Question

The data set entitled ‘Salmon’ contains 40 annual counts of the numbers of recruits
and spawners in a salmon population. The units are thousands of fish. Recruits are fish that
enter the catchable population. Spawners are fish that are laying eggs. Spawners die after
laying eggs.
The classic Beaverton-Holt model for the relationship between spawners and recruits is
R = 1 /(β1 + β2/S)
where R and S are the numbers of recruits and spawners, respectively. This model may be fit
using linear regression with the transformed variables R1 and S1 .
Now consider the problem of maintaining a sustainable fishery. The total population abundance
will only stabilize if R = S. The total population will decline if fewer recruits are produced
than the number of spawners who died producing them. If too many recruits are produced, the
population will also decline eventually because there is not enough food for them all. Thus,
only some middle level of recruits can be sustained indefinitely in a stable population. This
stable population level is the point where the 45-degree line intersects the curve relating R
and S.
(a) Fit the Bevearton-Holt model and find a point estimate for the stable population level
where R = S. Use the bootstrap to obtain a corresponding 95% confidence interval and a
standard error (estimated variance) for your estimate, from two methods: bootstrapping
the residuals and bootstrapping the cases. Histogram each bootstrap distribution, and
comment on the differences in your results.
(b) Do you believe there is any bias in your estimate? Defend your answer using the bootstrap

Year |recruits |spawners

1 | 68 | 56
2 77 62
3 299 445
4 220 279
5 142 138
6 287 428
7 276 319
8 115 102
9 64 51
10 206 289
11 222 351
12 205 282
13 233 310
14 228 266
15 188 256
16 132 144
17 285 447
18 188 186
19 224 389
20 121 113
21 311 412
22 166 176
23 248 313
24 161 162
25 226 368
26 67 54
27 201 214
28 267 429
29 121 115
30 301 407
31 244 265
32 222 301
33 195 234
34 203 229
35 210 270
36 275 478
37 286 419
38 275 490
39 304 430
40 214 235

Answer 1

Here, we fit the regression R1 = B1 + B2 S1

boot(data = data , statistic = getRegr, R = 1000)

below shown the result obtained from R

Bootstrap Statistics :
original bias std. error
t1* 66.2668909 0.5314801728 8.46766690
t2* 0.5113531 -0.0007441989 0.03122438

The table showes the bootstrap estimates and standard errors of B1 and B2

66.266   8.47

0.5113   0.03

conf
[1,] 0.95 40.99 987.1 0.005045347 0.007284913

Histogram of the estimate is

b. Yes, there is a bias in the estimates.

Computing ANOVA we can test the following