Transcript 1 - Egypt

How much statistics do we
need to know?
Confidence intervals
P value
Power of a study
Our field of research is biology
and not physics or chemistry.
The relative and the absolute
CERTAIN and RANDOM events
The certain event will ALWAYS occur every time the necessary set of conditions is
realized. The occurrence of the random event will remain a PROBABILITY because the
known set of conditions does not reflect the entire collection of reasons necessary
and sufficient for its occurrence.
This is mainly due to our variability and the reasons
behind it
are not always known nor fully
understood.
In consequence, we have to understand and to
benefit from the laws of probability because the
best we can do is to keep predicting the most
suitable ways to treat our patients.
We need to appreciate variability
Group B
Group A
49
50
51
50
99
1
50
 Although the mean age of both groups is 50, yet no one can argue
that both groups are comparable.
 The mean alone is not just meaningless but can be even
misleading, when it gives the wrong impression that both groups
are comparable or the same.
 Mean has to be coupled with an index of variability to the show
the dispersion of values from their mean.
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Looking to describe the variation
49
50
51
The variation around the mean:
49-50 = -1
50-50 = 0
51-50 = +1
50
99
1
50
The same applies for Group B;
where the variations around the mean=
99-50 = 49
1-50 = -49
50-50 = 0
Unfortunately, the sum of variations:
-1+0+1 = 0
Consequently, the mean variation:
(0) /3 = 0
Squaring variations before summing
SS = (-1)2 + (0)2 + (1)2 = 2
A practical mean variation (S2)
2 / (3-1) = 1
The sum of variations= 49-49+0=0
The mean variation = 0/3=0
Variance (S2) = SS/n-1
[(99-50)2 + (1-50)2 + (50-50)2] / (3-1)
= 2401 + 2401 + 0 = 4802/2 = 2401
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How far are we from the truth?
True
Mean
M3
M2
M1
1/n
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How to express variability: variance (S2)
•
How variable are the individual values of
your study:
SD = √S2
•
How far (variable) is your (m) from (M):
SEM = √S2/n
SEM = SD/√n
The aim of research is to understand
and hopefully to manipulate reality.
• In biology, variability is the rule rather
than being the exception, however
• Variability is not a Chaos and has
characteristic distributions that we
can understand, analyze and most
importantly, that we can predict.
The Normal distribution
The intervals of confidence
Birth weight classes (gm.)
(a) Centers
(b) Range*
(66 births; 69.5%)
2100
2300
2500
2700
2900
3100
3300
3500
3700
3900
4100
4300
4500
2000-2200
2200-2400
2400-2600
2600-2800
2800-3000
3000-3200
3200-3400
3400-3600
3600-3800
3800-4000
4000-4200
4200-4400
4400-4600
Total
Birth weight frequency
Absolute Relative (%)
(number)
2
2.1
4
4.2
6
6.3
4
4.2
10
10.5
18
18.9
21
22.1
17
17.9
5
5.3
4
4.2
3
3.2
0
0
1
1.1
95
100
Total weight
(gm.)
4200
9200
15000
10800
29000
55800
69300
59500
18500
15600
12300
0
4500
303700
(92 births; 96.8%)
The mean birth weight m = 3200 gm. and the SD = 500 gm.
2/3 of birth weights are included in the interval: m + 1SD: 2700-3700 gm.
95% of birth weights are included in the interval: m + 2SD: 2200-4200 gm.
Nearly all birth weights are comprised within a distance of + 3 SD from the mean
The Normal distribution
1 SD
2 SD
m
3 SD
The myth of 30!
The Normal distribution is
followed by the majority of
biological variables.
The more you include patients
in your study, the more the
distribution takes the Normal
characteristics.
Variables were found to follow
Normality, if the number of
patients >30/group (Qn.) or if
expected counts >5 (Ql.).
The CI of a subject
We know that 95% of values
of a Normal distribution lie
within the interval m+2SD;
this is the CI at 95%.
Any baby of the same
population has a 95%
chance to be born with a
birth weight between 3200
+ 2 (500); between 2200 and
4200 gm.; this is the
predictive value of the CI.
2 SD
m
The IC of a subject
Any baby whose birth
weight is outside this
interval has a very little
chance (less than 5%) to be
considered as being part of
this population.
This
baby
can
be
considered
as
being
coming from a different
population (e.g. a diabetic
mother); this is the
comparative value of the
CI.
2 SD
2.5%
5000gm.
2.5%
m
The cross roads: Testing hypothesis
Does this baby belong to
this population ?
H0 = yes
A general agreement is to
reject H0 if its probability is
as small as 5% or less.
H1= this baby does not
belong to this population.
We have to remember that
still, we can be wrong
about this conclusion; H0 is
true
This is the P value of your
conclusion.
H0 =IC at 95%
H1
5000gm.
H1
m
The P value
The probability to conclude upon a difference that
does not exist.
The smaller is our P value, the more confident we are
in our results but never the more truer.
P values should not be compared neither in the same
study nor between studies,
On the other hand, comparisons should be made by
the intervals of confidence.
Evaluating a single patient
A patient is considered to be
potentially diabetic if his
FBS>90 mg/dl.
We
have
to
do a
confirmatory test (PP) to
exclude the probability (P
value) that his FBS is just a
normal
but
extreme
variation of the same
population (non-diabetics)
and
hence,
P
value
measures
how
our
conclusion can be untrue.
2 SD
2.5%
50
100 mg/dl.
2.5%
60
70
80 90 100
Fasting blood sugar in mg/dl
m
Evaluation of a whole study
• A person is considered as
being part of the population
when he lies within the IC of
a subject at 95%:
• m + 2 SD
2 SEM
• A study is comparable to
another; i.e. both studies
belong
to
the
same
populations, whenever our
mean lies within the IC of
mean at 95%.
• m + 2 SEM.
2 SD
m
Studies are compared by IC of mean
AND NEVER by P values
4+1 = 2-6 d.
6+1 = 4-8 d.
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4
5
6
7
8
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10
A study has shown that the mean ACC time of On-pump
CABG was significantly shorter than of an equal group
operated Off-pump; P= 0.05.
• This conclusion is 95% correct.
• If you repeat this study 100 times, the ACC of Onpump CABG will be shorter in only 95 times.
• If we repeat this study 100 times, a shorter ACC
will be due to a true difference between both
procedures in 95 times and, in 5 times, this
difference will be just an extreme variation of
calculation between comparable procedures.
Mean ACC: On-pump = 30+5 min. Off-pump 40 + 10 min.
• A patient whose ACC is 40 min. will more likely
belong to On-pump rather than Off-pump group.
• A patient whose ACC is 100 min. will more likely
belong to On-pump rather than Off-pump group.
• This study cannot help us to discriminate with an
acceptable level of confidence between the ACC
of an On-pump and an Off-pump patient.
• ICs of a subject at 95% (mean+2SD): 20-40 min.
& 20-60 min.
Patients are discriminated by IC of subject
On-pump patients = 30 + 2(5) = 20-40 minutes.
Off-pump patients = 40 + 2(10) = 20-60 minutes.
10
20
30
40
50
60
70
We wished to compare our off-pump results (ACC 40+10 min.
in 100 patients) with those reported in another same size Offpump study that reported a slightly longer ACC of 45+10 min.
This result is comparable to ours because of the
small difference between both means: 40 & 45 min.
Results are comparable for being homogeneous
and carried out on similar number of patients.
The ACC times of our Off-pump group is different
from that calculated by the second study because
our mean (40 min.) drops outside the IC of their
mean.
Studies are compared by IC of means
mean + 2 SEM
Our study = 40 + 2 (10 / √100) = 38 - 42 minutes.
Second study = 45 + 2 (10 / √100) = 43 - 47 minutes
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Conclusion
• Mean has to be coupled with a measure of
variability: SD, SEM or CI.
• There are CI for subjects and CI for means, at
different percentages.
• CI represent the set of true (but unknown) data
that are compatible with our results.
• Data should be compared with CI and never by P
values.
The P value and the value of P
• P value is just a probability and the question is
how is it obtained or the story of 2 brothers.
• Bias behind repeatedly testing significance:
• A) Comparing identical twins.
• B) No one complains of getting “pass” on the
second exam.
• C) Planning repeated analysis: Post-Hoc tests.