Download Lecture 08. Mean and average quadratic deviation

Mean and average
quadratic deviation
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




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


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




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
SIMULATION
We take a simple random sample with replacement
of 25 cards from the box as follows. Mix the box of
cards; choose one at random; record it; replace it;
and then repeat the procedure until we have
recorded the numbers on 25 cards. Although
survey samples are not generally drawn with
replacement, our simulation simplifies the analysis
because the box remains unchanged between
draws; so, after examining each card, the chance
of drawing a card numbered 1 on the following
draw is the same as it was for the previous draw, in
this case a 60% chance.
SIMULATION
Let’s say that after drawing the 25 cards this way,
we obtain the following results, recorded in 5
rows of 5 numbers:
SIMULATION
Based on this sample of 25 draws, we want to guess
the percentage of 1’s in the box. There are 14
cards numbered 1 in the sample. This gives us a
sample percentage of p=14/25=.56=56%. If this is
all of the information we have about the population
box, and we want to estimate the percentage of 1’s
in the box, our best guess would be 56%. Notice
that this sample value p = 56% is 4 percentage
points below the true population value π = 60%.
We say that the random sampling error (or simply
random error) is -4%.
ERROR ANALYSIS
An experiment is a procedure which
results in a measurement or
observation. The Harris poll is an
experiment which resulted in the
measurement (statistic) of 57%. An
experiment whose outcome depends
upon chance is called a random
experiment.
ERROR ANALYSIS
On repetition of such an experiment one
will typically obtain a different
measurement or observation. So, if the
Harris poll were to be repeated, the
new statistic would very likely differ
slightly from 57%. Each repetition is
called an execution or trial of the
experiment.
ERROR ANALYSIS
Suppose we made three more series of draws,
and the results were + 16%, + 0%, and +
12%. The random sampling errors of the four
simulations would then average out to:
ERROR ANALYSIS

Note that the cancellation of the positive and
negative random errors results in a small average.
Actually with more trials, the average of the
random sampling errors tends to zero.
ERROR ANALYSIS
So in order to measure a “typical size” of a random
sampling error, we have to ignore the signs. We
could just take the mean of the absolute values
(MA) of the random sampling errors. For the four
random sampling errors above, the MA turns out to
be
ERROR ANALYSIS
The MA is difficult to deal with theoretically because
the absolute value function is not differentiable at
0. So in statistics, and error analysis in general, the
root mean square (RMS) of the random sampling
errors is generally used. For the four random
sampling errors above, the RMS is
ERROR ANALYSIS
The RMS is a more conservative
measure of the typical size of the
random sampling errors in the
sense that MA ≤ RMS.
ERROR ANALYSIS
For a given experiment the RMS of all possible
random sampling errors is called the standard
error (SE). For example, whenever we use a
random sample of size n and its percentages p to
estimate the population percentage π, we have
Dynamic analysis
 Health
of people and activity of
medical establishments change in
time.
 Studying of dynamics of the
phenomena is very important for the
analysis of a state of health and
activity of system of public health
services.
Example of a dynamic line
Year
1994
1995
1996
1997
1998
1999
2000
Bed occupancy (days)
340.1
340.9
338.0
343.0
341.2
339.1
344.2
Parameters applied for analysis
of changes of a phenomenon
of growth –relation of all
numbers of dynamic lines to
the previous level accepted for
100 %.
 Rate
Parameters applied for
analysis of changes of a
phenomenon
gain – difference
between next and previous
numbers of dynamic lines.
 Pure
Parameters applied for
analysis of changes of a
phenomenon
of gain – relation of the
pure gain to previous
number.
 Rate
Parameters applied for
analysis of changes of a
phenomenon
of visualization —
relation of all numbers of
dynamic lines to the first level,
which one starts to 100%.
 Parameter
Measures of Association
Measures of Association




Absolute risk
- The relative risk and odds ratio provide a measure of risk
compared with a standard.
Attributable risk or Risk difference is a measure of absolute
risk. It represents the excess risk of disease in those exposed
taking into account the background rate of disease. The
attributable risk is defined as the difference between the
incidence rates in the exposed and non-exposed groups.
Population Attributable Risk is used to describe the excess
rate of disease in the total study population of exposed and
non-exposed individuals that is attributable to the exposure.
Number needed to treat (NNT)
- The number of patients who would need to be treated to
prevent one adverse outcome is often used to present the
results of randomized trials.
Relative Values
As a result of statistical research during
processing of the statistical data of
disease, mortality rate, lethality, etc.
absolute numbers are received, which
specify the number of the phenomena.
Though absolute numbers have a
certain cognitive values, but their use is
limited.
Relative Values
In order to acquire a level of the phenomenon,
for comparison of a parameter in dynamics or
with a parameter of other territory it is
necessary to calculate relative values
(parameters, factors) which represent result
of a ratio of statistical numbers between itself.
The basic arithmetic action at subtraction of
relative values is division.
In medical statistics themselves the
following kinds of relative parameters
are used:





Extensive;
Intensive;
Relative intensity;
Visualization;
Correlation.
The extensive parameter, or a
parameter of distribution,
characterizes a parts of the
phenomena (structure), that is it
shows, what part from the general
number of all diseases (died) is
made with this or that disease
which enters into total.
Using this parameter, it is possible to
determine the structure of patients
according to age, social status, etc. It is
accepted to express this parameter in
percentage, but it can be calculated and in
parts per thousand case when the part of
the given disease is small and at the
calculation in percentage it is expressed as
decimal fraction, instead of an integer.
The general formula of its calculation is the
following:
part × 100
total




The intensive parameter characterizes frequency
or distribution.
It shows how frequently the given phenomenon
occurs in the given environment.
For example, how frequently there is this or that
disease among the population or how frequently
people are dying from this or that disease.
To calculate the intensive parameter, it is
necessary to know the population or the
contingent.

General formula of the calculation is the
following:
phenomenon×100 (1000; 10 000; 100 000)
environment
General mortality rate
number of died during the year × 1000
number of the population
SIMULATION
Let’s say that after drawing the 25 cards this way,
we obtain the following results, recorded in 5
rows of 5 numbers: