Descriptive Statistics

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Descriptive Statistics

• Tabular and Graphical Displays
• Frequency Distribution - List of intervals of
  values for a variable, and the number of
  occurrences per interval
• Relative Frequency - Proportion (often reported
  as a percentage) of observations falling in the
  interval
• Histogram/Bar Chart - Graphical representation of
  a Relative Frequency distribution
• Stem and Leaf Plot - Horizontal tabular display
  of data, based on 2 digits (stem/leaf)

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Constructing Pie Charts

• Select a small number of categories (say 5 or 6
  at most) to avoid many narrow slivers
• If possible, arrange categories in ascending or
  descending order for categorical variables

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Monthly Philly Rainfall 1825-1869 (1/100 in)
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Constructing Bar Charts

• Put frequencies on one axis (typically vertical,
  unless many categories) and categories on other
• Draw rectangles over categories with
  heightfrequency
• Leave spaces between categories

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Constructing Histograms

• Used for numeric variables, so need Class
  Intervals
• Let Range Largest - Smallest Measurement
• Break range into (say) 5-20 intervals depending
  on sample size
• Make the width of the subintervals a convenient
  unit, and make break points so that no
  observations fall on them
• Obtain Class Frequencies, the number in each
  subinterval
• Obtain Relative Frequencies, proportion in each
  subinterval
• Construct Histogram
• Draw bars over each subinterval with height
  representing class frequency or relative
  frequency (shape will be the same)
• Leave no space between bars to imply adjacency of
  class intervals

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Interpreting Histograms

• Probability Heights of bars over the class
  intervals are proportional to the chances an
  individual chosen at random would fall in the
  interval
• Unimodal A histogram with a single major peak
• Bimodal Histogram with two distinct peaks (often
  evidence of two distinct groups of units)
• Uniform Interval heights are approximately equal
• Symmetric Right and Left portions are same shape
• Right-Skewed Right-hand side extends further
• Left-Skewed Left-hand side extends further

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Stem-and-Leaf Plots

• Simple, crude approach to obtaining shape of
  distribution without losing individual
  measurements to class intervals. Procedure
• Split each measurement into 2 sets of digits
  (stem and leaf)
• List stems from smallest to largest
• Line corresponding leaves aside stems from
  smallest to largest
• If too cramped/narrow, break stems into two
  groups low with leaves 0-4 and high with leaves
  5-9
• When numbers have many digits, trim off
  right-most (less significant) digits. Leaves
  should always be a single digit.

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Comparing Groups

• Side-by-side bar charts
• 3 dimensional histograms
• Back-to-back stem and leaf plots
• Goal Compare 2 (or more) groups wrt variable(s)
  being measured
• Do measurements tend to differ among groups?

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Summarizing Data of More than One Variable

• Contingency Table Cross-tabulation of units
  based on measurements of two qualitative
  variables simultaneously
• Stacked Bar Graph Bar chart with one variable
  represented on the horizontal axis, second
  variable as subcategories within bars
• Cluster Bar Graph Bar chart with one variable
  forming major groupings on horizontal axis,
  second variable used to make side-by-side
  comparisons within major groupings (displays all
  combinations in factorial expt)
• Scatterplot Plot with quantitaive variables y
  and x plotted against each other for each unit
• Side-by-Side Boxplot Compares distributions by
  groups

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Example - Ginkgo and Acetazolamide for Acute
Mountain Syndrome Among Himalayan Trekkers
Contingency Table (Counts)
Percent Outcome by Treatment
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Sample Population Distributions

• Distributions of Samples and Populations- As
  samples get larger, the sample distribution gets
  smoother and looks more like the population
  distribution
• U-shaped - Measurements tend to be large or
  small, fewer in middle range of values
• Bell-shaped - Measurements tend to cluster around
  the middle with few extremes (symmetric)
• Skewed Right - Few extreme large values
• Skewed Left - Few extreme small values

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Measures of Central Tendency

• Mean - Sum of all measurements divided by the
  number of observations (even distribution of
  outcomes among cases). Can be highly influenced
  by extreme values.
• Notation Sample Measurements labeled Y1,...,Yn

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Median, Percentiles, Mode

• Median - Middle measurement after data have been
  ordered from smallest to largest. Appropriate for
  interval and ordinal scales
• Pth percentile - Value where P of measurements
  fall below and (100-P) lie above. Lower
  quartile(25th), Median(50th), Upper
  quartile(75th) often reported
• Mode - Most frequently occurring outcome.
  Typically reported for ordinal and nominal data.

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Measures of Variation

• Measures of how similar or different individuals
  measurements are
• Range -- Largest-Smallest observation
• Deviation -- Difference between ith individuals
  outcome and the sample mean

• Variance of n observations Y1,...,Yn is the
  average squared deviation

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Measures of Variation

• Standard Deviation - Positive square root of the
  variance (measure in original units)

• Properties of the standard deviation
• s ? 0, and only equals 0 if all observations are
  equal
• s increases with the amount of variation around
  the mean
• Division by n-1 (not n) is due to technical
  reasons (later)
• s depends on the units of the data (e.g. 1000s
  vs )

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Empirical Rule

• If the histogram of the data is approximately
  bell-shaped, then
• Approximately 68 of measurements lie within 1
  standard deviation of the mean.
• Approximately 95 of measurements lie within 2
  standard deviations of the mean.
• Virtually all of the measurements lie within 3
  standard deviations of the mean.

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Other Measures and Plots

• Interquartile Range (IQR)-- 75thile - 25thile
  (measures the spread in the middle 50 of data)
• Box Plots - Display a box containing middle 50
  of measurements with line at median and lines
  extending from box. Breaks data into four
  quartiles
• Outliers - Observations falling more than 1.5IQR
  above (below) upper (lower) quartile

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Dependent and Independent Variables

• Dependent variables are outcomes of interest to
  investigators. Also referred to as Responses or
  Endpoints
• Independent variables are Factors that are often
  hypothesized to effect the outcomes (levels of
  dependent variables). Also referred to as
  Predictor or Explanatory Variables
• Research ??? Does I.V. ? D.V.

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Example - Clinical Trials of Cialis

• Clinical trials conducted worldwide to study
  efficacy and safety of Cialis (Tadalafil) for ED
• Patients randomized to Placebo, 10mg, and 20mg
• Co-Primary outcomes
• Change from baseline in erectile dysfunction
  domain if the International Index of Erectile
  Dysfunction (Numeric)
• Response to Were you able to insert your P
  into your partners V? (Nominal Yes/No)
• Response to Did your erection last long enough
  for you to have succesful intercourse? (Nominal
  Yes/No)

Source Carson, et al. (2004).
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Example - Clinical Trials of Cialis

• Population All adult males suffering from
  erectile dysfunction
• Sample 2102 men with mild-to-severe ED in 11
  randomized clinical trials
• Dependent Variable(s) Co-primary outcomes listed
  on previous slide
• Independent Variable Cialis Dose (0, 10, 20 mg)
• Research Questions Does use of Cialis improve
  erectile function?

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Contingency Tables

• Tables representing all combinations of levels of
  explanatory and response variables
• Numbers in table represent Counts of the number
  of cases in each cell
• Row and column totals are called Marginal counts

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2x2 Tables - Notation
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Example - Firm Type/Product Quality

• Groups Not Integrated (Weave only) vs
  Vertically integrated (Spin and Weave) Cotton
  Textile Producers
• Outcomes High Quality (High Count) vs Low
  Quality (Count)

Source Temin (1988)
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Scatterplots

• Identify the explanatory and response variables
  of interest, and label them as x and y
• Obtain a set of individuals and observe the pairs
  (xi , yi) for each pair. There will be n
  pairs.
• Statistical convention has the response variable
  (y) placed on the vertical (up/down) axis and the
  explanatory variable (x) placed on the horizontal
  (left/right) axis. (Note economists reverse axes
  in price/quantity demand plots)
• Plot the n pairs of points (x,y) on the graph

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France August,2003 Heat Wave Deaths

• Individuals 13 cities in France
• Response Excess Deaths() Aug1/19,2003 vs
  1999-2002
• Explanatory Variable Change in Mean Temp in
  period (C)
• Data

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France August,2003 Heat Wave Deaths
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Sample Statistics/Population Parameters

• Sample Mean and Standard Deviations are most
  commonly reported summaries of sample data. They
  are random variables since they will change from
  one sample to another.
• Population Mean (m) and Standard Deviation (s)
  computed from a population of measurements are
  fixed (unknown in practice) values called
  parameters.

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Example 1.3 - Grapefruit Juice Study
To import an EXCEL file, click on FILE ? OPEN ?
DATA then change FILES OF TYPE to EXCEL
(.xls) To import a TEXT or DATA file, click on
FILE ? OPEN ? DATA then change FILES OF TYPE to
TEXT (.txt) or DATA (.dat) You will be prompted
through a series of dialog boxes to import dataset
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Descriptive Statistics-Numeric Data

• After Importing your dataset, and providing names
  to variables, click on
• ANALYZE ? DESCRIPTIVE STATISTICS? DESCRIPTIVES
• Choose any variables to be analyzed and place
  them in box on right
• Options include

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Example 1.3 - Grapefruit Juice Study

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Descriptive Statistics-General Data

• After Importing your dataset, and providing names
  to variables, click on
• ANALYZE ? DESCRIPTIVE STATISTICS? FREQUENCIES
• Choose any variables to be analyzed and place
  them in box on right
• Options include (For Categorical Variables)
• Frequency Tables
• Pie Charts, Bar Charts
• Options include (For Numeric Variables)
• Frequency Tables (Useful for discrete data)
• Measures of Central Tendency, Dispersion,
  Percentiles
• Pie Charts, Histograms

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Example 1.4 - Smoking Status
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Vertical Bar Charts and Pie Charts

• After Importing your dataset, and providing names
  to variables, click on
• GRAPHS ? BAR ? SIMPLE (Summaries for Groups of
  Cases) ? DEFINE
• Bars Represent N of Cases (or of Cases)
• Put the variable of interest as the CATEGORY AXIS
• GRAPHS ? PIE (Summaries for Groups of Cases) ?
  DEFINE
• Slices Represent N of Cases (or of Cases)
• Put the variable of interest as the DEFINE SLICES
  BY

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Example 1.5 - Antibiotic Study
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Histograms

• After Importing your dataset, and providing names
  to variables, click on
• GRAPHS ? HISTOGRAM
• Select Variable to be plotted
• Click on DISPLAY NORMAL CURVE if you want a
  normal curve superimposed (see Chapter 4).

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Example 1.6 - Drug Approval Times
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Side-by-Side Bar Charts

• After Importing your dataset, and providing names
  to variables, click on
• GRAPHS ? BAR ? Clustered (Summaries for Groups
  of Cases) ? DEFINE
• Bars Represent N of Cases (or of Cases)
• CATEGORY AXIS Variable that represents groups to
  be compared (independent variable)
• DEFINE CLUSTERS BY Variable that represents
  outcomes of interest (dependent variable)

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Example 1.7 - Streptomycin Study