Day 7

We’re approaching “the meat” of our first course section. Next week we’ll be removing a few training wheels and looking analyzing real data with the techniques we’ve learned up until now.

Review

Recall the 5 Number Summary:

\[ \begin{array}{|c|c|c|c|c|c|c|} \hline \text{Min} & \text{Q}_1 & \text{Median} & \text{Q}_3 & \text{Max} \\ \hline 90 & 124 & 165 & 182 & 196\\ \hline \end{array} \]

And how it’s represented in a boxplot:

What are the componenets of the 5 number summary on this boxplot?

What are the cirlces?

Boxplots are best used for comparing two data sets:

Scatterplots

When we have two quantitative variables we use scatterplots

Define one variable as \(x\) and one as \(y\)

Given \(x_i\) is the \(i^{th}\) data point in the \(x\) set

\(y_i\) is the \(i^{th}\) data point in the \(y\) set

The data set should be:

\[(x_1,y_1),(x_2,y_2),...,(x_n,y_n)\]

What is this called?

For any two variables we can define their relationship as a:

Positive association if large values of one variable are associated with large values of another

Negative association if large values of one variable are associated with small values of another

Two variables can have a linear relationship if the data tend to cluster around a straight line when plotted on a scatterplot

For each of the following scatterplots, state the type of association that is exhibited:

Simple Random Sample

A sample chosen by a method where every selection from the population made is equally likely to make up the sample

Stratified Sampling

Divide the population into similar groups (i.e., group students by college)
Randomly sample from those groups (strata)

Cluster Sampling

Divide the population into clusters (i.e., split Manhattan, KS by street block)
Randomly sample from the clusters

Systematic Sampling

Randomly choose a start point in a “lined-up” population
Sample every \(k^{th}\) item
i.e., Starting from the \(4^{th}\) batch of ice cream produced on a given day, Call Hall will check the quality of every \(4^{th}\) batch that comes off the production line

Voluntary Response Samples

Customer support reviews

Sample of Convenience

Class height

Questions?

Goals for Today:

Define Observational Studies vs. Designed Experiments
Learn some considerations and errors in study design
Look at some examples and analyze them

Correlation, Observation, and Regression

Observational Studies

An observational study is one in which we study something as it exists

Observe individuals and measure variables of interest
- Gather medical records to study relationship between smoking and heart disease

Does NOT attempt to influence the response

Purpose: Describe and compare EXSITING groups or situations (Cannot control anything!)

A hunting ground in Wisconsic has every \(3^{rd}\) hunter on site turn over any deer they kill to have their brain matter sampled. \(1452\) samples are taken and tested for Chronic Wasting Disease.

Why is this an observational study?

What kind of sampling technique was used?

Designed Experiments

An experiment occurs when we control the conditions under which observations are taken

Deliberately imposes treatments on individuals in order to observe response
- Lab rats are given either a low carbohydrate diet or a high carbohydrate diet to see the effect on weight

Does influence response

Purpose: Look to see whether the treatment causes a change in the response
- Understand cause and effect

A field of sorghum is split into \(4\) sections. \(3\) of those sections are given different fertilizers with varying levels of nitrogen. The fourth section is not given any fertilizer. The fields are harvested at the same time and their yield is recorded.

Experimental Unit: Subject, animal, or object used in the experiment

What’s our experimental unit in the experiment above?

Treatment: Experimental condition applied to experimental unit

What was the treatment in the sorghum experiment?

Response: The thing we measure to determine the effect of the treatments

What was the response above?

Study Considerations

Let’s go back to our cholesterol study

Are there any problems with the data we’ve collected and the inference we pull from it?

Age	67	52	57	56	60	42	58	53	52	39	54	50	62	38	54	48	64	69	53	62
Cholersterol	229	196	241	283	253	265	218	282	205	219	304	196	263	175	214	245	325	239	264	209

Confounding

Two variables in a study/experiment

Effects are indistinguishable
Can’t tell which variable caused an effect
Observational studies don’t show causality
Experiments can show causality
- Typically have to have that intent designed into them

Drinking wine is linked to better health

But what you drink is confounded with other variables
- Diet, wealth, lifestyle, etc

Collinearity

Two variables are linearly dependent

“A form of extreme confounding”

The variables contain the same information to an extent

On observing students in an elementary school, what is the single best variable to collect in order to predict their height?

Bias

The design of statistical study is biased when one outcome is systematically preferred to others

Impossible to correct for

It is a systematic error caused by bad sampling design

The outcomes of the shopping mall surveys will repeatedly miss the truth about the population in the same ways
- Overrepresents: Middle-class and Elderly
- Underrepresents: Poor

Problems in Sampling

Making up data is always bad

Samples of convenience are easy, cheap, and easy to intentionally bias

Voluntary response surveys can work well if designed well, but they’re very easy to design poorly

Undercoverage

Did we reach all of the intended groups in our population?

Nonresponse

Did we complete our survey with all respondants?

Response Bias

Were the respondants likely to avoid answering a question truthfully

Question Wording

Do we invoke bias through the way we construct our questions?

Order of Questions

Do previous questions have an effect on the response to later questions?

Case-control Studies

1 in every 1000 male newborns are affected by Jacob’s Syndrome, a rare genetic condition where a male recives an extra Y chromosome.

If you were to sample 10000 men in a perfectly designed SRS, how many useful data points would you expect to have for a study on Jacob’s Syndrome?

Case-control studies are a form of observational study that adjusts for this unique problem

Select the case-subjects (those with the trait/condition) of interest
- Take a random sample of those individuals

Select a control group without the condition (ideally with similarities to the case subjects)
- Take a random sample of those individuals

Researchers look for exposure factors in the subjects’ past that differ

Retrospective

Looking back into the past
Identifying outcome of interest prior to the risk factor of interest

May be confounded

Recall bias: memories are not always accurate

Cohort Studies

Millions of adults in the US have Type 2 Diabetes. As such, research into treatment is a wildly lucrative industry. But much of the treatment associated with the illness are very long term, with a necessity for regular “check-ins”

Subjects sharing a common demographic characteristic are enrolled and observed at regular intervals over an extended period of time

Starts with a group of similar individuals
Observations made over regular intervals
- Generally prospective
- Expensive
- Better suited for common outcomes
- Extremely Informative

Examples

Gallup polls considers a proper poll to be a representative sample of 1000 people. A local news station resolves to follow this rule of thumb and call 250 individuals from 4 different age groups as part of the design for a political poll.

\[ \begin{array}{|c|c|c|c|c|} \hline \text{Age Group} & 18-26 & 27-40 & 41-52 & 52+ \\ \hline \end{array} \]

They resolve to only count calls that are answered, rather than marking unanswered calls as non-respondant. They collect their results, develop their graphics, and run their “Political Statistics” segment as planned.