Analysis of Experiment 2
Last updated: 2020-04-30
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Knit directory: social_immunity/
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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 |
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 argumentError in get(S3[i, 1L], mode = "function", envir = parent.frame()) :
invalid first argument
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] 0.187Stats 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 workflowr_1.6.0
loaded via a namespace (and not attached):
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[64] shinystan_2.5.0 curl_4.3 yaml_2.2.1
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[112] shinyjs_1.1