-12

This vignette explains how to extract FBD (fossilized birth-death) parameters (net diversification, relative extinction (turnover), and relative fossilization) estimated from relaxed clock Bayesian inference analyses produced by the program Mr. Bayes, as used in .

## FBD Parameters Statistics and Plots

Below we demonstrate how to extract evolutionary rate summary statistics from each node from a Bayesian clock (time-calibrate) summary tree produced by Mr. Bayes, store them in a data frame, produce summary tables, and plots.

library(EvoPhylo)

### 1. Import combined log file from all runs.

This is produced by using combine_log(). Alternatively, users can also use LogCombiner from the BEAST2 software package. The first argument passed to combine_log() should be a path to the folder containing the log files to be imported and combined.

## Import all log (.p) files from all runs and combine them, with burn-in = 25%
## and downsampling to 2.5k trees in each log file
posterior3p <- combine_log("LogFiles3p", burnin = 0.25, downsample = 1000)

Below, we use the posterior dataset posterior3p that accompanies EvoPhylo.

data(posterior3p)

The posterior data must first be transformed from wide to long to be used with the functions described below; FBD_reshape() accomplishes this.

## Reshape imported combined log file from wide to long with FBD_reshape
posterior3p_long <- FBD_reshape(posterior3p)

## Show first 5 lines of combined log file
head(posterior3p_long, 5)
##       Gen       LnL      LnPr  TH.all.   TL.all. prop_ancfossil.all.   sigma.1.
## 1 8750000 -1449.425 -143.1907 5.271118 11.969460                   0 0.07660715
## 2 8761000 -1458.367 -174.9627 4.775064 11.070210                   0 0.05850396
## 3 8771000 -1449.445 -163.9216 5.927716 12.628460                   0 0.05182430
## 4 8782000 -1453.218 -153.2030 4.451376  9.931809                   0 0.14644520
## 5 8792000 -1461.906 -132.1172 5.095504 11.083810                   0 0.14143120
##     sigma.2.  sigma.3.      m.1.     m.2.     m.3. tk02var.1. tk02var.2.
## 1 1.33351500 0.8523453 0.3695799 1.544579 1.362332  0.3197728  0.3848931
## 2 0.06463618 0.1380557 0.5083868 1.495777 1.108471  0.2710006  0.3609312
## 3 0.67980130 0.7776142 0.4275609 1.569911 1.144364  0.2853423  0.1831945
## 4 0.65005980 0.2999867 0.6445027 1.329942 1.148377  0.4670378  0.3483061
## 5 0.52745340 1.3928490 0.4993570 1.368074 1.445410  0.2115789  0.2723863
##   tk02var.3. clockrate.all. Time_bin net_speciation relative_extinction
## 1  0.2075079     0.01192715        1     0.04987983           0.6785586
## 2  0.3622265     0.01086355        1     0.04675159           0.9174022
## 3  0.6146289     0.01349259        1     0.01064803           0.9677827
## 4  0.4949015     0.01016002        1     0.07373453           0.8976315
## 5  0.3463915     0.01160514        1     0.04990040           0.7887825
##   relative_fossilization
## 1            0.055629950
## 2            0.006523517
## 3            0.010535440
## 4            0.001264865
## 5            0.036796000

### 2. Summarize FBD parameters by time bin

Summary statistics for each FBD parameter by time bin can be quickly summarized using FBD_summary():

## Summarize parameters by time bin and analysis
t3.1 <- FBD_summary(posterior3p_long)
t3.1
FBD parameters by time bin
parameter Time_bin n mean sd min Q1 median Q3 max
net_speciation 1 10000 0.04 0.02 0.00 0.03 0.04 0.06 0.17
net_speciation 2 10000 0.03 0.02 0.00 0.02 0.03 0.04 0.12
net_speciation 3 10000 0.02 0.01 0.00 0.01 0.02 0.03 0.12
net_speciation 4 10000 0.05 0.02 0.00 0.03 0.05 0.06 0.12
relative_extinction 1 10000 0.79 0.15 0.08 0.71 0.82 0.90 1.00
relative_extinction 2 10000 0.93 0.05 0.55 0.90 0.93 0.96 1.00
relative_extinction 3 10000 0.95 0.05 0.18 0.93 0.96 0.98 1.00
relative_extinction 4 10000 0.03 0.10 0.00 0.00 0.00 0.01 0.97
relative_fossilization 1 10000 0.04 0.05 0.00 0.01 0.02 0.05 0.72
relative_fossilization 2 10000 0.07 0.04 0.00 0.04 0.06 0.09 0.36
relative_fossilization 3 10000 0.01 0.02 0.00 0.00 0.01 0.02 0.54
relative_fossilization 4 10000 0.04 0.11 0.00 0.00 0.00 0.02 0.99
## Export the table
write.csv(t3.1, file = "FBD_summary.csv")

### 3. Plot the distribution of each FBD parameter

Each of (or all) the FBD parameter distributions can be plotted by time bin using various plotting alternatives with FBD_dens_plot():

## Plot distribution of the desired FBD parameter by time bin with
## kernel density plot
FBD_dens_plot(posterior3p_long, parameter = "net_speciation",
type = "density", stack = FALSE)

## Plot distribution of the desired FBD parameter by time bin with
## stacked kernel density plot
FBD_dens_plot(posterior3p_long, parameter = "net_speciation",
type = "density", stack = TRUE)

## Plot distribution of the desired FBD parameter by time bin with
## a violin plot
FBD_dens_plot(posterior3p_long, parameter = "net_speciation",
type = "violin", stack = FALSE, color = "red")

## Plot distribution of all FBD parameter by time bin with a violin plot
p1 <- FBD_dens_plot(posterior3p_long, parameter = "net_speciation",
type = "violin", stack = FALSE, color = "red")
p2 <- FBD_dens_plot(posterior3p_long, parameter = "relative_extinction",
type = "violin", stack = FALSE, color = "cyan3")
p3 <- FBD_dens_plot(posterior3p_long, parameter = "relative_fossilization",
type = "violin", stack = FALSE, color = "green3")

library(patchwork)
p1 + p2 + p3 + plot_layout(nrow = 1)

## Save your plot to your working directory as a PDF
ggplot2::ggsave("Plot_regs.pdf", width = 12, height = 4)

### 4. Test for assumptions

In this step, users can perform tests for normality and homoscedasticity in data distribution for each of the FBD parameters under consideration. The output will determine whether parametric or nonparametric tests will be performed subsequently.

##### Tests for normality and homoscedasticity for each FBD parameter for all time bins
t3.2 <- FBD_tests1(posterior3p_long)
### Export the output table for all tests
write.csv(t3.2, file = "FBD_Tests1_Assum.csv")

The results of the Shapiro-Wilk normality test for each parameter can be output as seperate tables or as a single combined table.

# Output as separate tables
t3.2$shapiro Shapiro-Wilk normality test parameter statistic p-value Time bin 1 net_speciation 0.9917 0 Time bin 2 net_speciation 0.9385 0 Time bin 3 net_speciation 0.9227 0 Time bin 4 net_speciation 0.9898 0 Overall net_speciation 0.9568 0 Residuals net_speciation 0.9874 0 parameter statistic p-value Time bin 1 relative_extinction 0.8927 0 Time bin 2 relative_extinction 0.9247 0 Time bin 3 relative_extinction 0.8044 0 Time bin 4 relative_extinction 0.3775 0 Overall relative_extinction 0.7036 0 Residuals relative_extinction 0.8238 0 parameter statistic p-value Time bin 1 relative_fossilization 0.5764 0 Time bin 2 relative_fossilization 0.8853 0 Time bin 3 relative_fossilization 0.6210 0 Time bin 4 relative_fossilization 0.4637 0 Overall relative_fossilization 0.5473 0 Residuals relative_fossilization 0.5531 0 # OR as single merged table t3.2$shapiro$net_speciation$bin <- row.names(t3.2$shapiro$net_speciation)
t3.2$shapiro$relative_extinction$bin <- row.names(t3.2$shapiro$relative_extinction) t3.2$shapiro$relative_fossilization$bin <- row.names(t3.2$shapiro$relative_fossilization)

k1all <- rbind(t3.2$shapiro$net_speciation,
t3.2$shapiro$relative_extinction,
t3.2$shapiro$relative_fossilization,
make.row.names = FALSE)
k1all
Shapiro-Wilk normality test
parameter statistic p-value bin
net_speciation 0.9917 0 Time bin 1
net_speciation 0.9385 0 Time bin 2
net_speciation 0.9227 0 Time bin 3
net_speciation 0.9898 0 Time bin 4
net_speciation 0.9568 0 Overall
net_speciation 0.9874 0 Residuals
relative_extinction 0.8927 0 Time bin 1
relative_extinction 0.9247 0 Time bin 2
relative_extinction 0.8044 0 Time bin 3
relative_extinction 0.3775 0 Time bin 4
relative_extinction 0.7036 0 Overall
relative_extinction 0.8238 0 Residuals
relative_fossilization 0.5764 0 Time bin 1
relative_fossilization 0.8853 0 Time bin 2
relative_fossilization 0.6210 0 Time bin 3
relative_fossilization 0.4637 0 Time bin 4
relative_fossilization 0.5473 0 Overall
relative_fossilization 0.5531 0 Residuals
## Bartlett's test for homogeneity of variance
t3.2$bartlett Bartlett’s test parameter statistic p-value net_speciation 3815.464 0 relative_extinction 18159.213 0 relative_fossilization 25654.975 0 ## Fligner-Killeen test for homogeneity of variance t3.2$fligner
Fligner-Killeen test
parameter statistic p-value
net_speciation 3748.140 0
relative_extinction 12599.843 0
relative_fossilization 4808.545 0

Deviations from normality can be displayed graphically using FBD_normality_plot():

## Visualize deviations from normality and similarity of variances
FBD_normality_plot(posterior3p_long)

## Save your plot to your working directory as a PDF
ggplot2::ggsave("Plot_normTests.pdf", width = 8, height = 6)

### 5. Test for significant FBD shifts between time bins

Significant shifts in FBD parameters across time bins can be easily tested using parametric (Student’s t-test) and nonparametric (Mann-Whitney test) pairwise comparisons with FBD_tests2(). Both are automatically calculated and the preferred pairwise comparison will be chosen by the user depending on the results of the assumption tests step #4 (above).

##### Test for significant differences between each time bin for each FBD parameter
t3.3 <- FBD_tests2(posterior3p_long)
### Export the output table for all tests
write.csv(t3.3, file = "FBD_Tests2_Sign.csv")

## Pairwise t-tests
# Output as separate tables
t3.3$t_tests Significant tests parameter Time_bin1 Time_bin2 n1 n2 p-value p-value adj net_speciation 1 2 10000 10000 0 0 net_speciation 1 3 10000 10000 0 0 net_speciation 1 4 10000 10000 0 0 net_speciation 2 3 10000 10000 0 0 net_speciation 2 4 10000 10000 0 0 net_speciation 3 4 10000 10000 0 0 parameter Time_bin1 Time_bin2 n1 n2 p-value p-value adj relative_extinction 1 2 10000 10000 0 0 relative_extinction 1 3 10000 10000 0 0 relative_extinction 1 4 10000 10000 0 0 relative_extinction 2 3 10000 10000 0 0 relative_extinction 2 4 10000 10000 0 0 relative_extinction 3 4 10000 10000 0 0 parameter Time_bin1 Time_bin2 n1 n2 p-value p-value adj relative_fossilization 1 2 10000 10000 0 0 relative_fossilization 1 3 10000 10000 0 0 relative_fossilization 1 4 10000 10000 0 0 relative_fossilization 2 3 10000 10000 0 0 relative_fossilization 2 4 10000 10000 0 0 relative_fossilization 3 4 10000 10000 0 0 # OR as single merged table k3.3a <- rbind(t3.3$t_tests$net_speciation, t3.3$t_tests$relative_extinction, t3.3$t_tests$relative_fossilization, make.row.names = FALSE) k3.3a Pairwise t-tests parameter Time_bin1 Time_bin2 n1 n2 p-value p-value adj net_speciation 1 2 10000 10000 0 0 net_speciation 1 3 10000 10000 0 0 net_speciation 1 4 10000 10000 0 0 net_speciation 2 3 10000 10000 0 0 net_speciation 2 4 10000 10000 0 0 net_speciation 3 4 10000 10000 0 0 relative_extinction 1 2 10000 10000 0 0 relative_extinction 1 3 10000 10000 0 0 relative_extinction 1 4 10000 10000 0 0 relative_extinction 2 3 10000 10000 0 0 relative_extinction 2 4 10000 10000 0 0 relative_extinction 3 4 10000 10000 0 0 relative_fossilization 1 2 10000 10000 0 0 relative_fossilization 1 3 10000 10000 0 0 relative_fossilization 1 4 10000 10000 0 0 relative_fossilization 2 3 10000 10000 0 0 relative_fossilization 2 4 10000 10000 0 0 relative_fossilization 3 4 10000 10000 0 0 ## Mann-Whitney tests (use if Tests in step #4 fail assumptions) # Output as separate tables t3.3$mwu_tests
Mann-Whitney tests
parameter Time_bin1 Time_bin2 n1 n2 p-value p-value adj
net_speciation 1 2 10000 10000 0 0
net_speciation 1 3 10000 10000 0 0
net_speciation 1 4 10000 10000 0 0
net_speciation 2 3 10000 10000 0 0
net_speciation 2 4 10000 10000 0 0
net_speciation 3 4 10000 10000 0 0
parameter Time_bin1 Time_bin2 n1 n2 p-value p-value adj
relative_extinction 1 2 10000 10000 0 0
relative_extinction 1 3 10000 10000 0 0
relative_extinction 1 4 10000 10000 0 0
relative_extinction 2 3 10000 10000 0 0
relative_extinction 2 4 10000 10000 0 0
relative_extinction 3 4 10000 10000 0 0
parameter Time_bin1 Time_bin2 n1 n2 p-value p-value adj
relative_fossilization 1 2 10000 10000 0 0
relative_fossilization 1 3 10000 10000 0 0
relative_fossilization 1 4 10000 10000 0 0
relative_fossilization 2 3 10000 10000 0 0
relative_fossilization 2 4 10000 10000 0 0
relative_fossilization 3 4 10000 10000 0 0
# OR as single merged table
k3.3b <- rbind(t3.3$mwu_tests$net_speciation,
t3.3$mwu_tests$relative_extinction,
t3.3$mwu_tests$relative_fossilization,
make.row.names = FALSE)
k3.3b
Mann-Whitney tests
parameter Time_bin1 Time_bin2 n1 n2 p-value p-value adj
net_speciation 1 2 10000 10000 0 0
net_speciation 1 3 10000 10000 0 0
net_speciation 1 4 10000 10000 0 0
net_speciation 2 3 10000 10000 0 0
net_speciation 2 4 10000 10000 0 0
net_speciation 3 4 10000 10000 0 0
relative_extinction 1 2 10000 10000 0 0
relative_extinction 1 3 10000 10000 0 0
relative_extinction 1 4 10000 10000 0 0
relative_extinction 2 3 10000 10000 0 0
relative_extinction 2 4 10000 10000 0 0
relative_extinction 3 4 10000 10000 0 0
relative_fossilization 1 2 10000 10000 0 0
relative_fossilization 1 3 10000 10000 0 0
relative_fossilization 1 4 10000 10000 0 0
relative_fossilization 2 3 10000 10000 0 0
relative_fossilization 2 4 10000 10000 0 0
relative_fossilization 3 4 10000 10000 0 0

## References

Simões, Tiago R., and Stephanie E. Pierce. 2021. “Sustained High Rates of Morphological Evolution During the Rise of Tetrapods.” Nature Ecology & Evolution 5 (10): 1403–14. https://doi.org/10.1038/s41559-021-01532-x.