measures of association - biostatistics
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Understanding Measures of association briefly at http://www.helpwithassignment.com/statistics-assignment-helpTRANSCRIPT
Biostatistics
Lecture 10
Lecture 9 Review –Measures of association
Measures of association
– –
–
Risk differenceRisk ratio
Odds ratio
Calculation & interpretation
interval for each measure
of confidence
of association
2×2 table - Measures of associationOutcome - binary
Measure of Effect Formula
Risk difference p1-p0
Risk ratio p1 / p0
Odds ratio (d1/h1) / (d0/h0)
Differences in measures of association
• When there is no association between exposure and outcome,
–
–
–
risk difference = 0
risk ratio (RR) = 1
odds ratio (OR) = 1
•
•
Risk difference can be negative or positive
RR & OR are always positive
• For rare outcomes, OR ~ RR
• OR is always further from 1 than corresponding RR
– If RR > 1 then OR > RR
– If RR < 1 the OR < RR
Interpretation of measures of association
• RR & OR < 1, associated with a reduced risk / odds (mayprotective)
be
– RR = 0.8 (reduced risk of 20%)
• RR
–
& OR > 1, associated with an increased risk / odds
RR = 1.2 (increased risk of 20%)
• RR & OR – further the risk is from 1, stronger the association
between exposure and outcome (e.g. RR=2 versus RR=3).
Comparing the outcome measure of two exposure groups(groups 1 & 0)
s.e.(lo g RR )
=− + −
eeR
(lo g e
OR )
d h d h
Outcome variable – data type
Population parameter
Estimate of
population parameter
from sample
Standard error of loge(parameter)
95% Confidence interval of loge(population parameter)
Categorical
Population risk ratio
p1/p0 1 1 1 1d1 n1 d0 n0
log eRR
± 1.96 × s.e.(lo g R )
Categorical Population odds ratio
(d1/h1) / (d0/h0)
s.e. = 1
+ 1
+ 1
+ 1
1 1 0 0
log eOR
± 1.96 x s.e.(lo g eOR )
Calculation of p-values for comparing two groups
1 0z =
s.e.(lo g
( RR ))
s.e.(lo g
( OR ))
Outcome variable – data type
Population parameter Population parameter under null hypothesis
Test statistic
Categorical
π1-π0
Population risk ratio
Population odds ratio
π1-π0=0
Population risk ratio=1
Population odds ratio=1
p − p
s.e.( p1
− p 0
)
z = loge ( RR)
e
z = loge (OR)
e
Comparing the outcome measure of two exposure groups(TBM trial: dexamethasone versus placebo)
Outcome variable – data type
Population parameter under null hypothesis
Estimate of population parameter
from sample
95% confidence interval for population parameter
Two-sided p-value
Categorical Population risk
difference
= 0
p1-p0
= -0.095-0.175, -0.015 0.020
Categorical
Population risk ratio
= 1
p1/p0
= 0.77
0.62, 0.96 0.016
Categorical Population odds ratio
= 1
(d1/h1) / (d0/h0)= 0.66
0.46, 0.93 0.021
2×2 table – TBM trial example
Odds ratio for death = (d1/h1) / (d0/h0) = 0.465 / 0.704 = 0.66
Odds ratio for exposure to dexamethasone = (d1/d0) / (h1/h0) = 0.777 / 1.176 = 0.66
Odds ratio for not dying = (h1/d1) / (h0/d0) = 2.149 / 1.420 = 1.51 = (1/0.66)
Odds ratio for exposure to placebo = (d0/d1) / (h0/h1) = 1.287 / 0.850 = 1.51 = (1/0.66)
Death during 9 months post start of treatment
Treatment group Yes No Total
Dexamethasone
(group 1)
87 (d1) 187 (h1) 274 (n1)
Placebo
(group 0)
112 (d0) 159 (h0) 271 (n0)
Total 199 346 545
Measure of association
Study Design Risk difference
RiskRatio
OddsRatio
Randomised controlled trial
√ √ √Cohort Study
√ √ √Case-control Study
× × √
Lecture 10 – Controlling for confounding:stratification and regression
• A description of confounding
• How to control for confoundinganalysis by
– Stratification
– Regression modelling
in statistical
• A brief description of the role of multiplelinear or logistic regression in adjusting for confounding
Outcome and exposure variables(RECAP)
Outcomes are variables of interest (population health relevance) whose patterns and determinants we wish to learn about from data
•
• Exposures are the variables we think mightexplain observed variation in the outcomes
• Statistical analysis can be used to quantify theassociation between outcomes and exposures
What is confounding?
A confounding variable
1)2)
3)
is associated with the outcome variable;is associated with the exposure variable;
does not lie on the causal pathway.
Outcome variableExposure variable
Confounding variable
Failing to control for confounding may result in abiased estimate of the magnitude of the association between exposure and outcome
Example of confounding
Exposure variable Outcome variableAlcohol intake Heart disease
Confounding variablesCigarette smoking
Control of confounding
Design of Study
• Randomisation(randomised controlled trial: e.g. TBM trial)
• Restriction(only include those with one value of confounder)
• Matching
Control of confounding
Statistical analysis
• Stratification
• Regression modelling
Hypothetical example of a case-control studyAssociation between energy intake and heart disease
OddsOdds
of heart disease in high energy intake group = 730/600 = 1.22of heart disease in low energy intake group = 700/540 = 1.30
Odds ratio = 1.22 / 1.30 = 0.94
95% confidence interval: 0.80 up to 1.10
Heart disease
Energy intake Yes No Total
High(group 1)
730 (d1) 600 (h1) 1330 (n1)
Low(group 0)
700 (d0) 540 (h0) 1240 (n0)
Total 1430 1140 2570
Is this association confoundedby physical activity?
Exposure variable Outcome variableEnergy intake Heart disease
Confounding variablesPhysical activity
Stratify by physical activity…..
Calculate the stratum specific odds ratios…
Energy intake
High physical activity Low physical activity
Heart disease Heart disease
Yes No Yes No
High(group 1)
500 510 230 90
Low(group 0)
100 150 600 390
Stratify by physical activity…..
For high physical activity group: OR (95% CI) = 1.47 (1.11, 1.95)
For low physical activity group: OR (95% CI) = 1.66 (1.26, 2.19)
Energy intake
High physical activity Low physical activity
Heart disease Heart disease
Yes No Yes No
High(group 1)
500 510 230 90
Low(group 0)
100 150 600 390
Is this association confoundedby physical
???
activity?
Exposure variableEnergy intake
Outcome variableHeart disease
??????
Confounding variablesPhysical activity
Confounding – condition 1Association between physical activity and heart disease
** Look particularly in those who are not exposed to the factor of interest**
For low energy intake group: OR (95% CI) = 0.43 (0.33, 0.58)
For high energy intake group: OR (95% CI) = 0.38 (0.29, 0.50)
Physical activity
High energy intake Low energy intake
Heart disease Heart disease
Yes No Yes No
High(group 1)
500 510 100 150
Low(group 0)
230 90 600 390
Confounding – condition 2
Association between energy intake and physical activity
• In a case-control study: examine the association in the controls• In a cohort study: use the whole cohort
Confounding – condition 2Association between energy intake and physical activity for those
without heart disease (n=1140)
Proportion in high energy intake group who report high physical activity =510/600 = 0.85 (85%)
Proportion in low energy intake group who report high physical activity =150/540 = 0.28 (28%)
Odds Ratio = (510/90) / (150/390) = 14.7; 95% CI: 11.0 up to 19.7
Physical activity
Energy intake High Low Total
High(group 1)
510 90 600
Low(group 0)
150 390 540
Is this association confoundedby physical
???
activity?
Exposure variableEnergy intake
Outcome variableHeart disease
High energy intake:OR = 0.38 (95% CI: 0.29, 0.50) Low energy intake:OR = 0.43 (95% CI: 0.33, 0.58)
High energy intakeassociated
with high physical activity
Confounding variablesPhysical activity
So physical activity is a potential confounderControl for confounding - Stratified analyses
1) Start with stratum specific estimates differences, rate ratios
of odds ratios, risk ratios, risk
2) Calculate a weighted average of the‘pooled’ estimate
stratum-specific estimates
Usual method is Mantel-Haenszel method– Weights assigned according to amount of information in
each stratum
Calculate a pooled OR
(600×90)/1310) = 41.2
For low physical activity:
OR = 1.66
w= (d0×h1)/n =
For high physical activity:
OR = 1.47
w= (d0×h1)/n = (100×510)/1260) = 40.5
Energy intake
High physical activity(n=1260)
Low physical activity(n=1310)
Heart disease Heart disease
Yes No Yes No
High(group 1)
500 (d1) 510 (h1) 230 (d1) 90 (h1)
Low(group 0)
100 (d0) 150 (h0) 600 (d0) 390 (h0)
Calculate a pooled OR
(600×90)/1310) = 41.2
Mantel-Haenszel estimate of pooled odds ratio:
∑ (wi × ORi )OR =
MH ∑ wi
Stratum ‘i’
For low physical activity:
OR = 1.66
w= (d0×h1)/n =
For high physical activity:
OR = 1.47
w= (d0×h1)/n = (100×510)/1260) = 40.5
Calculate a pooled OR
(600×90)/1310) = 41.2
Mantel-Haenszel estimate of pooled odds ratio:
(40.5 ×1.47 ) + (41.2 ×1.66)OR = 1.57=
MH (40.5 + 41.2)
95% CI: 1.29 up to 1.91
Recall that the crude OR was 0.94 (95% CI 0.80-1.10)
Is there a difference between crudeand adjusted measures of effect?
For low physical activity:
OR = 1.66
w= (d0×h1)/n =
For high physical activity:
OR = 1.47
w= (d0×h1)/n = (100×510)/1260) = 40.5
Association between energy intake & heartdisease adjusting for physical activity
ORMH = 1.57
95% CI: 1.29, 1.91Exposure variableEnergy intake
Outcome variableHeart disease
High energy intake:OR = 0.38 (95% CI: 0.29, 0.50) Low energy intake:OR = 0.43 (95% CI: 0.33, 0.58)
High energy intakeassociated
with high physical activity
Confounding variablesPhysical activity
Multiple logistic regressionOutcome variable (y-variable) – binary
e.g. dead or alive; treatment failure or success;disease or no disease..
Measure of association – Odds ratio
Multiple logistic regression model –
loge(odds of outcome) = β0 + β1X1 + β2X2 + β3X3 +…. + βkXk
β1,…βk – loge(odds ratios)X1, …..Xk – k different exposure variables (do not need to
be binary but can be categorical with more than 2 categoriesor numerical)
Useful when there are many confounding variables…
Logistic regressionExample – Association between energy intake
and heart disease
Outcome variable (y-variable) – heart disease (coded as yes-1 & no-0)
Logistic regression model –loge(odds of outcome) = β0 + β1X1
β1 – loge(odds ratios)X1 – energy intake (high versus low)
Exposure Odds Ratio (expβi) 95% Confidence Interval
Energy intake(high vs low)
0.94 0.80, 1.10
Multiple logistic regressionExample – Association between energy intake and
heart disease
Outcome variable (y-variable) – heart disease (coded as yes-1 & no-0)
Multiple logistic regression model –loge(odds of outcome) = β0 + β1X1 + β2X2
β1, β2 – loge(odds ratios)X1 – energy intake (high versus low)X2 – physical activity (high versus low)
Exposure Odds Ratio (expβi) 95% Confidence Interval
Energy intake(high vs low)
1.57 1.29, 1.91
Physical activity(high vs low)
0.41 0.33, 0.49
Multiple linear regressionOutcome variable (y-variable) – numericale.g. blood pressure, forced expiratory volume in 1 sec (FEV1)
Linear regression model –
y = β0 + β1X1 + β2X2 + β3X3 +…. + βkXk
y – numerical outcome variable,
β1,…βk – increase in y for every unit increase in x
X1, …..Xk – k different exposure variables (can be numerical
or categorical with 2+ categories)
Useful when there are many confounding variables…
Lecture 10 - Objectives
• Understand confounding
• Calculate the Mantel-Haenszel estimate ofpooled odds ratio
the
• Understand the difference between linear andlogistic regression
Thank You
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