Chapter 12: Sampling

Transcription

1 Chapter 12: Sampling In all of the discussions so far, the data were given. Little mention was made of how the data were collected. This and the next chapter discuss data collection techniques. These methods depend strongly on randomization to insure that the data are as representative as possible of a population of interest. A population is a collection of units or individuals. In statistical applications, a researcher attempts to describe or draw conclusions about a population without examining all units in the population. Key ideas 1. Sampling: Draw some units from the population, and then measure the variables of interest on the sampled units. A sampling approach is pursued when the population is too large to allow measurement of all units. A sample is a subset of units drawn from the population. The principal objective is choose the units so that the sample is representative of the population 1. Some questions that might be answered by sampling: What is the average lead content of water in University of Montana drinking fountains? Has the survival time of cancer patients increased in the past 40 years? What is the ratio of elk calves to cows in Ravalli county? The procedure by which the sample is collected is called the sampling design. Polls are known as sample surveys. The most important attribute of a sampling design is the extent to which the sample is representative of the population from which it is drawn. If the sample is not representative of the population, it is biased. A biased sample is useless, or nearly so. Bias arises in many ways. The following examples of sample surveys illustrate a particular source of bias. 1 Even when things are done right, a small sample might not be representative by random chance. 84

2 (a) Voluntary response: KPAX conducts a poll where they ask viewers to call in and state whether or not they are happy in their marriage. 84% of the callers say they are unhappy in their marriage. Calls come from motivated callers. (b) Interviewer bias: A university instructor wants to know how students feel about their statistics course. As students come to her office hours, she asks them a few questions about the course. Most students will be reluctant to express negative feelings towards the course. (c) Convenience sampling: A survey was given to students regarding their opinions on a possible new business to open in the UC. Questionnaires were filled out by any student walking into the UC and willing to take the time to complete the questionnaire. Those individuals that were sampled were selected because it was convenient to the researcher. (d) Non-response bias: A questionnaire about hunting wolves is mailed to a random sample of households in Montana. Not all households will return the completed questionnaire. If there is something different about the way non-respondents would respond if they did respond, then the responses are biased. (e) Leading questions: There are now more wolves in Montana than anytime in the past 100 years. Don t you agree that an open season on wolves is necessary to restore elk populations? Leading questions, intentional or not, tend to prompt a particular answer to the respondent. 2. Randomization: Sample units are often selected from a list of all population units, and the selected units are chosen randomly. 2 Randomization minimizes bias related to the selection of sampling units. Examples of biased sampling designs: (a) 1936 Presidential election: Alf Landon v. Franklin Delano Roosevelt. 2.4 million people were polled by Literary Digest. 57% of the respondents said they would vote for Landon, yet Landon received 37% of the votes and Roosevelt received 62%. Researchers drew the sample from telephone books, magazine subscribers and club rosters. Why was the sampling design biased? 3 (b) 1948 Presidential election: Thomas Dewey v. Harry S. Truman. A phone survey conducted for The Chicago Tribune found that Dewey would beat Truman. The morning after the election, the Tribune announced that Dewey won. Yet Truman won easily. Why was the sampling design biased? 4 2 Some designs do not require a list. 3 The respondents were wealthier and more conservative than the general population. 4 People with phones were wealthier and more conservative than the general population. 85

3 (c) Would a telephone survey regarding the 2012 presidential election be unbiased if those called were randomly sampled from a phone book? 5 In summary, Collecting a sample by randomly selecting the sample units protects against selection bias. Selection bias is the result of important but perhaps unrecognized differences between population and sample units. Introducing randomness to the selection process usually is necessary for accurate inferences to be drawn about the population. Random sampling avoids bias because each population unit has an equal chance of being sampled Sample size: A fundamental question when planning a study is how many units are necessary to insure that the sample is representative of the population? Intuition suggests that the sample should be a certain fraction or percentage of the population. This is incorrect. In fact, the size of the population (as long as it s large) does not affect the accuracy of the results. For example, a random sample of 100 Montana residents will be about as representative of Montana incomes as a random sample of 100 U.S. residents will be representative of U.S. incomes. If the sample consists of the entire population, it is called a census. Collecting a census is usually not practical because of (a) Expense. (b) Most populations change over time so if it takes a long time to census the population, the population at the beginning of the survey may differ from the population at the end of the survey. (c) Complexity. Very large sampling efforts encounter problems such as undercounting, double-counting, and recording errors committed by the samplers. Parameters and Statistics The purpose of sampling is to obtain information about some aspect of the population. Often interest lies in estimating the mean or standard deviation of some variable or the proportion of population units with some characteristic. 5 Most land-line phone numbers are listed but few cell phones are listed. 6 This statement is not strictly true; there are unbiased designs with unequal but known positive probabilities of including each population unit. 86

4 Examples: The average December heating bill for Missoula residences, The proportion of UM students that are vaccinated against meningitis, The total number of cow elk between 6 and 12 years of age in Ruby River area of southwest Montana. These unknown population quantities are parameters. A sample is used to estimate parameters using statistics computed from the sample. Example: The average December heating bill y from a sample of 30 Missoula residences estimates the average December heading bill of all Missoula residences. Notation: (Sample) (Population) Name Statistic Parameter Mean y µ Standard Deviation s σ Proportion p p Correlation r ρ Regression coefficient b β More sampling terminology: Sampling unit: The sampling unit is an object on which the variables of interest are measured; sampling units might be people, households, mice, or plots of land. Sampling frame: The sampling frame is a list of units from which the sample is chosen. This will not always be the same as the population of interest. Example: if the population of interest is registered voters, then a phone book might be used as a sampling frame. Sampling variability: Sampling variability is variability in a statistic that is introduced by the random selection of units. Example: take three random samples of size n = 5 to estimate the average score on Exam I. The averages will be different because the samples are different. Variability among averages is sampling variability. Sample 1 Sample 2 Sample 3 Average If many samples of 5 scores were collected, then the sample averages will be distributed around the population mean µ. Variability of the sample means about µ is sampling 87

5 variability. The sampling variability of a statistic quantifies the precision of the statistic. Identify each of the following as a parameter or a statistic, and give the symbol used to represent it. 1. From a sample of 2290 U.S. voters, 65% report they will be voting for a certain Presidential candidate on election day The proportion of all U.S. voters that voted for Barack Obama in the 2008 presidential election The proportion of of all U.S. births that are boys The proportion of women serving in the 112th Congress The standard deviation of monthly incomes of 50 Missoula residents The average GPA of students at the University of Montana. 12 Sampling designs: 1. Simple random sample (SRS): A sample of size n is collected. If this sample is drawn so that every possible sample of size n has the same chance of being selected, it is a simple random sample. Example: Estimate the volume of timber or the number of woodpecker nests within a forest stand. Divide the area into equal-sized blocks. The blocks should be small enough to inventory reliably. The stand is divided into 36 blocks (right). Nine blocks will be selected and inventoried. To select a SRS, label the blocks in any order. Starting at row 7, write down nine consecutive two-digit numbers between 1 and These numbers identify the sample units Statistic: p = Parameter: p = Parameter: p = Parameter: p = Statistic: s. 12 Parameter: µ. 13 It the same number appears twice, replace one copy with another random number. 88

6 Go to the random number table and select a row at random to generate the sample. For example, row 7 is The sample consists of the first 9 two-digit numbers between 01 and 36: 18, 07, 05,.... In an SRS design, every combination of 9 blocks has the same probability of being selected. Unavoidably, some combinations of blocks will not be uniformly distributed in space. Selecting an SRS does not guarantee that a particular sample is perfectly representative of the population. It is not the sample that is biased or unbiased; it s the sampling design that is biased or unbiased. An unbiased sampling method is one that, on average, produces estimates that are not biased. 14 Regretably, a particular sample collected according to an unbiased sampling design may produce an inaccurate estimate of the parameter(s) of interest Stratified Random Sample: In the SRS example, suppose that trees within the stand vary predictably because of an environmental gradient (e.g., aspect). This extra information can be exploited to ensure a more representative sample if stratified random sampling is used. Stratified random sampling is used when the population is can be partitioned into a few sub-populations or strata that are more alike within sub-populations than between sub-populations. Each strata is randomly sampled and a parameter estimate is computed for each strata. Afterward, the estimates (or sub-samples) are pooled (combined) since it s usually of interest to compute estimates for the combined population as well as each strata. In the forest stand example, the stand might be divided into three bands or strata (from left to right with the elevation gradient) each containing 12 plots. The sample size is 9 as before, but under this design, the sample better represents each of the elevational strata. 14 That is, neither consistently too large nor too small. 15 By random chance. 89

7 Use row 29 from the random number table to select a random sample from the left stratum. Row 29 is: Identify the sample plots by determining the intervals containing the two-digit random numbers. For example, the random number 72 means that plot 10 is in the sample. Random Random Plot Numbers Plot Numbers Ignore numbers 96, 97, 98, Elevation In what situations is stratified random sampling better than simple random sampling? Systematic random sampling: An alternative to a stratified random sampling that works well when sampling a geographic area is systematic random sampling. Systematic sampling is easiest when the population size is a multiple of the sample size n, as it is here. The sampling interval is the population size divided by the sample size. In this case, it is 36/9 = 4. Next, randomly choose one of the first 4 plots randomly using the random number table The sample consists of this first plot chosen, and then every fourth plot after that. To take a systematic sample here, use row 14 of the random number table. Take the first number between 1 and 4 as a starting point: consists of plots 3, 7, 11,, The sample 16 When units are homogeneous within strata and heterogeneous between strata. Furthermore, when the population can be partitioned into a few meaningful strata, and the strata are straightforward to sample using simple random sampling. 90

8 Systematic sampling also works well for sampling plots along a transect, names from a list, and more generally, whenever there is a list that does not have systematic or repeating patterns. One advantage of systematic sampling is that a label does not have to be assigned to every population unit provided the units are already organized in some way. A phone book is an example of such a list. Other advantages: easy to use, and usually, the sample is uniformly distributed. Caution: be wary of patterns in the list that may lead to bias. 4. Cluster Sampling: Cluster sampling is conducted in two stages. In the first stage, a random sample of clusters (a cluster is a small group of units) is selected. Then, every unit in each selected cluster is sampled. Examples: (a) To survey households in Missoula, an SRS of 20 street blocks may be drawn. Then every household on the selected blocks are included in the sample. (b) To estimate the average height of trees in an area, randomly select 5 plots and measure every tree in each plot. Cluster sampling is preferable to stratified random sampling when the population can be organized as many small groups (clusters) Multistage Designs: All of the sampling methods discussed above require a list of every unit in the population. Sometimes lists are unavailable. In this case, multistage sample designs often are used. A multistage design successively breaks down the population as a layers. The layers are are successively sampled using the three methods discussed above. For example, a demographic study of the U.S. population might define the top strata as geographic region. Within each stratum (a geographic region), an SRS of counties (the second layer) is selected. Within each selected county, an SRS of blocks (a geographic unit defined by the U.S. Census Bureau) is sampled. The third layer are blocks. In this case, a list of every census block in the U.S. is not necessary. A list of all blocks within the sampled counties is necessary, however. 17 Clusters tend not to be substantially different, though the units within clusters may be different. It should be easy to sample all units within a cluster. An example is sampling demographic variables. It s convenient to define a cluster as a family. Stratified random sampling is used when the population can be organized as a few large distinct strata (by geographic region, for example) and there is interest in each stratum as a distinct sub-population. 91

9 Survey methods: The table below summarizes the advantages and disadvantages of some common survey methods. Strategies Advantages Disadvantages Personal High response Interviewer bias Interview rate Leading questions Cost/time Telephone Less expensive Good lists unavailable Interview Easy to monitor (undercoverage) Ought be fast Questionnaires Inexpensive Low response rate No interviewer bias possible non-response bias Direct Generally very accurate Time consuming Observation Observer error 92