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

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 3 minutes ago

asked 7 minutes ago

asked 7 minutes ago

asked 7 minutes ago

asked 8 minutes ago

asked 12 minutes ago

asked 17 minutes ago

asked 21 minutes ago

asked 22 minutes ago

asked 22 minutes ago

asked 22 minutes ago

asked 22 minutes ago