Bootstrapping Regression Models
Appendix to An R and S-PLUS Companion to Applied Regression
John Fox
January 2002
1
Basic Ideas
Bootstrapping is a general approach to statistical inference based on building a sampling distribution for a statistic by resampling from the data at hand. The term ‘bootstrapping,’ due to Efron (1979), is an allusion to the expression ‘pulling oneself up by one’s bootstraps’ – in this case, using the sample data as a population from which repeated samples are drawn. At first blush, the approach seems circular, but has been shown to be sound.
Two S libraries for bootstrapping are associated with extensive treatments of the subject: Efron and
Tibshirani’s (1993) bootstrap library, and Davison and
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∗
Next, we compute the statistic T for each of the bootstrap samples; that is Tb = t(S∗ ). Then the b ∗ distribution of Tb around the original estimate T is analogous to the sampling distribution of the estimator
T around the population parameter θ. For example, the average of the bootstrapped statistics,
∗
T = E ∗ (T ∗ ) =
R b=1 R
∗
Tb
∗
estimates the expectation of the bootstrapped statistics; then B ∗ = T − T is an estimate of the bias of T , that is, T − θ. Similarly, the estimated bootstrap variance of T ∗ ,
V ∗ (T ∗ ) =
R
∗
b=1 (Tb
∗
− T )2
R−1
estimates the sampling variance of T .
The random selection of bootstrap samples is not an essential aspect of the nonparametric bootstrap:
At least in principle, we could enumerate all bootstrap samples of size n. Then we could calculate E ∗ (T ∗ ) and V ∗ (T ∗ ) exactly, rather than having to estimate them. The number of bootstrap samples, however, is astronomically large unless n is tiny.2 There are, therefore, two sources of error in bootstrap inference: (1) the error induced by using a particular sample S to represent the population; and (2) the sampling error produced by failing to enumerate all bootstrap samples. The latter source of error can be controlled by making the number of bootstrap replications R sufficiently large.
2
Bootstrap Confidence Intervals
There are several approaches to
* Inferential statistics aims to draw conclusions about the population from the sample at hand. For example, it may try to infer the success rate of a drug in treating high temperature, by taking a sample of
Theoretical sampling, this involves data collection for comparative analysis (Glaser & Strauss 1967). The collected data then gives insights and the analysis leads to further data collection and analysis.
A sample can include any object or characteristic in a population. It is necessary to use samples for research, because it is impractical to study the whole population. The author is asking the student; How can we make inferences about whole populations from samples drawn from the population? By inferential statistics.
For practical reasons, variables are observed to collect data. The sampled data is then analyzed to elicit information for decision making in business and indeed in all human endeavors. However, sampled information is incomplete and not free from sampling error. Its use in decision-making processes introduces an element of chance. Therefore, it is important for a decision-maker to know the amount of chance associated with a statistical decision of it being wrong. To quantify the amount of chance due to sampling error, basic probability concepts are indispensable via modeling sampled populations and testing of research
The answer is be because the number “56” is not part of the original data so it cannot be used in bootstrapping.
Sampling error is the chance that the differences viewed in a measure of a sample group are due to chance and randomness. To avoid sampling errors, we run a statistical test that can help us find if
The sample size increases affect the estimate. As the sample size increases, the margin of error decreases.
The basis of sample surveying is to use a representative selection within a population to define the larger population as a whole. The sample survey should therefore reflect the characteristic make-up of the wider population and where
The main strength of Simple Random Sampling that it is most likely to produce representative samples and permits the use of inferential
For example, if researcher want to sample 10 houses from a street of 140 houses. 140/10=14, therefore every 14th house will be chosen after a random starting point between 1 and 14.
Since the population size is always larger than the sample size, then the sample statistic
There are two kinds of risk of sample, sampling risk and nonsampling risk, during the process of sampling. Sample always causes risk because the sample does not include all of the information of the whole population in the test. Thus, it is entirely possible that the sample does not show the correct result of examination to the test engagement team. For example, in the case of Wilson Corporation, the engagement team of Wilson tested 50 golfers as sample for calculating the distances for the new golf to make sure whether that Wilson’s golf balls provided an increase of distance. The engagement team is interested in determining whether the increase in distance is more than five yards. We assume that true average increase in distance is seven yards if the engagement tests all of the golf players’ data. So the team can get the correct
Welcome back! This power point is not as complex or as long as the previous power point. However, we’ll review very interesting concepts that you have heard before, such as estimation, hypothesis testing, and statistical significance. These are foundational concepts that will be used when we conduct inferential statistical techniques. I hope that you find the powerpoints helpful.
To determine sample size for proportion you must know the desired level of confidence (1 -∝), which determines the critical Z value, The acceptable sampling error (e), and the true proportion of ‘successes’,(π) . π can be estimated with past data, a pilot sample, or conservatively use π = 0.5
As the sample size becomes larger, the sampling distribution of the sample mean approaches a