This document linked from Unit 4B: Inference for Relationships

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]]>In each of the following three problems, you are presented with a brief description of a study involving two variables. Based on the role-type classification of the two variables, you’ll be asked to determine which of the four cases represents the data structure of the problem.

This document is linked from Role-Type Classification.

]]>2. How is the **number of calories** in a hot dog related to (or affected by) the **type of hot dog** (beef, meat or poultry)? In other words, are there differences in the number of calories among the three types of hot dogs?

4. Are the **smoking habits** of a person (yes, no) related to the person’s **gender**?

6. What is the relationship between driver’s **age** and sign legibility **distance** (the maximum distance at which the driver can read a sign)?

8. Can you predict a person’s **favorite type of music** (classical, rock, jazz) based on his/her **IQ level**?

This document is linked from Role-Type Classification.

]]>In the previous unit, we learned to perform inference for a **single** categorical or quantitative **variable** in the form of **point estimation**, **confidence** **intervals** or **hypothesis** **testing**.

The inference was actually

- about the
**population proportion**(when the variable of interest was**categorical**) and - about the
**population mean**(when the variable of interest was**quantitative**).

Our next (and final) goal for this course is to perform **inference** about **relationships** between **two** **variables** in a population, based on an observed relationship between variables in a sample. Here is what the process looks like:

We are interested in studying whether a **relationship** exists **between** the **variables** X and Y **in a population of interest**. We choose a random sample and collect data on both variables from the subjects.

Our goal is to determine whether these data provide strong enough evidence for us to **generalize** the **observed** **relationship** in the **sample** and **conclude** (with some acceptable and agreed-upon level of uncertainty) that a **relationship** between X and Y **exists** in the entire **population**.

The primary form of inference that we will use in this unit is **hypothesis testing** but we will discuss **confidence** **intervals** both to estimate unknown parameters of interest involving two variables and as an alternative way of determining the conclusion to our hypothesis test.

Conceptually, across all the inferential methods that we will learn, we’ll test some form of:

**Ho: There is no relationship between X and Y**

**Ha: There is a relationship between X and Y**

(We will also discuss point and interval estimation, but our discussion about these forms of inference will be framed around the test.)

Recall that when we discussed examining the relationship between two variables in the **Exploratory Data Analysis** unit, our discussion was framed around the **role-type classification**. This part of the course will be structured exactly in the same way.

In other words, we will look at hypothesis testing in the 3 sections corresponding to cases C→Q, C→C, and Q→Q in the table below.

Recall that case Q→C is not specifically addressed in this course other than that we may investigate the association between these variables using the same methods as case C→Q.

It is also important to remember what we learned about lurking variables and causation.

- If our explanatory variable was part of a
**well-designed experiment**then it may be**possible**for us to claim a**causal****effect**

- But if it was based upon an
**observational****study**, we must be**cautious**to**imply****only**a**relationship**or**association**between the two variables,**not**a direct**causal****link**between the explanatory and response variable.

Unlike the previous part of the course on Inference for One Variable, where we discussed in some detail the theory behind the machinery of the test (such as the null distribution of the test statistic, under which the p-values are calculated), in the inferential procedures that we will introduce in Inference for Relationships, we will discuss much less of that kind of detail.

The principles are the same, but the details behind the null distribution of the test statistic (under which the p-value is calculated) become more complicated and require knowledge of theoretical results that are beyond the scope of this course.

Instead, **within each of the inferential methods we will focus on:**

- When the inferential method is appropriate for use.

- Under what conditions the procedure can safely be used.

- The conceptual idea behind the test (as it is usually captured by the test statistic).

- How to use software to carry out the procedure in order to get the p-value of the test.

- Interpreting the results in the context of the problem.

- Also, we will continue to introduce each test according to the four-step process of hypothesis testing.

From this point forward, we will generally focus on

**TWO-SIDED tests**and**Supplement**with**confidence intervals**for the**effect of interest**to give further information

Using two-sided tests is **standard practice in clinical research** EVEN when there is a direction of interest for the research hypothesis, such as the desire to prove a new treatment is better than the current treatment.

Here are a few comments:

- Although fewer participants are required for one-sided tests, we are
**unable to draw appropriate conclusions**if the study demonstrates the new treatment is worse. (See Defending the Rationale for the Two-Tailed Test in Clinical Research for a detailed discussion of this and other issues.)

- Using a one-sided test for the purpose of gaining statistical significance is
**NOT A VALID APPROACH**. (See What are the differences between one-tailed and two-tailed tests? for more on this as well as a general overview of both types of tests.)

We are now ready to start with Case C→Q.

]]>While it is fundamentally important to know how to describe the distribution of a single variable, most studies pose research questions that involve exploring the relationship between **two** (or more) variables. These research questions are investigated using a sample from the population of interest.

Here are a few examples of such research questions with the two variables highlighted:

- Is there a relationship between
**gender**and**test scores**on a particular standardized test? Other ways of phrasing the same research question:- Is performance on the test related to gender?
- Is there a gender effect on test scores?
- Are there differences in test scores between males and females?

- How is the
**number of calories**in a hot dog related to (or affected by) the**type of hot dog**(beef, meat or poultry)? In other words, are there differences in the number of calories among the three types of hot dogs?

- Is there a relationship between the
**type of light**a baby sleeps with (no light, night-light, lamp) and whether or not the child develops**nearsightedness**?

- Are the
**smoking habits**of a person (yes, no) related to the person’s**gender**?

- How well can we predict a student’s freshman year
**GPA**from his/her**SAT score**?

- What is the relationship between driver’s
**age**and sign legibility**distance**(the maximum distance at which the driver can read a sign)?

- Is there a relationship between the
**time**a person has practiced driving while having a learner’s permit, and**whether or not this person passed the driving test**?

- Can you predict a person’s
**favorite type of music**(classical, rock, jazz) based on his/her**IQ level**?

In most studies involving two variables, each of the variables has a role. We distinguish between:

- the
**response**variable — the outcome of the study; and - the
**explanatory**variable — the variable that claims to explain, predict or affect the response.

As we mentioned earlier the variable we wish to predict is commonly called the **dependent variable**, the **outcome **variable, or the **response **variable. Any variable we are using to predict (or explain differences) in the outcome is commonly called an **explanatory variable**, an **independent** **variable**, a **predictor** variable, or a **covariate**.

**Comment:**

- Typically the
**explanatory**variable is denoted by X, and the**response**variable by Y.

Now let’s go back to some of the examples and classify the two relevant variables according to their roles in the study:

Is there a relationship between **gender** and **test scores** on a particular standardized test? Other ways of phrasing the same research question:

- Is performance on the test related to gender?
- Is there a gender effect on test scores?
- Are there differences in test scores between males and females?

We want to explore whether the outcome of the study — the score on a test — is affected by the test-taker’s gender. Therefore:

**Gender** is the **explanatory** variable

**Test score** is the **response** variable

Is there a relationship between the **type of light** a baby sleeps with (no light, night-light, lamp) and whether or not the child develops **nearsightedness**?

In this study we explore whether the nearsightedness of a person can be explained by the type of light that person slept with as a baby. Therefore:

**Light type** is the **explanatory** variable

**Nearsightedness** is the **response** variable

How well can we predict a student’s freshman year **GPA** from his/her **SAT score**?

Here we are examining whether a student’s SAT score is a good predictor for the student’s GPA freshman year. Therefore:

**SAT score** is the **explanatory** variable

**GPA of freshman year** is the **response** variable

Is there a relationship between the **time** a person has practiced driving while having a learner’s permit, and **whether or not this person passed the driving test**?

Here we are examining whether a person’s outcome on the driving test (pass/fail) can be explained by the length of time this person has practiced driving prior to the test. Therefore:

**Time** is the **explanatory** variable

**Driving test outcome** is the **response** variable

Now, using the same reasoning, the following exercise will help you to classify the two variables in the other examples.

**Question : **Is the role classification of variables always clear? In other words, is it always clear which of the variables is the explanatory and which is the response?

**Answer: **No. There are studies in which the role classification is not really clear. This mainly happens in cases when both variables are categorical or both are quantitative. An example is a study that explores the relationship between students’ SAT Math and SAT Verbal scores. In cases like this, any classification choice would be fine (as long as it is consistent throughout the analysis).

If we further classify each of the two relevant variables according to **type** (categorical or quantitative), we get the following 4 possibilities for **“role-type classification”**

- Categorical explanatory and quantitative response (Case CQ)
- Categorical explanatory and categorical response (Case CC)
- Quantitative explanatory and quantitative response (Case QQ)
- Quantitative explanatory and categorical response (Case QC)

This role-type classification can be summarized and easily visualized in the following table (note that the explanatory variable is always listed first):

This role-type classification serves as the infrastructure for this entire section. In each of the 4 cases, different statistical tools (displays and numerical measures) should be used in order to explore the relationship between the two variables.

This suggests the following important principle:

**PRINCIPLE: **When confronted with a research question that involves exploring the relationship between two variables, the first and most crucial step is to determine which of the 4 cases represents the data structure of the problem. In other words, the first step should be classifying the two relevant variables according to their role and type, and only then can we determine what statistical tools should be used to analyze them.

Now let’s go back to our 8 examples and determine which of the 4 cases represents the data structure of each:

Is there a relationship between **gender** and **test scores** on a particular standardized test? Other ways of phrasing the same research question:

- Is performance on the test related to gender?
- Is there a gender effect on test scores?
- Are there differences in test scores between males and females?

We want to explore whether the outcome of the study — the score on a test — is affected by the test-taker’s gender.

**Gender** is the **explanatory** variable and it is **categorical**.

**Test score** is the **response** variable and it is **quantitative**.

Therefore this is an example of **case C**→**Q**.

Is there a relationship between the **type of light** a baby sleeps with (no light, night-light, lamp) and whether or not the child develops **nearsightedness**?

In this study we explore whether the nearsightedness of a person can be explained by the type of light that person slept with as a baby.

**Light type** is the **explanatory** variable and it is **categorical**.

**Nearsightedness** is the **response** variable and it is **categorical**.

Therefore this is an example of **case C**→**C**.

How well can we predict a student’s freshman year **GPA** from his/her **SAT score**?

Here we are examining whether a student’s SAT score is a good predictor for the student’s GPA freshman year.

**SAT score** is the **explanatory** variable and it is **quantitative**.

**GPA of freshman** year is the **response** variable and it is **quantitative**.

Therefore this is an example of **case Q**→**Q**.

Is there a relationship between the **time** a person has practiced driving while having a learner’s permit, and **whether or not this person passed the driving test**?

Here we are examining whether a person’s outcome on the driving test (pass/fail) can be explained by the length of time this person has practiced driving prior to the test.

**Time** is the **explanatory** variable and it is **quantitative**.

**Driving test outcome** is the **response** variable and it is **categorical**.

Therefore this is an example of **case Q**→**C**.

Now you complete the rest…

The remainder of this section on exploring relationships will be guided by this role-type classification. In the next three parts we will elaborate on cases C→Q, C→C, and Q→Q. More specifically, we will learn the appropriate statistical tools (visual display and numerical measures) that will allow us to explore the relationship between the two variables in each of the cases. Case Q→C will **not** be discussed in this course, and is typically covered in more advanced courses. The section will conclude with a discussion on causal relationships.