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Measures of Association & rate standardization

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Measures of Association & rate standardization

白其卉

中研院生醫所

For lecture

2002/12/02

Association Index: measures of association

-- Index of measured relationship of risk factor (RF) and disease (Dx)

表示危險因子與疾病間的相關程度之指標

- Analytic epidemiology (epi)
- Cohort study (prospective)
- Fixed cohort
- Dynamic cohort

- Case-control study (retrospective)

- Cohort study (prospective)
- Major aims in analytic epi
- verify the relationship of RF & Dx

- Absolute risk
- Attribute risk
- Relative risk
- Risk ratios (RR)
- Odds ratios (OR)

- Risk Difference

- Magnitude of risk of a disease in a population
- Descriptive unit of magnitude of risk:
- Rate: incldence, prevalence, morbidity, mortality…

- Descriptive unit of magnitude of risk:
- Limitation:
- Absolute risk is not comparative.
- Absolute risk does not allow one to establish association between exposure and disease

Status 或 event 的測量指標

* Odds

常見的流行病學測量指標

- Definition:
- 某段觀察時間內，單位時間中所有可能發生某特定事件的人發生該事件的率。
I：新病例數

P：觀察人數（有可能發生者才算）

T：觀察期間

- 某段觀察時間內，單位時間中所有可能發生某特定事件的人發生該事件的率。

2 / ( 7 + 2 + 4.5 + 3 + 6 ) = 2 / 22.5 = 0.089 / 年

- Definition:
- 某世代族群或某固定族群的人，經過某段觀察時間後，發生某事件﹙疾病﹚的人口佔該世代族群人口總數的百分比。
- 該事件﹙疾病﹚的發生情形，用以推估在族群中的任一人發生該事件或得該疾病的機率，也稱危險度﹙risk﹚。
- 當疾病或事件的發生機率很小時，累積發生率幾乎可等於發生率乘以年齡層距之總和。
- 2/（5-2/2）= 0.5

No

Disease

Disease

A

B

A + B =N1

Exposed

Unexposed

C

D

C + D =N0

A + C

=M1

B + D

=M0

N

- 相差危險性
- P1 - P0

- 相對危險性
- P1/P0

- 相對差異量(危險變化量)
- (P1 - P0) / P1

- 對比值
- P1 (1- P0) / P0 (1- P1)

- 發病機率
- P1 = A/N1
- P0 = C/N0

Disease risk in exposed

Risk exposure

Disease risk in nonexposed

Risk nonexposure

- Definition:
- The ratio of the risk of disease in persons exposed compared to the risk of disease in persons unexposed

- Common formula of relative risk (RR)

RR=

=

- Diseased risk: cumulate rate(incidence), rate(incidence) density
- relative risk; rate ratio; risk ratio…

Incidenceexposed

No

Disease

Relative Risk =

Disease

Incidenceunexposed

A

B

a + b =N1

Exposed

Unexposed

C

D

C + D =N0

Incidence Densityexposed

A + C

B + D

N

Rate Ratio =

Cumulative Incidenceexposed

Incidence Densityunexposed

A/N1

Risk Ratio =

Risk Ratio =

Cumulative Incidenceunexposed

C/N0

Ex

Dx

Ex.

- RR > 1 - the risk of disease in the exposed group is greater than the risk in the unexposed group
- RR = 1 - the risk of disease is the same in the exposed and unexposed
- RR < 1 - the risk of disease in the exposed group is less than the risk in the unexposed

No

Disease

Disease

A

B

N1

Exposed

Unexposed

C

D

N0

A + C

B + D

N

Equal follow-up time:

不考慮time

- Cumulate incidence
- 研究對象發病的機率(risk)
- CI1 = A/N1
- CI0 = C/N0
- RRCI = CI1/CI0
- = (A/N1)/(C/N0)
- RDCI = CI1- CI0
- = (A/N1) - (C/N0)

ORCI = CI1(1-CI1) / CI0 (1-CI0)

= AD/BC

Point estimation & 95% confidence interval

Hypothetical testing

- In a prospective study, 8000 persons, including 3000 hypertensive and 5000 normtensive patients, were followed for 10 years. There were 84 and 87 CHD cases in hypertensive and normtensive persons, respectively.
- RR=?

Ex-1

CI in exposed (HT) patients = 84/3000 = 28.0

CI in non-exposed (non-HT) patients = 87/5000 = 17.4

Relative risk = 28.0/17.4 = 1.61

Ex-1

Observed

Person-years

Disease

A

L1

N1

Exposed

Unexposed

C

L0

N0

A + C

= m1

L1 + L0

= L

N

Unequal follow-up time: 考慮 time

- Incidence density
- 單位人時的發病狀況(rate)
- ID1 = A/L1
- ID0 = C/L0
- RRID = ID1/ID0
- = (A/L1)/(C/L0)
- RDID = ID1- ID0
- = (A/L1) - (C/L0)

No odds

Point estimation & 95% confidence interval

Hypothetical testing

P0=L1/L

q0=1-p0

- B肝帶原之追蹤研究共追蹤帶原者450人年及非帶原者380人年, 在追蹤期間有35名對象發生肝細胞癌,其中15名為帶原者,20名為非帶原者歐.
- RR=?

Ex-2

Person-

Years

HCC

450

15

HBsAg(+)

20

380

HBsAg(-)

35

830

Overall rate = 35/830 = 0.042 cases/PY = 4.2 cases/100 PY

IDexposed= 15/450 = 3.3 cases/100 PY

IDunexposed = 20/380 = 5.3 cases/100 PY

Rate Ratio = 3.3 cases/100 PYO/5.3 cases/100 PYO = 0.62

Ex-2

Dx/non-Dx in exposured

Exposure/unexposure in Dx

Dx ratioexposed

Ex ratiodisease

Dx/non-Dx in unexposured

Dx ratiounexposed

Exposure/unexposure in nonDx

Ex rationon-Dx

- Definition:
- The ratio of the ratio of exposure in diseased persons compared to the ratio of exposure in non-diseased persons

- Common formula of odds ratio (OR)

OR.cohort =

=

=

OR.cs-cn=

Random sample; case-control study

No

Disease

No

Disease

Disease

Disease

A

a

b

B

A + B

a + b

Exposed

Exposed

C

c

d

D

C + D

c + d

Unexposed

Unexposed

a+c

A + C

b + d

B + D

N

a+b+c+d

Population; Cohort study

Odds(Ex)=(A/N1)/(1-A/N1)

Odds(nonEx)=(C/N0)/(1-C/N0)

Odds Ratio

= odds(Ex)/odds(nonEx)

= A*D/B*C

Odds(Dx)=(a/a+c)/(c/a+c)

Odds(nonEx)=(b/b+d)/(d/b+d)

Odds Ratio

= odds(Dx)/odds(nonDx)

= a*d/b*c ~ A*D/B*C

Dx

Ex

- OR > 1 - the odds of exposure in the diseased group is greater than the risk in the non-diseased group
- OR = 1 - the odds of exposure is the same in the diseased and non-diseased
- OR < 1 - the odds of exposure in the diseased group is less than the risk in the non-diseased group

Point estimation & 95% confidence interval

Hypothetical testing

- In case-control study of 74 Endometrial Cancer patients & their 664 controls, 56 cases and 274 controls received estrogen therapy ever.
- OR=?

EX-3

No

Cancer

Cancer

56

330

274

Estrogens

No

Estrogens

18

390

408

664

74

738

Odds ratio = 56*390/18*274 = 4.42

EX-3

Why does

Odds Ratio represent Relative Risk ?

If case & control from fixed cohort

- Assumption:
- Case and control are random sample from diseased & non-diseased population (representative)
- Probability (ex CI) is very small (rare disease)

Persons-

Years

Disease

A

L1=N1*t1

Exposed

C

L0 = N0*t0

Unexposed

A + C

L1 + L0 = L

If case & control from dynamic cohort

- Assumption:
- Case and control are random sample from diseased & non-diseased population (representative)
- Density sampling (time-matching)

If t0=t1,

Odds(Ex)=(A/N1)

Odds(nonEx)=(C/N0)

Odds Ratio

= (A/N1)/ (C/N0)

= (A/L1)/ (C/L0) = RRID

- Try to determine the relationship between two random variablesX and Y
- Y ~ continuous variables ~ N(u,var)
- X ~ continuous variables usually ~ N(u,var)

- Y= a+bXi+e
- Interpretation of intercept & slope (coefficient).
- Obtain RDCI in cohort study

- Try to determine the relationship between risk factors Xi and disease probability(Y)
- Y ~ binary variables ~ B(p)
- Y = logit P =log (P/1-P)
- P/(1-P) ~ disease odds (log odds)

- Odds ratios = P1q0 / p0q1
- Log (odds) = logit P1– logit P0 = beta
- Logit P0 = alpha

- Y ~ binary variables ~ B(p)

- Y = logit P = Exp (a+bXi+e)
- Log transform of disease probability in each risk category is expressed as a linear function of regression
- P1 =exp(a+b)/1+exp(a+b)
- P0 =exp(a)/1+exp(a)

- The exposure risk(X) to disease risk(Y) fit the multiplicative hypothesis.
- Logit P(x1,x2) = a+b1x1+b2x2+rx1x2
- b1 = log (r10)
- b2 = log (r01)
- r = log (r11/r10r01)
= logit p11-logit p10- logit p01+logit p00

- Logit P(x1,x2) = a+b1x1+b2x2+rx1x2
- Obtain ORCI in cohort study, and OR in case-control study

- Try to determine the relationship of between risk factors Xi and disease probability(Y) under considering time to event
- Y~binary (dead/alive) ~ B(p)
- All risk factors are assumed to be constant over time
- hi(t) = Exp (a+bXi+e) h0 (t)
- h0 (t): baseline hazard function; ID0
- hi (t): hazard function in exposed group; IDi

- Obtain RRID in cohort study (hazard ratio)

Cohort studies may be classified according to both the

type of data that are collected at baseline and the nature of the

eventual outcome measure. The combination determines the

appropriate strategy for analysis. Below we consider four

common combinations, mention the usual method of analysis,

and give an example of each. (Table 14.2)

Back

Adjusted rate -Standardization

透過調整人口結構的不同,用以比較人口組成不同的團體比率

Status 或 event 的測量指標

- 當將兩個族群的率拿來加以比較時，常因其各自的加權量不同而有所差異，造成解釋上的困難
- 標準化比率是為了比較兩個以上團體的比率所推算出來的假想總合比率。
- 用途:
- 比較人口組成不同的團體比率。
- 進行國際比較
方法:

Direct method

Indirect method

- SIR（Standardized incidence ratio)
- 標準化發生率比

SIR =

ai：指標族群發生個案﹙事件﹚數

bi：對照組發生個案﹙事件﹚數

N1i：指標族群分層觀察人年（人）數

N0i：對照族群分層觀察人年（人）數

：標準族群各年齡層人數

：標準族群各年齡層總人數

Exposed (N1)

Non-Exposed (N0)

PY

death

MR

PY

death

MR

Young

3000

30

0.01

1000

5

0.005

Aged

1000

30

0.03

9000

225

0.025

Total

4,000

60

0.015

10,000

230

0.023

SIR：

Wi = N0i + N1i4000+10000

指標組群發生率：(4000×0.01＋10000×0.03)/14000＝0.024重新計算死亡率

對照族群發生率NE：(4000×0.005＋10000×0.025)/14000＝0.019

率比：0.024/0.019＝1.26已暴露和未暴露的比值

- SMR（Standardized mortality ratio)
- 標準化死亡比

在族群之間作比值

SMR=

~ Observed/Expected

ai：指標族群分層死亡（罹病）人數

bi：對照族群分層死亡（罹病）人數

N0i：對照族群分層觀察人年（人）數

：指標族群各年齡層人數

Exposed (N1)

Non-Exposed (N0)

PY

death

MR

PY

death

MR

Young

3000

30

0.01

1000

5

0.005

Aged

1000

30

0.03

9000

225

0.025

Total

4,000

60

0.015

10,000

230

0.023

SMR：觀察值/期望值