Two-by-two tables

Consider the LEAP study Results from section 1.1

The table below summarizes the test results of the Oral Food Challenge (OFC) for children who were enrolled in a Peanut Consumption (treatment) group or Peanut avoidance Control group.A child was recoded as passing the oral food challenge if no allergic reaction occurred, and failing if an allergic reaction occurred to peanuts

Fail OFC

Pass OFC

Sum

Peanut avoidance (control)

36

227

263

Peanut Consumption (treatment)

5

262

267

Sum

41

489

530

(a)What is the total number of children included in the study?

(b)What is the total number of participants in the control group?

(c)What is the total number of participants in the treatment group?

(d)What is the total number of children who failed the OFC test?

(e))What is the total number of children who passed the OFC test?

(f)How many children enrolled in the peanut avoidance group failed the OFC test?

(g)How many children enrolled in the peanut avoidance group passed the OFC test?

(h)How many children enrolled in the peanut consumption group failed the OFC test?

(i)How many children enrolled in the peanut consumption group passed the OFC test?

Probability Calculations LEAP study data:

Solved Examples (Marginal Probabilities)

(i)Find the probability that the child in a control group

Answer: 263/530= 0.4962

(ii)Find the probability that the child fails the test (we are including all children here)

Answer: 41/530= 0.0774

For you to do:

(a)Find the probability that the child in a treatment group =

(b)Find the probability that the child passes the test

Solved Examples: (Conditional Probabilities)

(i)Given the child is in a peanut avoidance (control) group, find the probability that he/she fails the OFC test:

Answer: 36/263 = 0.1369 *

(*notice we used 263 in the denominator, since we are only looking at the control groupthat is the given or known information, so we use 263 and not 530)

(ii)If it is known (means the same as "given") that the child is in a peanut consumption (treatment) group, find the probability that he/she passes the test.

Answer:262/267 = 0.9813

(*notice we used 267 in the denominator, since we are only looking at the treatment groupthat is the given or known information, so we use 267 and not 530)

For you to do:

(a)Given the child is in a peanut avoidance (control) group, find the probability that he/she passes the OFC test:

(b)If it is known (means the same as "given") that the child is in a peanut consumption (treatment) group, find the probability that he/she fails the OFC test.

(c) Given the child failed the OFC test, find the probability that he/she was enrolled in a peanut avoidance (control) group: (hint use 41 in the denominator)

(d)Given the child failed the OFC test, find the probability that he/she was enrolled in a peanut consumption (control) group: (hint use 41 in the denominator)

(e) Given the child passed the OFC test, find the probability that he/she was enrolled in a peanut avoidance (control) group:

(f)Given the child passed the OFC test, find the probability that he/she was enrolled in a peanut consumption (treatment) group:

Define Relative Risk:

Consider the LEAP study results:

Fail OFC

Pass OFC

Sum

Peanut avoidance (control)

36 (a)

227(b)

263 (a+b)

Peanut Consumption (treatment)

5(c)

262(d)

267 (c+d)

Sum

41 (a+c)

489 (b+d)

530 (a+b+c+d)

Write the Relative Risk formula (page 48) comparing groups 1 to 2 and 2 to 1, in terms of a, b, c, and d

Use the LEAP study data to calculate the relative risk of failing the oral food challenge comparing the individuals in the avoidance group to individuals in the control group

Use the LEAP study data to calculate the relative risk of failing the oral food challenge comparing the individuals in the control group to the individuals in the avoidance group

Interpret the results from ( ) and ()

Relative Risk (optional extra practice)

Consider the following table related to smoking and cardiovascular disease:

Cardiovascular Disease

Yes (present)

No (absent)

Sum

Non-smokers

32(a)

88(b)

120(a+b)

Smokers

40(c)

71(d)

111 (c+d)

Sum

72 (a+c)

159 (b+d)

231 (a+b+c+d)

Use the data in the table above to calculate the relative risk of the Cardiovascular Disease comparing the smokers to non-smokers

Use the data in the table above to calculate the relative risk of the Cardiovascular Disease comparing the non-smokers to smokers

The Odds Ratios

When we have case control studies the odds ratio are more appropriate to estimate the association between the exposure and th4 presence of a disease:

Consider a general 2X2 table of exposure vs disease status:

Disease

Yes (present)

No (absence)

Sum

Exposed

(a)

(b)

(a+b)

Unexposed

(c)

(d)

(c+d)

Sum

(a+c)

(b+d)

(a+b+c+d)

Odds Ratio (disease exposed vs disease unexposed) = (a*d)/(b*c)

Calculate the Odds ratio for the disease, exposed vs unexposed:

Disease present (yes)

Disease absent

(no)

Exposed to a virus

21(a)

79(b)

100(a+b)

Not exposed

13(c)

87(d)

100(c+d)

34(a)+(c)

166(b)+(d)

200 (a+b+c+d)

Answer:

Diagnostic Tests:

30.Define false negative:

31.Define false positive.

32.Define sensitivity of a test.

33.Define specificity of a test.

34.Define prevalence.

35.Define positive predictive value (PPV).

36. Define negative predictive value (PPV).

Consider the following results of a mammography test:

(Refer to the Week 3 notes, Part 3)

Breast Cancer Present

Breast Cancer Absent

Total

Mammogram positive

311(a)

6976(b)

7287(a+b)

Mammogram negative

38(c)

92675(d)

92713(c+d)

339(a)+(c)

99651(b)+(d)

100,000 (a+b+c+d)

Find the following:

Find the probability of a false negative test result

Find the probability of a false positive test result

Find the sensitivity

Find the specificity

Find the disease prevalence.

Find the positive predictive value

Find the negative predictive value