Biostatistics and Computer Applications

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Biostatistics and Computer Applications
Dafeng Hui Introduction Descriptive
Statistics SAS
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Introduction

• Introduction to the basic concepts of statistics
  as applied to problems in biological science.
• Goal of the course
• Understand statistical concepts (population,
  sample, t-test, slope, significant etc.)
• Identify appropriate methods for your data (e.g.,
  paired t-test or independent t-test, one-way
  block or two-way ANOVA)
• Select correct SAS procedures to analyze data
  (you may use different SAS procedure for one
  purpose, which more is more suitable)
• Scientific reading and interpretation.

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BiostatisticsComputer Applications

• Why study Biostatistics?
• Statistical methods are widely used in biological
  field
• Examples are from biological field, practical and
  useful
• Focus on application instead of mathematical
  derivation
• Help to evaluate the paper in an intelligent
  manner.
• Statistics - the science and art of obtaining
  reliable results and conclusions from data that
  is subject to variation.
• Biostatistics (Biometry)- the application of
  statistics to the biologic sciences.

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BiostatisticsComputer Applications

• Why Computer Applications?
• Statistical methods are mostly difficult and
  complicated (ANOVA, regression etc)
• Advances in computer technology and statistical
  software development make the application of
  statistical method much easier today than before
  
• Software such as SAS needs time to learn.

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Is Biostatistics hard to study?

• Factors make it hard for some students to learn
  statistics
• The terminology is deceptive. To understand
  statistics, you have to understand the
  statistical meaning of terms such as significant,
  error and hypothesis are distinct from ordinary
  uses of these words.
• Statistics requires mastering abstract concepts.
  It is not easy to think about theoretical
  concepts such as populations, probability
  distributions, and null hypotheses.

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Is Biostatistics hard? (cont)

• Statistics is at the interface of mathematics and
  science. To really grasp the concepts of
  statistics, you need to be able to think about it
  from both angles.
• The derivation of many statistical tests involves
  difficult math. However, you can learn to use
  statistical tests and interpret the results even
  if you do not fully understand how they work. You
  only need to know enough about how the tool works
  so that you can avoid using them in inappropriate
  situations.
• Basically, you can calculate statistical tests
  and interpret results even if you dont
  understand how the equations were derived, as
  long as you know enough to use the statistical
  tests appropriately.

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Questions about this class

• Is this class to be hard?
• No. Concept is easy and procedure is clear.
• Why do we spend time on theoretical stuff?
• Helpful to understand the application
• Do we need to know all the stuff?
• You may not need all, but be prepared

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Role of statistics in Biological Science
Statistics 1.Mathematical model /
hypothesis 2.Study design 3.Descriptive
statistics 4.Inferential statistics

• Science
• 1.Idea or Question
  
  
  
• 2.Collect data/make observations
• 3.Describe data / observations
• 4.Assess the strength of evidence for / against
  the hypothesis

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Contents of the course

• Descriptive statistics
• Graph, table, mean and standard deviation
• Inferential statistics
• Probability and distribution
• Hypothesis test
• Analysis of Variation
• Correlation and regression analysis
• Other special topic

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Basic Concept

• Data
• numerical facts, measurements, or observations
  obtained from an investigation, experiment aimed
  at answering a question
• Statistical analyses deal with numbers
• Variable
• a characteristic that can take on different
  values for different persons, places or things
• Statistical analyses need variability otherwise
  there is nothing to study
• Examples
• Concentration of a substance, pH values obtained
  from atmospheric precipitation, birth weight of
  babies whose mothers are smokers, etc.

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Basic Concept (cont.)

• Type of Variable
• Continuous variable
• Between any two values of a variable, there is
  another possible value
• Examples height, weight, concentration
• Discrete variable
• Value can be only integer
• Example number of people, plant etc.

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Basic Concept (cont.)

• Population
• Population a set or collection of objects we are
  interested in. (finite, infinite)
• Parameter a descriptive measure associated with
  a variable of an entire population, usually
  unknown because the whole population cannot be
  enumerated.
• For example,
• Plant height under warming conditions
• Graduates in US Smokers in the world.

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Basic Concept (cont.)

• Sample
• Sample a small number of subjects from a
  population to make inference about the
  population
• Random sample A sample of size n drawn from a
  population of size N in such a way that every
  possible sample of size n has the same chance of
  being selected.
• Statistic a descriptive measure associated with
  a random variable of a sample.

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Basic Concept (cont.)

• Population and Sample
• Sample?Population, Statistic?Parameter

population
Parameter predict properties of
sample
Generalize to a population
sample
statistic
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Descriptive Statistics

• Graphical Summaries
• Frequency distribution
• Histogram
• Stem and Leaf plot
• (Barplot, Boxplot)
• Numerical Summaries
• Location - mean, median, mode.
• Spread - range, variance, standard deviation
• (Shape skewness, kurtosis)

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Frequency Distribution (discrete var.)

• Example Number of sedge plant, Carex flacca,
  found in 800 sample quadrats (1m2) in an
  ecological study of grasses
• 1, 4, 1, 0, 0, 1, 0, 0, 2, 3, 1, 2, 3, 1, 0, 2,
  0, 1, 2,
• .
• 1, 2, 3, 2, 1, 1, 0, 5, 0, 0, 1, 0, 1, 0, 2, 4,
  7, 2, 1,0
• How is the plant number in a quadrat distributed?

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Frequency Distribution (discrete var.)

• Table 1. The frequency, relative frequency,
  cumulative frequencies of plant sedge in a
  quadrat.

• frequency - number of times value occurs in
  data.(probability for population).
• relative frequency - the of the time that the
  value occurs (frequency/n).
• cumulative relative frequency - the of the
  sample that is equal to or smaller than the value
  (cumulative frequency/n).

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Frequency Distribution (Conti. Var.)

• Grouping of continuous outcome
• Examples weight, height.
• Better understanding of what data show rather
  than individual values
• Example Fiber length of a cotton (n106)
• Data
• 27.5,28.6,29.4,30.5,31.4,29.8,27.6,28.7,27.6
• 31.8,32.0,27.8

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Frequency Distribution
Table 2. Frequency and relative frequency
distribution of fiber length (mm) of a cotton
variety (n106)
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Frequency Distribution (cont. var.)

• Calculate Range Rmax(X)-min(x)5.13
• Set Number of intervals g and interval range i
• Some rules exist, but generally create 8-15
  equal sized intervals, g11
• i R/(g-1)0.5
• Set intervals
• L1min(X)-i /227.0, L2L1i 27.5,
• Count number in each interval

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Frequency Distribution
Table 2. Frequency and relative frequency
distribution of fiber length (mm) of a cotton
variety (n106)
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Histogram (Bar graph) and polygon

• Histogram graph of frequencies
• Can be used to visually compare frequencies
• Easier to assess magnitude of differences rather
  than trying to judge numbers
• Frequency polygon - similar to histogram

Fig. 1. Frequency distribution of plants in a
quadrat.
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Histogram (Bar graph) and polygon
Fig. 2. Frequency distribution in fiber length of
a cotton.
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Stem-and-Leaf Displays

• Another way to assess frequencies
• Does preserve individual measure information, so
  not useful for large data sets
• Stem is first digit(s) of measurements, leaves
  are last digit of measurements
• Most useful for two digit numbers, more
  cumbersome for three digits

20 X 30 XXX 40 XXXX 50 XX 60 X
2 1 3 244 4 2468 5 26 6 4
Stem leaf
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Summary

• In practice, descriptive statistics play a major
  role
• Always the first 1-2 tables/figures in a paper
• Statistician needs to know about each variable
  before deciding how to analyze to answer research
  questions
• In any analysis, 90 of the effort goes into
  setting up the data
• Descriptive statistics are part of that 90

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Descriptive StatisticsMeasures of Location

• Descriptive measure computed from population data
  - parameter
• Descriptive measure computed from sample data -
  statistic
• Most common measures of location
• Mean
• Median
• Mode
• Geometric Mean, harmonic mean

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Arithmetic mean (population)

• Suppose we have N measurements of a particular
  variable in a population.We denote these N
  measurements as
• X1, X2, X3,,XN
• where X1 is the first measurement, X2 is the
  second, etc.
• Definition
• More accurately called the arithmetic mean, it is
  defined as the sum of measures observed divided
  by the number of observations.

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Arithmetic mean (sample)

• Sample Suppose we have n measurements of a
  particular variable in a population with N
  measurements.The n measurements are
• X1, X2, X3,,Xn
• where X1 is the first measurement, X2 is the
  second, etc.
• Definition

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Arithmetic mean (sample)

• Some Properties of the Arithmetic Mean
• 1. ,
• Prove 1.
• 2.

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Median

• Frequently used if there are extreme values in a
  distribution or if the distribution is non-normal
• Definition
• That value that divides the ordered array into
  two equal parts
• If an odd number of observations, the median Md
  will be the (n1)/2 observation
• ex. median of 11 observations is the 6th
  observation
• If an even number of observations, the median Md
  will be the midpoint between the middle two
  observations
• ex. median of 12 observations is the midpoint
  between 6th and 7th

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Mode

• Definition
• Value that occurs most frequently in data set
• Example
• 2 3 4 5 3 4 5 6 7 5 3 2 5, mode Mo5
• If all values different, no mode
• May be more than one mode
• Bimodal or multimodal
• Not used very frequently in practice

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Example Central Location
Suppose the ages of the 10 trees you are studying
are 34,24,56,52,21,44,64,34,
42,46 Then the mean age of this group is To
find the median, first order the
data 21,24,34,34,42,44,46,52,56,64 The mode
is 34 years Mo34 (occurred twice).
Mean are commonly used
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Geometric mean

• Used to calculate mean growth rate
• Definition
• Antilog of the mean of the log xi

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Geometric mean

• Example Root growth at 25oC, calculate mean
  growth rate (mm/d).

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Descriptive Statistics Measures of Dispersion

• Look at these two data sets
• Set 1 100, 30, 20, 7, 20, 30, 100
• Set 2 10, 3, 2, 7, -2, -3, -10
• If we calculate mean
• Set 1.
• Set 2.
• How to measure dispersion (spread, variability)?

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Descriptive Statistics Measures of Dispersion

• Common measures
• Range
• Variance and Standard deviation
• Coefficient of variation
• Many distributions are well-described by measure
  of location and dispersion

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Range

• Range is the difference between the largest and
  smallest values in the data set
• RMax(Xi)-Min(Xi)
• Heavily influenced by two most extreme values and
  ignores the rest of the distribution
• Set 1 100, 30, 20, 7, 20, 30, 100
• Set 2 10, 3, 2, 7, -2, -3, -10
• R1200
• R220

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Variance and standard deviation (population)

• Suppose we have N measurements of a particular
  variable in a population X1, X2, X3,,XN,
• The mean is , as ,
  we define
• as variance, unit is X unit2
• as standard deviation

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Variance and standard deviation (sample)

• Suppose we have n measurements of a particular
  variable in a sample X1, X2, X3,,Xn,
• The mean is , we define
• ?
• as mean squares, or sample variance
• ?
• as standard deviation

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Variance and standard deviation

• Corrected Sum of Squares (CSS)
• Degree of freedom
• n-1 used because if we know n-1 deviations, the
  nth deviation is known
• Deviations have to sum to zero

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Example

• Suppose the ages of the 10 trees you are studying
  are 34,24,56,52,21,44,64,34,42,46, We calculated
  
• Calculate range, variation, standard deviation
  and CV.

R64-2143 y, s21692.1/9188.01 y2, s13.72 y.
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Coefficient of Variation

• Relative variation rather than absolute variation
  such as standard deviation
• Definition of C.V.
• Useful in comparing variation between two
  distributions
• Used particularly in comparing laboratory
  measures to identify those determinations with
  more variation

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Example

• Set 1 100, 30, 20, 7, 20, 30, 100
• Set 2 10, 3, 2, 7, -2, -3, -10
• Calculate , s2, s and CV.
• Set s2 s CV
• 1 1 3773.7 61.4 61.4
• 2 1 44.7 6.7 6.7

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Descriptive Statistics (Summmary)

• Graphical Summaries
• Frequency distribution
• Histogram
• Stem and Leaf plot
• Boxplot
• Numerical Summaries
• Location - mean, median, mode.
• Dispersion - range, variance, standard deviation
• Shape (lab)

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Software

• Statistical software
• SAS
• SPSS
• Stata
• BMDP
• MINITAB
• Graphical software
• Sigmaplot
• Harvard Graphics
• PowerPoint
• Excel

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SAS

• Statistical Analysis System (SAS)
• World leader in business-intelligence software
  and services
• Founded in 1976, SAS serves more than 39,000
  business, government and university sites in 118
  countries.

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SAS introduction

• SAS has grown far beyond its origins as a
  "statistics package" and has positioned itself as
  "enterprise software", i.e. a complete system to
  manage, analyze, and present information,
  especially in a business environment.
• Standard statistical software

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Design and function of SAS

• Provides tools to scientists, so they do not need
  to spend time on the data analysis, but data
  collection and results interpretation.
• Four data-driven tasks data access, data
  management, data analysis and data presentation

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SAS programming

• SAS windows
• A simple SAS program
• DATA step
• PROC step
• SAS procedures for descriptive statistics
• UNIVARIATE, MEANS, SUMMARY

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• Merry Christmas!

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Box Plots (explain later)

• Descriptive method to convey information about
  measures of location and dispersion
• Box-and-Whisker plots
• Construction of boxplot
• Box is IQR
• Line at median
• Whiskers at smallest and largest observations
• Other conventions can be used, especially to
  represent extreme values

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Box Plots
Drug