I' Statistical Tests:

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I. Statistical Tests

• Why do we use them?
• We need to make inferences from incomplete or
  partial information
• If we had full information about the population,
  we wouldnt need statistical inferences or tests
• What do they involve?
• The basic logic of testing a null hypothesis
  against a an alternative (research) hypothesis?
• What assumptions do they rely on?
• How do we do them? What are the steps?
• Setting up the test choosing the test statistic
• Computing the sample value of statistics
• Compare values with prediction under H(null)

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I. Statistical Tests

• General Forms of Statistical tests
• Comparison/Difference tests
• Covariation/Association tests (covered after the
  exam)
• Single Sample tests
• Test sample statistics against predicted or
  hypothetical parameter values
• Only relevant when we have a specific prediction
  or known parameter (rarely) or when we are
  testing that the true value of the statistic is
  zero
• The simplest form of statistical test

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I. Statistical Tests

• Pairwise Comparison tests
• Comparing criterion values (dependent variable)
  when we consider only two groups to compare
• Testing the theoretical prediction that the real
  difference is zero i.e., the two groups are
  equivalent on the dependent variable the means
  are sampled from the same population of sample
  means.
• Use Z-test when population variance is known (or
  when samples are very large)
• Use t-test when population variance is unknown
  and must be estimated from the sample
• Assumptions of these tests?

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II. Statistical Tests with More than Two groups

• Calculation of multiple pairwise tests
• Can become unwieldy and labor-intensive if we
  have no specific predictions about the pattern
• Overall Type 1 error rate increases rapidly as
  the number of tests increases
• Note distinction between per-test error rate and
  overall study-wide error rate
• Development of overall (omnibus) statistical test
  of differences among multiple groups (any number
  of groups) ? ANOVA

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C. The Logic of ANOVA (ANalysis Of
VAriance)

• Similar logic to t-test comparison of two means
  except that we use squared deviations rather than
  simple differences
• Derive two independent estimates of the
  population variance for the scores
• One based on the variations among the group means
  (between-group estimate)
• One based on the variations within each of the
  groups (within-group estimates)
• Each group is taken as a random sample from the
  same population (under the null hypothesis)

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C. The Logic of ANOVA (ANalysis Of
VAriance)

• Compare the two separate variance estimates
  (between-group vs. within-group)
• If the estimate from the group means is larger
  than the pooled estimate from the separate
  groups, then the groups are really different (not
  from the same population)
• Use an F-statistic computed as the ratio of two
  variances and which has a known distribution
• Distribution depends on two parameters
  representing the degrees of freedom in the two
  variance estimates in the ratio

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C. The Logic of ANOVA (ANalysis Of
VAriance)

• If the F-statistic is substantially larger than
  1.0, then reject the null hypothesis of equality
  (no-difference)
• Base this decision on the probability
  distribution of the F-statistic so that chance of
  Type 1 error is .05 or less.
• What is the null hypothesis and the alternative
  hypothesis of the ANOVA F-test?
• How can we interpret the outcome when results
• Reject the null hypothesis?
• Retain the null hypothesis?

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C. The Logic of ANOVA (continued)

• The ANOVA F-test is an omnibus statistical test
• It only suggests there is some real difference
• It doesnt tell which differences are real
• Need to follow up the omnibus test with post hoc
  comparisons to identify which are signif.
• Control overall error rate of the whole set of
  comparisons
• Adjust each individual pairwise comparison to
  keep the overall error rate at desired level
  (e.g., .05)
• Many different post hoc comparison systems
• Here select only one Tukeys HSD procedure

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Post Hoc Tests