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Case-Control Studies

Overview of Lesson

Odds vs Probability

Overview of Case-Control Studies

Advantages

Disadvantages

Analysis

Calculating Odds Ratio (OR)

Probability vs Odds

Probability

Range of values: 0 – 1

Expressed in %

Probability is defined as the fraction of desired outcomes in the context of every possible outcome. Probabilities have a value between 0 and 1, where 0 is no chance of the desired outcome and 1 is guaranteed desired outcome. Probabilities are usually given as percentages.

Example: When flipping a coin, the coin will either land on heads or tails. Let’s say my desired outcome is heads. Each time I flip the coin, I will either get my desired outcome (heads) or an undesired outcome (tails). Because there are only two possible outcomes – heads or tails -- the probability of each flip is Heads/(Heads + Tails). Numerically, that would be 1/2, or 50%. Thus, each time I flip a coin, the probability of getting heads is 50%.

Let’s say I had a three-sided coin that has two heads and one tail. Thus, with every flip there are three possible outcomes – heads, heads, or tails. Two of those outcomes are my desired outcome. Thus I have a 2/3 -- or 67% probability – of getting heads each time I flip my three-sided coin.

Note: Probability calculation includes the numerator in the denominator of the calculation, because probability considers the context of the entire event.

Probability of getting a yellow M&M?

8 red

4 blue

3 orange

4 yellow

2 green

3 brown

If you were to close your eyes and pick one of the M&Ms from this spilled bag, what is the probability that you would get a yellow M&M?

Calculating Probability

The probability of getting a yellow m&m is calculated by taking the total number of yellow m&ms (4) and dividing it by the total number of m&ms (24) – which is all possible outcomes

If you said the probability of me getting a yellow m&m was 17% (or 16.7%), you’d be correct!

Odds

Range of values: 0 to ∞

Ratio: desired outcome to undesired outcome

Odds can have any value from zero to infinity and they represent a ratio of desired outcomes versus undesired outcomes. Odds can be expressed in two different ways: ‘odds in favor’ and ‘odds against’. ‘Odds in favor’ are the odds describing if an event will occur, while ‘odds against’ will describe if an event will not occur. With odds, the numerator is NOT a part of the denominator.

So, let’s go back to the coin example. For a regular coin, each flip will either be heads or tails – one desired outcome, one undesired outcome. Thus, the odds in favor of heads for a coin flip is 1:1. The odds against heads is also 1:1.

For my three-sided coin with two heads, the odds for heads is 2:1, whereas the odds against heads are 1:2

Odds of getting a yellow M&M?

8 red

4 blue

3 orange

4 yellow

2 green

3 brown

So, back to the m&ms… If I were to close my eyes and pick one, what are the odds that I’d get a yellow one? What are the odds that I wouldn’t?

Odds of getting yellow

The odds in favor of getting yellow are 4:20 or 1:5. Notice how the yellow M&Ms are in the numerator with the “in favor of” calculation.

Odds against getting yellow

The odds against getting yellow are 20:4 or 5:1. Notice how the yellow M&Ms are in the denominator for the against calculations.

M&Mmmmmmmm

Events that have a high probability, also have high odds. And, there is a mathematical relationship between the two, which goes beyond the scope of this mini-lesson and my understanding of math. But the reason for this information is to help you better understand the calculations used in this lesson on case-control studies.

Let’s learn about Case-Control Studies!

Starts in Present with D

In case-control studies, people get into the study based on their disease status and THEN information on past exposures is collected. People who are selected because they are D+ are called cases, and those who are selected because they are D- are called controls. Cases can be either incident (newly diagnosed) or prevalent (existing at a point in time) cases of the disease.

A case-control study looks at the odds of being exposed among cases and controls to tell us the likelihood of exposure based on disease status.

One famous epidemiologist calls ca-co studies “trohoc studies” because they are backwards cohort studies. The advantages of case-control studies are the disadvantages of f-up studies and the disadvantages of case-control studies are the advantages of f-up studies. Let’s take a look, shall we?

Advantages of Ca-Co

Suitable for rare and common diseases

Economical (Time and $$)

Allow testing of multiple hypotheses

Extremely efficient

Very valuable in controlling confounding and evaluating information

Case-control studies offer a solution to the difficulty of studying diseases with very long induction or latency periods, as investigators can identify affected (D+) and unaffected (D-) individuals and then look backward in time to assess their antecedent exposures, rather than having to wait a number of years for the disease to develop as with f-up designs. This also makes them ideally suited for investigating rare diseases, which would otherwise need to follow tremendously large numbers of individuals in order to accumulate a sufficient number of D+s. This process is also what makes case-control studies efficient, in terms of both time and costs, relative to other analytic approaches.

Ca-Co also allow for the evaluation of a wide range of potential etiologic exposures that might relate to a specific disease as well as the inter-relationships among these factors. Therefore case-control studies can be used to test specific hypothesis in the absence of an a priori hypothesis and explore a range of exposures among affected and non-affected individuals.

This study design is especially useful in the early stages of the development of knowledge about a particular disease or outcome of interest.

Disadvantages

Susceptible to bias

Study only one disease

Not good for rare exposures

Temporal relationship between E and D unknown

Cannot measure incidence

Sometimes difficult to find a control group

The major disadvantage of case-control studies is that both the exposure and the disease have already occurred at the time the participants enter into the study which makes this design especially susceptible to bias.

A fundamental problem with ca-co studies is that there is no way to prove to anyone that you have selected a proper control group. As a result, this design is particularly susceptible to bias in the selection of either the cases or the controls into the study on the basis of their exposure status, as well as from differential reporting or recording of exposure information between study group based on the disease status.

Another disadvantage in that you can only study one disease. It also is not good for rare exposures, and the time sequence between E and D may be uncertain

Ca-co studies also cannot measure incidence (or mortality) as the investigation begins with the selection of subjects based on D status.

Odds Ratio (OR)

A case-control study compares the odds of exposure among cases (D+) relative to the odds of controls (D-) who are E+. Let’s take a look at the 2x2 table to show how the each of these odds are calculated…

Odds of E+ among Cases

	Case (D+)	Control (D-)
E+	a	b
E-	c	d
Total	a+c	b + d

So, what are the odds in favor of being exposed if you are a case? I’ll give you two choices:

a/(a+c)

a/c

Scroll down for the answer.

#2 is correct. If you said #1, think back to the M&M example, where the odds in favor of selecting a yellow M&M was the number of yellow M&Ms divided by all of the M&Ms EXCEPT the yellow ones….

Odds of E+ among Controls

	Case (D+)	Control (D-)
E+	a	b
E-	c	d
Total	a+c	b + d

So, what are the odds in favor of being exposed if you are a control? If you said anything other than b/d, please review the odds information or email me directly for assistance.

Calculaing OR

OR = odds of E among Ca/odds of E among Co

= (a/c) / (b/d)

= (a x d) / (b x c)

= ad/bc

So to calculate OR, we could calculate the odds of E+ among cases and the odds of E- among controls, then calculate a ratio of the two odds. Or, we can do the math magic on the slide and boil it down to a very simple equation of ad/bc.

Note: It’s really not math magic. When dividing fractions by fractions, we flip the denominator and multiply the two fractions. Here’s a link to a refresher. https://www.wikihow.com/Divide-Fractions-by-Fractions

Concordant Cells

	Case (D+)	Control (D-)
E+	a	b
E-	c	d
Total	a+c	b + d

a and d are called “concordant cells”

Discordant Cells

	Case (D+)	Control (D-)
E+	a	b
E-	c	d
Total	a+c	b + d

B and C are called discordant cells

OR = Concordant/Discordant

	Case (D+)	Control (D-)
E+	a	b
E-	c	d
Total	a+c	b + d

So, OR = concordant cells divided by discordant cells. Fun times!

Interpretation of OR

When OR = 1.00

Exposure does not affect odds of disease

This is the NULL

When OR < 1.00

Negative association

Odds of exposure lower among cases

When OR > 1.00

Positive association

Odds of exposure higher among cases

An odds ratio of

• 1.0 (or close to 1.0) indicates that the odds of exposure among cases are the same as, or similar to, the odds of exposure among controls. Thus, the exposure is not associated with the disease.

• Greater than 1.0 indicates that the odds of exposure among cases are greater than the odds of exposure among controls. The exposure might be a risk factor for the disease.

• Less than 1.0 indicates that the odds of exposure among cases are lower than the odds of exposure among controls. The exposure might be a protective factor against the disease.

As with interpreting RR, the further away from 1, the stronger the association, be it positive or negative.

Example

We want to determine how much more common occupational benzene exposure is among persons with leukemia. We conducted a case-control study in which 85 of the 125 workers with leukemia were exposed to benzene on the job. Conversely, among the 125 controls, only 40 had been exposed.

Step 1: Set up data table

	Case (D+)	Control (D-)	Total
E+	85	40	125
E-	40	84	125
Total	125	125	250

Note: among the cases, we knew that 85 had been exposed. That is our cell a. To get our cell c, we subtract that number from the total number of cases, which we were given. We were given the value for cell b (exposed among controls), and determined cell d by subtracting that number from the total number of controls, which we were given.

Step 2: Calculate OR (ad/bc)

OR = ad/bc

= (85 x 85) / (40 x 40)

= 7225/1600

= 4.5

Step 3: Interpret

The odds of benzene exposure among cases is 4.5 times as great as the odds of exposure among controls.

Word of Caution

Very common

Very commonly wrong

Case-Control studies make up about 80 – 85% of all published studies. Unfortunately, many of these studies really are not case-control studies, despite being touted and published as such. This is because the case-control design is difficult to grasp and perform. Most commonly we see data from cross-sectional studies being used as a case-control study.