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08. Probabilistic sampling⚓︎

Module items⚓︎

R Script file⚓︎

Copy the code below ➜ Paste into [[RStudio console]] ➜ Hit Enter 

source(url("https://raw.githubusercontent.com/ttezcann/ssric-reg/refs/heads/main/regression/docs/assets/r-scripts/0_packages_data.R")); 
download.file("https://raw.githubusercontent.com/ttezcann/ssric-reg/refs/heads/main/regression/docs/assets/r-scripts/08-sampling.R", "08-sampling.R"); 
file.edit("08-sampling.R")

Lab assignment⚓︎

Sampling

Sample lab assignment⚓︎

Sample: Sampling

Learning outcomes⚓︎

  1. Learn the basic terms and concepts related to sampling
  2. Identify the differences between probability and non-probability sampling
  3. Compare and contrast probability sampling methods
  4. Learn how to choose non-random sample
  5. Learn how to choose simple random sample
  6. Learn how to choose systematic random sample
  7. Identify advantages and disadvantages of different sampling methods

Suggested reading⚓︎

  • 📖
    Aldridge, Alan, and Kenneth Levine. 2001. “Selecting Samples.” Pp. 61–82 in Surveying the social world: principles and practice in survey research, Understanding social research. Buckingham: Philadelphia, PA: Open University Press.

[[Sampling concepts]]⚓︎

Is the soup salty?

You’ve made soup, and you want to know if it’s salty. What would you do?

soup

Show the answer
  • Stir the soup ➜ This ensures everything is mixed evenly.
  • Take a spoonful ➜ This small portion represents the whole pot.
  • Taste the spoonful ➜ You draw a conclusion about the entire soup.
  • So then,
    • The soup ➜ the population (everyone or everything you want to study).
    • Stirring ➜ ensuring representativeness (making sure the population is not biased or unevenly distributed).
    • The spoonful ➜ the sample (a smaller subset of the population).
    • Tasting ➜ analyzing data (using the sample to make conclusions).
  • [[Population]]: The universe of units from which the sample is to be selected.
  • [[Sample]]: The segment of population that is selected for investigation.
  • [[Sampling]]: The process of selecting units from a population so that we can generalize our results over the sample.
  • [[Randomness]]: Everyone in the population has an equal chance of being selected

A circular diagram illustrating the relationship between a population and a sample. A large oval labeled “Population” contains many small person icons, and a smaller oval labeled “Sample” contains fewer person icons. Arrows show two directions: selecting a sample from the population and generalizing conclusions from the sample back to the population.

The mission⚓︎

  • We will generate four samples and choose one variable.
    • Our variable is family income, coninc.
    • If the mean family income difference between population (GSS data) and samples diverge a lot, if the mean difference is more than $3,000, we can say that sampling method does not represent the population.

  • First, let's create a descriptive table of family income, coninc.

    1
    descr(gss$coninc, out = "v", show = "short")  # (1)!
    1. We put coninc. Highlighting and running this code will generate the table below.

    Basic descriptive statistics

    Variable Label N Missings (%) Mean SD
    coninc Respondents' family income 3563 10.61 47738.65 47738.65

    Descriptive table interpretation sample

    The respondents' family income variable shows that the average family income is 47,738 dollars, with standard deviation 41,328 dollars.

  • So then, if the different samples fall in 50,738 - 44,738 dollars range, we'll assume our sampling methods work.

    flowchart LR
A["$44,738"] --- B["$47,738"] --- C["$50,738"]

style A fill:stroke:#333,stroke-width:1px
style B fill:stroke:#333,stroke-width:5px
style C fill:stroke:#333,stroke-width:1px

  • These are the sampling methods we'll use:

flowchart TD
    A("GSS data<br/>3,986 respondents")

    B("40% <br/>Non-random sampling<br/>1,594 respondents")
    C("30% <br/>Simple random sampling<br/>1,196 respondents")
    D("20% <br/>Systematic random sampling<br/>798 respondents")

    A --> B
    A --> C
    A --> D

[[Sampling methods]]⚓︎

  • Main distinction between sampling methods is the random selection of participants.
    • In [[non-probability sampling]], a sampling frame is not often available, and participants are not selected randomly.
      • This is called [[non-random sampling]].
    • In [[probability sampling]], a sampling frame is used and participants are selected randomly.
  • The choice of a sampling method depends on the type of study being conducted.

[[Types of probability sampling]]⚓︎

  • Considering quantitative research aims to use [[probability sampling]],
    • Mainly, there are four types of probability sampling, though we'll use the first two for simplicity.
      • [[Simple random sampling]]
      • [[Systematic random sampling]]
      • Stratified random sampling
      • Multi-stage cluster sampling
  • For comparison, we'll also use [[non-random sampling]].

[[Non-random sampling]]⚓︎

  • In non-random sampling, the samples are gathered in a process that does not give all the individuals in the population equal chances of being selected.

    • The most easily accessible individuals. In this type of sampling, researcher selects any participant who is readily available to participate in the study (e.g., people in your network, people in your classroom).
      • Their participation happens based on availability.
    • Useful when piloting a research. This type of sampling can be cost and time effective in the case of pilot studies.

    A diagram showing a large oval containing many person icons in different colors. Inside is a smaller highlighted circle indicating a subset. An arrow points from this subset to a separate circle which contains fewer person icons, illustrating that the sample is taken from an easily accessible portion of the population rather than randomly.

GSS example: 40% Non-random sampling⚓︎

  1. We'll choose 40% of the GSS respondents, but not in a random way. Instead we'll choose the first 40% of them in the dataset in the order they appear, and create a new dataset for them.

  2. First, create a new dataset for the first 40% of the GSS respondents.

    1
    gssfirst40per <- head(gss, round(0.40 * nrow(gss))) # (1)!
    1. gssfirst40per is the name of the new dataset. This will appear under gss in Data Environment; head is the order of the selection for the first respondents (tail is for the last); 0.40 is 40%.

  3. Now, let's check the descriptive table of coninc variable for the first 40% of the GSS respondents.

    1
    descr(gssfirst40per$coninc, out = "v", show = "short")  # (1)!

    1. Note that we use gssfirst40per instead of gss. Highlighting and running this code will generate the table below.

      Basic descriptive statistics

    Variable Label N Missings (%) Mean SD
    coninc Respondents' family income 1415 11.23 53688.98 44151.37

    Non-random sampling descriptive table interpretation sample

    The overall respondents' family income mean among GSS respondents is 47,738 dollars, while the family income mean with a 40% non-random sample is 53,688 dollars. In this case since the mean family incomes differ by more than 3,000 dollars, we can conclude that these mean family incomes are substantially different.

    Non-random sampling descriptive table interpretation template

    The overall variable label mean among GSS respondents is [mean], while the variable label mean with a sampling method is [mean].

    In this case since the mean variable label differ by more than 3,000 dollars, we can conclude that these mean variable label are substantially different.

    OR

    In this case since the variable label do not differ by more than 3,000 dollars, we can conclude that these variable label are not substantially different.

    Is a non-random sample preferable method for obtaining generalizable results?

    It is not preferable; samples that are non-random cannot represent a whole population because all the individuals in the population do not receive equal chances of being selected.

    Substantially different results were obtained about the family income means because non-random samples are not capable of representing the population as a whole.

[[Simple random sampling]]⚓︎

  • In simple random sampling, each individual has an equal probability of selection, without considering any other criteria. If the sample frame is know, we can:

    • List all individual and number them consecutively, and
    • Use random numbers to select individuals.

  • There are five steps in selecting a simple random sample:

  1. Obtain a complete list of individuals.
  2. Give each individual a unique number starting at one
  3. Decide on the required sample size.
  4. Select numbers for the sample size from a table of random numbers.

  5. Select the individuals that correspond to the randomly chosen numbers.

    Number Name Number Name Number Name
    1 Adams, H. 18 Iulianotti, G. 35 Quinn, J.
    2 Anderson, J. 19 Ivono, V. 36 Reddan, R.
    3 Baker, E. 20 Jabornik, T. 37 Risteski, B.
    4 Bradsley, W. 21 Jacobs, B. 38 Sawers, R.
    5 Bradley, P. 22 Kennedy, G. 39 Saunders, M.
    6 Carra, A. 23 Kassem, S. 40 Tarrant, A.
    7 Cidoni, G. 24 Ladd, F. 41 Thomas, G.
    8 Daperis, D. 25 Lamb, A. 42 Uttay, E.
    9 Devlin, B. 26 Mand, R. 43 Usher, V.
    10 Eastside, R. 27 McIlraith, W. 44 Varley, E.
    11 Einhorn, B. 28 Natoli, P. 45 Van Rooy, P.
    12 Falconer, T. 29 Newman, L. 46 Walters, J.
    13 Felton, B. 30 Ooj, W. 47 West, W.
    14 Garratt, S. 31 Oppenheim, F. 48 Yates, R.
    15 Gelder, H. 32 Peters, P. 49 Wyatt, R.
    16 Hamilton, I. 33 Palmer, T. 50 Zappulla, T.
    17 Hartnell, W. 34 Quick, B.

GSS example: 30% Simple random sampling⚓︎

  1. We'll choose 30% of the GSS respondents in a random way.

  2. First, create a new dataset for the 30% of the GSS respondents that will be chosen randomly.

    1
    gssrandom30per <- gss[sample(nrow(gss), round(0.30 * nrow(gss))), ] # (1)!
    1. gssrandom30per is the name of the new dataset. This will appear under gss in Data Environment; 0.30 is 30%.

  3. Now, let's check the descriptive table of coninc variable for the 30% of the GSS respondents.

    1
    descr(gssrandom30per$coninc, out = "v", show = "short")  # (1)!

    1. Note that we use gssrandom30per instead of gss. Highlighting and running this code will generate the table below.

      Basic descriptive statistics

    Variable Label N Missings (%) Mean SD
    coninc Respondents' family income 1067 10.79 46641.97 40957.59

  4. The overall respondents' family income mean among GSS respondents is 47,738 dollars, and we found 46,641 dollars. This is close!

    Here's the issue

    If you run the data creation code and the descriptive table code again, you will find different results. It's because every single time the data creation code is run, RStudio chooses different 30% of the GSS respondents. There's inconsistency issue.

    30% simple random sampling descriptive table interpretation sample

    The overall family income mean among GSS respondents is $47,738, while the family income mean with a 30% simple random sample is 46,641 dollars.

    In this case since the mean family incomes differ by more than 3,000 dollars, we can conclude that these mean family incomes are substantially different. In this case since the mean family incomes do not differ by more than 3,000 dollars, we can conclude that these mean family incomes are not substantially different.

    30% simple random sampling descriptive table interpretation template

    The overall variable label mean among GSS respondents is [mean], while the variable label mean with a sampling method is [mean].

    In this case since the mean variable label differ by more than 3,000 dollars, we can conclude that these mean variable label are substantially different.

    OR

    In this case since the variable label do not differ by more than 3,000 dollars, we can conclude that these variable label are not substantially different.

    Is a 30% simple random sample preferable to a non-random sample?

    It is preferable; 30% simple random sampling is better than and superior to non-random sampling as there is a random selection. However, with each time the code is run, there is a different result.

    Even though this method is better than non-random sampling, there is still inconsistency, as a different 30% of the data is selected each time.

[[Systematic random sampling]]⚓︎

  • Systematic random sampling is same as simple random sampling, except: Choosing individuals from a random starting point, and choosing every nth individual (e.g., every 3rd individual).

    A row of twelve numbered person icons labeled 1 through 12 is shown across the top. Four of these—numbers 2, 5, 8, and 11—are highlighted with boxes. Curved arrows point downward from the highlighted individuals to a smaller row below, which shows only the four selected person icons labeled 2, 5, 8, and 11, illustrating a selection or sampling of specific individuals from a larger group.

  • There are five steps in selecting a systematic random sample:

  1. Determine population size (e.g. 100).
  2. Determine sample size required (e.g. 20).
  3. Calculate sampling fraction (population / sample) (100/20=5)
  4. Select random staring point within first 5 cases (e.g. 3).

  5. Select every 5th case.
    01 21 41 61 81
    02 22 42 62 82
    03 23 43 63 83
    04 24 44 64 84
    05 25 45 65 85
    06 26 46 66 86
    07 27 47 67 87
    08 28 48 68 88
    09 29 49 69 89
    10 30 50 70 90
    11 31 51 71 91
    12 32 52 72 92
    13 33 53 73 93
    14 34 54 74 94
    15 35 55 75 95
    16 36 56 76 96
    17 37 57 77 97
    18 38 58 78 98
    19 39 59 79 99
    20 40 60 80 100

GSS example: 20% Systematic random sampling⚓︎

  1. We'll choose 20% of the GSS respondents in a random way, with assigning the starting point and the choosing interval.

  2. First, create a new dataset for the 20% of the GSS respondents that will be chosen randomly and systematically.

    1
    gss20persystematic <- gss[seq(4, nrow(gss), 5), ] # (1)!
    1. gss20persystematic is the name of the new dataset. This will appear under gss in Data Environment; 4 is the starting point, so the sample begins with the 4th respondent; 5 means every fifth respondent is selected after that. Because 1 out of every 5 cases is chosen, the sample includes about 20% of the dataset.

  3. Now, let's check the descriptive table of coninc variable for the 30% of the GSS respondents.

    1
    descr(gss20persystematic$coninc, out = "v", show = "short")  # (1)!
    1. Note that we use gssrandom30per instead of gss. Highlighting and running this code will generate the table below.

    Basic descriptive statistics

    Variable Label N Missings (%) Mean SD
    coninc Respondents' family income 706 11.42 47838.77 41591.10

  4. The overall respondents' family income mean among GSS respondents is 47,738 dollars, and we found 47,838 dollars. This is extremely close!

    20% Systematic random sampling descriptive table interpretation sample

    The overall family income mean among GSS respondents is $47,738, while the family income mean with a 20% systematic random sample is 47.838 dollars.

    In this case since the mean family incomes do not differ by more than 3,000 dollars, we can conclude that these mean family incomes are not substantially different.

    30% simple random sampling descriptive table interpretation template

    The overall variable label mean among GSS respondents is [mean], while the variable label mean with a sampling method is [mean].

    In this case since the mean variable label differ by more than 3,000 dollars, we can conclude that these mean variable label are substantially different.

    OR

    In this case since the variable label do not differ by more than 3,000 dollars, we can conclude that these variable label are not substantially different.

    Is a 10% systematic random sample preferable to a 25% simple random sample?

    It is preferable; systematic random sampling method provides consistency, and even if there were fewer people used in this sampling, this is better than 25% simple random sampling.

    Sampling is less about the size of the sample and more about the representativeness of the sample selected.