Transcript No Slide Title

Statistics (Biostatistics, Biometrics) is
‘the science of learning from sample data’
to make inference about
populations that were samples
Incorrect definition of statistics?
Data (singular: datum): are numerical values
of a variable on an observation that
constitute the building blocks of statistics
Variable: A characteristic or an attribute that
takes on different values.
It can be:
Response (dependent) or Explanatory
(independent) variable
It can be:
1
Quantitative or Qualitative variable
Recognition of the kinds of variables is crucial in
choosing appropriate statistical analysis
Examples of Response and Explanatory variables
(when appropriate)?
Examples of quantitative and qualitative variables?
2
Quantitative variables:
Convey amount
1. Discrete (counts--frequencies): number of
patients, species
- there are gaps in the values (zero or positive
integers)
2. Continuous (measurements):
a. ratio (with a natural zero origin: mass,
height, age, income, etc.)
b. interval (no natural zero origin: calendar
dates, ºF or ºC, etc.)
- no gaps in the values
- measuring precision is limited by the
precision of the measuring device
which results in recording it as discrete
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Qualitative (categorical) variables:
Convey attributes, cannot be measured in the
usual sense but can be categorized
1. Nominal: mutually exclusive and
collectively exhaustive (gender, race,
religion, etc.)
- can be assigned numbers but cannot be
ordered
2. Ordinal: categorical differences exist,
which can be:
- numbered and ordered but the distances
between values not equal (cold, cool,
warm, hot; low, medium, high;
sick, normal, healthy; depressed, normal,
happy; etc.)
Note: both nominal and ordinal variables are
discrete
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Source of Data and Type of Research:
1. Observational studies
Examples?
2. Experimental studies
Examples?
Field
Mesocosm
Green house
Laboratory
Recognition of the kind of the study, and the
way the study units are selected and treated
are crucial in:
- making statistical inference
- establishing correlational or causal
relationship
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Data can be used to perform:
1. Descriptive Statistics
2. Statistical Modeling
3. Inferential Statistics--Hypothesis Testing
6
Inferential Statistics
• Inference from one observation to a
population
e.g.1, comparing body temperature of one
bird with the mean body temp. of a bird
species
- test statistics?
a) population parameters are known
b) population parameters not known
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Inferential Statistics
• Inference from several observations to a
population
e.g.2, comparing body temperature of three
birds with the mean body temp. of a bird
species
- test statistics?
a) population parameters are known
b) population parameters not known
8
Inferential Statistics
• Comparing two or more sets of observations
to test whether or not they belong to different
populations
e.g., 3, comparing starting salaries of females
and males in several organizations to see if
their starting salaries differ
- do we know the populations’ parameters?
- can they be estimated?
- what would be the test statistics?
- what would be the scope of inference?
- why do we need Inferential Statistics to do
so?
9
What do we mean by ‘inference’?
An inference is a conclusion that patterns in
the data are present in some broader context
A statistical inference is the one justified by
a probability model linking the data to the
broader context
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• Inference from a sample to its parent
population, or from samples to compare their
parent populations, can be drawn from
observational studies
- such inference is optimally valid if the
sampling is random
- results from observational studies cannot
be used establish causal relationships,
but are still valuable in suggesting
hypothesis and the direction of controlled
experiments
• Inference to draw causal relationships can be
drawn from randomized, controlled
experiments, and not from observational
studies
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Statistical inference permitted by study designs
(Adapted from Ramsey and Schafer, 2002)
Inferences to
populations
Can be drawn
Causal
inferences
can be drawn
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Population (probability) Distribution:
• Discrete
1.Binomial (Bernoulli)
2.Multinomial: a generalized form of
binomial where there are > 2 outcomes
3.Uniform: similar to Multinomial except
that the probability of occurrence is
equal
4.Hypergeometric: similar to binomial
except that the probability of occurrence
is not constant from trial to trial sampling
without replacement—dependent trials)
5.Poisson: similar to binomial but p is very
small and the # of trials (s) is very large
such that s.p approaches a constant
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Population (probability) Distribution:
• Discrete (previous slide)
• Continuous
1. Uniform: similar to the discrete uniform
but the # of possible outcomes is infinite
2. Normal (Gaussian):
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Population (probability) Distribution:
• Discrete
1.Binomial (Bernoulli):
a. in a series of trials a variable (x) takes on
only 2 discrete outcomes (probability of
occurrence --p)
b. trials are independent (sampling with
replacement) and constant p from trial to
trial (e.g., probability distribution
of having 1, 2, or 3 girls in 10 families
each with 3 kids)
c. probability of occurrence can be equal or
unequal
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1. Discrete Binomial Distribution
Probability (p) distribution for the number of smokers in a
group of 5 people ( n = 5, p of smoking = 0.2, 0.5, or 0.8)
• Tabular presentation
# of smoker(s)
p = 0.2
p =0 .5
p = 0.8
0
0.328
0.031
0.000
1
0.410
0.156
0.006
2
0.205
0.312
0.051
3
0.051
0.312
0.205
4
0.006
0.156
0.410
5
0.000
0.031
0.328
• Graphical presentation
p = 0.2
p = 0.5
0 1 2 3 4 5 0 12 3 4 5
p = 0.8
0 1 2 3 4 5
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# of smokers
3. Discrete Uniform Distribution
Probability (p) distribution for one toss of a die
•Tabular presentation
toss
p
1
1/6
2
1/6
3
1/6
4
1/6
5
1/6
6
1/6
•Graphical presentation
p
1/6
1
2
3
4
5
6
toss
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• Continuous
1. Uniform
2. Normal (Gaussian): A bell-shaped symmetric
distribution
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The Normal distribution is central to the theory and
practice of parametric inferential statistics because
a.Distributions of many biological and environmental
variables are approximately normal
b.When the sample size (# of independent trials) is
large, or p and q are similar, other distributions such
as Binomial and Poisson distributions can be
approximated by the Normal Dist.
c.The distribution of the means of samples taken from
a population
i. is normal when samples are taken from a normal
population
ii. approaches normality as the size of samples (n)
taken from non-normal populations increases
(Central Limit Theorem--CLT)
- implication of CLT?
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• The normal distribution is a mathematical function
(may you observe it in real life??) defined by the
following equation:
Y = 1 / (  2) e - (Xi - m)
2 / 22
, where:
X
Y : height of the curve for a given Xi
e : 2.718
 : 3.142
m : arithmetic mean, measure of central tendency
 : measure of the dispersion (variability) of the
observations around the mean
The last two characteristics are the two unkown
parameters that shape the distribution
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m: the mean, a measure of central tendency
- calculated as the arithmetic average
- is a parameter
- is constant
- does not indicate the variability within a population,
-
when not known, estimated by y (a statistic) from
unbiased sample(s)
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2: Variance, a measure of variability (dispersion)
calculated as: ( y i - m ) 2 / N (definition formula)
or [ y i2 – ( yi)2/N ] / N (calculation formula)
- its unit is the squared unit of the variable of
interest
- also called mean square of error (why?)
- estimated best by S 2 (a statistic) from unbiased
sample(s) calculated as: ( y i - y ) 2 / (n-1)
or [ y i2 - ( yi)2/n ] / (n-1)
-- what do we call [ y i2 - ( yi)2/ n ]?
-- is it a good measure of variability?
-- what do we call ‘n-1’?
So, S 2 = SS / df
(reason to call it ………………….. – MSe)
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- to get a measure of variability in the unit of the
variable of interest , we take the square root of the
variance and call it Standard Deviation
-- is denoted by  (a parameter) for a population,
estimated best by S or SD (a statistic) from
unbiased sample(s)
-- it is a rough (not exact) measure of average
absolute deviations from the mean
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• Important properties of a normal distribution:
Because the normal distribution is symmetric
characterized by a mean m and a standard deviation
of  the followings are true:
a. the total area under the normal curve is 1 or 100%
b. half of the population (50%) is greater and half
(50%) is smaller than m
c. 68.27% of the observations are within m  1
d. 95.44% of the observations are within m  2
e. 99.74% of the observations are within m  3
- what do above statements mean in terms of
probability (e.g., probabilistic status of one
observation falling on the mean, on any boundaries,
or anywhere within or outside a boundary)?
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Standard Normal Distribution
Is the distribution of the Z values where:
Z = (Xi - m) / 
How many Normal Distributions you may
find?
How many Standard Normal Distributions you
may find?
What are the properties of the Standard
Normal Distribution?
A Z table in a stat book shows the proportion
of the population beyond the calculated Z
value
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To study populations, it is usually not
feasible to measure the entire population
of N members.
Why not sampling the entire population?
Real (finite and infinite) and Imaginary
populations?
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Therefore, we draw sample(s) to represent
the parent populations
Sample is a subset of a population, drawn
and analyzed to infer conclusions regarding
the population. Its size is usually denoted by
n.
Sampling can be done:
1. With replacement
2. Without replacement (the norm in
practice)
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• To infer valid (unbiased) conclusions
regarding a population from a sample, the
sample must represent the entire
population.
• For a sample to represent the entire
population, it is best to be drawn
RANDOMLY.
• A sample is random when each and every
member of the population has an equal and
independent chance of being sampled
(exceptions?).
• Random sampling, on average:
- represents the parent population
- prevents known and unknown biases to
affect the selection of observations, and
thus,
- allows the application of the laws of
probability in drawing a statistical
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inferences
Descriptive Statistics
• Data organization, summarization and presentation
1. Tabular (tally, simple, relative, relative-cumulative,
and cumulative frequencies)
2. Graphical (histograms, polygons)
Suppose we have the following random sample of
creativity scores:
Case
Score
Case
Score
1
2
3
4
5
6
7
8
9
10
11
12
26.7
12.0
24.0
13.6
16.6
24.3
17.5
18.2
19.1
19.3
23.1
20.3
13
14
15
16
17
18
19
20
21
22
23
24
29.7
20.5
12.0
20.6
17.2
21.3
12.9
21.6
19.8
22.1
22.6
19.8
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Steps in data organization, summarization and
presentation (Geng and Hills, 1985)
1. Determine the range (R):
largest - smallest , R = 29.7 - 12 = 17.7
2. Determine the number of classes (k) into which data
are to be grouped
a. 8 - 20 classes is often recommended
- too few--information loss
- too many--too expensive (time, etc.)
b. can be calculated based on Sturges’ rule as:
k = 1 + 3.3 log n, where n = number of cases, in
our case 1 + (3.3)(log 24) = 5.55, and thus k
should be at least 6, and this is what we use
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Steps in data organization, summarization and
presentation (Geng and Hills, 1985)
3. Select a class interval (difference between upper and
lower class boundaries--R/k may be used, we use 3)
4. Select the lower boundary of the lowest class and
add the interval successively to it until all data are
classified
- to avoid falling of data on the boundaries, they are
usually expressed to half unit greater than the
measurement accuracy
- in our case measurement accuracy is 0.1 and so
the class boundaries would for example be
expressed as 11.05-14.05.
5. Arrange the table as follows:
31
6
5
4
3
2
1
Class
32
IIII
III
IIIIIIII
IIIIIII
I
I
13.05
16.05
19.05
22.05
25.05
28.05
11.55-14.55
14.55-17.55
17.55-20.55
20.55-23.55
23.55-26.55
26.55-29.55
1
1
7
8
3
4
1.000
24
23
0.958
0.042
0.042
22
15
7
4
0.917
0.625
0.292
0.167
Cum.
freq.
0.292
0.333
0.125
0.167
Tallied Simple Relative Rel. Cum.
freq.
freq.
freq.
freq.
Midpoint
Boundaries
Histogram of Class Scores
Frequency
8
6
4
2
1
2
3
4
5
6
Score Classes
33
Polygon of Class Scores
Frequency
8
6
4
2
1
2
3
4
5
6
Score Classes
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0.8
0.6
0.4
0.2
1
2
3
4
5
Relative Frequency
Polygon of Class Scores
6
Score Classes
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0.8
0.6
0.4
0.2
1
2
3
4
5
Relative Frequency
Histogram of Class Scores
6
Score Classes
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0.8
0.6
0.4
0.2
1
2
3
4
5
Rel. Cum. Frequency
Polygon of Class Scores
6
Score Classes
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Cumulative Frequency
Polygon of Class Scores
24
18
12
6
1
2
3
4
5
6
Score Classes
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Stem-and-Leaf Diagram
A cross between a table and a graph
12
9
13
14
15
26
17
18
19
8
20
6
21
22
6
23
24
25
26
27
28
00
6
6
25
2
13
35
36
12
1
03
7
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Steps in Creating Stem-and-Leaf Diagram
1. Arrange the data in increasing order
2. First, write the whole numbers as the stem
3. Then, write the numbers after decimals, in increasing
order, as leaves
Advantages:
1. Ease of construction
2. Depiction of individual numbers, min., max., range,
median, and mode
3. Depiction of center, spread, and shape of distribution
Disadvantages:
1. Difficulty in comparing distributions when they have
a very different ranges
2. Difficulty in comprehension and construction when the
sample size is very large
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Measures of a Dataset that are Important in Descriptive
/ Inferential Statistics
1. Measure of Central Location (Tendency)
1.1. Mode: the value with highest frequency
- there may be no mode, one mode, or several
modes
- not influenced by extremes
- cannot be involved in algebraic manipulation
- not very informative
1.2. Median: the middle value when data are arranged in
order of magnitude
- not influenced by extremes (i.e., useful in
economics when extremes should be disregarded
- not involved in algebraic manipulation
- if n is odd, is the middle value when data are
ordered
- if n is even, is the average of the two middle values
when data are ordered
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1.3. Mean: arithmetic average, denoted by:
a. m = ( Xi / N): a parameter, for a population,
which is best estimated by:
b. x : a statistic, from a sample or samples
- most frequently used in statistics and subject of
algebraic manipulation
- is the best estimate of m if the sample is unbiased
(representative of its population); the sample is
unbiased if it is drawn randomly, not otherwise
- the mode, the median, and the mean are the same
when the distribution is perfectly symmetrical
- the units for the mode, median, and the mean are the
same as the unit of the variable of interest
- the mean does not indicate the variability of a dataset;
e.g., consider the following three sets of data with a
common mean:
22, 24, 26
20, 24, 28
16, 24, 32
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2.
Measure of variability--dispersion
2.1. Range: the largest value - the smallest value (R)
- not very informative, often affected by extremes
2.2. Variance (mean square*), represented by two
symbols [ 2, sigma square(d), and S 2]:
a.  2: represents the variance of a population; it is a
parameter, constant for a given population,
quantified as the sum of the squared deviations
of individual members from their mean divided
by the population size (finite):
 2 = [(X1 - m)2 + (X2 - m)2 + … + (XN - m)2] / N
=  (Xi - m)2 / N
 2 is estimated best by:
43
2.2.b. S 2: the variance of a sample; it is a statistic,
which varies from sample to sample taken from a
given population, and is calculated as the
sum of the squared differences between the
sampled individuals and their mean divided by
the sample size minus one:
S 2 =  ( x i - x ) 2 / n-1,
which is the definition
formula and can be reduced to a
computation formula as:
S 2 = [ x i2 - ( x i ) 2 / n ] / n-1
- easier to calculate
- scientific calculators provide the components
- more accurate because no rounding of numbers
is involved
44
Notes:
- S 2 is the best estimate of  2 if the sample is
unbiased (representative of its population); the
sample is unbiased if it is drawn randomly, not
otherwise
- the unit of the variance is the square of the unit of the
variable of interest
- the quantity “ x i2 - ( x i ) 2 / n” is called sum of
squares or SS, it is a minimum
- the quantity “( x i ) 2 / n” is called the correction
factor or C
- the quantity “n-1” is called degrees of freedom
or df
- Thus, Variance = sum of squares / degrees of
freedom, or S 2 = SS / df
* this is why the variance also is called “mean square”
45
2.3. Standard Deviation: square root of the Variance
a.  : a parameter, for a population:
b. S or SD: a statistic, for a sample
- unit of standard deviation is the same as that of the
variable of interest
2.4. Coefficient of Variation (CV ): is a relative term (%)
- calculated as (SD / x ) × 100
- used to compare the results of several studies done
differently (different experimenters, procedures,
etc.) on the same variable
46
2.5. Skewness: measure of deviations from symmetry
a. symmetrical
(skewness = 0)
b. Skewed to the left
(skewness > 0)
c. skewed to the right
(skewness < 0)
2.6. Kurtosis : measure of peakedness or tailedness
a. mesokurtic
(kurtosis = 0)
b. leptokurtic
(kurtosis > 0)
c. platykurtic
(kurtosis < 0)
47
Sampling Means Distribution
Is the distribution of the means of all
possible samples of size n taken from a
population.
• If sampled from a normal distribution will
be normal
• If sampled from a non-normal dist.
becomes more normal than the parent dist.
(Central Limit Theorem- CLT*)
- as sample size is increased, the sampling
mean distribution approaches normality
*Fuzzy CLT:
Data influenced by many small, unrelated,
random effects are approximately normally
distributed.
48
• The sampling mean distribution has a:
- mean, equal to m (mean of population)
- variance (measuring average squared
deviation of the sampled means from m),
-- calculated as variance of the means, or
2/n (based on LLN)
-- estimated best by S2 /n
- - denoted
as
σ2
y
or
s2 ,
y
respectively
-- square root of the above variance
is called …………………………
or …………..……………..
or …………..……………..
49
• Standard Error:
Is a rough measure of the average absolute
deviation of sampling means from m (typical
error made when estimating m from the
mean of a sample of size n)
can be calculated as:  / n
best estimated by S / n
denoted as σ2y or s2y or SE
50
Adapted from Ramsey and Schafer, 2002
51
Adapted from Ramsey and Schafer, 2002
52