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# Appendix_S1.pdf

Authors:
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Appendix S1
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William A. Hoffmann, R. Wyatt Sanders, Michael G Just, Wade A. Wall, Matthew G. Hohmann.
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Better lucky than good: How savanna trees escape the fire trap in a variable world. Ecology
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Formalizing the latent variable interpretation of fire severity
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In the latent-variable model of logistic regression (Tutz 2011), also known as a threshold model
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(Snijders and Bosker 1999), a binary outcome is considered to be governed by whether an
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underlying continuous variable exceeds a threshold value. Applied to the current study, a stem
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survives a fire if the stem is larger than a threshold size, and the threshold differs among
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individuals owing to variability in fire severity. The threshold for a particular individuals is not
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directly observable, thus it is a latent quantity. That is, stem survival of individual i is determined
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by two quantities: size of stem (Xi) and fire severity (si). We define fire severity as the threshold
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stem size at which the stem would be able to resist the fire. Fire severity within a site is variable,
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and can be expressed as
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si = T + cɛi (eqn. S1)
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Where T is mean severity and ɛi is a random variable having a standard logistic distribution
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and c is a scalar. Note that this description differs in a trivial way from the typical formulation of
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the latent variable model, but below we demonstrate that our approach is mathematically
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equivalent to the typical model, thus allowing us to adopt standard approaches for fitting our
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model and analyzing the results.
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Stem survival (Yi) is a binary variable defined to be Yi = 1 if a stem survives fire and Yi = 0
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if the stem is topkilled. According to our model, a stem survives if stem diameter exceeds a
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threshold, defined as the fire severity to which the individual was subjected. That is
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1if
0 otherwise
c
ii
i
YXT

(eqn. S2)
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2
By rearranging terms, we have
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1if - 0
0 otherwise
-
c
ii
i
X
YT
(eqn. S3),
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and
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1if 0
0 otherwise
ii
i
XT
Ycc
 
(eqn S4).
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Because ei is symmetric, this can be rewritten as
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1if 0
0 otherwise
ii
i
XT
Ycc
 
(eqn. S5),
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The term
ii
XT
cc

is a latent variable, which we denote as Li. This variable can be
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expressed as
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Li = β1Xi + β0 + ɛi (eqn. S6)
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Where
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c
(eqn. S7)
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and
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0T
c

(eqn. S8)
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Equation S6 is the form typically utilized in latent-variable logistic regression (Snijders and
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Bosker 1999, Tutz 2011), and β1 and βo are estimated as the logistic slope and intercept fitted
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with logistic regression (Snijders and Bosker 1999, Tutz 2011). Based on equations S7 and S8,
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the mean severity (T) is
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3
(eqn. S9)
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It is noteworthy that this mean severity (or mean threshold for stem survival) is equal to the stem
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diameter at which 50% of stems are predicted to survive fire.
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The variance of the standard logistic distribution is
2
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(Snijders and Bosker 1999), so from
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equation 1, the variance in fire severity is
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 
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3
()Var s c Var c

(eqn. S10).
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From equations S7 and S10,
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 
122
1
3
Var s
 
(eqn. S11).
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To partition the variance in fire severity into among-site variation and within-site variation, the
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latent-variable model can be extended to include site effects in a random-intercept model,
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Li = β1Xij + β0 + uj + ɛij (eqn. S12)
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where uj ~ N(0,σu2) is the between-site variation in severity. The parameters for this model can
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be estimated with a mixed-model GLM with random intercepts. The proportion of the variance
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attributable to between-site variation in severity is estimated as
2
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ˆ
ˆ/3
s
s

(Snijders and Bosker
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1999, page 224), where
2
ˆs
is the variance component for the random effect obtained from the
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GLM output.
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Statistical details for figure 2B of main text.
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Single-factor logistic model (binomial glm) in which data from all transects are pooled.
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Model: tk~logdia
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Coefficients:
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Estimate Std. Error z value Pr(>|z|)
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(Intercept) 11.3146 0.5143 22.00 <2e-16 ***
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logdia -6.4175 0.3269 -19.63 <2e-16 ***
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Mixed model GLM with site as a random factor.
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Model: tk~logdia+(1|site)
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Random effects:
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Groups Name Variance Std.Dev.
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site (Intercept) 8.6 2.948
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Number of obs: 1306, groups: site, 16
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Fixed effects:
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Estimate Std. Error z value Pr(>|z|)
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(Intercept) 19.5014 1.3826 14.11 <2e-16 ***
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logdia -11.1980 0.7023 -15.95 <2e-16 ***
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Mean fire interval (yr)
0 2 4 6 8 10
Escape probability (%)
0
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40
60
80
100 All Constant
Variable Growth
Variable interval
Variable severity
Variable growth & interval
Variable growth & severity
Variable interval & severity
All variable
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Figure S1. Full factorial of simulation scenarios. These results are for 100-year simulations for
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10,000 individuals in each scenario and mean fire interval.
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