Studies and Experiments¶

• Data matrices
• Rows corresponding to cases or observations
• Columns corresponding to variables
• Types of variables
• Numerical
• Categorical

Today we'll talk how we get data at a fundamental level using observational studies or controlled experiments. In either case, we need a strategy to find a sample on which to perform the study.

This is all based on sections 1.3 and 1.4 of our text, although in a somewhat different order.

Observational studies vs controlled experiments¶

• An observational study is one where the data collection does not interfere with how the data arises. Examples include
• Surveys,
• Review of records,
• Simple measurements
• In an experiment, researchers actively work with the samples applying treatments and observing effects.
• Often (if not always), the intention is to explore a relationship or even a causal effect between two variable.
• For that reason, it's often important to distinguish between an explanatory variable and a response variable.

Quick examples from the news¶

Note that both of these techniques should be contrasted with anecdotal evidence like - "My sister took 8 years to graduate from OU, so OU must be really hard."

Some sampling terminology¶

You can't clearly articulate a research question without first clearly identifying the population that you're working with, together with some related terms:

• Population refers to the complete set of entities under consideration,
• Sample refers to some subset of the population,
• Parameter refers to some summary characteristic of the population, and
• Statistic refers to some summary characteristic computed from a sample.

In the coffee example:

• The population might be all the coffee drinkers in the UK.
• The sample consists of the 8,412 people who were surveyed.
• A parameter of interest would be the proportion of all individuals in the population with stiff blood vessels.
• A statistic would be the computed proportion of individuals from the sample with stiff blood vessels.

In the cancer example:

• The population would be all women under the age of 59 with this specific cancer.
• The sample would be the 672 women under the age of 59 who participated.
• A parameter would be the survival rate among the population after 42 months.
• A statistic would be the survival rate among the sample after 42 months.

Often we are interested in the relationship between two variables - specifically, is there a correlation or even a causal relationship between two variables. In the context of a study, we should clearly identify:

• Independent or explanatory variables
• Dependent or response variables
• Confounding or lurking variables

Generally, an explanatory variable is one that a researcher suspects might affect the response variable. Correlation, however, does not always imply causation.

Example: We suspect that folks who use more sunscreen have a higher incidence of skin cancer. What are the explanatory and response variables - as well as any confounding variables?

Observational studies¶

Again, the basic idea is that the data collection does not interfere with how the data arises.

Suppose we'd like to know the average height of women enrolled at UNCA. According to the UNCA Factbook, there were 2147 women enrolled in the Fall of 2017. I'm not even sure how many were enrolled this past year; it might be hard to round up all of them.

So, here's a more practical approach: I had 42 women enrolled in my statistics classes that semester who filled out my online survey survey. The average height of those women was 5'5''. We might hope that could be a good estimate to the average height of all women at UNCA.

This is a fundamental idea in statistics: we wish to study some parameter for a large population but doing so is a too unwieldy or down right impossible. Thus, we choose manageable sample from the population and estimate the parameter with the corresponding statistic computed from the sample. In the example above,

• The population is the set of all women enrolled at UNCA,
• the parameter is the average height of women in the population,
• the sample is the set of all women enrolled in this class, and
• the statistic is the average height of women in the sample, which happens to be 5'5''.

A number of key questions arise:

• How well can we expect the statistic to approximate the parameter?
• What properties should the sample have in order to maximize the accuracy of our approximation?

The first question will consume much of last two-thirds of our semester under the general topic of inference. As we'll learn, we need a random sample of sufficient size.

This approach of sampling is in contrast to the idea of just grabbing the whole population - often called a census.

Potential issues:

• Volunteers
• Convenience
• Bias (response and non-response)

Designed experiments¶

The experimental approach to the cancer example might go like so: Select 672 women under 59. Randomly assign them into one of two groups - one who takes the drug and one that does not. Examine the groups after 42 months.

Some key principles of experiments¶

• Control: We split the group of patients into two groups:
• A treatment group that receives the experimental drug
• A control group that doesn't receive the drug; they might receive a placebo.
• Randomization: The groups should be chosen randomly to prevent bias and to even out confounding factors.
• Replication: The results should be reproducible
• Blocking: We might break the control and treatment groups in to smaller groups or blocks.
• Reduces variability in the groups
• Allows us to identify confounding factors
• Example: We might block by gender, age, or degree of risk.

If we find differences between the groups we can examine whether they are statistically significant or not.

Sampling strategies¶

Good random sampling is surprisingly hard to achieve in practice. Here are a few strategies to implement sampling.

Simple random samples: This is the ideal and is so simple - choose $n$ people from the population independently and with equal probability. It's quite hard to achieve in practice, though.

Other strategies:

• Convenience
• Ex: Survey the folks on your dorm floor
• Stratified samples:
• Ex: Survey college students and stratify by class
• Clustered samples
• Ex: Door-to-door survey picking one block out of each neighborhood
• Systematic sampling
• Ex: Survey every tenth person you see