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)

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

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

Constructing Bar Charts Put frequencies on one axis (typically vertical, unless many categories) and categories on other Draw rectangles over categories with height=frequency Leave spaces between categories

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

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

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.

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?

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

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

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

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

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.

Measures of Variation Measures of how similar or different individual’s measurements are Range -- Largest-Smallest observation Deviation -- Difference between ith individual’s outcome and the sample mean: Variance of n observations Y1,...,Yn is the “average” squared deviation:

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 $)

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.

Other Measures and Plots Interquartile Range (IQR)-- 75th%ile - 25th%ile (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

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.

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 partner’s 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).

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?

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

2x2 Tables - Notation n1+n2 (n1+n2)-(X1+X2) X1+X2 Outcome Total n2 Group 2 n1 n1-X1 X1 Group 1 Group Absent Present

Example - Firm Type/Product Quality 172 134 38 Outcome Total 84 79 5 Vertically Integrated 88 55 33 Not Group Low Quality High 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)

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

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:

France August,2003 Heat Wave Deaths

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.

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

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:

Example 1.3 - Grapefruit Juice Study

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

Example 1.4 - Smoking Status

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

Example 1.5 - Antibiotic Study

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).

Example 1.6 - Drug Approval Times

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)

Example 1.7 - Streptomycin Study