1 epi 5240: introduction to epidemiology measures used to compare groups october 5, 2009 dr. n....
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EPI 5240:Introduction to Epidemiology
Measures used to compare groups October 5, 2009
Dr. N. Birkett,Department of Epidemiology &
Community Medicine,University of Ottawa
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Session Overview
• Methods of Comparing groups– Risk/rate ratios– Odd ratios– Difference measures
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ONE BIG WARNING!!!!!Some books (e.g. the Greenberg one used in the summer course) rotate their 2X2 tables from the normal approach.
That is, they have the outcomes as the rows and the exposure as the columns.
BE WARNED. This could cause confusion. My tables use the more common approach.
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Comparing groups (1)
• Two main outcome measures– Incidence (either risk or rate)– Prevalence
• How do you determine if an exposure is related to an outcome?– Need to compare the measure in the two groups.
• Differences• Ratios (we’ll start with this one).
– Ratio measures have NO units.– All ratio measures have the same interpretation
• 1.0 = no effect• < 1.0 protective effect• > 1.0 increased risk
– Values over 2.0 are of strong interest
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Comparing groups:Cohorts (2)
YES NO
YES 1,000 9,000 10,000
NO 100 9,900 10,000
1,100 18,900 20,000
Disease
Exp
RISK RATIO
Risk in exposed: = 1000/10000Risk in Non-exposed = 100/10000
If exposure increases risk, you would expect the risk in the exposed to be larger than risk in the unexposed. How much larger can be assessed by the ratio of one to the other: Risk in expRisk ratio (RR) = ----------------------- Risk in non-exp
= (1000/10000)/(100/10000)
= 10.0
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Comparing groups:Cohorts (3)
YES NO
YES a b a+b
NO c d c+d
a+c b+d N
Disease
Exp
RISK RATIO
Risk in exposed: = a/(a+b)Risk in Non-exposed = c/(c+d)
If exposure increases risk, you would expect a/(a+b) to be larger than c/(c+d). How much larger can be assessed by the ratio of one to the other: Risk in expRisk ratio (RR) = ----------------------- Risk in non-exp
= (a/(a+b))/(c/(c+d)
a/(a+b)= -------------- c/(c+d)
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Comparing groups:Cohorts (4)
YES NO
High 42 80 122
Low 43 302 345
85 382 467
Death
Pollutantlevel
Risk in exposed: = 42/122 = 0.344Risk in Non-exposed = 43/345 = 0.125
Exp riskRisk ratio (RR) = ---------------------- Non-exp risk
= 0.344/0.125
= 2.76
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95% CI’s for CIR (1)
• For a mean value, the 95% CI is given as:
where
• Assumes mean has a normal (Gaussian) distribution
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• Might try using the same approach to obtain ’95% CI’ for CIR using:
• BUT: CIR is NOT normally distributed– Range from 0 to +∞– Null value = 1.0– Implies a non-symmetric distribution
95% CI’s for CIR (2)
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CIR
0 2 4 6 8
0.0
0.1
0.2
0.3
0.4
0.5
0.6
Plot of ‘CIR’ distribution when H0 is true
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• Instead, use ‘log(CIR)’ where the log is taken to the ‘natural’ base ‘e– Often written ln(CIR)
• ln(CIR) is approximately normally distributed– Range from -∞ to +∞– Null value = 0.0
• 95% CI is given by:
• Need to find formula for ‘se(ln(CIR))’
95% CI’s for CIR (3)
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Plot of ‘ln(CIR)’ distribution when H0 is true
CIR
-6 -4 -2 0 2 4 6
0.0
0.1
0.2
0.3
0.4
0.5
0.6
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if exposed and unexposed are independent
95% CI’s for CIR (4)
After some math, this gives the following result (next slide)
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95% CI’s for CIR (5)
YES NO
YES a b a+b
NO c d c+d
a+c b+d N
Disease
Exp
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95% CI’s for CIR (6)We’re close now. Just take the ‘anti-logs’ (usually called the ‘exp’ function
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Comparing groups:Cohorts (5)
YES NO
High 42 80 122
Low 43 302 345
85 382 467
Death
Pollutantlevel
Risk ratio (RR) or CIR = 2.76
80 302var(ln(CIR)) = ------------ + ------------- = 0.03597 42*122 43*345
se(ln(CIR)) = sqrt(0.03597) = 0.190
Upper 95% CI = 2.76 * exp(+1.96*0.190) = 4.00Lower 95% CI = 2.76 * exp(-1.96*0.190) = 1.90
Conclusion:CIR is:
2.76 (1.90 to 4.00)
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Comparing groups: Cohorts (6)
• Hypothesis testing (H0: CIR=1)
– Much less common than 95% CI’s– Normal approximation test is generally OK
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Comparing groups:Cohorts (7)
YES NO
YES 1,000 9,000 10,000
NO 100 9,900 10,000
1,100 18,900 20,000
Disease
Exp
RISK DIFFERENCE
Risk in exposed: = 1000/10000Risk in Non-exposed = 100/10000
If exposure increases risk, you would expect the risk in the exposed to be larger than risk in the unexposed. How much larger can be assessed by the difference between the two:
Risk difference (RD)
= (Risk in Exp) – (Risk in Non-exp)
1000 100 900= ---------- - ----------- = ----------- = 0.90 10,000 10,000 10,000
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Comparing groups:Cohorts (8)
YES NO
YES a b a+b
NO c d c+d
a+c b+d N
Disease
Exp
RISK DIFFERENCE
Risk in exposed: = a/(a+b)Risk in Non-exposed = c/(c+d)
If exposure increases risk, you would expect a/(a+b) to be larger than c/(c+d). How much larger can be assessed by the difference between the two:
Risk difference (RD)
= (Risk in Exp) – (Risk in Non-exp)
a c= ---------- - ----------- a + b c + d
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Comparing groups:Cohorts (9)
YES NO
High 42 80 122
Low 43 302 345
85 382 467
Death
PollutantLevel
Risk in exposed: = 42/122 = 0.344Risk in Non-exposed = 43/345 = 0.125
Risk difference (RD) = (Risk in Exp) - (Risk in Non-exp)
= 0.344 - 0.125
= 0.219
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We assume that the incidence follows a binomial distributionCan be considered as approximately normal if incidence isn’t too small).
95% CI’s for Risk Diff (1)
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95% CI’s for Risk Diff (2)
YES NO
High 42 80 122
Low 43 302 345
85 382 467
Death
Pollutantlevel
RD = 0.219
42*80 43*302var(RD) = ------------ + ------------- = 0.00217 1223 3453
se(RD) = sqrt(0.00217) = 0.047
Upper 95% CI = 0.219 + 1.96*0.047 = 0.310Lower 95% CI = 0.219 - 1.96*0.047 = 0.127
Conclusion:RD is:
0.219 (0.127 to 0.310)
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Comparing groups: Cohorts (10)
• Which comparative measure do you use?• Depends on the circumstances.• Risk Ratio RELATIVE risk measure• Risk Difference ABSOLUTE risk
measure• Post-menopausal estrogens & endometrial
cancer– RR = 2.3– RD = 2/10,000
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Comparing groups:Cohorts (11)
Disease Person-years
YES 1,000 9,500
NO 100 9,950
1,100 19,450
Exp
RATE RATIO
Rate in exposed: = 1000/9500Rate in Non-exposed = 100/9950
If exposure increases rate of getting disease, you would expect the rate in exposed to be larger than the rate in unexposed. How much larger can be assessed by the ratio of one to the other: Rate in ExpRate ratio (RR) = ------------------------ Rate in Non-exp
= (1000/9500)/(100/9950)
= 10.5
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Comparing groups: Cohorts (12)
DISEASE Person-time
YES A Y1
NO B Y2
A + B Y1 + Y2
Exp
RATE RATIO
Rate in exposed: = A/Y1
Rate in Non-exposed = B/Y2
If exposure increases rate of getting disease, you would expect A/Y1 to be larger than B/Y2. How much larger can be assessed by the ratio of one to the other: Rate in ExpRate ratio (RR) = ------------------------ Rate in Non-exp
= (A/Y1))/(B/Y2)
A/Y1
= -------------- B/Y2
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Comparing groups:Cohorts (13)
Rate in exposed: = 42/101 = 0.416Rate in Non-exposed = 43/323.5 = 0.133
Rate in ExpRate ratio (RR) = ------------------------ Rate in Non-exp
= 0.416/0.133
= 3.13
Pollutantlevel
Dead Person-years
High 42 101
Low 43 323.5
85 424.5
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• Use the same approach to obtain ’95% CI’ for IDR as we used for CIR:
• BUT: IDR is NOT normally distributed– Range from 0 to +∞– Null value = 1.0– Implies a non-symmetric distribution
95% CI’s for IDR (1)
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• Instead, use ‘log(IDR)’ where the log is taken to the ‘natural’ base ‘e– Often written ln(IDR)
• ln(IDR) is approximately normally distributed– Range from -∞ to +∞– Null value = 0.0
• 95% CI is given by:
• Need to find formula for ‘se(ln(IDR))’
95% CI’s for IDR (2)
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if exposed and unexposed are independent
95% CI’s for IDR (3)
After some math, this gives the following result (next slide)
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95% CI’s for IDR (4) DISEASE Person-time
YES a Y1
NO c Y2
a+c Y1 + Y2
Exp
DOES NOT DEPENDON PERSON-TIME!!
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95% CI’s for IDR (5)We’re close now. Just take the ‘anti-logs’ (usually called the ‘exp’ function
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Comparing groups:Cohorts (14)
Pollutantlevel
Dead Person-years
High 42 101
Low 43 323.5
85 424.5
Rate ratio (RR) or IDR = 3.13
1 1var(ln(IDR)) = ------ + ----- = 0.047 42 43
se(ln(IDR)) = sqrt(0.047) = 0.217
Upper 95% CI = 3.13 * exp(+1.96*0.217) = 4.79Lower 95% CI = 3.13 * exp( -1.96*0.217) = 2.05
Conclusion:IDR is:
3.13 (2.05 to 4.79)
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Comparing groups: Cohorts (15)
• Hypothesis testing (H0: IDR=1)
– Much less common than 95% CI’s– Normal approximation test is generally OK
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Comparing groups:Cohorts (16)
Disease Person-years
YES 1,000 9,500
NO 100 9,950
1,100 19,450
Exp
RATE DIFFERENCE
Rate in exposed: = 1000/9500Rate in Non-exposed = 100/9950
If exposure increases rate of getting disease, you would expect the rate in exposed to be larger than the rate in unexposed. How much larger can be assessed by the difference between the two:
Rate difference
= (Rate in Exp) – (Rate in Non-exp)
1000 100= --------- - --------- = 0.095 cases/PY 9500 9950
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Comparing groups:Cohorts (17)
DISEASE Person-time
YES A Y1
NO B Y2
A + B Y1 + Y2
Exp
RATE DIFFERENCE
Rate in exposed: = A/Y1
Rate in Non-exposed = B/Y2
If exposure increases rate of getting disease, you would expect A/Y1 to be larger than B/Y2. How much larger can be assessed by the difference between the two:
Rate difference
= (Rate in Exp) – (Rate in Non-exp)
A B= ------ - ------- Y1 Y2
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Comparing groups: Cohorts (18)
Rate in exposed: = 42/101 = 0.416Rate in Non-exposed = 43/323.5 = 0.133
Rate difference (RD) = (Rate in Exp) – (Rate in Non-exp)
= 0.416 - 0.133
= 0.283 cases/person-year
Pollutantlevel
Dead Person-years
High 42 101
Low 43 323.5
85 424.5
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We assume that the incidence follows a Poisson distributionCan be considered as approximately normal if incidence isn’t too small).
95% CI’s for Rate Diff (1)
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95% CI’s for Rate Diff (2)
RD = 0.283 cases/PY
42 43var(RD) = --------- + ----------- = 0.00453 1012 323.52
se(RD) = sqrt(0.00453) = 0.067
Upper 95% CI = 0.283 + 1.96*0.067 = 0.415Lower 95% CI = 0.283 - 1.96*0.067 = 0.152
Conclusion:Rate Diff is:
0.283 (0.152 to 0.415) Cases/PY
Pollutantlevel
Dead Person-years
High 42 101
Low 43 323.5
85 424.5
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Comparing groups: Cohorts (19)
Some Issues• What does RR (or RD) mean
– Can mean risk or rate ratio. Some people think this is pedantic rather than correct
– Need to tell which from context.– Sometimes referred to as Relative Risk (generic
term).
• Are risk differences or ratios preferred?– RR’s are much more common– Both have a role to play.
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• CAN NOT COMPUTE A RISK RATIO!• Can not estimate incidence from a case-control
study.• Can not compute risk differences.• Why? We choose the subjects based on their
outcome status. Usually, that means making the number of cases and controls equal. Hence, the ‘incidence’ in the case-control study is fixed at 0.50. In real world, it is most likely much lower (1/100,000).
• Let’s look at an example.
Comparing groups:Case-control (1)
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Comparing groups:Case-control (2)
YES NO
YES 1,000 9,000 10,000
NO 100 9,900 10,000
1,100 18,900 20,000
Disease
Exp
RISK RATIO
Risk in exposed: = 0.1Risk in Non-exposed = 0.01
RR = 0.1/0.01
= 10.0
Case Control
YES 1,000 524 1,524
NO 100 576 676
1,100 1,100 2,200
Exp
‘RISK RATIO’
‘Risk’ in exposed: = 0.656‘Risk’ in Non-exposed = 0.148
‘RR’ = 0.656/.148
= 4.44
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• CAN NOT COMPUTE A RISK RATIO!
• So, what do we do?– Cornfield & Haenzel provided solution in
1960. They looked at the ODDS of exposure. The ratio of the odds of exposure in the cases and controls is almost the same as the RR, if the disease is rare.
Comparing groups:Case-control (3)
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Comparing groups:Case-control (4)
YES NO
YES 900 400 1,300
NO 100 600 700
1,000 1,000 2,000
Disease
Exp
ODDS RATIO
Odds of exposure in cases = 900/100Odds of exposure in controls = 400/600
If exposure increases rate of getting disease, you would to find more exposed cases than exposed controls. That is, the odds of exposure for case would be high. How much larger can be assessed by the ratio of one to the other: Exp odds in casesOdds ratio (OR) = ----------------------------- Exp odds in controls
= (900/100)/(400/600)
= 13.5
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Comparing groups:Case-control (5)
YES NO
YES a b a+b
NO c d c+d
a+c b+d N
Disease
Exp
ODDS RATIO
Odds of exposure in cases = a/cOdds of exposure in controls = b/d
If exposure increases rate of getting disease, you would to find more exposed cases than exposed controls. That is, the odds of exposure for case would be high (a/c > b/d). How much larger can be assessed by the ratio of one to the other: Exp odds in casesOdds ratio (OR) = ----------------------------- Exp odds in controls= (a/c)/(b/d)
ad= ---------- bc
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Yes No
High 42 18
Low 43 67
85 85
PollutantLevel
Odds of exp in cases: = 42/43 = 0.977Odds of exp in controls: = 18/67 = 0.269
Odds ratio (OR) = Odds in cases/odds in controls
= 0.977/ 0.269 = (42*67)/(43*18)
= 3.64
Comparing groups:Case-control (6)Disease
NOTE:Risk ratio = 2.76Rate ratio = 3.13
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• Use the same approach to obtain ’95% CI’ for OR as we used for CIR/IDR:
• BUT: OR is NOT normally distributed– Range from 0 to +∞– Null value = 1.0– Implies a non-symmetric distribution
95% CI’s for OR (1)
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• Instead, use ‘log(OR)’ where the log is taken to the ‘natural’ base ‘e– Often written ln(OR)
• ln(OR) is approximately normally distributed– Range from -∞ to +∞– Null value = 0.0
• 95% CI is given by:
• Need to find formula for ‘se(ln(OR))’
95% CI’s for OR (2)
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95% CI’s for OR (3) Case Control
YES a b
NO c d
a+c a+d
Exp
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95% CI’s for OR (4)We’re close now. Just take the ‘anti-logs’ (usually called the ‘exp’ function
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Odds ratio (OR) = 3.63
1 1 1 1var(ln(OR)) = ----- + ---- + ---- + ---- = 0.118 42 18 43 67
se(ln(OR)) = sqrt(0.118) = 0.343
Upper 95% CI = 3.63 * exp(+1.96*0.343) = 7.11Lower 95% CI = 3.63 * exp( -1.96*0.343) = 1.85
Conclusion:OR is:
3.63 (1.85 to 7.11)
Comparing groups:Case-control (6)
Yes No
High 42 18
Low 43 67
85 85
PollutantLevel
Disease
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Comparing groups:Case-Control (7)
• Hypothesis testing (H0: OR=1)
– Much less common than 95% CI’s– Normal approximation test is generally OK
• JUST USE THE STANDARD Chi-square TEST!
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• You can compute an OR for a cohort. Why would you do so?– OR’s are the key outcome measure for logistic
regression, one of the most common analysis methods used in epidemiology
– Unless disease is common, the OR and the RR from the cohort will be very similar.
• But, where possible, rate ratios are preferred.
Comparing groups:Case-control (8)
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• Cohort studies– Relative risk– Relative rate– Risk/rate differences
• Case-control study– Odds-ratio
Summary: comparisons