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Review of observational study design and basic statistics for contingency tables. Coffee Chronicles BY MELISSA AUGUST, ANN MARIE BONARDI, VAL CASTRONOVO, MATTHEW - PowerPoint PPT Presentation
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Review of observational study Review of observational study design and basic statistics for design and basic statistics for
contingency tablescontingency tables
According to scientists, too much coffee may cause... 1986 --phobias, --panic attacks 1990 --heart attacks, --stress, --osteoporosis 1991 -underweight babies, --hypertension 1992 --higher cholesterol 1993, 08 --miscarriages 1994 --intensified stress 1995 --delayed conception But scientists say coffee also may help prevent... 1988 --asthma 1990 --colon and rectal cancer,... 2004—Type II Diabetes (*6 cups per day!) 2006—alcohol-induced liver damage 2007—skin cancer
Coffee Chronicles BY MELISSA AUGUST, ANN MARIE BONARDI, VAL CASTRONOVO, MATTHEW
JOE'S BLOWS Last week researchers reported that coffee might help prevent Parkinson's disease. So is the caffeine bean good for you or not? Over the years, studies haven't exactly been clear:
Medical StudiesMedical Studies
Evaluate whether a risk factor (or preventative factor) increases (or decreases) your risk for an outcome (usually disease, death or intermediary to disease).
The General Idea…
Exposure Disease?
Observational vs. Observational vs. Experimental StudiesExperimental Studies
Observational studies – the population is observed without any interference by the investigator
Experimental studies – the investigator tries to control the environment in which the hypothesis is tested (the randomized, double-blind clinical trial is the gold standard)
Limitation of observational Limitation of observational research: confoundingresearch: confounding
Confounding: risk factors don’t happen in isolation, except in a controlled experiment. – Example: In a case-control study of a salmonella outbreak,
tomatoes were identified as the source of the infection. But the association was spurious. Tomatoes are often eaten with serrano and jalapeno peppers, which turned out to be the true source of infection.
– Example: Breastfeeding has been linked to higher IQ in infants, but the association could be due to confounding by socioeconomic status. Women who breastfeed tend to be better educated and have better prenatal care, which may explain the higher IQ in their infants.
Confounding: A major problem Confounding: A major problem for observational studiesfor observational studies
Exposure Disease
Confounder
?
Why Observational Studies?Why Observational Studies?
CheaperFasterCan examine long-term effectsHypothesis-generatingSometimes, experimental studies are not
ethical (e.g., randomizing subjects to smoke)
Possible Observational Possible Observational Study DesignsStudy DesignsCross-sectional studies
Cohort studies
Case-control studies
Cross-Sectional (Prevalence) Cross-Sectional (Prevalence) StudiesStudies
Measure disease and exposure on a random sample of the population of interest. Are they associated?
Marginal probabilities of exposure AND disease are valid, but only measures association at a single time point.
The 2x2 TableThe 2x2 Table
Exposure (E) No Exposure (~E)
Disease (D) a b (a+b)/T = P(D)
No Disease (~D) c d (c+d)/T = P(~D)
(a+c)/T = P(E) (b+d)/T = P(~E)
Marginal probability of disease
Marginal probability of exposure
N
Example: cross-sectional Example: cross-sectional studystudy
Relationship between atherosclerosis and late-life depression (Tiemeier et al. Arch Gen Psychiatry, 2004).
Methods: Researchers measured the prevalence of coronary artery calcification (atherosclerosis) and the prevalence of depressive symptoms in a large cohort of elderly men and women in Rotterdam (n=1920).
Example: cross-sectional Example: cross-sectional studystudy
P(“D”)= Prevalence of depression (sub-thresshold or depressive disorder): (20+13+12+9+11+16)/1920 = 4.2%
P(“E”)= Prevalence of atherosclerosis (coronary calcification >500):(511+12+16)/1920 = 28.1%
The 2x2 table:The 2x2 table:
Coronary calc >500539
Coronary calc <=500 1381
81 1839 1920
Any depression
None
28 511
53 1328
P(depression)= 81/1920 = 4.2%
P(atherosclerosis) = 539/1920 = 28.1%
P(depression/atherosclerosis) = 28/539 = 5.2%
Difference of proportions Z-test:Difference of proportions Z-test:
Coronary calc >500539
Coronary calc <=500 1381
81 1839 1920
Any depression
None
28 511
53 1328
038.1381
53;052.
539
28// unblockeddepressionrosisatheroscledepression pp
18.;33.10101.
014.
1381)042.1)(042(.
539)042.1)(042(.
038.052.
).(.
pdifferencees
differenceZ
Coronary calc >500539
Coronary calc <=500 1381
81 1839 1920
Any depression
None
28 511
53 1328
2.19) (0.86, CI 95% ;37.1038.
052.RR
Or, use relative risk (risk ratio):Or, use relative risk (risk ratio):
Interpretation: those with coronary calcification are 37% more likely to have depression (not significant).
Or, use chi-square test:Or, use chi-square test:
Coronary calc >500539
Coronary calc <=500 1381
81 1839 1920
Any depression None
28 511
53 1328
Observed:
Expected:
Coronary calc >500539
Coronary calc <=500 1381
81 1839 1920
Any depression None
539*81/1920=
22.7
539-22.7=
516.381-22.7=
58.31381-58.3=
1322.7
Chi-square test:Chi-square test:
expected
expected) - (observed 22
18.
77.17.1322
)7.13221328(
3.516
)3.516511(
3.58
)3.5853(
7.22
)7.2228(
2
2222
1
p
Note: 1.77 = 1.332
Chi-square test also works for Chi-square test also works for bigger contingency tables (RxC):bigger contingency tables (RxC):
Chi-square test also works for Chi-square test also works for bigger contingency tables (RxC):bigger contingency tables (RxC):
Coronary calcification No
depression
Sub-threshhold depressive symptoms
Clinical
depressive disorder
0-100 865 20 9
101-500 463 13 11
>500 511 12 16
Coronary calcification
No depression
Sub-threshhold depressive symptoms
Clinical
depressive disorder
0-100 865 20 9 894
101-500 463 13 11 487
>500 511 12 16 539
1839 45 36 1920
Observed: Expected:
Coronary calcification No
depression
Sub-threshhold depressive symptoms
Clinical
depressive disorder
0-100 894*1839/1920=
856.3
849*45/1920=
21
894-(21+856.3)=16.7
101-500 487*1839/1920=
466.5
487*45/1920=
11.4
487-(466.5+11.4)=9.1
>500 1839-(856.3+466.5)=
516.2
45-(21+11.4)=
12.6
36-(16.7+9.1)=
10.2
Chi-square test:Chi-square test:
expected
expected) - (observed 22
096.
877.72.10
)2.1016(
6.12
)6.1212(
2.516
)2.516511(
1.9
)1.911(
4.11
)4.1113(
5.466
)5.466463(
7.16
)7.169(
21
)2120(
3.856
)3.856865(
222
222
2222
4
p
Cause and effect?Cause and effect?
atherosclerosis
depression in elderly
?Biological changes
?Lack of exercise Poor Eating
Confounding?Confounding?
atherosclerosis
depression in elderly
Advancing Age
?Biological changes
?Lack of exercise Poor Eating
Cross-Sectional StudiesCross-Sectional Studies
Advantages: – cheap and easy– generalizable– good for characteristics that (generally) don’t change
like genes or gender
Disadvantages – difficult to determine cause and effect– problematic for rare diseases and exposures
2. Cohort studies2. Cohort studies::
Sample on exposure status and track disease development (for rare exposures)
Marginal probabilities (and rates) of developing disease for exposure groups are valid.
Example: The Framingham Example: The Framingham Heart StudyHeart Study
The Framingham Heart Study was established in 1948, when 5209 residents of Framingham, Mass, aged 28 to 62 years, were enrolled in a prospective epidemiologic cohort study.
Health and lifestyle factors were measured (blood pressure, weight, exercise, etc.).
Interim cardiovascular events were ascertained from medical histories, physical examinations, ECGs, and review of interim medical record.
Example 2: Johns Hopkins Precursors StudyExample 2: Johns Hopkins Precursors Study(medical students 1948 through 1964)(medical students 1948 through 1964)
http://www.jhu.edu/~jhumag/0601web/study.html
From the John Hopkin’s Magazine website (URL above).
Cohort StudiesCohort Studies
Target population
Exposed
Not Exposed
Disease-free cohort
Disease
Disease-free
Disease
Disease-free
TIME
Exposure (E) No Exposure (~E)
Disease (D) a b
No Disease (~D) c d
a+c b+d
)/()/(
)~/(
)/(
dbbcaa
EDP
EDPRR
risk to the exposed
risk to the unexposed
The Risk Ratio, or Relative Risk (RR)
400 400
1100 2600
0.23000/4001500/400 RR
Hypothetical DataHypothetical Data
Normal BP
Congestive Heart Failure
No CHF
1500 3000
High Systolic BP
Advantages/Limitations:Advantages/Limitations:Cohort StudiesCohort Studies
Advantages:– Allows you to measure true rates and risks of disease for the
exposed and the unexposed groups.– Temporality is correct (easier to infer cause and effect).– Can be used to study multiple outcomes. – Prevents bias in the ascertainment of exposure that may occur
after a person develops a disease. Disadvantages:
– Can be lengthy and costly! 60 years for Framingham.– Loss to follow-up is a problem (especially if non-random).– Selection Bias: Participation may be associated with exposure
status for some exposures
Case-Control StudiesCase-Control Studies
Sample on disease status and ask retrospectively about exposures (for rare diseases) Marginal probabilities of exposure for cases and
controls are valid.
• Doesn’t require knowledge of the absolute risks of disease
• For rare diseases, can approximate relative risk
Target population
Exposed in past
Not exposed
Exposed
Not Exposed
Case-Control StudiesCase-Control Studies
Disease
(Cases)
No Disease
(Controls)
Example: the AIDS epidemic Example: the AIDS epidemic in the early 1980’sin the early 1980’s
Early, case-control studies among AIDS cases and matched controls indicated that AIDS was transmitted by sexual contact or blood products.
In 1982, an early case-control study matched AIDS cases to controls and found a positive association between amyl nitrites (“poppers”) and AIDS; odds ratio of 8.6 (Marmor et al. 1982). This is an example of confounding.
Case-Control Studies in Case-Control Studies in HistoryHistory
In 1843, Guy compared occupations of men with pulmonary consumption to those of men with other diseases (Lilienfeld and Lilienfeld 1979).
Case-control studies identified associations between lip cancer and pipe smoking (Broders 1920), breast cancer and reproductive history (Lane-Claypon 1926) and between oral cancer and pipe smoking (Lombard and Doering 1928). All rare diseases.
Case-control studies identified an association between smoking and lung cancer in the 1950’s.
Case-control exampleCase-control example
A study of the relation between body mass index and the incidence of age-related macular degeneration (Moeini et al. Br. J. Ophthalmol, 2005).
Methods: Researchers compared 50 Iranian patients with confirmed age-related macular degeneration and 80 control subjects with respect to BMI, smoking habits, hypertension, and diabetes. The researchers were specifically interested in the relationship of BMI to age-related macular degeneration.
ResultsResults
Table 2 Comparison of body mass index (BMI) in case and control groups
Case n = 50(%) Control n = 80 (%) p Value
Lean BMI <20 7 (14) 6 (7.5) NS
Normal 20 BMI <25 16 (32) 20 (25) NS
Overweight 25 BMI <30 21 (42) 36 (45) NS
Obese BMI 30 6 (12) 18 (22.5) NS
NS, not significant.
Overweight Normal
ARMD 27 23
No ARMD 54 26
Corresponding 2x2 Table
What is the risk ratio here?
Tricky: There is no risk ratio, because we cannot calculate the risk of disease!!
50
80
The odds ratio…The odds ratio…
We cannot calculate a risk ratio from a case-control study.
BUT, we can calculate a measure called the odds ratio…
Odds vs. RiskOdds vs. Risk
If the risk is… Then the odds are…
½ (50%)
¾ (75%)
1/10 (10%)
1/100 (1%)
Note: An odds is always higher than its corresponding probability, unless the probability is 100%.
1:1
3:1
1:9
1:99
The proportion of cases and controls are set by the investigator; therefore, they do not represent the risk (probability) of developing disease.
bc
ad
dcba
dcddccbabbaa
ORDEP
DEP
DEPDEP
)/()/()/()/(
)~/(~)~/(
)/(~)/(
Exposure (E) No Exposure (~E)
Disease (D) a b
No Disease (~D) c d
The Odds Ratio (OR)
a+b=cases
c+d=controls
Odds of exposure in the cases
Odds of exposure in the controls
dbca
dcba
bc
adOR
Exposure (E) No Exposure (~E)
Disease (D) a b
No Disease (~D) c d
The Odds Ratio (OR)
Odds of disease for the exposed
Odds of exposure for the controls
Odds of exposure for the cases.
Odds of disease for the unexposed
=
Odds of exposure in the controls
Odds of exposure in the cases
Bayes’ Rule
Odds of disease in the unexposed
Odds of disease in the exposed
What we want!
)~/(~
)~/()/(~
)/(
DEP
DEPDEP
DEP
)(~
)(~)~/(~)(~
)()/(~)(
)(~)~/()(
)()/(
DP
EPEDPDP
EPEDPDP
EPEDPDP
EPEDP
)~/(~
)~/()/(~
)/(
EDP
EDPEDP
EDP
Proof via Bayes’ Rule (optional)
dbca
dcba
bc
adOR
Overweight Normal
ARMD a b
No ARMD c d
The Odds Ratio (OR)
Odds of ARMD for the overweight
Odds of overweight for the controls
Odds of overweight for the cases.
Odds of ARMD for the normal weight
57.54*23
26*27
26542327
OR
Overweight Normal
ARMD 27 23
No ARMD 54 26
The Odds Ratio (OR)
57.54*23
26*27
26542327
OR
Overweight Normal
ARMD 27 23
No ARMD 54 26
The Odds Ratio (OR)
Can be interpreted as: Overweight people have a 43% decrease in their ODDS of age-related macular degeneration. (not statistically significant here)
The odds ratio is a good The odds ratio is a good approximation of the risk ratio approximation of the risk ratio
if the disease is rare.if the disease is rare.
RROR If the disease is rare (affecting <10% of the population), then:
WHY?
If the disease is rare, the probability of it NOT happening is close to 1, and the odds is close to the risk. Eg:
50.10:1
20/1
474.9/1
19/1
RR
OR
The rare disease assumptionThe rare disease assumption
RROR EDPEDP
EDPEDP
EDPEDP
)~/()/(
)~/(~)~/(
)/(~)/(
1
1
When a disease is rare: P(~D) = 1 - P(D) 1
The odds ratio vs. the risk ratioThe odds ratio vs. the risk ratio
1.0 (null)
Odds ratio
Risk ratio Risk ratio
Odds ratio
Odds ratio
Risk ratio Risk ratio
Odds ratio
Rare Outcome
Common Outcome
1.0 (null)
When is the OR is a good When is the OR is a good approximation of the RR?approximation of the RR?
General Rule of Thumb:
“OR is a good approximation as long
as the probability of the outcome in the
unexposed is less than 10%”
Prevalence of age-related macular degeneration is about 6.5% in people over 40 in the US (according to a 2011 estimate). So, the OR is a reasonable approximation of the RR.
Advantages/Limitations:Advantages/Limitations:Case-control studiesCase-control studies
Advantages:– Cheap and fast– Efficient for rare diseases
Disadvantages:– Getting comparable controls is often tricky– Temporality is a problem (did risk factor cause disease
or disease cause risk factor?– Recall bias
Inferences about the odds Inferences about the odds ratio…ratio…
Properties of the OR (simulation)(50 cases/50 controls/20% exposed)
If the Odds Ratio=1.0 then with 50 cases and 50 controls, of whom 20% are exposed, this is the expected variability of the sample ORnote the right skew
Properties of the lnOR
dcba
1111
Standard deviation =
Hypothetical DataHypothetical Data
0.8)10)(6(
)24)(20(OR
25.8) - (2.47(8.0)e ,(8.0)e CI %95 24
1
10
1
6
1
20
196.1
24
1
10
1
6
1
20
196.1
Amyl Nitrite Use No Amyl Nitrite
AIDS 20 10
Does not have AIDS
6 24
30
30
Note that the size of the smallest 2x2 cell determines the magnitude of the variance
When can the OR mislead?When can the OR mislead?
Example:Example:Does dementia predict death?Does dementia predict death?Dementia: The leading predictor of death in
a defined elderly population. Neurology 2004; 62: 1156-1162
Among patients with dementia: 291/355 (82%) died
Among patients without dementia: 947/4328 (22%) died
Dementia studyDementia study
Authors report OR = 16.23 (12.27, 21.48)But the RR = 3.72 Fortunately, they do not dwell on the OR,
but it could mislead if not interpreted correctly…
Better to give OR or RR?Better to give OR or RR?From an RCT (prospective!) of a new diet drug, the authors
showed the following table:
Odds Ratios for losing at least 5kg were:4.0 (low dose vs. placebo)
20.9 (medium dose vs. placebo)31.5 (high dose vs. placebo)
Better to give OR or RR?Better to give OR or RR?
Corresponding RRs are:59%/29%=2 (low dose vs. placebo)
87%/29%=3 (medium dose vs. placebo)91%/29%=3 (high dose vs. placebo)
Summary of statistical tests Summary of statistical tests for contingency tablesfor contingency tables
Table Size Test or measures of association
2x2 risk ratio (cohort or cross-sectional studies)
odds ratio (case-control studies)
Chi-square
difference in proportions
Fisher’s Exact test (cell size less than 5)
RxC Chi-square
Fisher’s Exact test (expected cell size <5)
Fisher’s Exact TestFisher’s Exact Test
Fisher’s “Tea-tasting Fisher’s “Tea-tasting experiment”experiment”
Claim: Fisher’s colleague (call her “Cathy”) claimed that, when drinking tea, she could distinguish whether milk or tea was added to the cup first.
To test her claim, Fisher designed an experiment in which she tasted 8 cups of tea (4 cups had milk poured first, 4 had tea poured first).
Null hypothesis: Cathy’s guessing abilities are no better than chance.
Alternatives hypotheses:
Right-tail: She guesses right more than expected by chance.
Left-tail: She guesses wrong more than expected by chance
Fisher’s “Tea-tasting Fisher’s “Tea-tasting experiment”experiment”
Experimental Results:
Milk Tea
Milk 3 1
Tea 1 3
Guess poured first
Poured First
4
4
Fisher’s Exact TestFisher’s Exact TestStep 1: Identify tables that are as extreme or more extreme than what actually happened:
Here she identified 3 out of 4 of the milk-poured-first teas correctly. Is that good luck or real talent?
The only way she could have done better is if she identified 4 of 4 correct.
Milk Tea
Milk 3 1
Tea 1 3
Guess poured firstPoured First
4
4
Milk Tea
Milk 4 0
Tea 0 4
Guess poured firstPoured First
4
4
Fisher’s Exact TestFisher’s Exact TestStep 2: Calculate the probability of the tables (assuming fixed marginals)
Milk Tea
Milk 3 1
Tea 1 3
Guess poured first
Poured First
4
4
Milk Tea
Milk 4 0
Tea 0 4
Guess poured first
Poured First
4
4
229.)3(
84
41
43 P
014.)4(
84
40
44 P
Step 3: to get the left tail and right-tail p-values, consider the probability mass function:
Probability mass function of X, where X= the number of correct identifications of the cups with milk-poured-first:
229.)3(
84
41
43 P
014.)4(
84
40
44 P
514.)2(
84
42
42 P
229.)1(
84
43
41 P
014.)0(
84
44
40 P
“right-hand tail
probability”: p=.243
“left-hand tail probability” (testing the alternative
hypothesis that she’s
systematically wrong): p=.986
SAS also gives a “two-sided p-value” which is calculated
by adding up all probabilities in the distribution that are less than or equal to
the probability of the observed table (“equal or more extreme”). Here:
0.229+.014+.0.229+.014= .4857
Summary of statistical tests Summary of statistical tests for contingency tablesfor contingency tables
Table Size Test or measures of association
2x2 risk ratio (cohort or cross-sectional study)
odds ratio (case-control study)
Chi-square
difference in proportions
Fisher’s Exact test (cell size less than 5)
RxC Chi-square
Fisher’s Exact test (expected cell size <5)
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