Published

May 20, 2020

# Statistical Rethinking: Week 6

## Quick summary of the week

The week was a whirlwind tour of:

1. Maximum entropy and introduction to GLMs.
2. The problems that come when using link functions.
3. The perils of relative effects when studying binomial regression and how complicated it is to directly calculate probabilities with GLMs: all the parameters interact among themselves.

This week was an introduction to GLMs and the principle of Maximum Entropy. Once we adventure outside the Gaussian, things start to become interesting. However, interesting can quickly devolve into chaotic and arbitrary modelling decisions. Against this, Richard started to introduce the principle of Maximum Entropy: when choosing how to approximate an unknown distribution, pick the most conservative distribution that satisfies your assumptions. This guiding principle works just as well for our likelihood choice, our prior choice and the resulting posterior distribution.

Also, once we work with likelihoods other than the normal, we must work with “link” functions: functions that link one of the likelihood’s parameters to a linear combination of our predictors. However, this change is not completely benign: prior setting now has become even more unnatural. Flat priors on the parameter space can now imply very different things in the outcome space. Thus, the heightened importance of prior predictive simulation.

# Homework 6

            discipline gender applications awards
1    Chemical sciences      m           83     22
2    Chemical sciences      f           39     10
3    Physical sciences      m          135     26
4    Physical sciences      f           39      9
5              Physics      m           67     18
6              Physics      f            9      2
7           Humanities      m          230     33
8           Humanities      f          166     32
9   Technical sciences      m          189     30
10  Technical sciences      f           62     13
11   Interdisciplinary      m          105     12
12   Interdisciplinary      f           78     17
13 Earth/life sciences      m          156     38
14 Earth/life sciences      f          126     18
15     Social sciences      m          425     65
16     Social sciences      f          409     47
17    Medical sciences      m          245     46
18    Medical sciences      f          260     29

These data have a very similar structure to the UCBAdmit data discussed in Chapter 11. I want you to consider a similar question: What are the total and indirect causal effects of gender on grant awards? Consider a mediation path (a pipe) through discipline. Draw the corresponding DAG and then use one or more binomial GLMs to answer the question.

Let’s start by writing the corresponding DAG:

Therefore, to find out the total effect of Gender on awards we must keep the pipe open. That is, do not include Discipline on our analysis. Given that we are dealing with unorder count data, we will use a Binomial regression.

However, before we fit our statistical model, we will create index variables for both gender and discipline.

  gender gender_int n
1      f          1 9
2      m          2 9
           discipline discipline_int n
1   Chemical sciences              1 2
2 Earth/life sciences              2 2
3          Humanities              3 2
4   Interdisciplinary              4 2
5    Medical sciences              5 2
6   Physical sciences              6 2
7             Physics              7 2
8     Social sciences              8 2
9  Technical sciences              9 2

Before setting the prior, we know that getting grants is difficult. So we should expect the probability to be pretty low for both genders. A log-odds of 0 is a probability of 1/2. Thus, zero is too large for being the center of our prior.

[1] 0.5
[1] 0.2689414

A quarter of a probability seems more plausible.

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Before we venture into finding out our chains’ health, let’s do some prior predictive simulation.

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Which are very mildly regularizing priors after all. Let’s check our chains’ health:

The chains look healthy because:

1. They are stationary.
2. They mix well.
3. Different chains converge to explore same regions of the parameter space.
          mean         sd      5.5%     94.5%    n_eff     Rhat4
g[1] -1.726559 0.08193524 -1.853391 -1.592376 1267.740 1.0018245
g[2] -1.529378 0.06271755 -1.631909 -1.429660 1427.682 0.9997384

The Rhat looks OK, too. The parameters seem to be pretty accurately estimated, given their absolute value compare to their standard deviation. Both are negative, reflecting that getting a grant is difficult.

For the men, the average probability seems to be around:

Average probability of getting a grant across disciplines for men is between: 0.16 and 0.19

For the women:

Average probability for men is between: 0.13 and 0.17

Let’s compute statistical inference on this difference. The relative effect of being a man:

                       mean         sd        5.5%     94.5%      histogram
diff_log_odds    0.19718116 0.10220652 0.033756350 0.3572811 ▁▁▁▂▃▅▇▇▅▃▂▁▁▁
diff_probability 0.02694168 0.01383279 0.004597959 0.0488329     ▁▁▂▅▇▇▃▁▁▁

On the log odds scale, that is, in the relative scale, we estimate that on average, across disciplines, the effect is positive. On the probability scale, that is, in the absolute scale, we estimate the difference to be around 1% and 5%.

Posterior predictions

However, this models helps us answer what is the total effect of gender on grants by answering the statistical question of what is the average probability of getting a grant across.

Conditional on our DAG, if we want to get the direct effect of gender on grants, we must adjust for the discipline to which the scholars belong to. That is, we must ask a different statistical question: within discipines, what is the average grant assignment for men and women?

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The chains seem to be healthy:

1. They are stationary.
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3. Different chains converge well.

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           mean        sd       5.5%      94.5%    n_eff     Rhat4
g[1] -0.6917646 0.1712243 -0.9685794 -0.4198413 468.5596 1.0003605
g[2] -0.5433700 0.1614290 -0.8033075 -0.2911890 428.5772 0.9998649
d[1] -0.5345053 0.2367730 -0.9240730 -0.1570969 828.8811 1.0002828
d[2] -0.8106297 0.2090504 -1.1513516 -0.4722240 604.9735 0.9995131
d[3] -1.0316070 0.1973300 -1.3539111 -0.7074005 595.0431 0.9996754
d[4] -1.0674090 0.2302974 -1.4420317 -0.6807685 816.7345 0.9998054
d[5] -1.1268116 0.1901408 -1.4227910 -0.8209012 561.1217 0.9998611
d[6] -0.8316702 0.2206251 -1.1789399 -0.4739253 747.6443 0.9998957
d[7] -0.5966858 0.2669562 -1.0279350 -0.1775297 879.7959 1.0002679
d[8] -1.2460656 0.1799261 -1.5312353 -0.9668919 524.0918 1.0008232
d[9] -1.0041924 0.2136897 -1.3537964 -0.6724457 629.1481 1.0014307

The Rhat seems to be ok. Also, men’s parameter seems to still be larger than the women’s parameters.

                   mean       sd        5.5%     94.5% histogram
diff_log_odds 0.1483946 0.106863 -0.01813196 0.3170597 ▁▁▁▅▇▅▁▁▁

Now, the difference, as measured on the log scale, the relative effect, seems to have decreased. However, within departments, men still seem to have a higher chance of receiving a grant.

However, we cannot directly calculate the difference in the absolute scale as readily as we did before. Why? The floor and ceiling effects that Richard discussed in class. That is, the departments influence the difference between genders; the differents base rates of acceptance among departments influence our prediction.

In comparison to our earlier predictions, the predicted difference is, in general, much lower. Let’s compare our models:

                      PSIS       SE    dPSIS      dSE     pPSIS    weight
model_only_gender 130.4475 9.669158 0.000000       NA  5.130467 0.6277665
model_discipline  131.4928 8.728316 1.045294 9.536532 14.111772 0.3722335
                     WAIC       SE    dWAIC      dSE     pWAIC    weight
model_discipline  126.272 7.789795 0.000000       NA 11.501352 0.8617014
model_only_gender 129.931 9.176973 3.658987 9.072812  4.872187 0.1382986

Somewhat surprisingly, the PSIS and the WAIC give different answers. If we check the differences and their standard error, we see that the differences are not precisely estimated. Hinting that the models give somewhat equivalent predictions.

Finally, let’s check the actual observed rates:

In reality, the relationship is not as straightforward as we predicted. That is, model one (that only takes into account gender) indeed is hinting much more to what we observe: a large gap in the acceptance rates between genders, even among departments.

. Suppose that the NWO Grants sample has an unobserved confound that influences both choice of discipline and the probability of an award. One example of such a confound could be the career stage of each applicant. Suppose that in some disciplines, junior scholars apply for most of the grants. In other disciplines, scholars from all career stages compete. As a result, career stage influences discipline as well as the probability of being awarded a grant. Add these influences to your DAG from Problem 1.

What happens now when you condition on discipline?

We are interested on the effect of Gender on Awards. If we only include Gender, there’s one backdoor path: the effect of Carrer Stage on Awards that gets picked up by discipline. However, that information won’t flow towards gender: Disciplin forms a collider that it is closed as long as we don’t adjust our estimates with discipline:

 {}

Therefore, according to our DAG, we can estimate the total effect of gender on awards.

However, once we adjust for discipline, we open the collider. Thus, we cannot reliably estimate the direct effect of Gender on awards unless we adjust for Carrer Stage.

{ CarrerStage }

The data in data(Primates301) were first introduced at the end of Chapter 7. In this problem, you will consider how brain size is associated with social learning. There are three parts

                                         name          genus          species
1                 Allenopithecus_nigroviridis Allenopithecus     nigroviridis
2                         Allocebus_trichotis      Allocebus        trichotis
3                           Alouatta_belzebul       Alouatta         belzebul
4                             Alouatta_caraya       Alouatta           caraya
5                            Alouatta_guariba       Alouatta          guariba
6                           Alouatta_palliata       Alouatta         palliata
7                              Alouatta_pigra       Alouatta            pigra
8                               Alouatta_sara       Alouatta             sara
9                          Alouatta_seniculus       Alouatta        seniculus
10                               Aotus_azarai          Aotus           azarai
11                   Aotus_azarai_boliviensis          Aotus           azarai
12                            Aotus_brumbacki          Aotus        brumbacki
13                            Aotus_infulatus          Aotus        infulatus
14                            Aotus_lemurinus          Aotus        lemurinus
15               Aotus_lemurinus_griseimembra          Aotus        lemurinus
16                            Aotus_nancymaae          Aotus        nancymaae
17                            Aotus_nigriceps          Aotus        nigriceps
18                          Aotus_trivirgatus          Aotus      trivirgatus
19                           Aotus_vociferans          Aotus       vociferans
20                        Archaeolemur_majori   Archaeolemur           majori
21                          Arctocebus_aureus     Arctocebus           aureus
22                    Arctocebus_calabarensis     Arctocebus     calabarensis
23                           Ateles_belzebuth         Ateles        belzebuth
24                           Ateles_fusciceps         Ateles        fusciceps
25                           Ateles_geoffroyi         Ateles        geoffroyi
26                            Ateles_paniscus         Ateles         paniscus
27                              Avahi_cleesei          Avahi          cleesei
28                              Avahi_laniger          Avahi          laniger
29                         Avahi_occidentalis          Avahi     occidentalis
30                             Avahi_unicolor          Avahi         unicolor
31                    Brachyteles_arachnoides    Brachyteles      arachnoides
32                       Bunopithecus_hoolock   Bunopithecus          hoolock
33                             Cacajao_calvus        Cacajao           calvus
34                     Cacajao_melanocephalus        Cacajao   melanocephalus
35                    Callicebus_donacophilus     Callicebus     donacophilus
36                      Callicebus_hoffmannsi     Callicebus       hoffmannsi
37                          Callicebus_moloch     Callicebus           moloch
38                      Callicebus_personatus     Callicebus       personatus
39                       Callicebus_torquatus     Callicebus        torquatus
40                          Callimico_goeldii      Callimico          goeldii
41                       Callithrix_argentata     Callithrix        argentata
42                          Callithrix_aurita     Callithrix           aurita
43                         Callithrix_emiliae     Callithrix          emiliae
44                       Callithrix_geoffroyi     Callithrix        geoffroyi
45                    Callithrix_humeralifera     Callithrix     humeralifera
46                         Callithrix_jacchus     Callithrix          jacchus
47                           Callithrix_kuhli     Callithrix            kuhli
48                          Callithrix_mauesi     Callithrix           mauesi
49                     Callithrix_penicillata     Callithrix      penicillata
50                         Callithrix_pygmaea     Callithrix          pygmaea
51                            Cebus_albifrons          Cebus        albifrons
52                               Cebus_apella          Cebus           apella
53                            Cebus_capucinus          Cebus        capucinus
54                            Cebus_olivaceus          Cebus        olivaceus
55                        Cebus_xanthosternos          Cebus    xanthosternos
56                          Cercocebus_agilis     Cercocebus           agilis
57                       Cercocebus_galeritus     Cercocebus        galeritus
58                       Cercocebus_torquatus     Cercocebus        torquatus
59                  Cercocebus_torquatus_atys     Cercocebus        torquatus
60                  Cercopithecus_albogularis  Cercopithecus      albogularis
61                     Cercopithecus_ascanius  Cercopithecus         ascanius
62                    Cercopithecus_campbelli  Cercopithecus        campbelli
63              Cercopithecus_campbelli_lowei  Cercopithecus        campbelli
64                       Cercopithecus_cephus  Cercopithecus           cephus
65                Cercopithecus_cephus_cephus  Cercopithecus           cephus
66           Cercopithecus_cephus_ngottoensis  Cercopithecus           cephus
67                        Cercopithecus_diana  Cercopithecus            diana
68                Cercopithecus_erythrogaster  Cercopithecus    erythrogaster
69  Cercopithecus_erythrogaster_erythrogaster  Cercopithecus    erythrogaster
70                   Cercopithecus_erythrotis  Cercopithecus       erythrotis
71                      Cercopithecus_hamlyni  Cercopithecus          hamlyni
72                      Cercopithecus_lhoesti  Cercopithecus          lhoesti
73                        Cercopithecus_mitis  Cercopithecus            mitis
74                         Cercopithecus_mona  Cercopithecus             mona
75                    Cercopithecus_neglectus  Cercopithecus        neglectus
76                    Cercopithecus_nictitans  Cercopithecus        nictitans
77                   Cercopithecus_petaurista  Cercopithecus       petaurista
78                     Cercopithecus_pogonias  Cercopithecus         pogonias
79                      Cercopithecus_preussi  Cercopithecus          preussi
80                      Cercopithecus_solatus  Cercopithecus          solatus
81                        Cercopithecus_wolfi  Cercopithecus            wolfi
82                     Cheirogaleus_crossleyi   Cheirogaleus        crossleyi
83                         Cheirogaleus_major   Cheirogaleus            major
84                        Cheirogaleus_medius   Cheirogaleus           medius
85                         Chiropotes_satanas     Chiropotes          satanas
86                       Chlorocebus_aethiops    Chlorocebus         aethiops
87                    Chlorocebus_pygerythrus    Chlorocebus      pygerythrus
88          Chlorocebus_pygerythrus_cynosurus    Chlorocebus      pygerythrus
89                        Chlorocebus_sabaeus    Chlorocebus          sabaeus
90                       Chlorocebus_tantalus    Chlorocebus         tantalus
91                         Colobus_angolensis        Colobus       angolensis
92               Colobus_angolensis_palliatus        Colobus       angolensis
93                            Colobus_guereza        Colobus          guereza
94                          Colobus_polykomos        Colobus        polykomos
95                            Colobus_satanas        Colobus          satanas
96                         Colobus_vellerosus        Colobus       vellerosus
98                         Erythrocebus_patas   Erythrocebus            patas
99                          Eulemur_coronatus        Eulemur        coronatus
100                  Eulemur_fulvus_albifrons        Eulemur           fulvus
101               Eulemur_fulvus_albocollaris        Eulemur           fulvus
102                   Eulemur_fulvus_collaris        Eulemur           fulvus
103                     Eulemur_fulvus_fulvus        Eulemur           fulvus
104                Eulemur_fulvus_mayottensis        Eulemur           fulvus
105                      Eulemur_fulvus_rufus        Eulemur           fulvus
106                   Eulemur_fulvus_sanfordi        Eulemur           fulvus
107                 Eulemur_macaco_flavifrons        Eulemur           macaco
108                     Eulemur_macaco_macaco        Eulemur           macaco
109                            Eulemur_mongoz        Eulemur           mongoz
110                       Eulemur_rubriventer        Eulemur      rubriventer
111                      Euoticus_elegantulus       Euoticus      elegantulus
112                             Galago_alleni         Galago           alleni
113                           Galago_gallarum         Galago         gallarum
114                             Galago_granti         Galago           granti
115                          Galago_matschiei         Galago        matschiei
116                             Galago_moholi         Galago           moholi
117                       Galago_senegalensis         Galago     senegalensis
118                       Galagoides_demidoff     Galagoides         demidoff
119                   Galagoides_zanzibaricus     Galagoides     zanzibaricus
120                          Gorilla_beringei        Gorilla         beringei
121                   Gorilla_gorilla_gorilla        Gorilla          gorilla
122                   Gorilla_gorilla_graueri        Gorilla          gorilla
123                          Hapalemur_aureus      Hapalemur           aureus
124                         Hapalemur_griseus      Hapalemur          griseus
125             Hapalemur_griseus_alaotrensis      Hapalemur          griseus
126                 Hapalemur_griseus_griseus      Hapalemur          griseus
127            Hapalemur_griseus_meridionalis      Hapalemur          griseus
128            Hapalemur_griseus_occidentalis      Hapalemur          griseus
129                           Hapalemur_simus      Hapalemur            simus
130                              Homo_sapiens           Homo          sapiens
131             Homo_sapiens_neanderthalensis           Homo          sapiens
132                          Hylobates_agilis      Hylobates           agilis
133                         Hylobates_klossii      Hylobates          klossii
134                             Hylobates_lar      Hylobates              lar
135                          Hylobates_moloch      Hylobates           moloch
136                        Hylobates_muelleri      Hylobates         muelleri
137                        Hylobates_pileatus      Hylobates         pileatus
138                               Indri_indri          Indri            indri
139                      Lagothrix_lagotricha      Lagothrix       lagotricha
140                               Lemur_catta          Lemur            catta
141                Leontopithecus_chrysomelas Leontopithecus      chrysomelas
142                Leontopithecus_chrysopygus Leontopithecus      chrysopygus
143                    Leontopithecus_rosalia Leontopithecus          rosalia
144                         Lepilemur_aeeclis      Lepilemur          aeeclis
145                    Lepilemur_ankaranensis      Lepilemur     ankaranensis
146                        Lepilemur_dorsalis      Lepilemur         dorsalis
147                        Lepilemur_edwardsi      Lepilemur         edwardsi
148                     Lepilemur_hubbardorum      Lepilemur      hubbardorum
149                        Lepilemur_leucopus      Lepilemur         leucopus
150                      Lepilemur_manasamody      Lepilemur       manasamody
151                        Lepilemur_microdon      Lepilemur         microdon
152                   Lepilemur_mitsinjoensis      Lepilemur    mitsinjoensis
153                      Lepilemur_mustelinus      Lepilemur       mustelinus
154                            Lepilemur_otto      Lepilemur             otto
155                   Lepilemur_randrianasoli      Lepilemur    randrianasoli
156                    Lepilemur_ruficaudatus      Lepilemur     ruficaudatus
157                  Lepilemur_sahamalazensis      Lepilemur   sahamalazensis
158                           Lepilemur_seali      Lepilemur            seali
159                 Lepilemur_septentrionalis      Lepilemur  septentrionalis
160                       Lophocebus_albigena     Lophocebus         albigena
161                      Lophocebus_aterrimus     Lophocebus        aterrimus
162                       Loris_lydekkerianus          Loris    lydekkerianus
164                          Macaca_arctoides         Macaca        arctoides
165                         Macaca_assamensis         Macaca       assamensis
166                        Macaca_brunnescens         Macaca      brunnescens
167                           Macaca_cyclopis         Macaca         cyclopis
168                       Macaca_fascicularis         Macaca     fascicularis
169                            Macaca_fuscata         Macaca          fuscata
170                              Macaca_hecki         Macaca            hecki
171                            Macaca_leonina         Macaca          leonina
172                              Macaca_maura         Macaca            maura
173                            Macaca_mulatta         Macaca          mulatta
174                            Macaca_munzala         Macaca          munzala
175                         Macaca_nemestrina         Macaca       nemestrina
176                 Macaca_nemestrina_leonina         Macaca       nemestrina
177                  Macaca_nemestrina_siberu         Macaca       nemestrina
178                              Macaca_nigra         Macaca            nigra
179                         Macaca_nigrescens         Macaca       nigrescens
180                           Macaca_ochreata         Macaca         ochreata
181                           Macaca_pagensis         Macaca         pagensis
183                            Macaca_silenus         Macaca          silenus
184                             Macaca_sinica         Macaca           sinica
185                           Macaca_sylvanus         Macaca         sylvanus
186                          Macaca_thibetana         Macaca        thibetana
187                           Macaca_tonkeana         Macaca         tonkeana
188                    Mandrillus_leucophaeus     Mandrillus      leucophaeus
189                         Mandrillus_sphinx     Mandrillus           sphinx
190                        Microcebus_berthae     Microcebus          berthae
191                  Microcebus_bongolavensis     Microcebus    bongolavensis
192                       Microcebus_danfossi     Microcebus         danfossi
193                    Microcebus_griseorufus     Microcebus      griseorufus
194                        Microcebus_jollyae     Microcebus          jollyae
195                  Microcebus_lehilahytsara     Microcebus    lehilahytsara
196                     Microcebus_lokobensis     Microcebus       lokobensis
197                    Microcebus_macarthurii     Microcebus      macarthurii
198                      Microcebus_mamiratra     Microcebus        mamiratra
199                   Microcebus_mittermeieri     Microcebus     mittermeieri
200                        Microcebus_murinus     Microcebus          murinus
201                       Microcebus_myoxinus     Microcebus         myoxinus
202                   Microcebus_ravelobensis     Microcebus     ravelobensis
203                          Microcebus_rufus     Microcebus            rufus
204                  Microcebus_sambiranensis     Microcebus    sambiranensis
205                       Microcebus_simmonsi     Microcebus         simmonsi
206                      Microcebus_tavaratra     Microcebus        tavaratra
207                      Miopithecus_talapoin    Miopithecus         talapoin
208                           Mirza_coquereli          Mirza        coquereli
209                                Mirza_zaza          Mirza             zaza
210                          Nasalis_larvatus        Nasalis         larvatus
211                         Nomascus_concolor       Nomascus         concolor
212                       Nomascus_gabriellae       Nomascus       gabriellae
213                       Nomascus_leucogenys       Nomascus       leucogenys
214                          Nomascus_nasutus       Nomascus          nasutus
215                             Nomascus_siki       Nomascus             siki
216                    Nycticebus_bengalensis     Nycticebus      bengalensis
217                        Nycticebus_coucang     Nycticebus          coucang
218                      Nycticebus_javanicus     Nycticebus        javanicus
219                     Nycticebus_menagensis     Nycticebus       menagensis
220                       Nycticebus_pygmaeus     Nycticebus         pygmaeus
221                   Otolemur_crassicaudatus       Otolemur   crassicaudatus
222                        Otolemur_garnettii       Otolemur        garnettii
223                              Pan_paniscus            Pan         paniscus
224            Pan_troglodytes_schweinfurthii            Pan      troglodytes
225               Pan_troglodytes_troglodytes            Pan      troglodytes
226                Pan_troglodytes_vellerosus            Pan      troglodytes
227                     Pan_troglodytes_verus            Pan      troglodytes
228                              Papio_anubis          Papio           anubis
229                        Papio_cynocephalus          Papio     cynocephalus
231                               Papio_papio          Papio            papio
232                             Papio_ursinus          Papio          ursinus
233                        Perodicticus_potto   Perodicticus            potto
234                           Phaner_furcifer         Phaner         furcifer
235                Phaner_furcifer_pallescens         Phaner         furcifer
237                         Piliocolobus_foai   Piliocolobus             foai
238                   Piliocolobus_gordonorum   Piliocolobus       gordonorum
239                       Piliocolobus_kirkii   Piliocolobus           kirkii
240                    Piliocolobus_pennantii   Piliocolobus        pennantii
241                      Piliocolobus_preussi   Piliocolobus          preussi
242                 Piliocolobus_rufomitratus   Piliocolobus     rufomitratus
243                 Piliocolobus_tephrosceles   Piliocolobus     tephrosceles
244                     Piliocolobus_tholloni   Piliocolobus         tholloni
245                         Pithecia_irrorata       Pithecia         irrorata
246                         Pithecia_pithecia       Pithecia         pithecia
247                              Pongo_abelii          Pongo           abelii
248                            Pongo_pygmaeus          Pongo         pygmaeus
249                          Presbytis_comata      Presbytis           comata
250                      Presbytis_melalophos      Presbytis       melalophos
251                          Procolobus_verus     Procolobus            verus
252                     Propithecus_coquereli    Propithecus        coquereli
253                      Propithecus_deckenii    Propithecus         deckenii
255                      Propithecus_edwardsi    Propithecus         edwardsi
256                   Propithecus_tattersalli    Propithecus      tattersalli
257                     Propithecus_verreauxi    Propithecus        verreauxi
258                         Pygathrix_cinerea      Pygathrix          cinerea
259                         Pygathrix_nemaeus      Pygathrix          nemaeus
260                   Rhinopithecus_avunculus  Rhinopithecus        avunculus
261                       Rhinopithecus_bieti  Rhinopithecus            bieti
262                    Rhinopithecus_brelichi  Rhinopithecus         brelichi
263                   Rhinopithecus_roxellana  Rhinopithecus        roxellana
264                       Rungwecebus_kipunji    Rungwecebus          kipunji
265                          Saguinus_bicolor       Saguinus          bicolor
266                      Saguinus_fuscicollis       Saguinus      fuscicollis
267         Saguinus_fuscicollis_melanoleucus       Saguinus      fuscicollis
268                        Saguinus_geoffroyi       Saguinus        geoffroyi
269                        Saguinus_imperator       Saguinus        imperator
270                         Saguinus_leucopus       Saguinus         leucopus
271                            Saguinus_midas       Saguinus            midas
272                           Saguinus_mystax       Saguinus           mystax
273                            Saguinus_niger       Saguinus            niger
274                          Saguinus_oedipus       Saguinus          oedipus
275                      Saguinus_tripartitus       Saguinus      tripartitus
276                       Saimiri_boliviensis        Saimiri      boliviensis
277                         Saimiri_oerstedii        Saimiri        oerstedii
278                          Saimiri_sciureus        Saimiri         sciureus
279                             Saimiri_ustus        Saimiri            ustus
280                    Semnopithecus_entellus  Semnopithecus         entellus
281                  Symphalangus_syndactylus   Symphalangus      syndactylus
282                          Tarsius_bancanus        Tarsius         bancanus
283                          Tarsius_dentatus        Tarsius         dentatus
284                           Tarsius_lariang        Tarsius          lariang
285                          Tarsius_syrichta        Tarsius         syrichta
287                    Trachypithecus_auratus Trachypithecus          auratus
288                  Trachypithecus_cristatus Trachypithecus        cristatus
289                  Trachypithecus_delacouri Trachypithecus        delacouri
290                  Trachypithecus_francoisi Trachypithecus        francoisi
291                       Trachypithecus_geei Trachypithecus             geei
292                   Trachypithecus_germaini Trachypithecus         germaini
293                     Trachypithecus_johnii Trachypithecus           johnii
294                     Trachypithecus_laotum Trachypithecus           laotum
295                   Trachypithecus_obscurus Trachypithecus         obscurus
296                    Trachypithecus_phayrei Trachypithecus          phayrei
297                   Trachypithecus_pileatus Trachypithecus         pileatus
298              Trachypithecus_poliocephalus Trachypithecus    poliocephalus
299                    Trachypithecus_vetulus Trachypithecus          vetulus
300                             Varecia_rubra        Varecia            rubra
301               Varecia_variegata_variegata        Varecia        variegata
subspecies spp_id genus_id social_learning research_effort  brain
1               <NA>      1        1               0               6  58.02
2               <NA>      2        2               0               6     NA
3               <NA>      3        3               0              15  52.84
4               <NA>      4        3               0              45  52.63
5               <NA>      5        3               0              37  51.70
6               <NA>      6        3               3              79  49.88
7               <NA>      7        3               0              25  51.13
8               <NA>      8        3               0               4  59.08
9               <NA>      9        3               0              82  55.22
10              <NA>     10        4               0              22  20.67
11       boliviensis     11        4              NA              NA     NA
12              <NA>     12        4               0              NA     NA
13              <NA>     13        4               0               6     NA
14              <NA>     14        4               0              16  16.30
15      griseimembra     15        4              NA              NA     NA
16              <NA>     16        4               0               5     NA
17              <NA>     17        4               0               1     NA
18              <NA>     18        4               0              58  16.85
19              <NA>     19        4               0              12     NA
20              <NA>     20        5              NA              NA     NA
21              <NA>     21        6              NA              NA   5.88
22              <NA>     22        6               0               1   6.92
23              <NA>     23        7               0              12 117.02
24              <NA>     24        7               0               4 114.24
25              <NA>     25        7               2              58 105.09
26              <NA>     26        7               0              30 103.85
27              <NA>     27        8              NA              NA     NA
28              <NA>     28        8               0              10   9.86
29              <NA>     29        8               0               6   7.95
30              <NA>     30        8              NA              NA     NA
31              <NA>     31        9               0              57     NA
32              <NA>     32       10               0              24 110.68
33              <NA>     33       11               0              11  76.00
34              <NA>     34       11               0               8  68.77
35              <NA>     35       12               0               1     NA
36              <NA>     36       12               0              NA     NA
37              <NA>     37       12               0              18     NA
38              <NA>     38       12               0              19     NA
39              <NA>     39       12               0               4     NA
40              <NA>     40       13               0              43  11.43
41              <NA>     41       14               0              16   7.95
42              <NA>     42       14               0              NA     NA
43              <NA>     43       14              NA              NA     NA
44              <NA>     44       14               0              NA     NA
45              <NA>     45       14               0               4     NA
46              <NA>     46       14               2             161   7.24
47              <NA>     47       14               0              NA     NA
48              <NA>     48       14               0              NA     NA
49              <NA>     49       14               0              NA   7.32
50              <NA>     50       14               0              36   4.17
51              <NA>     51       15               1              13  65.45
52              <NA>     52       15              17             249  66.63
53              <NA>     53       15               5              60  72.93
54              <NA>     54       15               0              18  69.84
55              <NA>     55       15              NA              NA  66.09
56              <NA>     56       16              NA              NA 116.43
57              <NA>     57       16               0              19  99.07
58              <NA>     58       16               1              32 105.99
59              atys     59       16              NA              NA  94.68
60              <NA>     60       17              NA              NA  70.10
61              <NA>     61       17               1              26  59.58
62              <NA>     62       17               0              11  57.39
63             lowei     63       17              NA              NA  55.64
64              <NA>     64       17               0               8  65.26
65            cephus     65       17              NA              NA     NA
66       ngottoensis     66       17              NA              NA     NA
67              <NA>     67       17               1              28  62.61
68              <NA>     68       17               0               3     NA
69     erythrogaster     69       17              NA              NA     NA
70              <NA>     70       17               0               3  65.40
71              <NA>     71       17               0               4  65.90
72              <NA>     72       17               0               7  74.20
73              <NA>     73       17               0              56  71.33
74              <NA>     74       17               0               8  61.84
75              <NA>     75       17               0              17  65.97
76              <NA>     76       17               0               7  71.13
77              <NA>     77       17               0               5  55.08
78              <NA>     78       17               0               8  59.56
79              <NA>     79       17               0               2     NA
80              <NA>     80       17               0               6     NA
81              <NA>     81       17               0               7  61.45
82              <NA>     82       18              NA              NA     NA
83              <NA>     83       18               0               3   5.81
84              <NA>     84       18               0              13   2.60
85              <NA>     85       19               0              21  48.33
86              <NA>     86       20               5              91  65.00
87              <NA>     87       20              NA              NA  62.58
88         cynosurus     88       20              NA              NA     NA
89              <NA>     89       20              NA              NA  64.91
90              <NA>     90       20              NA              NA     NA
91              <NA>     91       21               0              16  77.70
92         palliatus     92       21              NA              NA     NA
93              <NA>     93       21               0              42  74.39
94              <NA>     94       21               0              17  73.83
95              <NA>     95       21               0              10  74.90
96              <NA>     96       21              NA              NA  73.07
97              <NA>     97       22               0              52  44.85
98              <NA>     98       23               2              33  97.73
99              <NA>     99       24               0              11  20.65
100        albifrons    100       24              NA              NA  21.45
101     albocollaris    101       24              NA              NA  22.10
102         collaris    102       24              NA              NA     NA
103           fulvus    103       24               1              81  25.77
104      mayottensis    104       24              NA              NA     NA
105            rufus    105       24              NA              NA  25.40
106         sanfordi    106       24              NA              NA     NA
107       flavifrons    107       24              NA              NA     NA
108           macaco    108       24               0              32  24.51
109             <NA>    109       24               0              13  20.17
110             <NA>    110       24               0              13  26.23
111             <NA>    111       25               0               1   5.53
112             <NA>    112       26               0               2   5.58
113             <NA>    113       26              NA              NA     NA
114             <NA>    114       26               0              NA   4.07
115             <NA>    115       26              NA              NA   4.62
116             <NA>    116       26               0              14   3.71
117             <NA>    117       26               0              20   3.96
118             <NA>    118       27               0               5   2.65
119             <NA>    119       27               0              NA   3.51
120             <NA>    120       28              NA              NA 491.27
121          gorilla    121       28              13             517 490.41
122          graueri    122       28              NA              NA     NA
123             <NA>    123       29               0               5     NA
124             <NA>    124       29               0              40  14.09
125      alaotrensis    125       29              NA              NA  13.80
126          griseus    126       29              NA              NA     NA
127     meridionalis    127       29              NA              NA     NA
128     occidentalis    128       29              NA              NA  14.20
129             <NA>    129       29               0               8  27.14
130             <NA>    130       30              NA              NA     NA
131 neanderthalensis    131       30              NA              NA     NA
132             <NA>    132       31               0              16  91.16
133             <NA>    133       31               0               4  87.99
134             <NA>    134       31               0              86 101.87
135             <NA>    135       31               0              16     NA
136             <NA>    136       31               0               5  85.13
137             <NA>    137       31               0              16  84.69
138             <NA>    138       32               0               8  34.81
139             <NA>    139       33               0              34  96.50
140             <NA>    140       34               4             103  22.90
141             <NA>    141       35               0              46  11.84
142             <NA>    142       35               0              38     NA
143             <NA>    143       35               0              85  12.83
144             <NA>    144       36              NA              NA     NA
145             <NA>    145       36              NA              NA     NA
146             <NA>    146       36               0               1   6.70
147             <NA>    147       36               0               5   7.25
148             <NA>    148       36              NA              NA     NA
149             <NA>    149       36               0               2   6.87
150             <NA>    150       36              NA              NA     NA
151             <NA>    151       36               0               1   9.75
152             <NA>    152       36              NA              NA     NA
153             <NA>    153       36               0               5   9.56
154             <NA>    154       36              NA              NA     NA
155             <NA>    155       36              NA              NA     NA
156             <NA>    156       36               0               2   8.25
157             <NA>    157       36              NA              NA     NA
158             <NA>    158       36              NA              NA     NA
159             <NA>    159       36               0              NA     NA
160             <NA>    160       37               0              34  93.97
161             <NA>    161       37               0               6 101.59
162             <NA>    162       38              NA              NA   6.34
163             <NA>    163       38               0              14   5.87
164             <NA>    164       39               1              48 100.70
165             <NA>    165       39               0              17  90.46
166             <NA>    166       39              NA              NA     NA
167             <NA>    167       39               0              12  82.00
168             <NA>    168       39               7             174  63.98
169             <NA>    169       39              45             253 102.92
170             <NA>    170       39              NA              NA     NA
171             <NA>    171       39              NA              NA  85.60
172             <NA>    172       39               0              22     NA
173             <NA>    173       39              15             296  88.98
174             <NA>    174       39              NA              NA     NA
175             <NA>    175       39               3              51 105.59
176          leonina    176       39              NA              NA     NA
177           siberu    177       39              NA              NA     NA
178             <NA>    178       39               0              27  94.90
179             <NA>    179       39              NA              NA     NA
180             <NA>    180       39               0               3     NA
181             <NA>    181       39              NA              NA     NA
182             <NA>    182       39               0              34  74.87
183             <NA>    183       39               1              48  85.00
184             <NA>    184       39               0              12  69.70
185             <NA>    185       39               0              67  93.20
186             <NA>    186       39               1              42     NA
187             <NA>    187       39               2              26     NA
188             <NA>    188       40               0              18 148.00
189             <NA>    189       40               3              30 153.88
190             <NA>    190       41              NA              NA     NA
191             <NA>    191       41              NA              NA     NA
192             <NA>    192       41              NA              NA     NA
193             <NA>    193       41              NA              NA     NA
194             <NA>    194       41              NA              NA     NA
195             <NA>    195       41              NA              NA     NA
196             <NA>    196       41              NA              NA     NA
197             <NA>    197       41              NA              NA     NA
198             <NA>    198       41              NA              NA     NA
199             <NA>    199       41              NA              NA     NA
200             <NA>    200       41               0              66   1.63
201             <NA>    201       41               0              NA     NA
202             <NA>    202       41               0              NA     NA
203             <NA>    203       41               0               8   1.72
204             <NA>    204       41              NA              NA     NA
205             <NA>    205       41              NA              NA     NA
206             <NA>    206       41              NA              NA     NA
207             <NA>    207       42               0               4     NA
208             <NA>    208       43               0               3   5.80
209             <NA>    209       43              NA              NA     NA
210             <NA>    210       44               0              17  92.30
211             <NA>    211       45               0              21     NA
212             <NA>    212       45               0               4 119.38
213             <NA>    213       45               0               8     NA
214             <NA>    214       45              NA              NA     NA
215             <NA>    215       45              NA              NA     NA
216             <NA>    216       46              NA              NA  13.49
217             <NA>    217       46               0              37  10.13
218             <NA>    218       46              NA              NA     NA
219             <NA>    219       46              NA              NA   9.67
220             <NA>    220       46               0              19   7.23
221             <NA>    221       47               1              36  11.78
222             <NA>    222       47               1              12  11.50
223             <NA>    223       48               5             225 341.29
224   schweinfurthii    224       48              NA              NA 390.33
225      troglodytes    225       48             214             755 363.05
226       vellerosus    226       48              NA              NA     NA
227            verus    227       48              NA              NA 371.74
228             <NA>    228       49               4              43 167.42
229             <NA>    229       49               2             114 163.19
230             <NA>    230       49               1              78 146.17
231             <NA>    231       49               3               8 142.50
232             <NA>    232       49               5              22 178.00
233             <NA>    233       50               0              10  12.42
234             <NA>    234       51               0               1     NA
235       pallescens    235       51              NA              NA   6.68
236             <NA>    236       52               0              52  63.59
237             <NA>    237       52              NA              NA     NA
238             <NA>    238       52              NA              NA     NA
239             <NA>    239       52               1               7  57.25
240             <NA>    240       52               0              NA     NA
241             <NA>    241       52               0              NA     NA
242             <NA>    242       52              NA              NA     NA
243             <NA>    243       52              NA              NA  70.95
244             <NA>    244       52              NA              NA     NA
245             <NA>    245       53               0               7     NA
246             <NA>    246       53               0              28  32.26
247             <NA>    247       54              NA              NA 389.50
248             <NA>    248       54              86             321 377.38
249             <NA>    249       55               0              11  80.30
250             <NA>    250       55               0               6  64.85
251             <NA>    251       56               0               3  52.60
252             <NA>    252       57              NA              NA  30.19
253             <NA>    253       57              NA              NA  30.15
254             <NA>    254       57               0              28  39.80
255             <NA>    255       57              NA              NA  39.49
256             <NA>    256       57               0               9     NA
257             <NA>    257       57               1              41  26.21
258             <NA>    258       58              NA              NA     NA
259             <NA>    259       58               0              25  91.41
260             <NA>    260       59               0              11     NA
261             <NA>    261       59               0              NA     NA
262             <NA>    262       59               0              16     NA
263             <NA>    263       59               0              36 117.76
264             <NA>    264       60              NA              NA     NA
265             <NA>    265       61               0               9     NA
266             <NA>    266       61               2              81   7.94
267     melanoleucus    267       61              NA              NA     NA
268             <NA>    268       61               0              NA  10.14
269             <NA>    269       61               0              16     NA
270             <NA>    270       61               0               3   9.70
271             <NA>    271       61               0              17   9.78
272             <NA>    272       61               0              46  11.09
273             <NA>    273       61              NA              NA   9.48
274             <NA>    274       61               0             153   9.76
275             <NA>    275       61               0               5     NA
276             <NA>    276       62               0              36     NA
277             <NA>    277       62               1               4  25.07
278             <NA>    278       62               1              89  24.14
279             <NA>    279       62               0               4     NA
280             <NA>    280       63               2              98 110.93
281             <NA>    281       64               0              40 123.50
282             <NA>    282       65               0               8   3.16
283             <NA>    283       65               0               2   3.00
284             <NA>    284       65              NA              NA     NA
285             <NA>    285       65               0              10   3.36
286             <NA>    286       66               0              34 133.33
287             <NA>    287       67               0               2     NA
288             <NA>    288       67               0               8  57.86
289             <NA>    289       67               0              NA     NA
290             <NA>    290       67               0              45     NA
291             <NA>    291       67               0               7  81.30
292             <NA>    292       67              NA              NA     NA
293             <NA>    293       67               1               9  84.60
294             <NA>    294       67              NA              NA     NA
295             <NA>    295       67               0               6  62.12
296             <NA>    296       67               0              16  72.84
297             <NA>    297       67               0               5 103.64
298             <NA>    298       67              NA              NA     NA
299             <NA>    299       67               0               2  61.29
300             <NA>    300       68              NA              NA  31.08
301        variegata    301       68               0              57  32.12
body group_size gestation weaning longevity sex_maturity
1     4655.00      40.00        NA  106.15    276.00           NA
2       78.09       1.00        NA      NA        NA           NA
3     6395.00       7.40        NA      NA        NA           NA
4     5383.00       8.90    185.92  323.16    243.60      1276.72
5     5175.00       7.40        NA      NA        NA           NA
6     6250.00      13.10    185.42  495.60    300.00      1578.42
7     8915.00       5.50    185.92      NA    240.00           NA
8     6611.04         NA        NA      NA        NA           NA
9     5950.00       7.90    189.90  370.04    300.00      1690.22
10    1205.00       4.10        NA  229.69        NA           NA
11         NA         NA        NA      NA        NA           NA
12         NA         NA        NA      NA        NA           NA
13         NA         NA        NA      NA        NA           NA
14     734.00         NA    132.23   74.57    216.00       755.15
15         NA         NA        NA      NA        NA           NA
16     791.03       4.00        NA      NA        NA           NA
17     958.00       3.30        NA      NA        NA           NA
18     989.00       3.15    133.47   76.21    303.60       736.60
19     703.00       3.30        NA      NA        NA           NA
20         NA         NA        NA      NA        NA           NA
21     210.00         NA        NA      NA        NA           NA
22     309.00       1.00    133.74  109.26    156.00       298.91
23    8167.00      14.50    138.20      NA    336.00           NA
24    9025.00         NA    224.70  482.70    288.00      1799.68
25    7535.00      42.00    226.37  816.35    327.60      2104.57
26    8280.00      20.00    228.18  805.41    453.60      2104.57
27         NA         NA        NA      NA        NA           NA
28    1207.00       2.00    136.15  149.15        NA           NA
29     801.00       3.00        NA      NA        NA           NA
30         NA         NA        NA      NA        NA           NA
31   10537.31      19.60    221.75  734.82        NA      2876.24
32    6728.00       3.20    232.50  635.13        NA      2689.08
33    3165.00      23.70    180.00  339.29    324.00      1262.74
34    2935.00      30.00        NA      NA    216.00           NA
35     897.67       1.00        NA      NA        NA           NA
36    1067.61       1.00        NA      NA        NA           NA
37     958.13       2.95    164.00   58.85    303.60      1262.74
38    1390.80       2.35        NA      NA        NA           NA
39    1245.00       3.85        NA  121.66        NA      1683.65
40     484.00       6.85    153.99   66.53    214.80       413.84
41     345.00       9.50        NA      NA    201.60       701.52
42     429.00       6.00    140.00      NA        NA           NA
43     309.58         NA        NA      NA        NA           NA
44     342.00         NA        NA      NA        NA           NA
45     370.00       8.50        NA   99.01    180.00           NA
46     320.00       8.55    144.00   60.24    201.60       455.99
47     374.99         NA        NA      NA        NA           NA
48     443.79         NA        NA      NA        NA           NA
49     328.00       5.90        NA      NA        NA           NA
50     116.00       6.00    134.44   90.73    181.20       708.50
51    2735.00      25.00    158.29  270.32    528.00      1501.69
52    2936.00       7.90    154.99  263.12    541.20      1760.81
53    2861.00      18.15    161.06  514.07    657.60      2134.73
54    2931.00      11.45        NA  725.86    492.00      2525.48
55    2440.00         NA        NA      NA        NA           NA
56    7580.00         NA        NA      NA        NA           NA
57    7435.00      20.35    174.43      NA    252.00      2735.94
58    7485.00      26.85    168.98      NA    360.00      1318.86
59    8600.00      35.00    165.08      NA    321.60      1321.67
60    5620.00      32.50        NA      NA        NA           NA
61    3714.00      26.30    148.50  146.54    339.60      1718.73
62    3600.00      11.00    180.80  362.93    396.00           NA
63    3187.00         NA        NA      NA        NA           NA
64    3585.00      11.00    169.51  362.93    276.00      1521.90
65         NA         NA        NA      NA        NA           NA
66         NA         NA        NA      NA        NA           NA
67    4550.00      24.95        NA  362.93    447.60      2279.95
68    3444.88         NA        NA      NA        NA           NA
69         NA         NA        NA      NA        NA           NA
70    3250.00         NA        NA      NA        NA           NA
71    4425.00         NA        NA      NA        NA           NA
72    4710.00      17.40        NA      NA    192.00           NA
73    6109.00      16.00    138.39  688.08    325.20      2049.25
74    3719.00         NA        NA      NA    360.00           NA
75    5450.00       4.50    172.07  417.62    315.60      2076.39
76    5465.00      16.00    169.51      NA    276.00      1684.59
77    3609.00      14.00        NA      NA    228.00           NA
78    3580.00      15.00    169.51      NA    289.20      1684.59
79    5132.57       3.00        NA      NA        NA           NA
80    5256.91      10.00        NA      NA        NA           NA
81    3390.00         NA        NA      NA        NA           NA
82         NA         NA        NA      NA        NA           NA
83     400.00       1.00     70.00   47.14    180.00       420.91
84     140.00       1.00     61.79   60.65    231.60       413.84
85    3030.00      14.40    157.67      NA    216.00           NA
86    3720.00         NA        NA  217.76    379.20           NA
87    4324.00         NA        NA      NA        NA           NA
88         NA         NA        NA      NA        NA           NA
89    4312.00         NA        NA      NA        NA           NA
90         NA         NA        NA      NA        NA           NA
91    8625.00      10.90        NA      NA        NA           NA
92         NA         NA        NA      NA        NA           NA
93    8589.00       7.60    169.02  387.79    294.00      1929.19
94    9100.00      10.20    172.69  213.78    366.00      1629.84
95    8910.00      15.50    192.76      NA        NA           NA
96    7820.00      16.00        NA      NA        NA           NA
97    2555.00       1.00    166.48  197.70    291.60       834.72
98    9450.00      28.00    167.20  211.79    286.80      1246.07
99    1180.00       6.95    124.04      NA    220.80       701.52
100   2336.00         NA        NA      NA        NA           NA
101   2140.00         NA        NA      NA        NA           NA
102        NA         NA        NA      NA        NA           NA
103   2292.00       9.15    120.83  134.64    444.00       791.75
104        NA         NA        NA      NA        NA           NA
105   2220.00       9.50        NA      NA        NA           NA
106   2394.03       7.70        NA      NA        NA           NA
107        NA         NA        NA      NA        NA           NA
108   2390.00       9.20    127.49  143.28    360.00       660.75
109   1212.00       2.70    129.00  151.13    360.00      1060.70
110   1960.00       3.30    126.99  151.22        NA       566.36
111    274.00       1.00    133.45      NA    180.00           NA
112    252.00       6.00    133.00      NA    144.00       283.18
113    250.00         NA        NA      NA        NA           NA
114        NA         NA        NA      NA        NA           NA
115    210.00       1.00        NA      NA        NA           NA
116    148.00       1.00    122.29   90.46    198.00       420.91
117    194.00       3.50    126.98   93.93    204.00       330.37
118     75.00       5.50    111.00   43.47    168.00       345.24
119    143.00       1.00    120.00   59.27        NA       322.75
120 130000.00         NA        NA      NA        NA           NA
121 120950.00       6.00    257.00  920.35    648.00      3353.12
122        NA         NA        NA      NA        NA           NA
123   1562.41       3.00    142.50      NA        NA           NA
124    709.00       3.10    141.24  136.29    205.20      1003.17
125   1240.00         NA        NA      NA        NA           NA
126        NA         NA        NA      NA        NA           NA
127        NA         NA        NA      NA        NA           NA
128        NA         NA        NA      NA        NA           NA
129   2150.00       7.50    140.00      NA    144.00           NA
130  58540.63         NA    274.78  725.86   1470.00      5582.93
131        NA         NA        NA      NA        NA           NA
132   5850.00       4.20        NA      NA    528.00           NA
133   5795.00       3.00    207.59      NA        NA           NA
134   5595.00       3.20    212.91  725.86    480.00      3852.57
135   5860.81       2.15    241.20      NA        NA           NA
136   5821.00       3.20    206.70      NA        NA           NA
137   5470.00       3.25    200.16  635.13    432.00      2454.24
138   6335.00       3.10    136.50  331.34        NA      1605.69
139   7150.00      33.00    223.99  312.66    360.00      1729.33
140   2210.00      16.45    134.74  126.51    360.00       831.62
141    655.00       6.70        NA      NA        NA           NA
142    656.12       3.60        NA      NA        NA           NA
143    609.00       4.50    134.00   75.69    297.60       890.34
144        NA         NA        NA      NA        NA           NA
145        NA         NA        NA      NA        NA           NA
146    870.00       1.00        NA      NA        NA           NA
147    931.00       1.00        NA      NA        NA           NA
148        NA         NA        NA      NA        NA           NA
149    606.00       1.00    135.92  121.66    103.00       620.76
150        NA         NA        NA      NA        NA           NA
151    970.00       1.00        NA      NA        NA           NA
152        NA         NA        NA      NA        NA           NA
153    777.00       1.00    133.45   76.21    144.00       663.81
154        NA         NA        NA      NA        NA           NA
155        NA         NA        NA      NA        NA           NA
156    805.00       1.00    135.92  119.32        NA           NA
157        NA         NA        NA      NA        NA           NA
158        NA         NA        NA      NA        NA           NA
159    755.77       1.00    134.99  120.97        NA       377.57
160   6950.00      16.00    182.64  211.71    392.40      2525.48
161   6800.00      17.50        NA      NA    321.60           NA
162    267.00         NA        NA      NA        NA           NA
163    193.00       1.00    165.99  167.49    196.80       350.76
164  10300.00         NA    176.60  377.66    360.00      1570.01
165   9100.00      21.00        NA      NA        NA           NA
166        NA         NA        NA      NA        NA           NA
167   5470.00      20.20    161.06  205.24        NA      1650.01
168   4251.00      27.00    164.69  283.53    456.00      1319.50
169   9515.00      40.65    172.99  265.04    396.00      1460.77
170        NA         NA        NA      NA        NA           NA
171   5642.00         NA        NA      NA        NA           NA
172   7290.30         NA    167.19  497.16        NA           NA
173   6793.00      38.50    166.07  304.16    432.00      1101.07
174        NA         NA        NA      NA        NA           NA
175   8821.00      22.60    171.00  292.60    411.60      1427.17
176        NA         NA        NA      NA        NA           NA
177        NA         NA        NA      NA        NA           NA
178   7680.00      35.00    172.43  365.00    216.00      1984.51
179        NA      14.50        NA      NA        NA           NA
180   3400.00         NA        NA      NA        NA           NA
181   4534.66         NA        NA  272.20    360.00      1227.12
182   5084.00      33.50    161.56  332.25    360.00      1785.78
183   7500.00      21.00    172.00  362.93    480.00      1912.19
184   4440.00      20.10    180.90      NA    420.00      1894.11
185  12078.00      18.30    164.84  210.25    264.00      1542.25
186  10593.06      21.00    169.02  451.79        NA           NA
187  10035.53         NA        NA      NA        NA           NA
188  15000.00      17.00    179.22  486.66    400.80      1745.96
189  23600.00      13.90    173.99  348.01    555.96      2122.11
190     33.45         NA        NA      NA        NA           NA
191        NA         NA        NA      NA        NA           NA
192        NA         NA        NA      NA        NA           NA
193     70.24         NA        NA      NA        NA           NA
194        NA         NA        NA      NA        NA           NA
195        NA         NA        NA      NA        NA           NA
196        NA         NA        NA      NA        NA           NA
197        NA         NA        NA      NA        NA           NA
198        NA         NA        NA      NA        NA           NA
199        NA         NA        NA      NA        NA           NA
200     65.00       1.00     60.34   40.45    186.00       355.53
201     31.23       1.00     59.99      NA        NA           NA
202     58.60         NA        NA      NA        NA           NA
203     43.00       1.00     59.99   40.00    144.00           NA
204     49.06         NA        NA      NA        NA           NA
205        NA         NA        NA      NA        NA           NA
206     68.01         NA        NA      NA        NA           NA
207   1248.86      91.20    164.38  178.98    370.80      1733.36
208    312.00       1.00     88.58  136.00    183.60       343.74
209        NA         NA        NA      NA        NA           NA
210  14561.00      11.25    165.04  211.75    252.00      1894.11
211   6410.47       4.00    205.81  635.13    529.20      2454.24
212   7365.00       1.00        NA      NA        NA           NA
213   7320.00       1.00        NA      NA        NA           NA
214        NA         NA        NA      NA        NA           NA
215        NA         NA        NA      NA        NA           NA
216   1060.00         NA        NA      NA        NA           NA
217    653.00       1.00    191.09  181.21    318.00       660.82
218        NA         NA        NA      NA        NA           NA
219    634.00         NA        NA      NA        NA           NA
220    307.00       1.00    185.42      NA        NA           NA
221   1150.00       3.50    131.04  124.62    225.60       609.86
222    764.00       1.00    132.24  139.20    204.00       592.15
223  39100.00      85.00    235.24 1081.31    576.00      5465.72
224  38200.00         NA        NA      NA        NA           NA
225  52750.00      50.00    231.49 1260.81    720.00      3897.96
226        NA         NA        NA      NA        NA           NA
227  43950.00         NA        NA      NA        NA           NA
228  18150.00      40.00    178.96  596.60    302.40           NA
229  17150.00      48.20    172.99  450.42    540.00      2560.56
230  14150.00      36.90    180.00  363.96    450.00      1652.37
231  18026.05         NA    184.42      NA    480.00           NA
232  22300.00      47.00    185.92  877.09    540.00      1543.35
233    835.00       1.00    193.00  149.15    312.00       561.58
234    409.87       1.00    174.46      NA    144.00           NA
235    339.00         NA        NA      NA        NA           NA
236   8285.00      34.00    151.41  783.93        NA      1473.20
237        NA         NA        NA      NA        NA           NA
238        NA         NA        NA      NA        NA           NA
239   5630.00      33.60    165.00      NA        NA           NA
240  10896.00         NA        NA      NA        NA           NA
241   8865.71      40.00    195.00      NA        NA           NA
242   8030.75      24.50    195.00      NA        NA           NA
243   8409.00      34.00        NA      NA        NA           NA
244        NA         NA        NA      NA        NA           NA
245   2308.17       4.40        NA      NA        NA           NA
246   1760.00       2.70    161.13  113.15    248.40      1089.37
247  62815.00         NA        NA      NA        NA           NA
248  58542.00       1.00    259.42 1088.80    720.00      3318.62
249   6695.00       7.05        NA      NA        NA           NA
250   6560.00      14.00        NA      NA    192.00           NA
251   4450.00       6.30    167.84      NA        NA           NA
252   3729.00       5.50    140.99  180.96        NA           NA
253   3532.00         NA        NA      NA        NA           NA
254   6130.00       4.95    152.08  256.27        NA      1683.65
255   5682.00       6.00        NA      NA        NA           NA
256   3531.39       4.10        NA  152.13        NA           NA
257   2955.00       6.30    149.77  177.83    247.20       943.94
258        NA         NA        NA      NA        NA           NA
259   9720.00       9.30    182.88      NA    300.00           NA
260   9086.19      30.00    200.00      NA        NA           NA
261  11000.54      50.00    170.00      NA        NA       755.15
262  12267.15         NA    200.00      NA        NA           NA
263  14750.00      65.00    199.34      NA        NA           NA
264        NA         NA        NA      NA        NA           NA
265    465.00       6.70    158.16      NA        NA           NA
266    401.00       6.00    148.00   90.10    294.00       406.61
267        NA         NA        NA      NA        NA           NA
268    517.00       6.90        NA      NA        NA           NA
269    407.91       5.00        NA      NA    242.40           NA
270    525.00       7.50    142.50      NA        NA           NA
271    563.00       5.55    138.24   69.60    184.80       841.82
272    584.00       5.40    148.28      NA        NA       556.85
273    375.00         NA        NA      NA        NA           NA
274    431.00       7.05    166.49   49.85    277.20       680.38
275    385.05         NA        NA      NA        NA           NA
276    799.45      60.00    157.79      NA        NA           NA
277    789.00      25.10    161.00  362.93        NA           NA
278    799.00      34.85    164.09  177.41    324.00      1399.88
279    886.47         NA        NA  238.64        NA           NA
280  14742.00      19.00    197.70  402.10    300.00      1497.64
281  11295.00       3.80    230.66  635.38    456.00      3788.23
282    126.00       1.00    125.84   78.55    144.00       658.68
283    113.00       1.00        NA      NA        NA           NA
284        NA         NA        NA      NA        NA           NA
285    126.00       1.00    177.99   82.49    180.00           NA
286  15350.00      10.00    178.64  494.95    336.00      1894.11
287   9719.60      11.00        NA      NA        NA           NA
288   6394.00      27.40        NA  362.93    373.20           NA
289        NA         NA        NA      NA        NA           NA
290   8139.93         NA        NA  391.76        NA           NA
291  10150.00      11.00        NA      NA        NA           NA
292        NA         NA        NA      NA        NA           NA
293  11600.00      10.00        NA      NA        NA           NA
294        NA         NA        NA      NA        NA           NA
295   7056.00      10.00    146.63  362.93    300.00           NA
296   7475.00      12.90    180.61  305.87        NA           NA
297  11794.00       8.50        NA      NA        NA           NA
298        NA         NA        NA      NA        NA           NA
299   6237.00       8.35    204.72  245.78    276.00      1113.70
300   3470.00         NA        NA      NA        NA           NA
301   3575.00       2.80    102.50   90.73    384.00       701.52
maternal_investment
1                    NA
2                    NA
3                    NA
4                509.08
5                    NA
6                681.02
7                    NA
8                    NA
9                559.94
10                   NA
11                   NA
12                   NA
13                   NA
14               206.80
15                   NA
16                   NA
17                   NA
18               209.68
19                   NA
20                   NA
21                   NA
22               243.00
23                   NA
24               707.40
25              1042.72
26              1033.59
27                   NA
28               285.30
29                   NA
30                   NA
31               956.57
32               867.63
33               519.29
34                   NA
35                   NA
36                   NA
37               222.85
38                   NA
39                   NA
40               220.52
41                   NA
42                   NA
43                   NA
44                   NA
45                   NA
46               204.24
47                   NA
48                   NA
49                   NA
50               225.17
51               428.61
52               418.11
53               675.13
54                   NA
55                   NA
56                   NA
57                   NA
58                   NA
59                   NA
60                   NA
61               295.04
62               543.73
63                   NA
64               532.44
65                   NA
66                   NA
67                   NA
68                   NA
69                   NA
70                   NA
71                   NA
72                   NA
73               826.47
74                   NA
75               589.69
76                   NA
77                   NA
78                   NA
79                   NA
80                   NA
81                   NA
82                   NA
83               117.14
84               122.44
85                   NA
86                   NA
87                   NA
88                   NA
89                   NA
90                   NA
91                   NA
92                   NA
93               556.81
94               386.47
95                   NA
96                   NA
97               364.18
98               378.99
99                   NA
100                  NA
101                  NA
102                  NA
103              255.47
104                  NA
105                  NA
106                  NA
107                  NA
108              270.77
109              280.13
110              278.21
111                  NA
112                  NA
113                  NA
114                  NA
115                  NA
116              212.75
117              220.91
118              154.47
119              179.27
120                  NA
121             1177.35
122                  NA
123                  NA
124              277.53
125                  NA
126                  NA
127                  NA
128                  NA
129                  NA
130             1000.64
131                  NA
132                  NA
133                  NA
134              938.77
135                  NA
136                  NA
137              835.29
138              467.84
139              536.65
140              261.25
141                  NA
142                  NA
143              209.69
144                  NA
145                  NA
146                  NA
147                  NA
148                  NA
149              257.58
150                  NA
151                  NA
152                  NA
153              209.66
154                  NA
155                  NA
156              255.24
157                  NA
158                  NA
159              255.96
160              394.35
161                  NA
162                  NA
163              333.48
164              554.26
165                  NA
166                  NA
167              366.30
168              448.22
169              438.03
170                  NA
171                  NA
172              664.35
173              470.23
174                  NA
175              463.60
176                  NA
177                  NA
178              537.43
179                  NA
180                  NA
181                  NA
182              493.81
183              534.93
184                  NA
185              375.09
186              620.81
187                  NA
188              665.88
189              522.00
190                  NA
191                  NA
192                  NA
193                  NA
194                  NA
195                  NA
196                  NA
197                  NA
198                  NA
199                  NA
200              100.79
201                  NA
202                  NA
203               99.99
204                  NA
205                  NA
206                  NA
207              343.36
208              224.58
209                  NA
210              376.79
211              840.94
212                  NA
213                  NA
214                  NA
215                  NA
216                  NA
217              372.30
218                  NA
219                  NA
220                  NA
221              255.66
222              271.44
223             1316.55
224                  NA
225             1492.30
226                  NA
227                  NA
228              775.56
229              623.41
230              543.96
231                  NA
232             1063.01
233              342.15
234                  NA
235                  NA
236              935.34
237                  NA
238                  NA
239                  NA
240                  NA
241                  NA
242                  NA
243                  NA
244                  NA
245                  NA
246              274.28
247                  NA
248             1348.22
249                  NA
250                  NA
251                  NA
252              321.95
253                  NA
254              408.35
255                  NA
256                  NA
257              327.60
258                  NA
259                  NA
260                  NA
261                  NA
262                  NA
263                  NA
264                  NA
265                  NA
266              238.10
267                  NA
268                  NA
269                  NA
270                  NA
271              207.84
272                  NA
273                  NA
274              216.34
275                  NA
276                  NA
277              523.93
278              341.50
279                  NA
280              599.80
281              866.04
282              204.39
283                  NA
284                  NA
285              260.48
286              673.59
287                  NA
288                  NA
289                  NA
290                  NA
291                  NA
292                  NA
293                  NA
294                  NA
295              509.56
296              486.48
297                  NA
298                  NA
299              450.50
300                  NA
301              193.23

First, model the number of observations of social_learning for each species as a function of the log brain size. Use a Poisson distribution for the social_learning outcome variable. Interpret the resulting posterior

First, we create the log of brain size and standardize. Thus, we will have an intuitive interpretation for the intercept

To fit the model, we must set the priors. However, given that we must ensure that $$\lambda$$ must be positive, we must use a link function. Therefore, if we use normal priors on the log of $$\lambda$$, we are acutally saying that $$\lambda$$ is log normal.

[1] 1.656986
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The chains look healthy:

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3. Different chains converge to explore the same parameter space.
       mean         sd      5.5%     94.5%    n_eff    Rhat4
a -1.173153 0.12149622 -1.369680 -0.979759 337.0766 1.005123
b  2.760913 0.07808635  2.636306  2.885549 362.0357 1.005746

The Rhat also looks OK. The slope seems very accurately estimating: a whole standard deviation of more brain size seems associated with around 1.98 social_learning observations. Let’s use a posterior check:

Clearly, our modelling is off. There are more factors at play here.

Second, some species are studied much more than others. So the number of reported instances of social_learning could be a product of research effort. Use the research_effort variable, specifically its logarithm, as an additional predictor variable. Interpret the coefficient for log research_effort. Does this model disagree with the previous one?

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The chains look healthy:

1. They are stationary.
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3. Different chains converge to explore the same paths.

Let’s fit a longer chain:

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        mean         sd       5.5%      94.5%    n_eff     Rhat4
a -1.6775881 0.15234024 -1.9318421 -1.4427545 639.6881 1.0037398
b  0.4096989 0.08751584  0.2750084  0.5525733 733.6968 0.9998347
r  2.1130741 0.09568045  1.9593550  2.2640599 558.1447 1.0018470

The Rhats look OK, too. Surprisingly, the log of brain size decreased quite a bit. Let’s compare the overfitting risk of both models:

                          WAIC       SE    dWAIC      dSE     pWAIC
model_brain_research  564.7344 175.2123   0.0000       NA  53.60439
model_only_brain     1442.7986 486.5391 878.0641 406.9318 147.12908
weight
model_brain_research  1.000000e+00
model_only_brain     2.141893e-191
                          PSIS       SE    dPSIS      dSE    pPSIS
model_brain_research  546.0781 161.4227   0.0000       NA 44.27621
model_only_brain     1339.1373 415.0120 793.0592 335.8656 95.29845
weight
model_brain_research  1.000000e+00
model_only_brain     6.157072e-173

Both WAIC and PSIS agree: the model that incorporates the effect of research effort makes better predictions. Naturally, we are not saying that research effor influences learning; just that it influences the measurement of learning. Given that including it in the model reduced the predicted influence of brain size, it is likely that research effort is positively linked with brain size. That is, reseraches study more the primates with larger brain sizes.

WAIC and LOOk can help us to clarify what is going on here. Essentially, a cluster of species are driving the model: the ones that highly researched. If true, the models should predict equivalent predictions out of sample for each of the observations, except the ones with high research efforts. Both LOO and WAIC give us pointwise overfitting penalties; that is, pointwise measures of where the model is having trouble predicting. If we compare these quantities, we should identify the observations that are driving the difference in predictive performance:

The difference in predictions, and in accuracy, is entirely driven by a couple of observations. Finally, the DAG: