Content uploaded by William Arthur Hoffmann
Author content
All content in this area was uploaded by William Arthur Hoffmann on Sep 25, 2019
Content may be subject to copyright.
1
Appendix S1
1
William A. Hoffmann, R. Wyatt Sanders, Michael G Just, Wade A. Wall, Matthew G. Hohmann.
2
Better lucky than good: How savanna trees escape the fire trap in a variable world. Ecology
3
Formalizing the latent variable interpretation of fire severity
4
In the latent-variable model of logistic regression (Tutz 2011), also known as a threshold model
5
(Snijders and Bosker 1999), a binary outcome is considered to be governed by whether an
6
underlying continuous variable exceeds a threshold value. Applied to the current study, a stem
7
survives a fire if the stem is larger than a threshold size, and the threshold differs among
8
individuals owing to variability in fire severity. The threshold for a particular individuals is not
9
directly observable, thus it is a latent quantity. That is, stem survival of individual i is determined
10
by two quantities: size of stem (Xi) and fire severity (si). We define fire severity as the threshold
11
stem size at which the stem would be able to resist the fire. Fire severity within a site is variable,
12
and can be expressed as
13
si = T + cɛi (eqn. S1)
14
Where T is mean severity and ɛi is a random variable having a standard logistic distribution
15
and c is a scalar. Note that this description differs in a trivial way from the typical formulation of
16
the latent variable model, but below we demonstrate that our approach is mathematically
17
equivalent to the typical model, thus allowing us to adopt standard approaches for fitting our
18
model and analyzing the results.
19
Stem survival (Yi) is a binary variable defined to be Yi = 1 if a stem survives fire and Yi = 0
20
if the stem is topkilled. According to our model, a stem survives if stem diameter exceeds a
21
threshold, defined as the fire severity to which the individual was subjected. That is
22
1if
0 otherwise
c
ii
i
YXT
(eqn. S2)
23
2
By rearranging terms, we have
24
1if - 0
0 otherwise
-
c
ii
i
X
YT
(eqn. S3),
25
and
26
1if 0
0 otherwise
ii
i
XT
Ycc
(eqn S4).
27
Because ei is symmetric, this can be rewritten as
28
1if 0
0 otherwise
ii
i
XT
Ycc
(eqn. S5),
29
The term
ii
XT
cc
is a latent variable, which we denote as Li. This variable can be
30
expressed as
31
Li = β1Xi + β0 + ɛi (eqn. S6)
32
Where
33
11
c
(eqn. S7)
34
and
35
0T
c
(eqn. S8)
36
Equation S6 is the form typically utilized in latent-variable logistic regression (Snijders and
37
Bosker 1999, Tutz 2011), and β1 and βo are estimated as the logistic slope and intercept fitted
38
with logistic regression (Snijders and Bosker 1999, Tutz 2011). Based on equations S7 and S8,
39
the mean severity (T) is
40
3
0
1
T
(eqn. S9)
41
It is noteworthy that this mean severity (or mean threshold for stem survival) is equal to the stem
42
diameter at which 50% of stems are predicted to survive fire.
43
The variance of the standard logistic distribution is
2
3
(Snijders and Bosker 1999), so from
44
equation 1, the variance in fire severity is
45
22
3
()Var s c Var c
(eqn. S10).
46
From equations S7 and S10,
47
122
1
3
Var s
(eqn. S11).
48
To partition the variance in fire severity into among-site variation and within-site variation, the
49
latent-variable model can be extended to include site effects in a random-intercept model,
50
Li = β1Xij + β0 + uj + ɛij (eqn. S12)
51
where uj ~ N(0,σu2) is the between-site variation in severity. The parameters for this model can
52
be estimated with a mixed-model GLM with random intercepts. The proportion of the variance
53
attributable to between-site variation in severity is estimated as
2
22
ˆ
ˆ/3
s
s
(Snijders and Bosker
54
1999, page 224), where
2
ˆs
is the variance component for the random effect obtained from the
55
GLM output.
56
57
58
59
60
4
Statistical details for figure 2B of main text.
61
Single-factor logistic model (binomial glm) in which data from all transects are pooled.
62
Model: tk~logdia
63
Coefficients:
64
Estimate Std. Error z value Pr(>|z|)
65
(Intercept) 11.3146 0.5143 22.00 <2e-16 ***
66
logdia -6.4175 0.3269 -19.63 <2e-16 ***
67
68
Mixed model GLM with site as a random factor.
69
Model: tk~logdia+(1|site)
70
Random effects:
71
Groups Name Variance Std.Dev.
72
site (Intercept) 8.6 2.948
73
Number of obs: 1306, groups: site, 16
74
75
Fixed effects:
76
Estimate Std. Error z value Pr(>|z|)
77
(Intercept) 19.5014 1.3826 14.11 <2e-16 ***
78
logdia -11.1980 0.7023 -15.95 <2e-16 ***
79
80
81
82
83
5
Mean fire interval (yr)
0 2 4 6 8 10
Escape probability (%)
0
20
40
60
80
100 All Constant
Variable Growth
Variable interval
Variable severity
Variable growth & interval
Variable growth & severity
Variable interval & severity
All variable
84
Figure S1. Full factorial of simulation scenarios. These results are for 100-year simulations for
85
10,000 individuals in each scenario and mean fire interval.
86
87
88
89
90