2.3 Drawing Conclusions about Studies

Section Overview

Figure 2.3.1
Interactive 2.3.1 - Breakdown of Fig. 2.3.1
Example 2.3.1
Example 2.3.2
Pop Quiz 2.3.1

Scope of Inference

Researchers who conduct statistical studies often want to draw conclusions, or make inferences, that go beyond the data produced.

What types of inference can be made from a particular study? The answer depends on the design of the study. Random selection of individuals allows inference about the population. Random assignment of individuals to groups permits inference about cause and effect. Figure 2.3.1 outlines all of the the possibilities:

Figure 2.3.1

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Were subjects randomly assigned?	Were subjects randomly selected?	Population	Cause & Effect
Yes	Yes	Yes (green)	Yes (green)
Yes	No	No (orange)	Yes (green)
No	Yes	Yes (green)	No (orange)
No	No	No (orange)	No (orange)

The figure is explored in further detail in Interactive 2.3.1.

Interactive 2.3.1 Detailed Breakdown of Figure 2.3.1

Frame 1 of 5

Population → Sample; Sample → Treatment Groups. Callouts: Randomness, Relationship to Figure 2.3.1, Partiality, Observational Studies.
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Frame 2 of 5

Randomness — The grey dots represents randomness, blindness, impartiality: the surveyor is impartial to the individuals that go into the sample and selects them for the sample randomly and “blindly”.
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Frame 3 of 5

Relationship to Figure 2.3.1 — Each image in this Interactive is meant to correspond to a square in Figure 2.3.1. Inferences about the population correspond to the population/sample images and inferences about cause and effect correspond to the sample/treatment group images.
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Frame 4 of 5

Partiality — The red dots represent the opposite of the grey dots: intent and partiality. In this specific example, the sample was not randomly selected from the population.
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Frame 5 of 5

Observational Studies — Assigning individuals in a sample to treatment groups can only be done with experiments. It is not possible to establish causation with only observational studies, even if the sample was randomly selected from the population, hence the dotted line.
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Capture provenance

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Example 2.3.1

A hair stylist in Rusticville wants to know if people that get hair cuts at least once every two months will result in fewer split ends than people who get hair cuts less frequently and is considering the following four study designs:

a) Get all people in town with hair appointments in the next week to take part in an experimental study. Ask each participant if they cut their hair at least once every two months and if they have split ends. Based on the answer to the first question, put them in one of two groups: those that cut their hair at least once every two months and those that don’t.

b) Get all people in town with hair appointments in the next week to take part in an experimental study. Randomly assign half to get their hair cut once every two months over a period of 6 months and half to not have their hair cut over a period of 6 months.

c) Select a random sample of people in town to take part in an experimental study. Divide the patients into two groups based on the two questions as in (a)

d) Select a random sample of people in town to take part in an experimental study. Randomly assign them as in (b)

What can be concluded for each design?

Example 2.3.1 — Recorded answer

a) In this design, the individuals were not randomly selected, so no inference about the greater population can be made. Since no treatment was given, this was an observational study and thus no inference can be made about cause and effect, either. The stylist would only know that the people who got their hair cut at least once every two months had fewer split ends than people who didn’t, but not if the cut was actually the cause. See the square in the bottom right of Figure 2.3.1.

b) As in (a), the stylist cannot make any inference about the greater population because the individuals were not randomly selected. This time, however, subjects were randomly assigned to treatments, so the stylist can conclude that haircuts once every two months reduced split ends. This design would fall under the square in the top right of Figure 2.3.1.

c) In this design, the individuals were randomly selected, so the stylist can generalize the results to the greater population. As in (a), since no treatments were given to the subjects, this was an observational study and thus no inference can be made about cause and effect. This design would fall under the square in the bottom left of Figure 2.3.1.

d) As in (c), since the individuals were randomly selected, so the stylist can generalize the results to the greater population. And as in

Source gap (S0194): Source gap: the left column ends “And as in” and the right column begins “treatments,” with the intervening wording absent from the captured page. No words have been supplied.

treatments, the stylist can conclude that haircuts once every two months reduced split ends. This design would fall under the square in the top left of Figure 2.3.1.

Source gap (S0193): Source terminology inconsistency: designs (a) and (c) call their studies “experimental,” but their recorded answers describe them as observational because no treatments were given. Both original statements are preserved.

The Difficulty in Establishing Causation

Making a strong case for causation is often difficult. Below are some criteria we use for establishing causation:

The association between the explanatory variable and the response is strong
The association shows a consistent pattern
The cause precedes the effect in time

Example 2.3.2

We’ve long observed that the majority of people with lung cancer are also smokers. Does smoking cause lung cancer, or could the association between the two be due be due to a lurking variable? Use the criteria for establishing causation for this question.

Example 2.3.2 — Recorded answer

The association between the explanatory variable and the response is strong - Studies have shown that long-term or heavy smokers get lung cancer more frequently than those that smoke less often and with fewer cigarettes. Those that decide to quit significantly reduce their risk of getting lung cancer.
The association shows a consistent pattern - There have been countless studies over the years and throughout various countries. The results suggest that smoking is the most likely cause of lung cancer.
The cause precedes the effect in time - It usually takes at least a few decades for smokers to get lung cancer.

Pop Quiz 2.3.1

Question 1 of 5

In a study, half of the participants complete an easy test of mental ability and half complete a difficult test. What technique should the investigators use to ensure that any post-test differences in the group’s eating habits actually result from the differences in test difficulty?

A. ReplicationB. Random SamplingC. Random AssignmentD. None of the above

Question 2 of 5

Which of the following is incorrect?

A. Both random sampling and random assignment introduce chance variation into a statistical study.B. Random sampling of individuals allows inference about the population.C. Random assignment of individuals to groups allows inference about case and effect.D. At least one sentence above is incorrect.

Source gap (S0197): Source wording: choice C reads “case and effect.” That wording is retained.

Question 3 of 5

Which of the following is incorrect?

A. A well-designed experiment tells that changes in the explanatory variable cause changes in the response variable.B. Lack of realism in an experiment can prevent general results: the subjects or treatments or setting of an experiment may not realistically duplicate the conditions we really want to study.C. An experiment is recommended to verify causation between smoking and lung cancer.D. At least one sentence above is incorrect.

Question 4 of 5

If a random sample of 100 bananas is taken from a supermarket and 50 are put out at room temperature and 50 are put in the freezer and the bananas from the freezer are found to stay fresh longer, then:

A. Inferences can be made about the populationB. Inferences can be made about cause and effectC. Both A and BD. None of the above

Question 5 of 5

In question 4, if 500 bananas were taken from the supermarket, put into a pile, and every 5th banana was selected for the experiment, then

A. Inferences can be made about the populationB. Inferences can be made about cause and effectC. Both A and BD. None of the above

Source gap (S0196): The captures of Pop Quiz 2.3.1 show all five questions and their choices, but no checked answers or answer feedback.

Glossary and deeper reading · 3 refreshers

These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.

Random sampling
Random sampling uses a specified chance process to select individuals from a population. It supports population inference when the sampling frame and data collection are appropriate, but does not guarantee perfect balance or eliminate every source of bias. For example, a random-number generator can select student identifiers from an enrollment list.
Read more: Random sampling in the captured Statistics reading →
Population
A population is the full set of individuals or measurements a study aims to describe. It must be defined clearly before deciding whether a sample supports generalization. For example, all students enrolled in a particular school this year form a population.
Read more: Population in the captured Statistics reading →
Explanatory variable
An explanatory variable is used to predict or explain variation in a response variable. In an experiment, researchers deliberately assign values or treatments related to an explanatory variable. For example, assigned study time could be an explanatory variable when test score is the response.
Read more: Explanatory variable in the captured Statistics reading →