Lecture 27: Analysis

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Lecture 27 Analysis
09/10/99
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Goals

• Introduction to Data Analysis
• How to use a statistical Package
• Review of basic statistical concepts and methods

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Steps in Data Analysis

• Enter the Data
• Choose the Analysis
• Run the Analysis
• Interpret the results
• Explore more analyses
• Learn and Communicate

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Why Use Statistics?

• Because Eyeballing data can be misleading
• Because Decisions require objective proof
• We need a consistent method for establishing
  proof
• Statistics, combined with good experimental
  design, helps researchers overcome their biases,
  assumptions and expectations
• e.g., the Canals on Mars problem
• Statistics help us overcome
• Experimental Error
• Confusion of correlation with causation
• complexity of effects studied

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Types of Statistics

• Inferential statistics
• Concerned with making comparisons
• e.g., A is bigger than B
• e.g., There is a positive relationship (slope)
  between A and B
• Inferential statistics emphasize tests of
  significance
• Descriptive statistics
• concerned with describing patterns and properties
  of data
• e.g., there are three types of patient that
  differ on these dimensions
• e.g., these three items in the personality test
  appear to be measuring the same underlying
  construct (e.g., extraversion)

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Distributions
Yield Time order Method Yield 1 A 89.7 2 A
81.4 3 A 84.5 4 A 84.8 5 A 87.3 6 B 79.7 7 B
85.1 8 B 81.7 9 B 83.7 10 B 84.5

• Data
• Time Order X Yield
• Method A vs. Method B
• Does Method affect Result?

Method A Method B Difference 85.54 82.94 2..6
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How the Distributions Look

• A looks bigger
• Lots of variation

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Ways of Visualizing Distributions

• Dot Diagrams
• Histograms
• Data Runs (previous slide)
• Stem and Leaf Plots (Tukey)
• and others

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Ways of Describing Distributions

• Measures of Location
• Mean
• Median
• Mode
• Measures of Variability
• Standard Deviation (Variance)
• Kurtosis (peakiness)

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Randomization Test

• Imagine that we had the 10 observations shown
  earlier and randomly shuffled them into two
  groups of five. What are the chances that the
  difference between the means of the two groups
  would be greater than or equal to the observed
  difference?
• If we calculate this probability across all
  possible shuffles of the data, this is called a
  randomization test.

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Significance Tests

• Randomization test is a significance test
• compute a statistic to test a hypothesis (e.g.,
  means are significantly different)
• create reference distribution (for true
  hypothesis)
• calculate probability of observed discrepancy
  occurring by chance
• If probability is low enough, reject hypothesis
  and assert statistically significant difference

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Normal and t-distributions

• Normal distribution is bell-curve shaped
• t-distribution reflects difference between 2
  Normal distributions (approximates Normal as
  number of observations increases)
• Reasons why normal distribution is important
• Central Limit Theorem
• Robustness of statistical procedures to
  departures from normality

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Characterizing Normal Distributions

• mean
• variance
• probabilities under the distribution
• z-values
• z-values measure position in standard deviation
  units.
• t-values are sample approximations of z values
• probability that data point will differ by more
  than two standard deviations from the centre of a
  normal distribution is roughly 5

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t-tests

• See standard texts for calculation of t-statistic
• divide difference in means by measure of
  variability
• refer result to tables of t values
• select entry based on degrees of freedom and
  check out assigned probability of difference
• Paired vs. Independent samples t-test

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Entering Data in SPSS
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Putting in Numbers
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Setting the Variable Type
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Number with two decimal places
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Data Matrix
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Choosing an Analysis Method
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Analysis Options
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Sample Output
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Choosing a Variable
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Setting Options
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Chart Output
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Inferential Statistics and Analysis of Variance

• Why use statistics?
• Randomization and Randomization tests
• Tests of Significance
• Confidence Intervals
• Accounting for Error Variation
• Analysis of Variance

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Sample Data for Paired (dependent) t-test
Mnths_6 Mnths_24 124 114 94 88 115 102 110 2 116 2
139 2 116 2 110 2 129 2 120 2 105 2 88 2 120 2 12
0 2 116 2 105 2 ... ... ... ... 123 132
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T-test Results
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Anova Example Data Matrix
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Deviations from Grand Mean
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Total Sums of Squares
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One-way ANOVA
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Two-way Data Matrix
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Table of Means
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Two-way Anova Summary
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Application to Design

• Targetted experiments (with analyses) can help
  guide design
• Analysis of results can be used to support
  proposed design during reviews
• Efficient experiments dont have to be expensive

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Cautions

• Formal experiments are often overkill
• Many quick iterations are probably more effective
  than a few thorough iterations
• Design sufficient power in the experiment

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Summary

• Design often requires observation
• Methods of observation need to be carefully
  controlled (including sampling)
• Formal experiments provide control
• Careful analysis helps interpretation/insight
• Descriptive Statistics
• Inferential Statistics
• T-tests and ANOVAs are typically used to analyze
  experiments