hsrp 734: advanced statistical methods may 22, 2008
DESCRIPTION
HSRP 734: Advanced Statistical Methods May 22, 2008. Course Website. Course site in Public Health Sciences (PHS) website: http://www.phs.wfubmc.edu/public/edu_statMeth.cfm. Course Syllabus. HSRP 734: Advanced Statistical Methods. Categorical Data Analysis Logistic Regression - PowerPoint PPT PresentationTRANSCRIPT
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HSRP 734: Advanced Statistical Methods
May 22, 2008
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Course Website
• Course site in Public Health Sciences (PHS) website:
http://www.phs.wfubmc.edu/public/edu_statMeth.cfm
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Course Syllabus
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HSRP 734: Advanced Statistical Methods
• Categorical Data Analysis
• Logistic Regression
• Survival analysis
• Cox PH regression
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What is Categorical Data Analysis?
• Statistical analysis of data that are non-continuous
• Includes dichotomous, ordinal, nominal and count outcomes
• Examples: Disease incidence, Tumor response
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What is Logistic Regression?
A statistical method used to model dichotomous or binary outcomes (but not limited to) using predictor variables.
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What is Logistic Regression?
• Used when the research method is focused on whether or not an event occurred, rather than when it occurred
• Time course information is not used
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Logistic Regression quantifies “effects” using Odds Ratios
• Does not model the outcome directly, which leads to effect estimates quantified by means (i.e., differences in means)
• Estimates of effect are instead quantified by “Odds Ratios”
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The Logistic Regression Model
0 1 1 2 2 K K
P Yln
1-P YX X X
predictor variables
YP1
YPln is the log(odds) of the outcome.
dichotomous outcome
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The Logistic Regression Model
0 1 1 2 2 K K
P Yln
1-P YX X X
intercept
YP1
YPln is the log(odds) of the outcome.
model coefficients
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A Short ReviewA Short Review
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Philosophy of Science
• Idea: We posit a paradigm and attempt to falsify that paradigm.
• Science progresses faster via attempting to falsify a paradigm than attempting to corroborate a paradigm.
(Thomas S. Kuhn. 1970. The Structure of Scientific Revolutions. University of Chicago Press.)
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Philosophy of Science• The fastest way to progress in science under this paradigm of
falsification is through perturbation experiments.
• In epidemiology, – often unable to do perturbation experiments– it becomes a process of accumulating evidence
• Statistical testing provides a rigorous data-driven framework for falsifying hypothesis
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The P-Value
• What is the probability of having gotten a sample mean as extreme as 4.8 if the null hypothesis was true (H0: = 0)?
• P-value = probability of obtaining a result as or more “extreme” than observed if H0 was true.
• Consider for the above example, if p = 0.0089 (less than a 9 out of 1,000 chance)
• What if p = 0.0501 (5 out of 100 chance) ?
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Hypothesis Testing
1. Set up a null and alternative hypothesis
2. Calculate test statistic
3. Calculate the p-value for the test statistic
4. Based on p-value make a decision to reject or fail to reject the null hypothesis
5. Make your conclusion
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Hypothesis Testing
Your decision vs. Truth
Truth: H0 True Truth: H0 False
Decision: Fail to reject H0
Correct Decision Incorrect DecisionType II Error ()
Decision:Reject H0
Incorrect DecisionType I Error ()
Correct Decision(Power)
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Hypothesis Testing
• Type I error () = the probability of rejecting the null hypothesis given that H0 is true (the significance level of a test).
• Type II error (): the probability of not rejecting the null hypothesis given that H0 is false (not rejecting when you should have).
• Power = 1 -
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Power
• The power of a test is: The probability of rejecting a false null
hypothesis under certain assumed differences between the populations.
• We like a study that has “high” power (usually at least 80%).
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• Any difference can become significant if N is large enough
• Even if there is statistical significance is there clinical significance?
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Controversy around HT and p-value
“A methodological culprit responsible for spurious theoretical conclusions”
(Meehl, 1967; see Greenwald et al, 1996)
“The p-value is a measure of the credibility of the null hypothesis. The smaller the p-value is, the less likely one feels the null hypothesis can be true.”
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HT and p-value
• “It cannot be denied that many journal editors and investigators use p-value < 0.05 as a yardstick for the publishability of a result.”
• “This is unfortunate because not only p-value, but also the sample size and magnitude of a physically important difference determine the quality of an experimental finding.”
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HT and p-value
• Consider a new cancer drug that possibly shows significant improvements.
• Should we consider a p = 0.01 the same as a p = 0.00001 ?
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HT and p-value
• “[We] endorse the reporting of estimation statistics (such as effect sizes, variabilities, and confidence intervals) for all important hypothesis tests.”
– Greenwald et al (1996)
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Reporting Statistics
• Reporting I. Statistical Methods
The changes in blood pressure after oral contraceptive use were calculated for 10 women. A paired t-test was used to determine if there was a significant change in blood pressure and a 95% confidence was calculated for the mean blood pressure change (after-before).
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Reporting Statistics
• Reporting II. Results
Blood pressure measurements increased on average 4.8 mmHg with standard deviation of 4.57. The 95% confidence interval for the mean change was (1.53, 8.07).
There was evidence that blood pressure measurements after oral contraceptive use were significantly higher than before oral contraceptive use (p = 0.009).
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HSRP 734Lecture 1:
Measures of Disease Occurrence and Association
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Objectives:
1.Define and compute the measures of disease occurrence and association
2.Discuss differences in study design and their implications for inference
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Example
CT images rated
by radiologist
(Rosner p.65)
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Rated as normal
Rated as questionable
Rated as abnormal
Normal 39 6 13
Abnormal 5 2 44
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(Cell %)Row %Col %
Rated as normal
Rated as questionable
Rated as abnormal
Normal
39 (35.8%)
67%88.6%
6 (5.5%)10.3%75%
13 (11.9%)22.4%22.8%
58
Abnormal
5(4.6%)9.8%
11.4%
2(1.8%)3.9%25%
44(40.4%)86.3%77.2%
51
44 8 57 109
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Basic Probability
• Conditional probability
– Restrict yourself to a “subspace” of the sample space
Male Female
Young 20% 10%
Old 35% 35%
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Conditional probabilities
• Probability that something occurs (event B), given that event A has occurred (conditioning on A)
• Pr(B given that A is true) = Pr(B | A)
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Conditional probabilities
• Categorical data analysis• odds ratio = ratio of odds of two
conditional probabilities
• Conditional probabilities in survival analysis of the form :
Pr(live till time t1+t2 | survive up till time t1)
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Basic probability
• Example: automatic blood-pressure machine
• 84% hypertensive and 23% normotensives are classified as hypertensive
• Given 20% of adult population is hypertensive
• We now know:
Pr(machine says hypertensive | truly hypertensive)
• What is Pr(truly hypertensive| machine says hypertensive)?
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Basic probability
Machine diagnosed as hypertensive (D)
Hypertension (H) Yes No
Yes
No
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Basic probability
• Positive predictive value — Probability that a randomly selected subject from the population actually has the disease given that the screening test is positive
• Negative predictive value — Probability that a randomly selected subject from the population is actually disease free given that the screening test is negative
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Basic probability
• Sensitivity — Probability that the procedure is positive given that the person has the disease
• Specificity — Probability that the procedure is negative given that the person does not have the disease
Review examples 3.26, 3.27, and 3.28 in Rosner
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• Measures of Occurrence– Measure using proportions (e.g.,
prevalence, odds)– Rates (e.g., incidence, cumulative
incidence)
• Measure of Association– Based on odds (e.g., odds ratio)– Based on probabilities (e.g., risk ratio)
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Absolute Measures of Disease Occurrence
• Point prevalence = proportion of cases at a given point in time– cross-sectional measure
• Incidence = number of new cases within a specified time interval– prospective measure
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Absolute Measures of Disease Occurrence
• Example:
Consider four individuals diagnosed with lung cancer
• Proportion of death = 2/4 = 0.5• Rate of death = 2/(3+5+2+1) = 0.18 deaths per person year
Person Years of Follow-up Status
1 3 Dead
2 5 Alive
3 2 Alive
4 1 Dead
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Absolute Measures of Disease Occurrence
• Two kinds of quantities used in measurement:
– Proportion: the numerator of a proportion as a subset of the denominator, e.g., prevalence
– Rate: # events which occur during a time interval divided by the total amount of time, e.g., incidence rate
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Absolute Measures of Disease Occurrence
Remarks:
1) Diseases of long duration tend to have a higher prevalence
2) Incidence tends to be more informative than prevalence for causal understanding of the disease etiology
3) Incidence is more difficult to measure & more expensive
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Absolute Measures of Disease Occurrence
4) Prevalence & incidence can be influenced by the evolution of screening procedures and diagnostic tests
5) Both incidence and prevalence rates may be age dependent
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Absolute Measures of Disease Occurrence
• Odds = ratio of P(event occurs) to the P(event does not occur).
Example:
The probability of a disease is 0.20.
Thus, the odds are 0.20/(1-0.20) = 0.20/0.80 =0.25 = 1:4
That is, for every one person with an event, there are 4 people without the event.
p
podds
1
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Absolute Measures of Disease Occurrence
• Risk of disease in time interval [t0, t1)
P(t) = Pr(developing disease in interval of length
t = t1 - t0 given disease free at the start
of the interval)
• Average Prevalence = Incidence x Duration
duration = average duration of disease after onset
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Measures of Disease Association
• So far we have discussed
– Prevalence
– Incidence rate
– Cumulative incidence rate
– Risk of disease within an interval t
• All absolute measures
• Next, relative measures and associations
– Exposed (E) versus Unexposed ( )
E
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Measures of Disease Association
• Population versus sample
– Probabilities (population) are denoted by symbols such as
• = P(disease within the exposed population)
– Sample estimates are denoted by
1p
1p̂
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Measures of Disease Association
Exposed E
Not Exposed Total
Disease D
a b n1
No Diseasec d n0
Total m1 m0 n
D
E
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Conditional distribution
Exposed E
Not Exposed Margin
Disease D
No Disease
Margin 1
D
E
1p
11 p
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Conditional distribution
Exposed E
Not Exposed Margin
Disease D
No Disease
Margin 1
D
E
0p
01 p
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Measures of Association
• Odds ratio: Odds of disease among exposed divided by odds of disease among unexposed
0
0
1
1
1
1
pp
pp
OR
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Measures of Association
OR > 1 implies a positive association between disease and exposure
OR < 1 implies a negative association between disease and exposure
OR for disease = OR for exposure
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Measures of Association
• Risk ratio = ratio between P(disease for exposed) and P(disease for unexposed) , both P(.) measured within the same duration of time
1
0
pRR
p
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Measures of Association?
• Risk Difference (Excess Risk): RD = 1 - 0
RD not scale free
e.g., What is the meaning of these two equal differences
RR = 0.009. RD = 0.010-0.001 vs. RD = 0.210-0.201
• Attributable Risk for Exposed Persons:AR = (1 - 0) / 1 = 1 – 1 / RR
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• Measurements of risk and relative risk in different sampling designs
• Cross-sectional• Cohort• Case-control
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Measures of Disease Association
Exposed E
Not Exposed Total
Disease D
a b n1
No Diseasec d n0
Total m1 m0 n
D
E
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• Cross-Sectional SamplingRandomly sample n subjects from population at time t and determine disease and exposure status.
Important: n is fixed for this design.
1) a/m1 estimates prevalence of disease at t among exposed
2) b/m0 estimates prevalence of disease at t among unexposed
3) ad/bc estimates the OR for disease and exposure
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Odds Ratio
p1 = a/m1 = disease risk among exposedp0 = b/m0 = disease risk among unexposed
If p1 and p0 are small (rare disease) and the time interval is relatively short, it can be shown that OR ≈ RR
)1(
)1(
0
0
1
1
pp
pp
OR
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Cross-sectional Sampling
• Cross-sectional design not prospective
• Can only test for association between exposure and prevalence and not incidence
• Cannot test hypotheses about causality
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• Cohort SamplingSample n disease-free individuals from the population at time t0 and follow them until time t1.
Measure exposure history for each subject and observe which subjects develop disease in interval [t0, t1)
Important: m1, m0, and n are fixed
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Cohort study: Estimates of risk
1) p1 = a/m1 estimates risk of developing disease in interval among exposed
2) p0 = b/m0 estimates risk of developing disease in interval among unexposed
3) RR ≈ p1 / p0
4) OR = ad / bc
5) IR (incidence rate): i ≈ pi / t for i = 0, 1 (and small t)
6) RD (risk difference): RD ≈ 1 – 0 ≈ (p1 – p0) / t
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• Case-Control Sampling
Sample n1 cases and n0 disease free controls from target population during interval [t0, t1)
Important: n1, n0, and n are fixed
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1) a/m1 and b/m0 do not estimate population disease risks
2) a/n1 estimates Pr(prior exposure | disease incidence in [t0, t1)
3) c/n0 estimates Pr(prior exposure | no disease incidence in [t0, t1)
4) OR = ad / bc
5) RR ≈ OR for rare disease or short time intervals
6) IR (incidence rate) or disease risks cannot be estimated; RD (risk difference) cannot be estimated
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• Hypothetical exampleFrequency of disease and exposure in a target population
p1 = ? p0 = ?
RR = p1 / p0 = ? OR = ?
ExposureNot
ExposureTotal
Disease 8 32 40
No Disease 92 868 960
Total 100 900 1000
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• Hypothetical exampleFrequency of disease and exposure in a target population
p1 = 8 / 100 = 0.08; p0 = 32 / 900 = 0.036
RR = p1 / p0 = 0.08 / 0.036 = 2.25 OR = (8 x 868) / (92 x 32) = 2.36
ExposureNot
ExposureTotal
Disease 8 32 40
No Disease 92 868 960
Total 100 900 1000
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• Cohort Study50% of exposed individuals sampled25% of unexposed individuals sampled
p1 = 4 / 50 = 0.08; p0 = 8 / 225 = 0.036 RR = p1 / p0 = 0.08 / 0.036 = 2.25 OR = (4 x 217) / (46 x 8) = 2.36
Exposure Not Exposure Total
Disease 4 8 12
No Disease 46 217 263
Total 50 225 275
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• Case-Control Study100% of diseased individuals sampled25% of disease-free individuals sampled
p1 = 8 / 31 = 0.26 ≠ 0.08; p0 = 32 / 249 = 0.13 ≠ 0.036
RR = p1 / p0 = (8/31) / (32/249) = 2.01 ≠ 2.25 OR = (8 x 217) / (23 x 32) = 2.36
ExposureNot
ExposureTotal
Disease 8 32 40
No Disease 23 217 240
Total 31 249 280
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Odds ratio
• The odds ratio is equally valid for retrospective, prospective, or cross-sectional sampling designs
• That is, regardless of the design it estimates the same population parameter
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Take home messages
– Occurrence of disease measured by prevalence, or proportion
– Incidence measured by incidence rates, or proportion per unit time
– Risk is probability of developing disease over a specified period of time
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Take home messages
– Association of disease with exposure measured by odds ratios and risk ratios
– Odds ratios are valid for cross-sectional, cohort, and case-control designs, risk ratios are not
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HW #1
• Due May 29
• Can talk to others but turn in own work