Sampling and Randomness

• Sampling is about who you conduct research on
• The objectives are to understand general concepts associated with sampling and randomness
• Also, to have a basic understanding of probability.

Research Steps

• Identification of a research question
• Literature search
• Generation of hypotheses based on the literature
• Design of a study to test hypotheses
• Sample selection involving statistical concepts
• Data collection using statistics
• Results analysis using statistics
• Conclusion based on the results/statistics
• Presentation of results through publication/communication

Target Population vs Sample

• The population is everything or everyone that you want to understand
• The sample is a small group of the population that represents it
• It had always been assumed that the population could never really be accessed
• However, this may be changing with the digital revolution

What is a sample

• A group of participants, treatments, or situations selected from a defined population
• Population consists of a mean = μ and a standard deviation = σ (theoretical)
• Sample parameters consist of a mean (x) and a standard deviation = (s)
• It is important not to state that you have measured the population mean, or population standard deviation
• Hypothesis should relate to the population, using µ and σ in hypothesis statements
• Extrapolation back to the population depends on how the sample is drawn
• The process of selecting your sample is called Sampling

Sampling in the Digital Age

• ResearchKit at Stanford University obtained 11,000 participants in 24 hours
• Ordinarily that would take 50 medical centers a year
• The goal is to make it cheaper and easier for researchers to recruit participants for their studies

The Purpose of Sampling

• Access information on the population of interest in a reasonable / do-able fashion
• Helps eliminate selection bias
• Accounts for natural variable in ethnicity, sex, education, etc. that exist in the population of interest
• Checking in on subjects can be an aspect of testing

How to Determine Sample Size

• The sample size needed per group is:
• n= (𝒁α/2+ 𝒁β/Δ)2 × 2 × SD2
• Where:
• 𝒁α/2 is the z-score corresponding to the two-tailed alpha level
• 𝒁β is the z-score corresponding to the desired power
• SD is the common standard deviation of both groups
• Δ is the difference in means between the two groups that you want to detect
• Understanding the variance in your measure is needed before determining sample size

Example of Power Calculation

• Significance level (α) = 0.05 = 𝒁α/2 = 1.96 (from normal distribution = this is the chance of a type I error at 5%)
• Power (1-β)= 0.80 = Zp = 0.84 (Z value of 0.2 from the normal distribution, accepting 20% chance of a type II error)
• For a difference of (Δ) = 5 and a standard deviation of the population (SD) = 10
• Za/2+ZB = 1.96 + 0.84 = 2.80
• 1. Divide by the expected difference 2.80/5 = 0.56
• Square the result: (0.56)² = 0.3136
• Multipy by 2 and standard deviation squared = 2× (10)² = 200
• Calculate final sample size: n = 0.3136× 200 = 62.72
• Convert to discrete number: n = 63 per group!
• So, you will need 63 participants in each group.

Stages of sampling (finding participants)

• Define the population of concern / interest, understanding that population = mean and variance
• Specify a sampling procedure, either random, stratified, systematic, clustered, or convenience
• Specify a sampling method for selecting the objects, participants, etc. (process - computer, phone, mail, email)
• Determine the sample size based on size of population, resources, etc
• Implementation of the sampling plan, from the population
• Assignment to groups if there is an intervention

Sampling Bias (What to minimize!)

• Occurs when the individuals selected for a sample over-represent or under-represent certain population attributes that are related to the phenomenon under study
• This can be either conscious or unconscious
• Typically, an impartial mechanism is required to prevent / protect against sampling bias

Other Considerations Regarding Sampling Bias

• Research studies are usually carried out on sample of subjects rather than whole populations
• Ideally, random samples are drawn from the target population to which the results of the study would be generalized
• Degree of bias is based on understanding of the population from which the sample was drawn
• The ability to generalize to a larger population depends on understanding

Target Population vs Accessible Population

• Target population is the overall group of people (or objects) in which the researchers intend to generalize the findings about
• Accessible population is a portion of the target population that has a chance of being selected into the study

Inclusion and Exclusion Criteria

• Inclusion Criteria refers to the primary traits of the target and accessible populations that will qualify someone as a subject
• Exclusion Criteria refers to those factors that would exclude someone from being a subject

Exclusion of Samples and the Impacts

• There can be overuse; for instance, "we can't use women because of the menstrual cycle"
• There are also very few studies on exercise and pregnancy / exercise

Sampling Techniques

• Sampling can be classified into probability or nonprobability samples
• Probability sampling is achieved through the process of random selection
• It gives everyone an equal chance or equal "probability” of being chosen
• Nonprobability samples are made through nonrandom methods
• Sampling Error is the difference between the sample averages and population averages

Non-probabilistic Sampling

• Used to seek participants with a particular trait, feature, disease, condition, ethnic origin, etc.
• Used to seek participants who have a particular vantage point, life experience, who are of a certain age, share a lived experience, etc.
• Used to select people within a group (community or otherwise) to study it
• Used with people who are willing to speak about experiences

Nonprobability Sampling Techniques

• Convenience sampling is chosen on basis of availability, requests for volunteers, or from recommendations
• Snowball sampling is similar to convenience sampling but more selective
• Participants identify and recommend other potential participants who are deemed fitting to the research purpose
• Purposive sampling is sampling with a defined purpose using maximum variation, a homogeneous group, a typical case sampling, extreme case sampling, and total population Sample
• Expert Sampling

Randomness

• Mathematical – probability, equal likelihood of being selected
• Random Selection is involves:
  • Random selection from the population of interest
• Random Assignment is used to compare two groups
• Involves:
  • Random assignment to groups

Probability Sampling Techniques

• Randomized Sample
• Systematic Sample • Has a random start where every nth person or object
• Stratified random sample • Has a breakdown by characteristic/category • has subpopulations
• Clustered Sampling • Breaks the population into clusters and randomly selects the cluster

Simple Random Sampling

• Use of a random numbers table
  • Randomly pick a number and travel in any direction to get remaining numbers
• Computer-generated random numbers (e.g. Lotto quick pick)
• Use everyone's names

Samples

• Involve different sample sizes (n = 5; n = 10; n = 20)
• Involve random selection for external validity
• Involve random assignment for internal validity

Systematic Sampling

• Select another method of sampling from the population using a defined sample size number
• Define the kth item
• k= population size/sample size
• Pick every kth item

Example of Systematic Sampling

• To obtain a sample of 500 individuals from a town of 10,000 people, use systematic sampling
• k= population size/sample size
• k= 10,000/500
• k= 20 (e.g. 20th, 40th, 60th, 80th)
• To take a sample from a community of 100 houses, use every second house on the street where houses have odd numbers

Stratified Random Sampling

• Population is too large where the population is divided into “strata” based on Size and Colour
• Randomly select from each strata
• Proportionally select the number

Example of Stratified Random Sampling

• Taking a sample of 35 from a population of 220
• 65/220 = 0.30 and 0.30 * 100 = 30%
• Therefore, 30% of the sample of 35 should come from the blue circles
• As 0.30 * 35 = 10.5, 10 blue dots would be picked as part of the sample

Cluster Sampling

• Divide the population into “clusters”
• Randomly select the cluster to be used

Random Assignment

• Used when there is an Intervention with more than one group
• This occurs with Randomized Control Trials (RCTs) using true experimental Research
• Needed for Internal Validity where Cause < -> Effect
• Each participant/object has an equal chance of being chosen for each group

Asking Questions Pertaining to Your Study

• Who is your target population?
• Is it accessible?
• Who is your accessible population?
• What are your inclusion and exclusion criteria?
• How many participants do you want/need?
• Do they fit into the operational definitions of your variables?

Probability

• There is a difference in probability for events when they are predicted

Probability in Prediction

• P = 0 (impossible) vs P=1 (must happen)
• Probability = Number of favourable outcomes/Number of possible outcomes
• Coin flip (say at least 2 out of 3 flips will be heads) involves determining the # of favourable outcomes and the # of all possible outcomes

All Possible Outcomes

• 1st, 2nd, and 3rd tosses in 14 conditions (21+22+23) can determine all possible outcomes

Experiences

• Unlike predictions, every time the flips occur, the probability is 50%
• Gamblers' fallacy is the incorrect belief that If random events occur more frequently in the past, they are less likely to occur in the future
• Independent events do not change the probability of the next event
• The velocity of the flip, height the coin travels, surface the coin hits, and distance from the ground all make the outcome difference
• A computer can flip a coin so that heads or tails come up > 98 % of the time