Basic Statistical Concepts

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Basic Statistical Concepts

Chapter 2 Reading instructions


2.1 Introduction: Not very important


2.2 Uncertainty and probability: Read


2.3
Bias and variability
: Read


2.4
Confounding and interaction
: Read


2.5
Descriptive and inferential statistics
: Repetition


2.6
Hypothesis testing and p
-
values
: Read


2.7
Clinical significance and clinical equivalence
: Read


2.8 Reproducibility and generalizability: Read

Bias

and variability

Bias: Systemtic deviation from the true value






ˆ
E
Design, Conduct, Analysis, Evaluation

Lots of examples on page 49
-
51

Bias

and variability

Larger study does not decrease bias






n
ˆ
;



n
Drog X

-

Placebo

-
7

-
4

-
10

mm Hg

-
7

-
4

-
10

Drog X

-

Placebo

mm Hg

mm Hg

Drog X

-

Placebo

-
7

-
4

-
10

n=40

n=200

N=2000

Distribution of sample means:

= population mean



Population mean

bias

Bias

and variability

There is a multitude of sources for bias

Publication bias

Selection bias

Exposure bias

Detection bias

Analysis bias

Interpretation bias

Positive results tend to be published while negative of
inconclusive results tend to not to be published

The outcome is correlated with the exposure. As an example,
treatments tends to be prescribed to those thought to
benefit from them. Can be controlled by randomization

Differences in exposure e.g. compliance to treatment could
be associated with the outcome, e.g. patents with side
effects stops taking their treatment

The outcome is observed with different intensity depending
no the exposure. Can be controlled by blinding investigators
and patients

Essentially the I error, but also bias caused by model miss
specifications and choice of estimation technique

Strong preconceived views can influence how analysis results
are interpreted.

Bias and
variability

Amount of difference between observations

True biological:

Temporal:

Measurement error:

Variation
between

subject due
to biological factors (covariates)
including the treatment.

Variation over time (and space)

Often
within

subjects.

Related to instruments or observers

Design, Conduct, Analysis, Evaluation

Raw Blood pressure data

Baseline

8 weeks

Placebo

Drug X

DBP

(mmHg)

Subset of plotted data

Bias and
variability





X
Y
Unexplained
variation

Variation in
observations

=

Explained
variation

+

Bias and
variability

Drug A

Drug B

Outcome

Is there any difference between drug A and drug B?

Bias and
variability

Y=
μ
A
+βx

Y=
μ
B
+βx

μ
A

μ
B

x=age

Model:

ij
ij
i
ij
x
Y






Confounding

Predictors

of
treatm
ent

Predictors

of
outcome

Confounders

Treatment

allocation

A

B

Outcome

Example

Smoking Cigarettes is not so bad but watch out for
Cigars or Pipes (at least in Canada)

Variable

Non smokers

Cigarette

smokers

Cigar

or pipe
smokers

Mortality rate
*

20.2

20.5

35.5

Cochran, Biometrics 1968

*) per 1000 person
-
years %

Example

Smoking Cigaretts is not so bad but watch out for
Cigars or Pipes (at least in Canada)

Variable

Non
smokers

Cigarette

smokers

Cigar

or
pipe

smokers

Mortality

rate*

20.2

20.5

35.5

Average

age

54.9

50.5

65.9

Cochran, Biometrics 1968

*) per 1000 person
-
years %

Example

Smoking Cigaretts is not so bad but watch out for
Cigars or Pipes (at least in Canada)

Variable

Non
smokers

Cigarette

smokers

Cigar

or
pipe

smokers

Mortality

rate*

20.2

20.5

35.5

Average

age

54.9

50.5

65.9

Adjusted

mortality

rate*

20.2

26.4

24.0

Cochran, Biometrics 1968

*) per 1000 person
-
years %

Confounding

The effect of two or more factors can not be separated

Example:

Compare survival for
surgery and drug

R

Life long treatment with drug

Surgery at time 0


Surgery only if healty enough


Patients in the surgery arm may take drug


Complience in the drug arm May be poor

Looks ok but:

Survival

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

R

A

B

Gender

M

F

R

R

A

A

B

B

Balance on average

Always balance

Interaction

The outcome on one variable depends
on the value of another variable.

Example

Interaction between two drugs

R

A

A

B

B

Wash
out

A=AZD1234

B=AZD1234 +
Clarithromycin

Interaction

Mean
0
1
2
3
4
5
0
4
8
12
16
20
24
Time after dose
Plasma concentration (µmol/L)
linear scale
AZD0865 alone
Combination of clarithromycin
and AZD0865
19.75
(µmol*h/L)

36.62
(µmol*h/L)

AUC AZD1234:

AUC AZD1234 + Clarithromycin:

Ratio:

0.55 [0.51, 0.61]

AZD1234

AZD1234

Example: Drug interaction

Interaction

Example:

Treatment by center interaction

Treatment difference in diastolic blood pressure
-25
-20
-15
-10
-5
0
5
10
15
0
5
10
15
20
25
30
Ordered center number
mmHg
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.

Descriptive and inferential
statistics

Inferential statistics
forms a basis for a conclusion
regarding a prespecified objective addressing the
underlying population.

Hypothesis

Results

Confirmatory analysis:

Conclusion

Descriptive and inferential
statistics

Explorative statistics

aims to find interesting results that

Can be used to formulate new objectives/hypothesis for
further investigation in future studies.

Results

Hypothesis

Explorative analysis:

Conclusion?

Hypothesis testing, p
-
values and
confidence intervals

Objectives

Variable

Design

Statistical Model

Null hypothesis

Estimate

p
-
value

Confidence interval

Results

Interpretation

Hypothesis testing, p
-
values

Statistical model: Observations



n
n
R
X
X



,
1
X
from a class of distribution functions









:
P
Hypothesis test: Set up a null hypothesis: H
0:

0



and an alternative H
1
:

1



Reject H
0

if










0
|
c
S
P
X
n
c
R
S


X
p
-
value:

Rejection region

The smallest significance level for which the
null hypothesis can be rejected.

Significance level

Confidence intervals

A confidence set is a random subset
covering the true parameter value with probability at
least .





X
C


1
Let



rejected
not

:

if

0
rejected

:

if

1
,
*
0
*
0
*









H
H
X
(critical function)

Confidence set:







0
,
:





X
X
C
The set of parameter values correponding to hypotheses
that can not be rejected.

Example

y
ij

=
μ

+
τ
i

+
β

(
x
ij

-

x
∙∙
) +
ε
ij


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

Null hypoteses (subsets of ):

H
01
:
τ
1
=
τ
2
(DBP)

H
02
:
τ
1
=
τ
2
(SBP)

H
03
:
τ
2
=
τ
3
(DBP)

H
04
:
τ
2
=
τ
3
(SBP)

Objective: To compare sitting diastolic blood pressure (DBP) lowering effect of
hypersartan 16 mg with that of hypersartan 8 mg

Model:

treatment effect

i = 1,2,3

{16 mg, 8 mg, 4 mg}

Parameter space:

4
R
4
R


0
3
2
1






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:





c
T

X
X
:
001
.
0


P
-
value: The null hypothesis can pre rejected at



X
T
0

-
c

c

2
1



0

-
4.6

-
2.8

P
-
value says nothing about the
size of the effect!

No. of patients per group

Estimation of effect

p
-
value

10

1.94 mmHg

0.376

100

-
0.65 mmHg

0.378

1000

0.33 mmHg

0.129

10000

0.28 mmHg

<0.0001

100000

0.30 mmHg

<0.0001

A statistical significant difference does
NOT

need to be clinically relevant!

Example:

Simulated data. The difference between treatment and
placebo is 0.3 mmHg

Statistical and clinical
significance

Statistical significance:

Clinical significance:

Health ecominical relevance:

Is there any difference between
the evaluated treatments?

Does this difference have any
meaning for the patients?

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!

Health economically relevant?

Clinically significant?