Transcript Lecture 13. Measuring of correlation

Measuring of Correlation
Definition
Correlation is a measure of
mutual
correspondence
between two variables and is
denoted by the coefficient of
correlation.
Applications and
characteristics
a) The simple correlation coefficient, also
called the Pearson's product-moment
correlation coefficient, is used to indicate
the extent that two variables change with
one another in a linear fashion.
Applications and
characteristics
b) The correlation coefficient can
range from - 1 to + 1 and is unites
(Fig. A, B, C).
Applications and
characteristics
c)
When the correlation coefficient
approaches - 1, a change in one variable is
more highly, or strongly, associated with an
inverse linear change (i.e., a change in the
opposite direction) in the other variable.
Applications and
characteristics
d) When the correlation coefficient
equals zero, there is no
association between the changes
of the two variables.
Applications and
characteristics
(e) When the correlation coefficient
approaches +1, a change in one
variable is more highly, or
strongly, associated with a direct
linear change in the other
variable.
Applications and
characteristics
A correlation coefficient can be calculated
validly only when both variables are
subject to random sampling and each
is chosen independently.
Correlation coefficient
Correlation coefficient
Correlation coefficient
Types of correlation
There are the following types of
correlation (relation) between the
phenomena and signs in nature:
 а) the reason-result connection is the
connection between factors and
phenomena, between factor and result
signs.
 б) the dependence of parallel changes of a
few signs on some third size.
Quantitative types of connection
 functional
one is the connection, at
which the strictly defined value of the
second sign answers to any value of
one of the signs (for example, the
certain area of the circle answers to
the radius of the circle)
Quantitative types of connection

correlation - connection at which a few values of
one sign answer to the value of every average
size of another sign associated with the first one
(for example, it is known that the height and
mass of man’s body are linked between each
other; in the group of persons with identical
height there are different valuations of mass of
body, however, these valuations of body mass
varies in certain sizes – round their average
size).
Correlative connection


Correlative connection foresees the dependence
between the phenomena, which do not have
clear functional character.
Correlative connection is showed up only in the
mass of supervisions that is in totality. The
establishment of correlative connection foresees
the exposure of the causal connection, which will
confirm the dependence of one phenomenon on
the other one.
Correlative connection


Correlative connection by the direction (the
character) of connection can be direct and
reverse. The coefficient of correlation, that
characterizes the direct communication, is
marked by the sign plus (+), and the coefficient
of correlation, that characterizes the reverse one,
is marked by the sign minus (-).
By the force the correlative connection can be
strong, middle, weak, it can be full and it can be
absent.
Estimation of correlation by
coefficient of correlation
Force of connection
Complete
Line (+)
Reverse (-)
+1
Strong
From +1 to +0,7
Average
from +0,7 to +0,3 from –0,7 to –0,3
Weak
No connection
From -1 to -0,7
from +0,3 to 0
from –0,3 to 0
0
0
Types of correlative
connection
By direction
direct (+) – with the increasing of one sign
increases the middle value of another one;
 reverse (-) – with the increasing of one sign
decreases the middle value of another one;

Types of correlative
connection
By character
 rectilinear - relatively even changes of
middle values of one sign are
accompanied by the equal changes of the
other (arterial pressure minimal and
maximal)
 curvilinear – at the even change of one
sing there can be the increasing or
decreasing middle values of the other sign.
Average Values

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



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Mean:  the average of the data
 sensitive to outlying data
Median:  the middle of the data
 not sensitive to outlying data
Mode:  most commonly occurring value
Range:  the difference between the largest observation and
the smallest
Interquartile range:  the spread of the data
 commonly used for skewed data
Standard deviation:  a single number which measures how much
the observations vary around the mean
Symmetrical data:  data that follows normal distribution
 (mean=median=mode)
 report mean & standard deviation & n
Skewed data:  not normally distributed
 (meanmedianmode)
 report median & IQ Range
Average Values

Limit is it is the meaning of edge variant
in a variation row
lim = Vmin Vmax
Average Values

Amplitude is the difference of edge
variant of variation row
Am = Vmax - Vmin
Average Values

Average quadratic deviation
characterizes dispersion of the variants
around an ordinary value (inside
structure of totalities).
Average quadratic deviation
σ=
d
2
n 1
simple arithmetical method
Average quadratic deviation
d=V-M
genuine declination of variants from the true
middle arithmetic
Average quadratic deviation
d

σ=i
n
2
p
  dp 


 n 


method of moments
2
Average quadratic deviation
is needed for:
1. Estimations of typicalness of the middle
arithmetic (М is typical for this row, if σ is less
than 1/3 of average) value.
2. Getting the error of average value.
3. Determination of average norm of the
phenomenon, which is studied (М±1σ), sub
norm (М±2σ) and edge deviations (М±3σ).
4. For construction of sigmal net at the
estimation of physical development of an
individual.
Average quadratic deviation
This dispersion a variant around of
average characterizes an average
quadratic deviation (  )
2
nd


n
 Coefficient
of variation is the
relative measure of variety; it
is a percent correlation of
standard deviation and
arithmetic average.
Terms Used To Describe The
Quality Of Measurements
Reliability is variability between subjects
divided by inter-subject variability plus
measurement error.
 Validity refers to the extent to which a test
or surrogate is measuring what we think it
is measuring.

Measures Of Diagnostic Test
Accuracy
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Sensitivity is defined as the ability of the test to identify
correctly those who have the disease.
Specificity is defined as the ability of the test to identify
correctly those who do not have the disease.
Predictive values are important for assessing how
useful a test will be in the clinical setting at the individual
patient level. The positive predictive value is the
probability of disease in a patient with a positive test.
Conversely, the negative predictive value is the
probability that the patient does not have disease if he
has a negative test result.
Likelihood ratio indicates how much a given diagnostic
test result will raise or lower the odds of having a disease
relative to the prior probability of disease.
Measures Of Diagnostic Test
Accuracy
Expressions Used When
Making Inferences About Data

Confidence Intervals
- The results of any study sample are an estimate of the true value
in the entire population. The true value may actually be greater or
less than what is observed.



Type I error (alpha) is the probability of incorrectly
concluding there is a statistically significant difference in
the population when none exists.
Type II error (beta) is the probability of incorrectly
concluding that there is no statistically significant
difference in a population when one exists.
Power is a measure of the ability of a study to detect a
true difference.
Multivariable Regression
Methods


Multiple linear regression is used when the
outcome data is a continuous variable such as
weight. For example, one could estimate the
effect of a diet on weight after adjusting for the
effect of confounders such as smoking status.
Logistic regression is used when the outcome
data is binary such as cure or no cure. Logistic
regression can be used to estimate the effect of
an exposure on a binary outcome after adjusting
for confounders.
Survival Analysis
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
Kaplan-Meier analysis measures the ratio of
surviving subjects (or those without an event)
divided by the total number of subjects at risk for
the event. Every time a subject has an event, the
ratio is recalculated. These ratios are then used
to generate a curve to graphically depict the
probability of survival.
Cox proportional hazards analysis is similar to
the logistic regression method described above
with the added advantage that it accounts for
time to a binary event in the outcome variable.
Thus, one can account for variation in follow-up
time among subjects.
Kaplan-Meier Survival Curves
Why Use Statistics?
Cardiovascular Mortality in Males
1.2
1
0.8
SMR 0.6
0.4
0.2
0
'35-'44 '45-'54 '55-'64 '65-'74 '75-'84
Bangor
Roseto
Descriptive Statistics
Identifies patterns in the data
 Identifies outliers
 Guides choice of statistical test

Percentage of Specimens Testing
Positive for RSV (respiratory syncytial virus)
Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr May Jun
South 2
2
5
7
20
30
15
20
15
8
4
3
North- 2
east
West 2
3
5
3
12
28
22
28
22
20
10
9
2
3
3
5
8
25
27
25
22
15
12
2
2
3
2
4
12
12
12
10
19
15
8
Midwest
Descriptive Statistics
Percentage of Specimens Testing Postive for
RSV 1998-99
35
30
25
20
15
10
5
0
South
Northeast
West
Midwest
Jul
Sep
Nov
Jan
Mar
May
Jul
Distribution of Course Grades
14
12
10
Number of 8
Students 6
4
2
0
A
A- B+ B
B- C+ C
Grade
C- D+ D
D-
F
Describing the Data
with Numbers
Measures of Dispersion
•
•
•
RANGE
STANDARD DEVIATION
SKEWNESS
Measures of Dispersion
• RANGE
highest to lowest values
STANDARD DEVIATION
• how closely do values cluster around the
mean value
SKEWNESS
• refers to symmetry of curve
•
•
•
Measures of Dispersion
• RANGE
highest to lowest values
STANDARD DEVIATION
• how closely do values cluster around the
mean value
SKEWNESS
• refers to symmetry of curve
•
•
•
Measures of Dispersion
•
•
•
RANGE
• highest to lowest values
STANDARD DEVIATION
• how closely do values cluster around the
mean value
SKEWNESS
• refers to symmetry of curve
The Normal Distribution
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Mean = median =
mode
Skew is zero
68% of values fall
between 1 SD
95% of values fall
between 2 SDs
Mean, Median, Mode
.
1

2