Chapter 14 - Bootstrap Methods and Permutation Tests - WH Freeman
Chapter 14 - Bootstrap Methods and Permutation Tests - WH Freeman
Chapter 14 - Bootstrap Methods and Permutation Tests - WH Freeman
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<strong>14</strong>-46 CHAPTER <strong>14</strong> <strong>Bootstrap</strong> <strong>Methods</strong> <strong>and</strong> <strong>Permutation</strong> <strong>Tests</strong><br />
(b) Examine the bootstrap distribution of the slope b 1 of the least-squares regression<br />
line. The distribution shows some departures from normality. In<br />
what way is the bootstrap distribution nonnormal? What is the bootstrap<br />
estimate of bias? Based on what you see, would you consider use of bootstrap<br />
t or bootstrap percentile intervals?<br />
(c) Give the BCa 95% confidence interval for the slope β 1 of the population<br />
regression line. Compare this with the st<strong>and</strong>ard 95% confidence interval<br />
based on normality, the bootstrap t interval, <strong>and</strong> the bootstrap percentile<br />
interval. Using the BCa interval as a st<strong>and</strong>ard, which of the other intervals<br />
are adequately accurate for practical use?<br />
<strong>14</strong>.43 Table <strong>14</strong>.2 gives data on a sample of 50 baseball players.<br />
(a) Find the least-squares regression line for predicting salary from batting<br />
average.<br />
(b) <strong>Bootstrap</strong> the regression line <strong>and</strong> give a 95% confidence interval for the<br />
slope of the population regression line.<br />
(c) In the discussion of Example <strong>14</strong>.9 we found bootstrap confidence intervals<br />
for the correlation between salary <strong>and</strong> batting average. Does your interval<br />
for the slope of the population line agree with the conclusion of that<br />
example that there may be no relation between salary <strong>and</strong> batting average?<br />
Explain.<br />
<strong>14</strong>.44 We know that outliers can strongly influence statistics such as the mean <strong>and</strong><br />
the least-squares line. Example 7.7 (page 459) describes a matched pairs study<br />
of disruptive behavior by dementia patients. The differences in Table 7.2 show<br />
several low values that may be considered outliers.<br />
(a) <strong>Bootstrap</strong> the mean of the differences with <strong>and</strong> without the three low values.<br />
How do these values influence the shape <strong>and</strong> bias of the bootstrap<br />
distribution?<br />
(b) Give the BCa or tilting confidence interval from both bootstrap distributions.<br />
Discuss the differences.<br />
<strong>14</strong>.5 Significance Testing Using<br />
<strong>Permutation</strong> <strong>Tests</strong><br />
Significance tests tell us whether an observed effect, such as a difference between<br />
two means or a correlation between two variables, could reasonably<br />
occur “just by chance” in selecting a r<strong>and</strong>om sample. If not, we have evidence<br />
that the effect observed in the sample reflects an effect that is present in the<br />
population. The reasoning of tests goes like this:<br />
1. Choose a statistic that measures the effect you are looking for.<br />
2. Construct the sampling distribution that this statistic would have if the effect<br />
were not present in the population.<br />
3. Locate the observed statistic on this distribution. A value in the main body<br />
of the distribution could easily occur just by chance. A value in the tail would
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