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Load data and R packages

Inspect the raw data

Sample sizes by treatment

treatment n
Ringers 294
LPS 291

Sample sizes by treatment and hive

hive treatment n
Arts Ringers 70
Arts LPS 68
Garden Ringers 75
Garden LPS 75
Skylab Ringers 99
Skylab LPS 100
Zoology Ringers 50
Zoology LPS 48

Inspect the results

hive treatment outcome n
Arts LPS Stayed inside 56
Arts LPS Left voluntarily 5
Arts LPS Forced out 7
Arts Ringers Stayed inside 64
Arts Ringers Left voluntarily 5
Arts Ringers Forced out 1
Garden LPS Stayed inside 70
Garden LPS Left voluntarily 2
Garden LPS Forced out 3
Garden Ringers Stayed inside 73
Garden Ringers Left voluntarily 2
Garden Ringers Forced out 0
Skylab LPS Stayed inside 93
Skylab LPS Left voluntarily 2
Skylab LPS Forced out 5
Skylab Ringers Stayed inside 97
Skylab Ringers Left voluntarily 1
Skylab Ringers Forced out 1
Zoology LPS Stayed inside 38
Zoology LPS Left voluntarily 4
Zoology LPS Forced out 6
Zoology Ringers Stayed inside 42
Zoology Ringers Left voluntarily 2
Zoology Ringers Forced out 6

Raw results as a bar chart

Error in get(S3[i, 1L], mode = "function", envir = parent.frame()) : 
  invalid first argument

Version Author Date
1ce9e19 lukeholman 2020-04-21

Multinomial model of outcome

Run the models

Fit a multinomial logisitic model, with 3 possible outcomes describing what happened to each bee introduced to the hive: stayed inside, left of its own volition, or forced out by the other workers. To assess the effects of our predictor variables, we compare 5 models with different fixed factors, ranking them by posterior model probability.

Posterior model probabilites

Model post_prob
hive + observation_time_minutes 0.530
treatment + hive + observation_time_minutes 0.466
treatment * hive + observation_time_minutes 0.003

Fixed effects from the top model

The code chunk below wrangles the raw output of the summary() functions for brms models into a more readable table of results, and also adds ‘Bayesian p-values’ (i.e. the posterior probability that the true effect size has the same sign as the reported effect).

Parameter Estimate Est. Error Lower 95% CI Upper 95% CI p
% bees leaving voluntarily
Intercept -6.368 6.669 -19.689 6.679 0.168
Treatment: LPS 0.377 0.439 -0.480 1.245 0.197
hive1 0.111 1.522 -2.868 3.075 0.466
hive2 -0.183 0.679 -1.513 1.160 0.390
hive3 0.082 2.531 -4.835 5.113 0.491
Observation duration (minutes) 0.029 0.068 -0.103 0.165 0.337
% bees forced out
Intercept -5.179 6.708 -18.321 7.894 0.222
Treatment: LPS 1.105 0.433 0.291 1.989 0.004 **
hive1 -0.030 1.544 -3.059 2.971 0.490
hive2 -0.847 0.705 -2.266 0.490 0.112
hive3 0.049 2.554 -4.892 5.089 0.493
Observation duration (minutes) 0.015 0.069 -0.118 0.149 0.414

Table S3: Table summarising the posterior estimates of each fixed effect in the best-fitting model of Experiment 2. This was a multinomial model with three possible outcomes (stay inside, leave voluntarily, be forced out), and so there are two parameter estimates for the intercept and for each predictor in the model. ‘Treatment’ is a fixed factor with two levels, and the effect of LPS shown here is expressed relative to the ‘Ringers’ treatment. ‘Hive’ was a fixed factor with four levels; unlike for treatment, we modelled hive using deviation coding, such that the intercept term represents the mean across all hives (in the Ringers treatment), and the three hive terms represent the deviation from this mean for three of the four hives. Lastly, observation duration was a continuous variable expressed to the nearest minute. The \(p\) column gives the posterior probability that the true effect size is opposite in sign to what is reported in the Estimate column, similarly to a \(p\)-value.

Plotting estimates from the model

Derive prediction from the posterior

Make Figure 2

Error in get(S3[i, 1L], mode = "function", envir = parent.frame()) : 
  invalid first argument
Error in get(S3[i, 1L], mode = "function", envir = parent.frame()) : 
  invalid first argument

Version Author Date
99649a7 lukeholman 2020-04-25
1ce9e19 lukeholman 2020-04-21

The dashed lines mark \(LOR = 0\), indicating no effect, and \(LOR = \pm log(2)\), i.e. the point at which the odds are twice as high in one treatment as the other.

Hypothesis testing and effect sizes

This section calculates the posterior difference in treatment group means, in order to perform some null hypothesis testing, calculate effect size, and calculate the 95% credible intervals on the effect size. In all cases, the effect size expresses the effect of the “LPS-treated bee CHCs” treatment relative to the “Ringer-treated bee CHCs” control.

Stats associated with Figure 2B

Making Table S4

Metric Estimate Est.Error Q2.5 Q97.5 p
% bees staying inside
Absolute difference in % bees staying inside -11.949 4.942 -22.208 -2.753
Log odds ratio -0.836 0.331 -1.520 -0.199 0.0048 **
% bees leaving voluntarily
Absolute difference in % bees leaving voluntarily 1.335 2.623 -3.720 7.086
Log odds ratio 0.242 0.440 -0.621 1.106 0.2881
% bees forced out
Absolute difference in % bees forced out 10.614 4.627 2.538 20.464
Log odds ratio 1.079 0.433 0.270 1.967 0.0047 **

Table S4: This table gives statistics associated with each of the contrasts plotted in Figure 2B. Each pair of rows gives the absolute and standardised effect size (as log odds ratio; LOR) for the LPS treatment, relative to the Ringers treatment, for one of the three possible outcomes (stayed inside, left voluntarily, or forced out). A LOR of \(|log(x)|\) indicates that the outcome is \(x\) times more frequent in one treatment compared to the other, e.g. \(log(2) = 0.69\) indicates a two-fold difference in frequency. The \(p\) column gives the posterior probability that the true effect size has the same sign as is shown in the Estimate column; this metric has a similar interpretation to a one-tailed \(p\) value in frequentist statistics.

Evaluating evidence for the null hypothesis

Here, we derive the result present in prose in the Results, that the true effect size

[1] 0.5837
[1] 0.8323
[1] 0.0557
[1] 0.187

Stats associated with Figure 2C

Here, we derive the result presented in prose in the Results, regarding the effect of LPS on the proportion of bees that left the hive by force.

  Hypothesis  Estimate Est.Error   CI.Lower CI.Upper Evid.Ratio Post.Prob Star
1    (x) > 0 0.7281842 0.5984938 -0.2336917 1.712823   8.082652    0.8899     

R version 3.6.3 (2020-02-29)
Platform: x86_64-apple-darwin15.6.0 (64-bit)
Running under: macOS Catalina 10.15.4

Matrix products: default
BLAS:   /Library/Frameworks/R.framework/Versions/3.6/Resources/lib/libRblas.0.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/3.6/Resources/lib/libRlapack.dylib

locale:
[1] en_AU.UTF-8/en_AU.UTF-8/en_AU.UTF-8/C/en_AU.UTF-8/en_AU.UTF-8

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base     

other attached packages:
 [1] cowplot_1.0.0    tidybayes_2.0.1  bayestestR_0.5.1 kableExtra_1.1.0
 [5] gridExtra_2.3    forcats_0.5.0    stringr_1.4.0    dplyr_0.8.5     
 [9] purrr_0.3.3      readr_1.3.1      tidyr_1.0.2      tibble_3.0.0    
[13] ggplot2_3.3.0    tidyverse_1.3.0  bayesplot_1.7.1  brms_2.12.0     
[17] Rcpp_1.0.4.6     showtext_0.7-1   showtextdb_2.0   sysfonts_0.8    
[21] thematic_0.0.1  

loaded via a namespace (and not attached):
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  [4] rsconnect_0.8.16        rprojroot_1.3-2         markdown_1.1           
  [7] base64enc_0.1-3         fs_1.4.1                rstudioapi_0.11        
 [10] farver_2.0.3            bootstraplib_0.1.0.9000 rstan_2.19.3           
 [13] svUnit_0.7-12           DT_0.13                 fansi_0.4.1            
 [16] mvtnorm_1.1-0           lubridate_1.7.8         xml2_1.3.1             
 [19] bridgesampling_1.0-0    knitr_1.28              shinythemes_1.1.2      
 [22] jsonlite_1.6.1          workflowr_1.6.0         broom_0.5.4            
 [25] dbplyr_1.4.2            shiny_1.4.0             compiler_3.6.3         
 [28] httr_1.4.1              backports_1.1.6         assertthat_0.2.1       
 [31] Matrix_1.2-18           fastmap_1.0.1           cli_2.0.2              
 [34] later_1.0.0             htmltools_0.4.0.9003    prettyunits_1.1.1      
 [37] tools_3.6.3             igraph_1.2.5            coda_0.19-3            
 [40] gtable_0.3.0            glue_1.4.0              reshape2_1.4.4         
 [43] cellranger_1.1.0        jquerylib_0.1           vctrs_0.2.4            
 [46] nlme_3.1-147            crosstalk_1.1.0.1       insight_0.8.1          
 [49] xfun_0.13               ps_1.3.0                rvest_0.3.5            
 [52] mime_0.9                miniUI_0.1.1.1          lifecycle_0.2.0        
 [55] gtools_3.8.2            zoo_1.8-7               scales_1.1.0           
 [58] colourpicker_1.0        hms_0.5.3               promises_1.1.0         
 [61] Brobdingnag_1.2-6       parallel_3.6.3          inline_0.3.15          
 [64] RColorBrewer_1.1-2      shinystan_2.5.0         curl_4.3               
 [67] yaml_2.2.1              loo_2.2.0               StanHeaders_2.19.2     
 [70] sass_0.2.0              stringi_1.4.6           highr_0.8              
 [73] dygraphs_1.1.1.6        pkgbuild_1.0.6          rlang_0.4.5            
 [76] pkgconfig_2.0.3         matrixStats_0.56.0      evaluate_0.14          
 [79] lattice_0.20-41         labeling_0.3            rstantools_2.0.0       
 [82] htmlwidgets_1.5.1       processx_3.4.2          tidyselect_1.0.0       
 [85] plyr_1.8.6              magrittr_1.5            R6_2.4.1               
 [88] generics_0.0.2          DBI_1.1.0               withr_2.1.2            
 [91] pillar_1.4.3            haven_2.2.0             whisker_0.4            
 [94] xts_0.12-0              abind_1.4-5             modelr_0.1.5           
 [97] crayon_1.3.4            arrayhelpers_1.1-0      rmarkdown_2.2.0        
[100] grid_3.6.3              readxl_1.3.1            callr_3.4.3            
[103] git2r_0.26.1            threejs_0.3.3           webshot_0.5.2          
[106] reprex_0.3.0            digest_0.6.25           xtable_1.8-4           
[109] httpuv_1.5.2            stats4_3.6.3            munsell_0.5.0          
[112] viridisLite_0.3.0       shinyjs_1.1