Data Set: email

Consider the email data set, found in OS4 Section 2.2 and in the R library openintro.

Data Set: email, organization

The data in Table 1 represent a data matrix tibble data frame, which is The way to organize data for statistical analysis.

Each row represents a new observation (individual or case) and each column represents a new variable.

More observations are added to the data set by appending rows, and more variables are added by appending columns.

Data Set: email, words

Table 1 displays rows 1, 2, 3, and 50 of a data set containing emails received during early 2012.

It is important to know the units of each variable, and understand what each variable means.

For example, the first row represents the first email of this data set, which is not spam, contains 21,705 characters, was written in HTML format, and contains only numbers larger than one million.

Type of Variables

In Table 1, there are multiple types of variables.

• Numerical data is quantitative, e.g. it takes on numerical values and all mathematical operations \((+, -, *, /, <, >, =, ... )\) make sense with these values.

• Categorical data can be categorized, or placed into non-overlapping groups. Possible values of a categorical variables are called levels.

Types of Variables, sub-types of numerical

There are two general sub-types of numerical variables.

• Continuous data can take on any value within a specified range.

• Discrete data can take on only a countable number of values.

Types of Variables, a combo

Another good to know, but less common, variable type is ordinal.

• Ordinal data can be considered a hybrid of numerical and categorical variables: it is a categorical variable but the levels have a natural ordering.

Types of Variables, examples

• continuous: money, height, …
• discrete: sides of dice, number of words in book, …
• categorical: names of fertilizers, gender, …
• ordinal: race positions, …

U.S. Counties

Use the words we just learned to explain the following table.

Data Frames in R

Run these commands in R

suppressMessages(library(openintro))
data(email) # load dataset email
str(email)  # ensure we're looking at a df
head(email) # top 6 rows; try tail
email$spam  # just one variable
names(email) # column names
some_cols <- c("spam", "num_char", "format", "number")
email[ , some_cols] # specified variables/columns
email[c(1,2,3,50), some_cols] # Table 1

Population

• A population is the complete set of observatons of interest.

• It is often too difficult to survey an entire popluation of interest:

  • prohibitively expensive

  • physically impossible or nearly so

  • too time consuming

  • …

Identifying the Population

To which population do the following research questions refer?

• What is the average mercury content in swordfish?

• Over the last 5 years, what is the average time to complete a degree for Chico State undergraduate students?

• Does a new drug reduce the number of deaths in patients with severe heart disease?

Population Out of Reach

A sample represents a subset of the cases and is often a small fraction of the population. For instance, 60 swordfish in the population might be selected. Statistics based on this sample provide estimates (best guesses) about the population.

Sampling From a Population, good

In general, we always seek to randomly select a sample from a population. How might we do this?

Five graduates are randomly selected from the population to be included in the sample.

Sampling From a Population, bad

Bias occurs in samples that over/under represent specific sub-groups of the population. We seek to avoid bias in our samples.

Instead of sampling from all graduates equally, a nutrition major might inadvertently pick graduates with health-related majors disproportionately often.

The Most Common Forms of Bias

Consider the following anecdotes.

• A man on the news got mercury poisoning from eating a swordfish, so the average mercury concentration in swordfish must be dangerously high.

• I met two students who took more than 7 years to graduate from Chico State, so it must take longer to graduate at Chico State than at many other colleges.

• My friend’s dad had a heart attack and died after they gave him a new heart disease drug, so the drug must not work.

• In February 2010, some media pundits cited one large snow storm as valid evidence against global warming, so climate change is obviously fake.

Avoid Anecdotal Evidence

Anecdotal evidence typically is composed of unusual cases that stand out in our memory. For instance, we are more likely to remember two people we met who took 7 years to graduate, than the six others who graduated in four years. Instead of looking at the most unusual cases, we should examine a sample of many cases that represent the population.

Another Example of Bias

Say we are interested in estimating the average gas mileage of all cars.

A graphical depiction of sample bias.

Origins of Bias

Bias can come from a number of different places:

• Convenience Sampling occurs when a sample is chosen based on ease of access to the individuals. Examples?

• Non-response occurs when some respondents refuse to answer survey questions. Examples?

• Self-selection occurs when some individuals of a study eagerly participate while others are generally uninterested. Examples?

• Exclusion occurs when a sub-group of the population is generally unreachable. Examples?

• …

Proper Ways to Sample

• sample with replacement. Draw members from the population one at a time such that each member has equal probability of being selected, returning the selected member to the population for the next draw. If any member is selected more than once, put that selection back and select again. Repeat the process until you have a sample of the desired size.

• sampling without replacement. Draw members from the population one at a time such that each member has equal probability of being selected at each draw. Repeat the process until you have a sample of the desired size.

Statistics in General

Characteristics of both can be numeric, categorical, \(\ldots\)

Observational and Experimental Studies

There are two primary types of data collection: observational studies and experiments.

• observational study. The researcher simply monitors and collects data on things as they are, by observing. There is no manipulation of the study by the researcher.

• experiment. The researcher assigns the value of the explanatory/independent variable for each unit. In other words, the researcher controls which subjects go into which treatment groups.

Experimental Studies, examples

Experimental studies, through the direct manipulation from the researcher, can provide cause-and-effect relationships between the response/dependent and explanatory/independent variable.

• (insert context) … researchers collect a sample of individuals and split them into groups. The individuals in each group assign a treatment, one group per level of the explanatory/independent variable.

• To study the effect of tar contained in cigarettes researchers painted tobacco tar on the back of some mice but not others, and recorded if the painted mice had cancer at a higher rate than those not exposed to the tar TODO add cition Wynder:1953

Observational and Experimental Studies, identify

For each of the following situations, identify if it is an observational study or an experiment.

• Review medical or company records to attempt to identify fraud.

• Follow a group of many similar individuals to study why certain diseases might develop.

• Plant a specific type of native grass in select areas to see if the native species will out-compete an invasive species.

Independent/Explanatory and Dependent/Response Variables

In studying the relationship between two variables, the variables are often viewed as either a response/dependent variable or an explanatory/idependent variable.

To identify the explanatory variable in a pair of variables, ask yourself which of the two explains the other. Often, many variables will explain/predict the response/dependent variable.

Explanatory and Response Variables

• response/dependent variable. The response/dependent variable is the variable or characteristic of the data that we are wanting to learn about (to explain, to predict, or to estimate), or that depends on other variables.

• explanatory/independent variable. The explanatory/independent variable is the variable that does the explaining, or whose effect on the response variables is of interest.

Explanatory and Response, examples

Identify the explanatory/independent and response/dependent variables from the following.

• fertilizer and growth
• college grade point average and high school grade point average
• average federal spending and counties with high rates of poverty
• police department budget and crime rate
• …

Explanatory/independent and Response, caution

Labeling variables as explanatory/independent and response/dependent does not guarantee the relationship between the two is actually causal, even if there is an association identified between the two variables. We use these labels only to keep track of which variable we suspect affects the other.

flowchart LR
    A(Explanatory variables ) -- might affect --> B(Response variable )

Confounding Variables

Suppose an observational study tracked sunscreen use and skin cancer, and it was found that the more sunscreen someone used, the more likely the person was to have skin cancer. Does this mean sunscreen causes skin cancer? Or is there another variable we aren’t accounting for?

flowchart LR
    A(sun exposure) --> B(skin cancer)
    A --> C(use sunscreen)
    C -- ? --> B

Stents Study

TODO cite this Chimowitz:2011 collected data on 451 at-risk patients. Two time points were measured 30 days after enrollment and 365 days after enrollment.

Each volunteer patient was randomly assigned to one of two groups:

Treatment group 224 patients in the treatment group received a stent and medical management.

Control group 227 patients in the control group received the same medical management as the treatment group, but they did not receive stents.

Experiments

Studies where the researchers assign treatments to cases are called experiments.

When this assignment includes randomization, e.g. using a coin flip to decide which treatment a patient receives, it is called a randomized experiment. Randomized experiments are fundamentally important when trying to show a causal connection between two variables.

Stents Study, data

OS 4 Case study 1.1. Of the 224 patients in the treatment group, 45 had a stroke by the end of the first year. Using these two numbers, compute the proportion of patients in the treatment group who had a stroke by the end of their first year. Compute the proportion of patients in the control group who had a stroke by the end of their first year.

Stents Study, R is a calculator

We do the calculations in R.

45 / (45+179)   # treatment, had stroke

[1] 0.2008929

28 / (28 + 199) # control, had stroke

[1] 0.123348

Stents Study, careful with conclusions

Our findings are a bit surprising, and since this is a randomized experiment we may be ready to draw causal connections, but don’t!

• Did you happen to see the words “volunteer patient”?

• Are there other reasons, confounding variables, that might explain these findings?

• Does our sample generalize to all stroke patients?

• …

Experiments, principles

To better ensure proper randomized experiments, there are some basic principles that all experiments aim to perfect.

• Control. Researchers assign treatments to cases (or experimental units), and they do their best to control any other differences in the groups.

• Randomization. Researchers randomize patients into treatment groups to account for variables that cannot be controlled.

• Replication. The more cases researchers observe, the more accurately they can estimate the effect of the explanatory/independent variable(s) on the response/dependent variable.

• Measurement error. Science is beyond the easy to measure. Exact measurements are not always available, instead assumptions/guesses/best efforsts are often made.

Experiments, control

Examples of control in experiments.

• When patients take a drug in pill form, some patients take the pill with only a sip of water while others may have it with an entire glass of water. To control for the effect of water consumption, a doctor may ask all patients to drink a 12 ounce glass of water with the pill.

• Researchers assigning levels of certain fertilizers.

• …

Experiments, randomization

Examples of randomization in experiments.

• Some patients may be more susceptible to a disease than others due to their dietary habits. Randomizing patients into the treatment or control group helps even out such differences, and it also prevents accidental bias from entering the study.

• Order of foods/drinks eaten in taste tests.

• …

Experiments, replication

Examples of replication in experiments.

• In a single study, we replicate by collecting a sufficiently large sample.

• Ask multiple people to taste test a food/drink.

• …

Experiments, measurement error

Example of measurement error.

• Counting salmon populations in the Sacramento River.

• Measuring unemployment at the census tract, county, state, nation.

• yada yada yada physics.

• …

Identify Basic Principles, Rats

To study the effect of tar contained in cigarettes, TODO cite Wynder:1953 painted tobacco tar on the back of some mice but not others, and recorded if the painted mice had cancer at a higher rate than those not exposed to the tar.

Experiments, blocking can help

Researchers sometimes know or suspect that variables, other than the treatment, influence the response. Under these circumstances, they may first group individuals based on this variable into blocks and then randomize cases within each block to the treatment groups.

• Blocking. The grouping of similar individuals. They are grouped into blocks.

Experiments, blocking can help picture

Example of blocking

Experiments, blocking example

Figure 1 shows blocking using a variable depicting patient risk. Patients are first divided into low-risk and high-risk blocks, then each block is evenly separated into the treatment groups using randomization. This strategy ensures an equal representation of patients in each treatment group from both the low-risk and high-risk categories.

Experiments, specific keywords

Time to clear up some commonly used keywords that are specific to experiments.

• Treatment. A specific experimental condition applied to each case.

• Levels. Unique set of values that the Treatment (or any categorical variable) takes on.

• Experimental Units. The observation/case/subject to which a treatment is applied.

Experiments, specific keywords by example

Consider the dataset ChickWeight by TODO cite Crowder:1990.

Experiments, already? known keywords

And let’s just be clear about a few other keywords.

• Placebo. A fake treatment.

• Blind. When researchers keep their patients uninformed about their treatment.

• Double Blind. When both researchers and patients do not know which patient receives which treatment.

Experiments, homework 02

A researcher is conducting an experiment to see the effect of diet and exercise on weight gain in baby hamsters. Two different diets (low fat and high fat) and three different levels of exercise (high, moderate, or none) will be used. Eight baby hamsters will be raised under each combination of diet and exercise and their weight gain after 4 weeks will be measured.

• What are the experimental units here?
• What are the categorical variables? List the levels of each.
• How many different treatments are there? List them.
• How many experimental units will this experiment require?