Download lecture 7, 8 organising, summerising, understanding data and

Lecture 7,8
Summarizing Data, hypothesis
testing and expression of results
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Purpose of Statistics
• To describe & summarize information,
thereby reducing it to smaller, more
meaningful sets of data
• To make predictions or to generalize about
occurrences based on observations
• To identify associations, relationships or
differences between sets of observations
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Main Types of Statistics
• Descriptive Statistics
• Inferential Statistics
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Descriptive Statistics
• Descriptive Statistics involves
organizing, summarizing & displaying
data to make them more
understandable.
• Most common statistics used are
frequencies, percents, measures of
central tendency, summary tables,
charts & figures.
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Inferential Statistics
• Inferential Statistics: a set of
statistical techniques that provides
predictions about population
characteristics based on information
from a sample drawn from that
population.
• Inferential statistics report the
degree of confidence of the sample
statistic that predicts the value of
the population parameter.
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Describing data
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Describing data
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Measures of Central
Tendency
• When assessing the central tendency of
your measurements, you are attempting to
identify the “average” measurement
– Mean: best known & most widely used
average, describing the center of a frequency
distribution
– Median: the middle value/point of a set of
ordered numbers below which 50% of the
distribution falls
– Mode: the most frequent value or category in
a
distribution
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Thinking Challenge
$400,000
$70,000
$50,000
$30,000
... employees cite low pay -most workers earn only
$20,000.
$20,000
... President claims average
pay is $70,000!
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Comparison of
Central Tendency Measures
• Use Mean when distribution is
reasonably symmetrical, with few
extreme scores and has one
mode.
• Use Median with nonsymmetrical
distributions because it is not
sensitive to skewness.
• Use Mode when dealing with
frequency distribution for
nominal data
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Variability
• A quantitative measure of the degree to
which scores in a distribution are spread out
or are clustered together;
• Types of variability include:
– Standard Deviation: a measure of the
dispersion of scores around the mean
– Range: Highest value minus the lowest value
– Interquartile Range: Range of values
extending from 25th percentile to 75th
percentile
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Key Principles of Statistical
Inference
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Explaining precision or
certainty of effect
• Studies can only ever hazard a guess
at the effect of exposures or
interventions in the target
population.
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Explaining precision or
certainty of effect
• Imagine you tested a tablet on 100 adults to see
whether it reduced arthritic pain. You find that it
reduces pain in 63 of 100 people. That is, it has an
absolute risk reduction of 63%. Say you now
randomly sampled 100 different adults, and
repeated exactly the same experiment. Do you
think 63 of 100 would have pain relief? Probably
not. By chance, this time you might find that only
58 of the 100 people had pain relief. This second
study would have, therefore, reported an absolute
risk reduction of 58%. If you repeated the
experiment enough times, you might, by chance,
get a few really weird results, like 30% or 88%,
which were a long way away from the rest.
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Explaining precision or
certainty of effect
•
Luckily, there is a law of statistics which
begins to get around this difficulty.
•
It says that if you were to conduct more
and more studies on random samples
from the target population and keeping
averaging out their results, you will get
closer and closer to the ‘Truth.’
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Explaining precision or
certainty of effect
•
This is all well and good if you can average out
results from lots of studies (this is what metaanalyses do – providing they are carried out
properly).
•
But what if there are only one or two studies?
Fortunately, the second part of the law says
that the results from different studies are
distributed in a predictable way (in a normal or
Gaussian distribution) around the ‘True’ value.
So, even from a single study, we can predict
what the results of other, similar studies might
be, were we to conduct them.
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Explaining precision or
certainty of effect
•
•
Statisticians can, therefore, use this aspect of
the law to calculate confidence intervals around
a single study result . Thus, the confidence
interval may be accurately interpreted in the
following way:
The range of values in which you would
expect a specified percentage (usually 95%)
of study results from studies like this one to
fall, had you conducted them.
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CENTRAL LIMIT THEOREM
• A. For a randomly selected sample of size n (n should be
at least 25, but the larger n is, the better the
approximation) with a mean u and a standard
deviation o-.
•
_
– 1.
The distribution of sample means x is
approximately normal regardless
of
whether the population distribution is normal
• From statistical theory come these two additional theories
•
– 2.
The mean of distribution of sample
means is equal to the mean of the
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population distribution – that is  x  
3. The standard deviation of the distribution of
sample means calculated by the usual methods
is very close to the standard error of the
population mean calculated by
the
standard deviation of the population divided by
the square root of
the sample size –
• ẍ = SEẍ = /sr n
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STANDARD ERROR OF THE MEAN
The measure of variation of the distribution of sample means,
 n , referred to as the standard error of the mean, is denoted as
SE x  - that is

SE x   x 

n
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POINT ESTIMATES AND
CONFIDENCE INTERVALS

95% CI of   x  1.96 
n

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Important note
• We are confident that 95% of
samples mean are within 1.96 sd of
the population mean and then 95% of
all samples will provide A CI that
capture the actual population mean
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And
• We are confident that 95% of
samples mean differences are within
1.96 sd of the population mean
difference and then 95% of all
samples mean differences will
provide A CI that capture the actual
population mean difference
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Population estimates
AND Confidence interval
• If we have a sample of 25 people and
their average blood glucose is 161.84
( equals 58.15) what is the 95% CI
of mean=
=sample mean±1.96SE= 138.6-185.24
(upper and lower confidence limit)
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Interpretation of the
confidence interval
• The upper and lower limits of the confidence
interval can be used to assess whether the
difference between the two mean values is
clinically important.
• For example, if the lower limit is close to zero,
this indicates that the true difference may be
very small and clinically meaningless, even if the
test is statistically significant.
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Significance
Testing
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Null Hypothesis - Ho
• Ho proposes no difference or
relationship exists between the
variables of interest
• Foundation of the statistical test
• When you statistically test an
hypothesis, you assume that Ho
correctly describes the state of
affairs between the variables of
interest
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Example (Hypothesis
testing using p value)
• Drug A= 95±12.4 (n=10)
• Drug B=105±11.8 (n=10)
1. Null hypothesis is that mean
difference=0
2. alpha=0.05
3. Mean difference =10
4. SE for the difference=5.4
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SPSS Output for Two Sample
Independent t-test Example
Group Statistics
Pulse rate
Group
1
2
N
10
10
Mean
95.00
105.00
Std. Deviation
12.428
11.776
Std. Error
Mean
3.930
3.724
Independent Samples Test
Levene's Test for
Equality of Variances
t-tes t for Equality of
Means
Pulse rate
Equal variances
Equal variances
ass umed
not as sumed
.134
.719
-1.847
-1.847
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17.948
.081
.081
F
Sig.
t
df
Sig. (2-tailed)
Mean Difference
Std. Error Difference
95% Confidence Interval
of the Difference
Lower
Upper
-10.000
-10.000
5.414
5.414
-21.374
1.374
-21.377
1.377
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What does this mean
• 1. the difference is due to chance
• 2. the samples came from a population with
a true mean difference that is equal to 0
• 3. mean difference of 10 has a probability
of 8%
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P-values
• The p value measures how likely a particular
difference between groups is to be due to chance.
• A p-value of 0.1 means that, extrapolating from the
results of our study, there is a 10% chance that in
‘Truth,’ there is no effect.
• A p-value of 0.05 means that there is a 5% chance
the in ‘Truth’ there is no effect.
• Usually, we studies define p-values of <0.05 as being
statistically significant. There is nothing magic
about this cut-off. What the authors are saying
when they choose this cut-off is that they are happy
to report that there is an effect, providing the
chance that they are wrong (ie in ‘Truth’ there is no64
effect) is less than 5%.
Hypothesis testing Using
CI
•
•
•
•
•
•
Mean difference =10
SE=5.4
CI=2*5.4
10±10.8= -0.2 to 20.8
Accept the null hypothesis
P>0.05
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Confidence Interval
• The range of the actual difference
between the two drugs
• Mean difference in HBA1c reduction
1 (0.1-1.9)
• Absolute risk reduction 30% (3-70%)
• NNT 3 (1-10000)
• RRR 40% (20-120)
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Interpretation of the
confidence interval
• The upper and lower limits of the confidence
interval can be used to assess whether the
difference between the two mean values is
clinically important.
• For example, if the lower limit is close to zero,
this indicates that the true difference may be
very small and clinically meaningless, even if the
test is statistically significant.
67
SPSS Output for Two Sample
Independent t-test Example
Group Statistics
Pulse rate
Group
1
2
N
10
10
Mean
95.00
105.00
Std. Deviation
12.428
11.776
Std. Error
Mean
3.930
3.724
Independent Samples Test
Levene's Test for
Equality of Variances
t-tes t for Equality of
Means
Pulse rate
Equal variances
Equal variances
ass umed
not as sumed
.134
.719
-1.847
-1.847
18
17.948
.081
.081
F
Sig.
t
df
Sig. (2-tailed)
Mean Difference
Std. Error Difference
95% Confidence Interval
of the Difference
Lower
Upper
-10.000
-10.000
5.414
5.414
-21.374
1.374
-21.377
1.377
68
• Confidence intervals show a statistically
significant result if they “do not cross the
line of no effect.” What does this mean?
Basically, you can tell whether a result is
statistically significant by seeing whether
the confidence intervals include zero (for
comparative absolute measures of risk,
such as absolute risk reduction) or one
(for relative measures of risk, including
relative risks and odds ratios).
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Statistical Significance VS
Meaningful Significance
• Common mistake is to confuse statistical
significance with substantive
meaningfulness
• Statistically significant result simply
means that if Ho were true, the observed
results would be very unusual
• With N > 100, even tiny
relationships/differences are statistically
significant
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Statistical Significance VS
Meaningful Significance
• Statistically significant results say
nothing about clinical importance or
meaningful significance of results
• Researcher must always determine if
statistically significant results are
substantively meaningful.
• Refrain from statistical
“sanctification” of data
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Confidence Interval (0.3 – 4) •
Lower limit must be > 2 to be Clinical
significant
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HOW THE RESULTS ARE
EXPRESSED
Understanding Results
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Mean Difference
• Drug A reduced HBA1c by 1%
• Drug B reduced HBA1c by 2%
• Mean difference =
80
• The benefits or harms of a treatment can be
shown in various ways:
• Drug X produced an absolute reduction in deaths
by 7 per cent ("absolute risk reduction")
•
Drag X reduced the death rate by 28 per cent
("relative risk reduction")
• Drug X increased the patients' survival rate from
75 to 82 percent
• 14 people would need to be treated with drug X to
prevent one death ("number needed to treat")
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8 week
Treatment
New Drug
death
No death
1
a
c
Old drug
2
999999999
1000000000
b
d
999999998
1000000000
200000
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GERD Treatment: 2 x 2
Table
8 week
Treatment
Proton Pump
Inhibitor
Esophageal
bleeding
No bleeding
45
55
100
75
25
100
120
80
a
c
H2
Antagonist
b
d
200
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Relative Risk (RR)
• The relative risk of an outcome is the chances of
that outcome occurring in the treatment group
compared with the chances of it occurring in the
control group.
• If the chances are the same in both groups, the
relative risk is 1.
• The relative risk reduction (RRR) is the amount by
which the risk (of death) is reduced by drug X as
a comparative percentage of the control,
calculated as:
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Risk: The incidence of an event. •
EER: Experimental event rate. –
CER: Control event rate. –
EER
RR =
Relative Risk
CER(RR) •
Used: Randomized trial, Cohort study •
Not used: Case-control study •
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Relative Risk (Risk Ratio)
GERD Ttreatment:
Green et al. Br. J. Clin. Res. 6: 63-76, 1995.
RR =
RR =
a/(a+b)
c/(c+d)
45 / 99
73 / 97
RR = 0.60
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Relative Rate Reduction
(RRR)
Used to measure treatment effects •
CER – EER
RRR =
.100
•
CER
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Relative Rate Reduction
GERD Ttreatment:
Green et al. Br. J. Clin. Res. 6: 63-76, 1995.
EER = 45/99 = 0.45
CER = 73/97 = 0.75
RRR = CER - EER
CER
RRR = 0.75 – 0.45 . 100
0.75
RRR = 40%
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Absolute Rate Reduction
(ARR)
ARR is the reduction in the rate of •
an event (sometimes “undesirable”)
that comes about from applying a
treatment.
Absolute Rate Reduction (re: •
undesirable)
ARR = CER - EER
Absolute Rate Reduction (re: •
desirable) 89
ARR = EER - CER
Absolute Rate Reduction
GERD Ttreatment:
Green et al. Br. J. Clin. Res. 6: 63-76, 1995.
ARR = CER - EER
ARR = 73/97 – 45/99
ARR = 0.75 – 0.45
ARR = 0.30
30 per 100
90
Number Needed to Treat
(NNT)
NNT Interpret as the number of •
patients that must receive treatment
to have one success.
More effective = lower NNT •
1
NNT =
1
=
ARR
(EER-CER)
91
Number Needed to Treat
GERD Ttreatment:
Green et al. Br. J. Clin. Res. 6: 63-76, 1995.
1
NNT =
ARR
1
NNT =
0.30
NNT = 3.33
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95
96
97
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