We will give below an example of the limited sampling bias of mutual information. In this simulated example, the samples were taken truly randomly and correctly from the underlying probability distributions, and thus there was no sampling bias. However, due to limited sampling, the estimated probabilities red line of Figure 2 A and Figure 2 B differ markedly from 0. This shows that the mutual information estimate suffers from limited sampling bias.
The greater the number of samples, the smaller the fluctuations in the estimated probabilities, and consequently the smaller the limited sampling bias.
Similar problems apply also to measures of causal relationships such as Granger causality and transfer entropy. Note that the limited sampling bias arises because mutual information is a nonlinear function of the probabilities. The probabilities themselves would be unaffected by limited sampling bias, because they would average to the true probabilities over many repetitions of the experiment with a finite number of data.
Limited sampling bias can be corrected by computing its approximated value analytically and subtracting it out, or by using prior information about the underlying probability distributions to reduce their statistical sampling fluctuations Panzeri et al. Over recent years there has been growing interest in the effect of sampling bias and of limited sampling bias in neuroscience.
An important problem in sensory neuroscience is to understand how networks of neurons represent and exchange sensory information by means of their coordinated pattern of response to stimuli. A widely used empirical approach to this problem is to record extracellularly the action potentials emitted by neurons. Extracellular electrodes are often placed in a brain location selected because action potentials can be detected. It is recognized that this procedure may bias the sampling toward larger neurons emitting signals that are easier to detect and toward most active neurons Shoham et al.
This is somewhat related to the problem of 'convenience sampling' discussed above. Neuroscientists are more likely to report the behavior of those neurons that are most easily "conveniently" observed with the methods at their disposal. Correcting this sampling bias requires recording also from smaller and less active neurons and evaluating, using various types of anatomical and functional information, the relative distributions of different types of neural populations.
The implications of this sampling problem and ways to take it into account are discussed in Shoham et al. The limited sampling bias gives problems in the determination of the causal relation between sensory stimuli and certain features of the neuronal population responses, because it may artificially increase the mutual information available in complex characterizations of the neuronal responses such as those based on the precise times of action potentials over the information available in simpler characterization of the neuronal activity such as those which neglect the details of the temporal structure of the neuronal response.
The implications of this sampling problem and ways to correct for it are discussed in Panzeri et al. Even randomized samples will have some degree of sampling error because a sample is only an approximation of the population from which it is drawn. There are different categories of sampling errors. A population-specific error occurs when a researcher doesn't understand who to survey. Selection error occurs when the survey is self-selected, or when only those participants who are interested in the survey respond to the questions.
Researchers can attempt to overcome selection error by finding ways to encourage participation. A sample frame error occurs when a sample is selected from the wrong population data.
A non-response error occurs when a useful response is not obtained from the surveys because researchers were unable to contact potential respondents or potential respondents refused to respond. The prevalence of sampling errors can be reduced by increasing the sample size. As the sample size increases, the sample gets closer to the actual population, which decreases the potential for deviations from the actual population.
Consider that the average of a sample of 10 varies more than the average of a sample of Steps can also be taken to ensure that the sample adequately represents the entire population. Researchers might attempt to reduce sampling errors by replicating their study. This could be accomplished by taking the same measurements repeatedly, using more than one subject or multiple groups, or by undertaking multiple studies. Random sampling is an additional way to minimize the occurrence of sampling errors.
Random sampling establishes a systematic approach to selecting a sample. For example, rather than choosing participants to be interviewed haphazardly, a researcher might choose those whose names appear first, 10th, 20th, 30th, 40th, and so on, on the list. Assume that XYZ Company provides a subscription-based service that allows consumers to pay a monthly fee to stream videos and other types of programming via an Internet connection.
The firm wants to survey homeowners who watch at least 10 hours of programming via the Internet per week and that pay for an existing video streaming service.
XYZ wants to determine what percentage of the population is interested in a lower-priced subscription service. If XYZ does not think carefully about the sampling process, several types of sampling errors may occur.
A population specification error would occur if XYZ Company does not understand the specific types of consumers who should be included in the sample. For example, if XYZ creates a population of people between the ages of 15 and 25 years old, many of those consumers do not make the purchasing decision about a video streaming service because they may not work full-time.
On the other hand, if XYZ put together a sample of working adults who make purchase decisions, the consumers in this group may not watch 10 hours of video programming each week. Selection error also causes distortions in the results of a sample. A common example is a survey that only relies on a small portion of people who immediately respond.
There are different types of errors that can occur when gathering statistical data. Sampling errors are the seemingly random differences between the characteristics of a sample population and those of the general population.
The extent of this non-response error can be checked through follow-up surveys using alternate modes. As described previously, sampling errors occur because of variation in the number or representativeness of the sample that responds.
Sampling errors can be controlled and reduced by 1 careful sample designs, 2 large enough samples check out our online sample size calculator , and 3 multiple contacts to assure a representative response. Be sure to keep an eye out for these sampling and non-sampling errors so you can avoid them in your research. Sample Size. Market Research. Just a minute! It looks like you entered an academic email. This form is used to request a product demo if you intend to explore Qualtrics for purchase.
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We're hiring! View Careers. Qualtrics Life Read more. Show 1 more comment. Active Oldest Votes. Improve this answer. Add a comment. Form this identity we can see that in the context of estimators, bias is an error because it is a component of the mean square error.
What I present here is about the terms error and bias for estimators but I think the principles hold true for the words as they are used in statistics in general: One can decompose error into a systematic and an unsystematic component.
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