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Reference the current page of this Newspaper. Copy Citation. Univesal Viewer Copy. We collect a sample of data to better understand the characteristics of a population. A variable is a characteristic we measure for each individual or case. The overall quantity of interest may be the mean, median, proportion, or some other summary of a population. These population values are called parameters. We estimate the value of a parameter by taking a sample and computing a numerical summary called a statistic based on that sample.
Note that the two ps population, parameter go together and the two ss sample, statistic go together. Identify the variable to be measured and the parameter and statistic of interest. The variable is the level of mercury content in swordfish in the Atlantic Ocean.
It will be measured for each individual swordfish. The parameter of interest is the average mercury content in all swordfish in the Atlantic Ocean. If we take a sample of 50 swordfish from the Atlantic Ocean, the average mercury content among just those 50 swordfish will be the statistic.
Two statistics we will study are the mean also called the average and proportion. When we are discussing a population, we label the mean as the Greek letter, mu , while we label the sample mean as x read as x-bar. When we are discussing a proportion in the context of a population, we use the label p, while the sample proportion has a label of p read as p-hat.
Generally, we use x to estimate the population mean,. Likewise, we use the sample proportion p to estimate the population proportion, p. What about p? What is the parameter of interest? The characteristic that we record on each individual is the number of years until graduation, which is a numerical variable. The parameter of interest is the average time to degree for all Duke undergraduates, and we use to describe this quantity.
Is the variable numerical or categorical? Describe the statistic that should be calculated in this study. If these topics are still a bit unclear, dont worry.
Well cover them in greater detail in the next chapter. This is categorical because it will be a yes or a no. The statistic that should be recorded is the proportion of patients that die within the time frame of the study, and we would use p to denote this quantity. As comedian Jon Stewart pointed out, Its one storm, in one region, of one country.
Consider the following possible responses to the three research questions: 1. A man on the news got mercury poisoning from eating swordfish, so the average mercury concentration in swordfish must be dangerously high. I met two students who took more than 7 years to graduate from Duke, so it must take longer to graduate at Duke than at many other colleges.
My friends dad had a heart attack and died after they gave him a new heart disease drug, so the drug must not work. Each conclusion is based on data. However, there are two problems. First, the data only represent one or two cases. Second, and more importantly, it is unclear whether these cases are actually representative of the population.
Data collected in this haphazard fashion are called anecdotal evidence. Anecdotal evidence Be careful of making inferences based on anecdotal evidence. Such evidence may be true and verifiable, but it may only represent extraordinary cases. The majority of cases and the average case may in fact be very different.
Anecdotal evidence typically is composed of unusual cases that we recall based on their striking characteristics. Instead of focusing on the most unusual cases, we should examine a representative sample of many cases. Consider the following question from page 13 for the county data set: 1 Is federal spending, on average, higher or lower in counties with high rates of poverty?
If we suspect poverty might affect spending in a county, then poverty is the explanatory variable and federal spending is the response variable in the relationship. TIP: Explanatory and response variables To identify the explanatory variable in a pair of variables, identify which of the two is suspected of affecting the other. Caution: Association does not imply causation Labeling variables as explanatory and response does not guarantee the relationship between the two is actually causal, even if there is an association identified between the two variables.
We use these labels only to keep track of which variable we suspect affects the other. In many cases, the relationship is complex or unknown. It may be unclear whether variable A explains variable B or whether variable B explains variable A. For example, it is now known that a particular protein called REST is much depleted in people suffering from Alzheimers disease.
While this raises hopes of a possible approach for treating Alzheimers, it is still unknown whether the lack of the protein causes brain deterioration, whether brain deterioration causes depletion in the REST protein, or whether some third variable causes both brain deterioration and REST depletion.
That is, we do not know if the lack of the protein is an explanatory variable or a response variable. Perhaps it is both. There are two primary types of data collection: observational studies and experiments.
Researchers perform an observational study when they collect data without interfering with how the data arise.
For instance, researchers may collect information via surveys, review medical or company records, or follow a cohort of many similar individuals to study why certain diseases might develop.
In each of these situations, researchers merely observe or take measurements of things that arise naturally. When researchers want to investigate the possibility of a causal connection, they conduct an experiment. For all experiments, the researchers must impose a treatment. For most studies there will be both an explanatory and a response variable.
For instance, we may suspect administering a drug will reduce mortality in heart attack patients over the following year. To check if there really is a causal connection between the explanatory variable and the response, researchers will collect a sample of individuals and split them 12 Sometimes the explanatory variable is called the independent variable and the response variable is called the dependent variable.
However, this becomes confusing since a pair of variables might be independent or dependent, so we avoid this language. The individuals in each group are assigned a treatment.
When individuals are randomly assigned to a group, the experiment is called a randomized experiment. For example, each heart attack patient in the drug trial could be randomly assigned into one of two groups: the first group receives a placebo fake treatment and the second group receives the drug.
See the case study in Section 1. Should she carry out an observational study or an experiment? In addressing this question, we ask, Will the researcher be imposing any treatment? Because there is no treatment or interference that would be applicable here, it will be an observational study. Additionally, one consideration the researcher should be aware of is that, if customers know their tips are being recorded, it could change their behavior, making the results of the study inaccurate.
Generally, data in observational studies are collected only by monitoring what occurs, while experiments require the primary explanatory variable in a study be assigned for each subject by the researchers.
Making causal conclusions based on experiments is often reasonable. However, making the same causal conclusions based on observational data is treacherous and is not recommended. Observational studies are generally only sufficient to show associations. Does this mean sunscreen causes skin cancer? One important piece of information that is absent is sun exposure. Sun exposure is what is called a confounding variable also called a lurking variable, confounding factor, or a confounder. Confounding variable A confounding variable is a variable that is associated with both the explanatory and response variables.
Because of the confounding variables association with both variables, we do not know if the response is due to the explanatory variable or due to the confounding variable. Sun exposure is a confounding factor because it is associated with both the use of sunscreen and the development of skin cancer. People who are out in the sun all day are more likely to use sunscreen, and people who are out in the sun all day are more likely to get skin cancer. Research shows us the development of skin cancer is due to the sun exposure.
The variables of sunscreen usage and sun exposure are confounded, and without this research, we would have no way of knowing which one was the true cause of skin cancer.
The researched controlled for several confounding factors, such as age, physical activity, smoking, and many other factors. Can we conclude that the consumption of chocolate caused the people to live longer? This is an observational study, not a controlled randomized experiment. Even though the researches controlled for many possible variables, there may still be other confounding factors. Can you think of any that werent mentioned? While it is possible that the chocolate had an effect, this study cannot prove that chocolate increased the survival rate of patients.
That is, they acknowledged that there may be confounding factors. One possible confounding factor not considered was mental health. In context, explain what it would mean for mental health to be a confounding factor in this study. Mental health would be a confounding factor if, for example, people with better mental health tended to eat more chocolate, and those with better mental health also were less likely to die within the 8 year study period.
Notice that if better mental health were not associated with eating more chocolate, it would not be considered a confounding factor since it wouldnt explain the observed associated between eating chocolate and having a better survival rate. If better mental health were associated only with eating chocolate and not with a better survival rate, then it would also not be confounding for the same reason. Only if a variable that is associated with both the explanatory variable of interest chocolate and the outcome variable in the study survival during the 8 year study period can it be considered a confounding factor.
While one method to justify making causal conclusions from observational studies is to exhaust the search for confounding variables, there is no guarantee that all confounding variables can be examined or measured. Chocolate consumption and mortality following a first acute myocardial infarction: the Stockholm Heart Epidemiology Program. Journal of Internal Medicine , p In the same way, the county data set is an observational study with confounding variables, and its data cannot be used to make causal conclusions.
However, it is unreasonable to conclude that there is a causal relationship between the two variables. Suggest one or more other variables that might explain the relationship visible in Figure 1. Observational studies come in two forms: prospective and retrospective studies. A prospective study identifies individuals and collects information as events unfold. For instance, medical researchers may identify and follow a group of similar individuals over many years to assess the possible influences of behavior on cancer risk.
One example of such a study is The Nurses Health Study, started in and expanded in Retrospective studies collect data after events have taken place, e. Some data sets, such as county, may contain both prospectively- and retrospectively-collected variables. Local governments prospectively collect some variables as events unfolded e. We might try to estimate the time to graduation for Duke undergraduates in the last 5 years by collecting a sample of students.
All graduates in the last 5 years represent the population, and graduates who are selected for review are collectively called the sample. In general, we always seek to randomly select a sample from a population. The most basic type of random selection is equivalent to how raffles are conducted. For example, in selecting graduates, we could write each graduates name on a raffle ticket and draw tickets. The selected names would represent a random sample of graduates. Why pick a sample randomly?
Why not just pick a sample by hand? Consider the following scenario. Population density may be important. If a county is very dense, then this may require a larger fraction of residents to live in multi-unit structures.
Additionally, the high density may contribute to increases in property value, making homeownership infeasible for many residents. What kind of students do you think she might collect? Do you think her sample would be representative of all graduates? Perhaps she would pick a disproportionate number of graduates from health-related fields.
Or perhaps her selection would be well-representative of the population. When selecting samples by hand, we run the risk of picking a biased sample, even if that bias is unintentional or difficult to discern. If the student majoring in nutrition picked a disproportionate number of graduates from health-related fields, this would introduce selection bias into the sample.
Selection bias occurs when some individuals of the population are inherently more likely to be included in the sample than others. In the example, this bias creates a problem because a degree in health-related fields might take more or less time to complete than a degree in other fields.
Suppose that it takes longer. Since graduates from health-related fields would be more likely to be in the sample, the selection bias would cause her to overestimate the parameter. Sampling randomly resolves the problem of selection bias.
The most basic random sample is called a simple random sample, which is equivalent to using a raffle to select cases. This means that each case in the population has an equal chance of being included and there is no implied connection between the cases in the sample. A common downfall is a convenience sample, where individuals who are easily accessible are more likely to be included in the sample.
For instance, if a political survey is done by stopping people walking in the Bronx, this will not represent all of New York City. It is often difficult to discern what sub-population a convenience sample represents. Similarly, a volunteer sample is one in which peoples responses are solicited and those who choose to participate, respond. This is a problem because those who choose to participate may tend to have different opinions than the rest of the population, resulting in a biased sample. These ratings are based only on those people who go out of their way to provide a rating.
From our own anecdotal experiences, we believe people tend to rant more about products that fell below expectations than rave about those that perform as expected. For this reason, we suspect there is a negative bias in product ratings on sites like Amazon. However, since our experiences may not be representative, we also keep an open mind. It is difficult, and often times impossible, to completely fix this problem.
The act of taking a random sample helps minimize bias; however, bias can crop up in other ways. Even when people are picked at random, e.
This non-response bias can skew results. Even if a sample has no selection bias and no non-response bias, there is an additional type of bias that often crops up and undermines the validity of results, known as response bias. Response bias refers to a broad range of factors that influence how a person responds, such as question wording, question order, and influence of the interviewer. This type of bias can be present even when we collect data from an entire population in what is called a census.
Because response bias is often subtle, one must pay careful attention to how questions were asked when attempting to draw conclusions from the data. Lets assume that she manages to survey every student in the school.
How might response bias arise in this context? There are many possible correct answers to this question. For example, students might respond differently depending upon who asks the question, such as a school friend or someone who works in the cafeteria. The wording of the question could introduce response bias. Students would likely respond differently if asked Do you like the food in the cafeteria?
TIP: Watch out for bias Selection bias, non-response bias, and response bias can still exist within a random sample. Always determine how a sample was chosen, ask what proportion of people failed to respond, and critically examine the wording of the questions. When there is no bias in a sample, increasing the sample size tends to increase the precision and reliability of the estimate. When a sample is biased, it may be impossible to decipher helpful information from the data, even if the sample is very large.
Comment on the usefulness of this approach. Almost all statistical methods for observational data rely on a sample being random and unbiased. When a sample is collected in a biased way, these statistical methods will not generally produce reliable information about the population.
The idea of a simple random sample was introduced in the last section. Here we provide a more technical treatment of this method and introduce four new random sampling methods: systematic, stratified, cluster, and multistage.
Simple random sampling is probably the most intuitive form of random sampling. Consider the salaries of Major League Baseball MLB players, where each player is a member of one of the leagues 30 teams. For the season, N, the population size or total number of players, is Then we could randomly select numbers between 1 and without replacement using a random number generator or random digit table.
The players with the selected numbers would comprise our sample. Two properties are always true in a simple random sample: 1. Each case in the population has an equal chance of being included in the sample. Each group of n cases has an equal chance of making up the sample. The statistical methods in this book focus on data collected using simple random sampling. Note that Property 2 that each group of n cases has an equal chance making up the sample is not true for the remaining four sampling techniques.
As you read each one, consider why. Though less common than simple random sampling, systematic sampling is sometimes used when there exists a convenient list of all of the individuals of the population. Suppose we have a roster with the names of all the MLB players from the season.
To take a systematic random sample, number them from 1 to Select one random number between 1 and and let that player be the first individual in the sample. Then, depending on the desired sample size, select every 10th number or 20th number, for example, to arrive at the sample.
The same type of people that did not respond to the first survey are likely not going to respond to the second survey. Instead, she should make an effort to reach out to the households from the original sample that did not respond and solicit their feedback, possibly by going door-to-door.
Suppose we randomly select the number Then player , , , , 6, 12, , , and would make up the sample. In the top panel, simple random sampling was used to randomly select 18 cases. In the lower panel, systematic random sampling was used to select every 7th individual.
Provide an example of a sample that can come from a simple random sample but not from a systematic random sample. Answers can vary. If we take a sample of size 3, then it is possible that we could sample players numbered 1, 2, and 3 in a simple random sample. Such a sample would be impossible from a systematic sample.
Property 2 of simple random samples does not hold for other types of random samples. Sometimes there is a variable that is known to be associated with the quantity we want to estimate.
In this case, a stratified random sample might be selected. Stratified sampling is a divide-and-conquer sampling strategy. The population is divided into groups called strata. The strata are chosen so that similar cases are grouped together and a sampling method, usually simple random sampling, is employed to select a certain number or a certain proportion of the whole within each stratum.
In the baseball salary example, the 30 teams could represent the strata; some teams have a lot more money were looking at you, Yankees. Each team can serve as a stratum, and we could take a simple random sample of 4 players from each of the 30 teams, yielding a sample of players.
Stratified sampling is inherently different than simple random sampling. For example, the stratified sampling approach described would make it impossible for the entire Yankees team to be included in the sample. Why is it good for cases within each stratum to be very similar? We should get a more stable estimate for the subpopulation in a stratum if the cases are very similar.
These improved estimates for each subpopulation will help us build a reliable estimate for the full population. For example, in a simple random sample, it is possible that just by random chance we could end up with proportionally too many Yankees players in our sample, thus overestimating the true average salary of all MLB players. A stratified random sample can assure proportional representation from each team.
Next, lets consider a sampling technique that randomly selects groups of people. Cluster sampling is much like simple random sampling, but instead of randomly selecting individuals, we randomly select groups or clusters.
Unlike stratified sampling, cluster sampling is most helpful when there is a lot of case-to-case variability within a cluster but the clusters themselves dont look very different from one another. That is, we expect strata to be self-similar homogeneous , while we expect clusters to be diverse heterogeneous. Sometimes cluster sampling can be a more economical random sampling technique than the alternatives. For example, if neighborhoods represented clusters, this sampling method works best when each neighborhood is very diverse.
Because each neighborhood itself encompasses diversity, a cluster sample can reduce the time and cost associated with data collection, because the interviewer would need only go to some of the neighborhoods rather than to all parts of a city, in order to collect a useful sample. In the top panel, stratified sampling was used: cases were grouped into strata, and then simple random sampling was employed within each stratum.
In the middle panel, cluster sampling was used, where data were binned into nine cluster and three clusters were randomly selected. In the bottom panel, multistage sampling was used. Data were binned into the nine clusters, three of the cluster were randomly selected, and then six cases were randomly sampled in each of the three selected clusters. Multistage sampling, also called multistage cluster sampling, is a two or more step strategy.
The first step is to take a cluster sample, as described above. Then, instead of including all of the individuals in these clusters in our sample, a second sampling method, usually simple random sampling, is employed within each of the selected clusters. In the neighborhood example, we could first randomly select some number of neighborhoods and then take a simple random sample from just those selected neighborhoods.
As seen in Figure 1. Multistage sampling selects observations only from those clusters that were randomly selected in the first step. It is also possible to have more than two steps in multistage sampling. Each cluster may be naturally divided into subclusters. For example, each neighborhood could be divided into streets. To take a three-stage sample, we could first select some number of clusters neighborhoods , and then, within the selected clusters, select some number of subclusters streets.
Finally, we could select some number of individuals from each of the selected streets. It is believed that older students are more likely to work than younger students. What sampling method should be employed? Describe how to collect such a sample to get a sample size of Because grade level affects the likelihood of having a part-time job, we should take a stratified random sample. To do this, we can take a simple random sample of 15 students from each grade.
This will give us equal representation from each grade. Note: in a simple random sample, just by random chance we might get too many students who are older or younger, which could make the estimate too high or too low. Also, there are no well-defined clusters in this example.
We wouldnt want to use the grades as clusters and sample everyone from a couple of the grades. This would create too large a sample and would not give us the nice representation from each grade afforded by the stratified random sample. We learn that there are 30 villages in that part of the Indonesian jungle, each more or less similar to the next. Our goal is to test individuals for malaria.
A simple random sample would likely draw individuals from all 30 villages, which could make data collection extremely expensive. Stratified sampling would be a challenge since it is unclear how we would build strata of similar individuals. However, multistage cluster sampling seems like a very good idea. First, we might randomly select half the villages, then randomly select 10 people from each. This would probably reduce our data collection costs substantially in comparison to a simple random sample and would still give us reliable information.
Caution: Advanced sampling techniques require advanced methods The methods of inference covered in this book generally only apply to simple random samples. More advanced analysis techniques are required for systematic, stratified, cluster, and multistage random sampling. In the last section we investigated observational studies and sampling strategies.
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