Risk Ratio and Odds Ratio

Risk and Odds

• Risk is a probability as calculated from

𝑝𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 =𝑜𝑛𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒

𝐴𝐿𝐿 𝑝𝑜𝑠𝑠𝑖𝑏𝑙𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠

• Odds is opposed to probability, and is calculated from

𝑂𝑑𝑑𝑠 =𝑜𝑛𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒

𝑎𝑙𝑙 𝑂𝑇𝐻𝐸𝑅 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠• Both are measures of likelihood but differ in

• Denominator

• 0 ≤ Risk (or probability) ≤ 1 whereas 0 ≤ Odds ≤ ∞

Calculation Examples

• The probability and the odds of flipping a coin and getting a head• Outcome: H

• Other outcome: T

• All possible outcome: H, T

• 𝑝𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 =𝑜𝑛𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒

𝐴𝐿𝐿 𝑝𝑜𝑠𝑠𝑖𝑏𝑙𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠=

1

2= 0.5

• 𝑂𝑑𝑑𝑠 =𝑜𝑛𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒

𝑎𝑙𝑙 𝑂𝑇𝐻𝐸𝑅 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠=

1

1= 1: 1

• The probability and the odds of flipping a coin twice and getting two heads• Outcome: HH

• Other outcome: HT, TH, TT

• All possible outcome: HH, HT, TH, TT

• 𝑝𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 =𝑜𝑛𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒

𝐴𝐿𝐿 𝑝𝑜𝑠𝑠𝑖𝑏𝑙𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠=

1

4= 0.25

• 𝑂𝑑𝑑𝑠 =𝑜𝑛𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒

𝑎𝑙𝑙 𝑂𝑇𝐻𝐸𝑅 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠=

1

3= 1: 3

Another Example• The probability and the odds of having 2 short-hair kittens in a litter with 5 kittens

• Outcome: 2 short-hair kittens (S S)

• Other outcome: 3 long-hair kittens (L L L)

• All possible outcome: S S L L L

• 𝑝𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 =𝑜𝑛𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒

𝐴𝐿𝐿 𝑝𝑜𝑠𝑠𝑖𝑏𝑙𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠=

2

5= 0.4

• 𝑂𝑑𝑑𝑠 =𝑜𝑛𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒

𝑎𝑙𝑙 𝑂𝑇𝐻𝐸𝑅 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠=

2

3= 2: 3 ≈ 0.67

• The probability and the odds of having 4 short-hair kittens in a litter with 5 kittens• Outcome: 4 short-hair kittens (S S S S)

• Other outcome: 1 long-hair kittens (L)

• All possible outcome: S S S S L

• 𝑝𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 =𝑜𝑛𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒

𝐴𝐿𝐿 𝑝𝑜𝑠𝑠𝑖𝑏𝑙𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠=

4

5= 0.8

• 𝑂𝑑𝑑𝑠 =𝑜𝑛𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒

𝑎𝑙𝑙 𝑂𝑇𝐻𝐸𝑅 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠=

4

1= 4: 1 = 4

Graphical Representations of Risk and Odds

Odds =Risk =

Epidemiology

• Wiki definition: “Epidemiology is the study and analysis about the distribution and determinants of health and diseases in defined population”

• Many types of epidemiological studies• Randomized control study

• Cohort study

• Case-control study

https://en.wikipedia.org/wiki/Epidemiologyhttps://youtu.be/sdFYHSxq_qo

Epidemiological studies: Some assessments• Experimental study or Observational study

• If a researcher assigns mixture of participants to groups (i.e. randomization), it’s the experimental study

• If the researcher does not assign participants to any groups, but let participants’ characteristics determined which group they should fall in, it’s an observational study

• Directionality• Forward study – the exposure is known, then follow up to see what outcomes

occurred

• Backward study – the outcomes are occurred, then exposure is determined

• Timing• Prospective – the study starts before the outcome occurred

• Retrospective – the study starts after the outcomes occurred

https://youtu.be/sdFYHSxq_qo

Epidemiology Study Types: Cohort study

• An observational study; forward directionality; prospective timing, or can be a retrospective timing

• Conceptually,• Start with a population of disease-free individuals

• Identify individuals that are exposed to a risk factor(s) and those that are NOT exposed to the same risk factor(s)

• then follow up both groups over time to find out the risk of specific outcomes (e.g. diseases) occurring in each individual

• Relative Risk is used to determined association between the exposure and the outcomes

• Hypotheses (tentative!!!)• Ho: Proportions of the outcome in exposed and unexposed groups are equal

• H1: Proportions of the outcome in exposed and unexposed groups are not equal

https://youtu.be/sGfIKmKMRdg

Cohort study

time

Exposuree.g. smoking

smoke

not smoke

outcomee.g. lung cancer

Lung cancer

No Lung cancer

versus

Breast cancer and Hormone replacement therapy in the million-women study

OutcomeExposure

Breast cancer

No breast cancer

Used HRT 1934 140,936

Never used HRT 2894 389,863

The Lancet 362: 419-427

Question: what is the risk of using HRT on breast cancer occurrence in women?

OutcomeExposure

Breast cancer

No breast cancer

Used HRT a b

Never used HRT c d

Relative Risk (RR) and confident intervals

𝑙𝑖𝑚𝑖𝑡𝑠 = 𝑙𝑛(𝑅𝑅)±𝑧ൗ𝑏 𝑎

𝑎 + 𝑏+

ൗ𝑑 𝑐𝑐 + 𝑑

𝐶𝐼 = 𝑒𝑙𝑜𝑤𝑒𝑟 𝑙𝑖𝑚𝑖𝑡 , 𝑒𝑢𝑝𝑝𝑒𝑟 𝑙𝑖𝑚𝑖𝑡

OutcomeExposure

Breast cancer

No breast cancer

Used HRT a b

Never used HRT c d

𝑅𝑒𝑙𝑎𝑡𝑖𝑣𝑒 𝑅𝑖𝑠𝑘 =

𝑎𝑎 + 𝑏𝑐

𝑐 + 𝑑

Interpretation of RR

• If RR = 1 or CI includes 1, there is no risk for the outcome to the exposed group nor the unexposed group

• If RR is more than 1 and CI does not includes 1, the relative risk of the outcome in the exposed group was increased by 1 − 𝑅𝑅 × 100% relative to the unexposed group• Or the risk of the outcome has RR times more likely to occur in

the exposed group than in the unexposed group

• If RR is less than 1 and CI does not includes 1 in its range, the relation risk of the outcome in the exposed group was reduced by 1 − 𝑅𝑅 × 100% relative to the unexposed group

Breast cancer and Hormone replacement therapy in the million-women study

OutcomeExposure

Breast cancer

No breast cancer

Used HRT 1934 140,936

Never used HRT 2894 389,863

𝑙𝑖𝑚𝑖𝑡 𝑎 = ln 1.824 − 1.96ൗ140936 1934

1394 + 140936+

ൗ389863 28942894 + 389863

𝑙𝑖𝑚𝑖𝑡 𝑏 = ln 1.824 + 1.96ൗ140936 1934

1394 + 140936+

ൗ389863 28942894 + 389863

𝑅𝑖𝑠𝑘𝐻𝑅𝑇 =1934

1934 + 140936= 0.0135

𝑅𝑖𝑠𝑘𝑁𝑜 𝐻𝑅𝑇 =2894

2894 + 389863= 0.0074

𝑅𝑒𝑙𝑒𝑡𝑖𝑣𝑒 𝑅𝑖𝑠𝑘 =𝑅𝑖𝑠𝑘𝐻𝑅𝑇

𝑅𝑖𝑠𝑘𝑁𝑜 𝐻𝑅𝑇=0.0135

0.0074= 1.824

𝑙𝑖𝑚𝑖𝑡 𝑎 = 0.5573𝑙𝑖𝑚𝑖𝑡 𝑏 = 0.6120𝐶𝐼 = 𝑒 .5573, 𝑒 .6120

𝐶𝐼 = 1.746,1.958

Relative Risk (RR) and confident intervals

• RR=1.824 with CI=(1.746,1.958) RR is significantly different from 1, reject Ho then accept H1 stating that proportions of breast cancer in both groups are not equal

• Interpretation: Relative risk of breast cancer in women who used HRT is increased by |1-1.824|100%=82.4% relative to women who did not use HRT.

• Or the risk of breast cancer is 1.8 times more likely to occur in women who used HRT than in the women who did not use HRT.

OutcomeExposure

Breast cancer

No breast cancer

Used HRT 1934 140,936

Never used HRT 2894 389,863

Cardiovascular diseases among users of estrogen with progestin as compared to nonusers

• 𝑅𝑅 = Τ.000295 .001414 = 0.208 with CI=(0.103,0.419) RR is significantly different from 1, reject Ho ten accept H1 stating that proportions of breast cancer in both groups are not equal

• Interpretation: Relative risk of major coronary disease is reduced by|1-0.208|100% = 79.2% in users of estrogen with progestin relative to users who has not used any hormone

• Or the risk of major coronary disease is 0.2 times less likely to occur in user who of estrogen with progestin than in users who do not use any hormone.

OutcomeExposure

Major coronary disease

No disease Risk

Estrogen with progestin 8 27,153 𝟖𝟖 + 𝟐𝟕𝟓𝟏𝟑

= 𝟎. 𝟎𝟎𝟎𝟐𝟗𝟓

Not Used 431 304,313 𝟒𝟑𝟏𝟒𝟑𝟏 + 𝟑𝟎𝟒𝟑𝟏𝟑

= 𝟎. 𝟎𝟎𝟏𝟒𝟏𝟒

Adapted from NEJM 1996: 335-453

Epidemiology Study Types: Case-Control study• An observational study; backward directionality; retrospective

timing

• Conceptually,• Start with the case, i.e. a group of individuals having the outcomes

(e.g. disease), and the control, i.e. a group of individuals not having the outcomes,

• then look back in time in both groups to find out what exposure(s) in both case and control that lead to specific outcomes or diseases

• Odds ratio is used to determined association

• Hypotheses (tentative!!!)• Ho: Proportions of the exposure in case and control are equal

• H1: Proportions of the exposure in case and control are not equal

https://youtu.be/sGfIKmKMRdg

Case-Control study

time

Exposuree.g. smoking

Lung cancer

No Lung cancer

outcomee.g. lung cancer

Smoking

Not smoking

versus

Hay fever and eczema in 11 years old children

Case is children with hay fever and control is children without hay fever. Exposure is the children has experienced eczema or not. What is the odds of children having hay fever will develop eczema compared to children without hay fever?

Hay feverEczema

Yes (case)

No (control)

Yes 141 420

No 928 13525

BMJ. 2000 May 27; 320(7247): 1468.

Hay feverEczema

Yes (case)

No (control)

Yes a b

No c d

Odds ratio (OR) and confident intervals

𝑙𝑖𝑚𝑖𝑡𝑠 = 𝑙𝑛(𝑂𝑅)±𝑧1

𝑎+1

𝑏+1

𝑐+1

𝑑

𝐶𝐼 = 𝑒𝑙𝑜𝑤𝑒𝑟 𝑙𝑖𝑚𝑖𝑡 , 𝑒𝑢𝑝𝑝𝑒𝑟 𝑙𝑖𝑚𝑖𝑡

Hey feverEczema

Yes (case)

No (control)

Yes a b

No c d

𝑂𝑑𝑑𝑠 𝑅𝑎𝑡𝑖𝑜 =Τ𝑎 𝑐

ൗ𝑏 𝑑

Interpretation of OR• If OR = 1 or CI includes 1, the odds are equal for the case

group and the control group to experience the exposures

• If OR is more than 1 and CI does not includes 1, the odds of the exposure in the case group was higher relative to the control group

• If OR is less than 1 and CI does not includes 1, the odd of the exposure in the case group was lower relative to the control group• Normally, there should be switched the case and the control (NOT

shuffling data!) so that OR is greater than 1

Odds ratio (OR) and confident intervals

𝑙𝑖𝑚𝑖𝑡 𝑎 = 𝑙𝑛(4.893)−1.961

141+

1

420+

1

928+

1

13525

𝑙𝑖𝑚𝑖𝑡 𝑏 = 𝑙𝑛(4.893)+1.961

141+

1

420+

1

928+

1

13525

Hey feverEczema

Yes (case)

No (control)

Yes 141 420

No 928 13525

𝑂𝑑𝑑𝑠 𝑅𝑎𝑡𝑖𝑜 =ൗ141 928

ൗ420 13525

= 4.893

𝑙𝑖𝑚𝑖𝑡 𝑎 = 1.386𝑙𝑖𝑚𝑖𝑡 𝑏 = 1.790

𝐶𝐼 = 𝑒1.386, 𝑒1.790

𝐶𝐼 = 3.998,5.998

Hay fever and eczema in 11 years old children

• OR = 4.893 with CI=(3.998,5.998) OR is significantly different from 1, reject Ho then accept H1 stating that proportions of eczema developed in the case and the control are not equal

• Interpretation: children having hay fever has the odds of 4.893 times to develop eczema compared to children without hay fever

Hay feverEczema

Yes (case) No (control)

Yes 141 420

No 928 13525

BMJ. 2000 May 27; 320(7247): 1468.

Leukemia and parental smoking in pregnancy

• Case is patients with leukemia and control is patients without leukemia. Exposure is whether or not there is parental smoking during pregnancy.

• What is the odd of patients with leukemia (the case) to have been exposed to parental smoking in pregnancy?

LeukemiaSmoking

Yes (case) No (control)

Yes 87 147

No 201 508

https://youtu.be/Sec4fewyUig

Leukemia and parental smoking in pregnancy

• 𝑂𝑅 = Τ0.433 0.289 = 1.496 with CI=(1.096,2.042) OR is significantly different from 1, reject Ho then accept H1 stating that proportions of exposing to parental smoking in pregnancy in case and control are not equal

• Interpretation: In patients with leukemia (case group), the odds is 1.496 times to have been exposed to parental smoking in pregnancy

LeukemiaSmoking

Yes (case) No (control)

Yes 87 147

No 201 508

Odds ൗ𝟖𝟕 𝟐𝟎𝟏 = 𝟎. 𝟒𝟑𝟑 ൗ𝟏𝟒𝟕

𝟓𝟎𝟖 = 𝟎. 𝟐𝟖𝟗

https://youtu.be/Sec4fewyUig

Choosing a test [after a thought!]• If you want to know whether or not the observation deviates from the theory choose the test for goodness of fit• If you have only 2 outcomes, use binomial test; else, use 2 test

• If observations is less than 1000, you may find exact probability [you are using a computer, aren’t you?]; else, asymptotic probability will suffice it

• However, you want to find association between 2 nominal variables, a) You may choose 2 test or Fisher’s exact test for a test of independence if what you

really want to know is whether one or more categories in variable A affect one or more categories in variable B; or

b) You may choose 2 test for a test of homogeneity if you just want to know whether proportions of one category in variable A are equal in 2 or more groups (=variable B); or

c) You may choose to find relative risk if you want to know causality of the outcome (=one of two categories in variable A) in 2 different exposed groups from the cohort study; or

d) You may choose to find odds ratio if you want to know odds of the exposure (=one of two categories in variable A) in the case and control groups from the case-control study

Strength of association by crosstab 2 test

• For 2x2 tables, i.e. 2 binary nominal variables

• Phi that is defined as 𝜙 =𝜒2

𝑛

• Example: if 2 = 9.375 and n=150, then 𝜙 =9.375

150= 0.25

• For table larger than 2x2 tables

• Cramer’s V that is defined as 𝑉 =𝜒2

𝑛 min(𝑟−1,𝑐−1)

• Example: if 2 = 126.105, n=566, r=4 rows and c=3 columns, so r-1=4-1=3 and

c-1=3-1=2, then 𝑉 =126.105

566 2= 0.1114 = 0.334

Interpretation of Phi and Cramer’s V

• Reminder: Phi and Cramer’s V are the measures of association between two nominal variables, i.e. how strong the association is • Both Phi and Cramer’s V do not identify the pattern nor direction

• To assess the pattern of association, interpret the column percentages in the bivariate table

• Here the guideline

Measure of association Strength of association

Between 0.00 and 0.10 Weak

Between 0.11 and 0.30 Moderate

Greater than 0.30 Strong