hsrp 734: advanced statistical methods may 22, 2008

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

• Survival analysis

• Cox PH regression

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

What is Logistic Regression?

A statistical method used to model dichotomous or binary outcomes (but not limited to) using predictor variables.

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

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”

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

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

A Short ReviewA Short Review

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.)

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

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) ?

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

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)

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 -

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%).

• Any difference can become significant if N is large enough

• Even if there is statistical significance is there clinical significance?

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.”

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.”

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 ?

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)

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).

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).

HSRP 734Lecture 1:

Measures of Disease Occurrence and Association

Objectives:

1.Define and compute the measures of disease occurrence and association

2.Discuss differences in study design and their implications for inference

Example

CT images rated

by radiologist

(Rosner p.65)

Rated as normal

Rated as questionable

Rated as abnormal

Normal 39 6 13

Abnormal 5 2 44

(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

Basic Probability

• Conditional probability

– Restrict yourself to a “subspace” of the sample space

Male Female

Young 20% 10%

Old 35% 35%

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)

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)

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)?

Basic probability

Machine diagnosed as hypertensive (D)

Hypertension (H) Yes No

Yes

No

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

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

• 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)

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


• 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


• 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


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


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


• 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


• 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

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


• Population versus sample

– Probabilities (population) are denoted by symbols such as

• = P(disease within the exposed population)

– Sample estimates are denoted by

1p

1p̂


Exposed E

Not Exposed Total

Disease D

a b n1

No Diseasec d n0

Total m1 m0 n

D

E

Conditional distribution

Exposed E

Not Exposed Margin

Disease D

No Disease

Margin 1

D

E

1p

11 p

Conditional distribution

Exposed E

Not Exposed Margin

Disease D

No Disease

Margin 1

D

E

0p

01 p

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


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


• 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

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

• Measurements of risk and relative risk in different sampling designs

• Cross-sectional• Cohort• Case-control


Exposed E

Not Exposed Total

Disease D

a b n1

No Diseasec d n0

Total m1 m0 n

D

E

• 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

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

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

• 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

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

• 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

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

• 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

• 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


Total 100 900 1000

• 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


Total 50 225 275

• 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


Total 31 249 280

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

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

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

HW #1

• Due May 29

• Can talk to others but turn in own work

hsrp 734: advanced statistical methods may 22, 2008

Documents

statistical significance

false null hypothesis

statistical analysis

null hypothesismake

true h0

usedlogistic regression

hypothesis testingset

paradigm of falsification