Download Basic Statistical Concepts

Basic Statistical Concepts
Chapter 2 Reading instructions
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2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
Introduction: Not very important
Uncertainty and probability: Read
Bias and variability: Read
Confounding and interaction: Read
Descriptive and inferential statistics: Repetition
Hypothesis testing and p-values: Repetition*
Clinical significance and clinical equivalence: Read
Reproducibility and generalizability: Read
*) Read extra material
Bias and variability
E [µˆ ] − µ
Bias: Systemtic deviation from the true value
Design, Conduct, Analysis, Evaluation
Lots of examples on page 49-51
Bias and variability
Larger study does not decrease bias
µˆ n → µ + ϕ
;
n → ∞
Distribution of sample means:
-10
-7
-4
n=40
mm Hg
-10
Population mean
bias
= population mean
Drog X - Placebo
Drog X - Placebo
µ
ϕ
Drog X - Placebo
-7
-4
n=200
mm Hg
-10
-7
-4
N=2000
mm Hg
Bias and variability
Amount of difference between observations
True biological:
Variation between subject due
to biological factors (covariates)
including the treatment.
Temporal:
Variation over time (and space)
Often within subjects.
Measurement error:
Related to instruments or observers
Design, Conduct, Analysis, Evaluation
Bias and variability
Y = Xβ + ε
Variation in = Explained + Unexplained
observations
variation
variation
Bias and variability
Is there any difference between drug A and drug B?
Drug A
Drug B
Outcome
Bias and variability
Model: Yij = µ i + βxij + ε ij
Y=µB+βx
µB
Y=µA+βx
µA
x=age
Confounding
The effect of two or more factors can not be separated
Example: Compare survival for
surgery and drug
Survival
Life long treatment with drug
R
Surgery at time 0
Looks ok but:
•Surgery only if healty enough
•Patients in the surgery arm may take drug
•Complience in the drug arm May be poor
Time
Confounding
Can be sometimes be handled in the design
Example: Different effects in males and females
Imbalance between genders affects result
Stratify by gender
A
M
A
R
Gender
R
B
Balance on average
B
A
F
R
Always balance
B
Interaction
The outcome on one variable depends
on the value of another variable.
Example
Interaction between two drugs
A
B
A=AZD1234
Wash
out
R
B
B=AZD1234 +
Clarithromycin
A
Interaction
Plasma concentration (µmol/L)
linear scale
Mean
5
4
AZD1234
AZD0865
alone
3
Combination of clarithromycin
and AZD0865
AZD1234
2
1
0
0
4
8
12
16
20
24
Time after dose
AUC AZD1234:
19.75 (µmol*h/L)
AUC AZD1234 + Clarithromycin:
Ratio:
36.62 (µmol*h/L)
0.55 [0.51, 0.61]
Interaction
Example:
Treatment by center interaction
Treatment difference in diastolic blood pressure
15
10
5
mmHg
0
-5
0
5
10
15
20
25
30
-10
-15
-20
-25
Ordered center number
Average treatment effect: -4.39 [-6.3, -2.4] mmHg
Treatment by center: p=0.01
What can be said about the treatment effect?
Descriptive and inferential
statistics
The presentation of the results from a clinical trial
can be split in three categories:
•Descriptive statistics
•Inferential statistics
•Explorative statistics
Descriptive and inferential
statistics
Descriptive statistics aims to describe various
aspects of the data obtained in the study.
•Listings.
•Summary statistics (Mean, Standard Deviation…).
•Graphics.
Example?
Descriptive and inferential
statistics
Inferential statistics forms a basis for a conclusion
regarding a prespecified objective addressing the
underlying population.
Confirmatory analysis:
Hypothesis
Results
Conclusion
Descriptive and inferential
statistics
Explorative statistics aims to find ineresting results that
Can be used to formulate new objectives/hypothesis for
further investigation in future studies.
Explorative analysis:
Results
Hypothesis
Hypothesis testing, p-values and
confidence intervals
Objectives
Variable
Design
Estimate
p-value
Confidence interval
Statistical Model
Null hypothesis
Results
Interpretation
Hypothesis testing, p-values
Statistical model: Observations X = ( X 1 , K X n ) ∈ R
from a class of distribution functions
n
℘ = {Pθ : θ ∈ Θ}
Hypothesis test: Set up a null hypothesis: H0: θ ∈ Θ 0
and an alternative H1: θ ∈ Θ1
Reject H0 if
X ∈ Sc ⊆ R n
P(X ∈ S c | θ ∈ Θ 0 ) < α
Rejection region
Significance level
p-value: The smallest significance level for which the
null hypothesis can be rejected.
Confidence intervals
Let δ (X,θ * )
1 if H 0 : θ = θ * rejected
=
0 if H 0 : θ = θ * not rejected
(critical function)
Confidence set: C (X ) = {θ : δ (X,θ ) = 0}
The set of parameter values correponding to hypotheses
that can not be rejected.
A confidence set is a random subset C (X ) ⊆ Θ
covering the true parameter value with probability at
least 1 − α .
Example
Objective: To compare sitting diastolic blood pressure (DBP) lowering effect of
hypersartan 16 mg with that of hypersartan 8 mg
Variable: The change from baseline to end of study in sitting DBP
(sitting SBP) will be described with an ANCOVA model,
with treatment as a factor and baseline blood pressure
as a covariate
Model:
yij = µ + τi + β (xij - x··) + εij
Parameter space: R 4
(τ 1 + τ 2 + τ 3 = 0)
treatment effect
i = 1,2,3
{16 mg, 8 mg, 4 mg}
Null hypoteses (subsets of
H01: τ1 = τ2 (DBP)
H02: τ1 = τ2 (SBP)
H03: τ2 = τ3 (DBP)
H04: τ2 = τ3 (SBP)
R 4):
Example contined
Hypothesis
Variable
LS Mean
CI (95%)
p-value
1: 16 mg vs 8 mg
Sitting DBP
-3.7 mmHg
[-4.6, -2.8]
<0.001
2: 16 mg vs 8 mg
Sitting SBP
-7.6 mmHg
[-9.2, -6.1]
<0.001
3: 8 mg vs 4 mg
Sitting DBP
-0.9 mmHg
[-1.8, 0.0]
0.055
4 : 8 mg vs 4 mg
Sitting SBP
-2.1 mmHg
[-3.6, -0.6]
0.005
This is a t-test where the test statistic follows a t-distribution
Rejection region:
{X : T (X ) > c}
-c
0
c
P-value: The null hypothesis can pre rejected at α
-4.6
-2.8
0
T (X )
< 0.001
τ1 −τ 2
Statistical and clinical
significance
Statistical significance:
Is there any difference between
the evaluated treatments?
Clinical significance:
Does this difference have any
meaning for the patients?
Health ecominical relevance: Is there any economical
benefit for the society in
using the new treatment?
Statistical and clinical
significance
A study comparing gastroprazole 40 mg and mygloprazole 30 mg
with respect to healing of erosived eosophagitis after 8 weeks
treatment.
Drug
Healing rate
gastroprazole 40 mg
87.6%
mygloprazole 30 mg
84.2%
Cochran Mantel Haenszel p-value = 0.0007
Statistically significant!
Clinically significant?
Health economically relevant?