A generalized linear model (GLM) with a binomial distribution and logit link function was used to model
survival probabilities as a function of treatment and length. The statistics on the fitted model are given
below.
Coefficients
Intercept (PICO-tag)
PIT-tag
Length
Estimate
-1.2303
-1.5577
0.0779
SE
1.2300
0.4967
0.0227
Z-value
-1
-3.136
3.433
P-value
0.3172
0.0017
0.0006
The coefficients of survival probability are estimated on a log odds scale. Thus, an expression for
survival probability, p, as a function of the coefficients, β , requires the inverse logit function
𝑝=
1
1
1+ ∑ 𝛽 𝑋
𝑖 𝑖
𝑒
=
1
1+𝑒 − ∑ 𝛽𝑖 𝑋𝑖
,
where X is a vector of indicators [0, 1] for categorical (treatment) effects plus the unknown length, L, of
fish that yields a desired p . The parameterized model of survival of fish of different lengths, L, with the
PIT treatment is:
𝑝=
1
.
1+𝑒 − ∑ −1.2303+0.07784∗𝐿−1.5577∗𝑃𝐼𝑇
This expression can be rearranged for L as a function of p:
1
𝑝
− 1 = 𝑒 − ∑ −1.2303+0.07784∗𝐿−1.5577∗𝑃𝐼𝑇 ,
𝐿𝑃𝐼𝑇 =
1
𝑝
log( −1)−2.7869
−0.07784
.
This expression returns lengths 53.6 mm and 64.0 mm for survival probabilities 0.8 and 0.9, respectively.
Zeroing out the effect of PIT reduces the model to an expression of length for the PICO treatment:
𝐿𝑃𝐼𝐶𝑂 =
1
𝑝
log( −1)−1.2303
−0.07784
,
which returns a length of 44.0 mm when survival probability is 0.9.
Analogous reasoning is used to convert the estimates of a GLM of survival as a function of treatment
and weight. Note that a single GLM with treatment, length, and weight would was not fitted because
length and weight are highly collinear and therefore produce unstable estimates. The statistics from the
GLM of survival as a function of treatment and weight are given below.
Coefficients
Intercept (PICO-tag)
PIT-tag
Weight
Estimate
1.4009
-1.4150
0.9265
SE
0.5685
0.4943
0.2696
Z-value
2.464
-2.863
3.437
P-value
0.0137
0.0042
0.0006
The expression for weight as a function of survival probability for fish receiving the PIT treatment is:
𝑊𝑃𝐼𝑇 =
1
𝑝
log( −1)−0.0141
−0.9265
.
This expression returns weights of 1.51 g and 2.4 g for survival probabilities 0.8 and 0.9, respectively.
Zeroing out the effect of PIT reduces the model to an expression of length for the PICO treatment:
𝑊𝑃𝐼𝐶𝑂 =
1
𝑝
log( −1)−1.4009
−0.9265
,
which returns a weight of 0.86 g when survival probability is 0.9.