Transcript c+d

臨床醫學常用統計方法
郭浩然
成功大學醫學院環境醫學研究所
成大醫院職業及環境醫學部
Course Contents
 Basic
concepts in biostatistics.
 Common statistical methods.
 Interpretation of statistical tests.
What is statistics?
 Statistics
The science whereby inference are
made about specific random
phenomenon on the basis of
relatively limited material.
 Mathematical
statistics.
 Applied statistics.
What is biostatistics?
The branch of applied statistics that
concerns the application of
statistical methods to medical and
biological problems.
 Figures
 Data.
(numbers).
Evaluation of the Vita-Stat
 Vita-Stat
•
•
is an automatic blood
pressure measuring device
Origin of the problem:
A friend of Dr. Rosner has his diastolic
pressure measured as 115 mm Hg
several times and even up to 130 mm
Hg by a machine.
A measurement at a clinic was 90 mm
Hg.
Basic Steps of Data Analysis
 Error
checking.
 Data transformation.
 Descriptive analysis.
 Hypothesis testing.
 Univariate analysis.
 Multi-variate analysis.
Error Checking

Errors.
 Measurement
error.
 Transcription error.
 Coding error.
 Key-in error.

Errors Checking.
 Frequency
table or plot.
 Range check.
 Logic check.
 Repeat measurement.
Variables
 Dichotomous


Based on count data.
“Yes” vs. “no” in most cases.
 Categorical


data.
Nominal data.
Ordinal data.
 Continuous
data.
data.
Data Transformation
 Continuous
to categorical.
Related
to hypothesis.
Natural or biological scale.
Convenience scale.
Equal participants.
 Log
transformation.
Descriptive Statistics

Measurement of central location.
 Mean.
 Median.
 Mode.

Measurement of spread.
 Standard
deviation.
 Range.
 Q1
to Q3.
Coefficient
of variation.
Coefficient of Variation
100% x (s/mean)
 Relate the arithmetic mean and the
standard deviation.
 No unit.
Variance and Standard Deviation
 Variance:
The sum of the squares of the
deviations divided by (n-1).
 Standard
deviation:
The square root of variance.
 Standard error?
Measurements
 Frequency.
 Effect.
 Association.
 Stability.
頻率(frequency)的指標
 盛行率(prevalence)
 發病率(incidence)
 累積發病率(cumulative
 死亡率(mortality)
 致死率(case
fatality)
 存活率(survival)
 比例(ratio)
incidence)
Measurements of Effect
 Absolute
risk.
 Risk difference.
 Regression coefficient of a linear model.
 Relative risk.
 Rate ratio
 Odds ratio.
 Risk: cumulative incidence
Fourfold (Two by Two) Table


The ultimate form of data in an analytical
epidemiologic study.
By convention, the columns indicate the outcome
status and the rows indicate the exposure status.
exposure
yes
no
disease (outcome) PT
yes no total
a
b a+b
PTe
c
d c+d
PTu
a+c b+d T(a+b+c+d)
世代研究
outcome (disease)
PT
exposure
yes no total
yes
a
b
a+b
PTe
no
c
d c+d
PTu
a+c b+d a+b+c+d
 (incidence) rate difference= a/PTe – c/Ptu
 risk difference=a/(a+b) – c/(c+d)
 risk ratio=a(a+b)/c(c+d)
 (incidence) rate ratio= aPTe/cPTu
病例對照研究
disease
exposure case control total
yes
a
b
a+b
no
c
d
c+d
a+c b+d
a+b+c+d
 odds ratio=ac/bd=ad/bc
勝算比(odds ratio)
disease
exposure case control
yes
a
b
no
c
d
 odds = p/ (1-p)
 exposure odds in cases=a(a+c)/c(a+c)=ac
 exposure odds in controls=bd
 exposure odds ratio=ac/bd=ad/bc
橫斷式研究
outcome (disease)
exposure
yes no total
yes
a
b a+b
no
c
d c+d
a+c b+d a+b+c+d
 prevalence rate ratio=a(a+b)/c(c+d)
 odds ratio= ad/bc
Odds Ratio vs. Rate Ratio
outcome (disease)
exposure
yes no total
yes
a
b a+b
no
c
d c+d
a+c b+d a+b+c+d
 rate ratio=a(a+b)/c(c+d)
 odds rate ratio= ad/bc
 when a<<b and c<<d, OR=RR
R by C Contingency Table
Column
Row Col1 Col2
R1 C11
C12
R2 C21
C22
.
.
.
.
.
.
total Cm1
Cm2
.
.
.
.
.
.
.
total
.
Rm1
.
Rm2
.
.
.
.
. grand total
Linear Regression
Y = a + bX
Y: outcome
X: exposure (predictor; continuous or dichomotous)
a: intercept (the baseline risk in many cases)
b: regression coefficient
 Indicates the incremental change of Y associated
with each one-unit increase in X.
 When Y is the risk of an outcome, b indicates the
risk difference associated with one-unit increase in
X.
Logistic Regression
logit(p) = a + bX
logit(p) = p/(1-p)
X: exposure (predictor)
a: intercept (indicating baseline risk in many cases)
b: regression coefficient
 eb indicates the odds ratio associated with each
one-unit increase in X.
Measurements of Association
 All
measurements of effect.
 Correlation coefficient.
 Kappa statistics.
(Po-Pe) / (1-Pe)
Po=observed probability of concordance
Pe=expected probability of concordance
Correlation Coefficient ( r )
from –1 to 1.
 A negative value indicates a negative
association, and positive value
indicates a positive association.
 A value
 |r| < 0.4 indicates a poor correlation.
 0.4 < |r| < 0.75 indicates a fair to good
correlation.
 |r| > 0.75 indicates excellent correlation.
0
indicates no association.
Kappa Statistics (k)
2nd survey
1st survey yes no
yes
a
b a+b
no
c
d c+d
a+c b+d T
a1=(a+b)/T, a2=(c+d)/T, b1=(a+c)/T, b2=(b+d)/T
Pe=a1b1+a2  b2
Po= (a+d)/T
k=(Po-Pe)/(1-Pe)
Hypothesis Testing




To test the statistical significance of the study
result.
The test is on the basis of observed data, but
the hypothesis is formulated before the data
collection.
An approach to reach an objective
interpretation of data.
Statistical significance does not necessarily in
concordance to clinical significance.
Types of Hypotheses
 Null
hypothesis:
There is no association between the
predictor and the outcome tested.




difference in parameter =0
relative risk=1
regression slope=0
correlation coefficient=0
 Alternative
hypothesis:
There is an association between the
predictor and the outcome tested.
Null v.s. Alternative Hypotheses




The null hypothesis is tested.
The alternative hypothesis is not tested
directly.
We reject the null hypothesis if the statistical
test result is significant, but we do not accept
the alternative hypothesis by rejecting the null
hypothesis.
The validity of the alternative hypothesis is
built up by repetitive rejection of the null
hypothesis.
Types of Hypotheses
 One-tailed
(sided) hypothesis:
specify the direction of the association


relative risk>1
relative risk<1
 Two-tailed
(sided) hypothesis:
not specify the direction of the




difference in parameter≠0
relative risk≠1
regression slope≠0
correlation coefficient≠0
One vs. Two Tailed Hypotheses
 In
one tailed test, the p value is
twice for the same test statistics as
in one-sided testing.
 We should have a scientific basis in
conducting one tailed test.
 Whether one or two tailed test
should be conducted should also
be determined before the testing.
Types of Statistical Tests
 Parametric


tests.
Based on assumptions of the distribution of
the variables.
May lead to wrong conclusions.
 Non-parametric




tests.
Not based on assumptions of the distribution
of the variables.
Can always be applied.
May loss power.
Less sensitive to extreme values.
Variables
 Dichotomous
 based on count data
 “yes” vs. “no” in most cases
 OR and RR
 Categorical
 nominal data

data
ordinal data
 Continuous
data
Dichotomous Variable
2 by 2
 R by C
 Chi-square test
 McNemar’s test (for correlated proportion)

Dichotomous Variable
Patients
60
40
Controls
20
80
Smokers
Non-Smokers
N=200
 Case-Control Study: OR=60x80/20x40=6
 Cohort Study: RR= (60/80) ÷ (40/120)=2.25
 Chi-square test
Categorical
Contingency Table (chi-square test)
 Kappa statistics (for reproducibility)
 Rank correlation

Continuous
Two groups
 two-sample t test
 paired t test
 correlation coefficient
 Multiple groups
 ANOVA (analysis of variance)

Non-parametric tests

Chi-square test: Fisher’s exact test

Two sample-t test: Wilcoxon rank sum
test
 paired t test: Wilcoxon sign rank test
 Person correlation coefficient:
Spearman rank correlation
 ANOVA: Kruskal-Wallis test
Multi-variate (Multiple) Regession
 linear
regression
 logistic regression
 poisson regression
 Cox proportion hazard models
• adjust for effects of other predictors
• evaluate the effects of more than one
predictor at the same time
Right Time to Select Study Groups
prospective: A
retrospective: B,C,D
cross-sectional: B,C (A,D)
Exposure
Outcome
Time
A
B
C
D
Measurements of stability
p
value.
 Confidence interval.
Normal Distribution
 If
X follows a normal distribution with
a mean of μ and a variance of σ2 and
Z = (X-μ) ÷ σ, then Z follows a normal
distribution with a mean of 0 and a
variance of 1; Z～N(0,1).
 If x is a sample mean of X from n
subjects, then x follows a normal
distribution with a mean of μ and a
variance of σ2 /n; x～N(μ, σ2 /n).
 Z = (x-μ) ÷ √σ2/n = (x-μ) ÷σ√n .
t Distribution
 When
the variance of the underlying
normal distribution (σ2) is unknown, it can
be estimated by the variance of the
sample s, but the value (x-μ) ÷ s√n = t
does not follow a normal distribution.
 The value t follows a t distribution with a
degree of freedom df = n-1.
 The test on the basis of a t distribution is
called “t test.”
 The test that uses Z as the test statistic is
called “z test.”
Central Limit Theorem
 Let
X1, … Xn be a random sample from
a population with mean μ and variance
σ2. Then, for a large n, the mean of X
～ N(μ, σ2/n) even if the underlying
distribution of individual observations in
the population is not normal.
Critical-value Method
Critical-value Method
p(t≦tn-1, α) = p(t≧tn-1,1-α)
p(t≦t99, 0.05) = p(t≧t99, 0.95)
N=60, u=1.671
N=120, u=1.658
N=99, u=X
(1.671-X) ÷ (X-1.658) = (99-60) ÷ (120-99)
X=1.663
t99, 0.05 = -1.663
The p Value
 The
α level at which we would be
indifferent between accepting or rejecting
Ho given the sample data at hand.
 The α level at which the given value of the
test statistic would be on the borderline
between the acceptance and rejection
regions.
 The probability of obtaining a test statistic
as extreme as or more than the actual test
statistic obtained given that the null
hypothesis is true.
The p Value
 It
is a measurement of the statistical
significance of the test, but not the
clinical significance.
 It is a measurement of the stability of
the point estimate, not the magnitude of
the effect.
 Direct comparisons between p values
have little value.
Example-2
 Topic:
the feasibility of using drugs
to reduce the size in patients who
have a myocardial infarction within
the past 24 hours.
 The mean infract size in untreated
patients is 25 (ck-g-EQ/m2).
 In 8 patients treated with a drug, the
mean size was 16 with a standard
deviation (SD) of 10.
 Q: is the drug effective in reducing
infract size?
The p Value Method
t = (16-25) / 10÷√8 = -2.55
How to calculate the p value?
p(t≦tn-1, α) = p(t≧tn-1,1-α)
p(t≦t7, p) = p(t≧t7,1-p)
The p Value Method
(0.99-X) ÷ (X-0.975) = (2.998-2.55) ÷ (2.552.365)
X=0.981
p = 1-0.981 = 0.019
The p Value Method
 Choose
a statistical test.
 Calculate the test statistic on the basis of the
probability distribution function.
 Obtain the p value from the test statistic
probability table or a computer software.
 Compare the p value with the significance
level to determine whether the null hypothesis
should be rejected.
Confidence Interval Method
 Calculate
the variance.
 Construct the confidence interval
according to the α level.
 If the null value is outside the
confidence interval, the null
hypothesis should be rejected.
 Recommended by most biostatisticians and epidemiologists.
Example-3
 Topic:
family aggregation of cholesterol
level.
 A large study showed the mean
cholesterol level in children 2-14 years of
age was 175 mg/L.
 The mean mean cholesterol level in 100
children of patients with elevated
cholesterol level (≧250 mg/L) of same
ages was 207.3 mg/L with a SD of 30.0
mg/L.
 Q: is the underlying mean cholesterol
level in children of patients with elevated
cholesterol level different from 175 mg/L?
Confidence Interval Method
100% × (1-α) confidence interval (CI)
for the meanμ of a normal
distribution with unknown variance
is:
(x - tn-1, 1-α/2 s/√n, x + tn-1, 1-α/2 s/√n).
 A 95% CI for the meanμ of a normal
distribution with unknown variance
is:
(x - tn-1, 0.975 s/√n, x + tn-1, 0.975 s/√n).
A
Confidence Interval Method
CI: 207.3 ± 6.0 = (201.3, 213.3)
 The null value 175 is lower than the lower
limit of the 95% CI (175 < 201.3), and
therefore the null hypothesis is rejected.
 CI is a measure of stability and measure of
effect.
 CI leads to the same conclusion as the t
test:
p for t test < 0.05.
 95%