Statistical Core Didactic

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Statistical Core Didactic

• Introduction to
• Biostatistics
• Donald E. Mercante, PhD

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Randomized Experimental Designs
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Randomized Experimental Designs

• Three Design Principles
• 1. Replication
• 2. Randomization
• 3. Blocking

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Randomized Experimental Designs

• 1. Replication
• Allows estimation of experimental error, against
  which, differences in trts are judged.

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Randomized Experimental Designs

• Replication
• Allows estimation of exptl error, against which,
  differences in trts are judged.
• Experimental Error
• Measure of random variability.
• Inherent variability between subjects treated
  alike.

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Randomized Experimental Designs

• If you dont replicate . . .
• . . . You cant estimate!

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Randomized Experimental Designs

• To ensure the validity of our estimates of
  exptl error and treatment effects we rely on ...

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Randomized Experimental Designs

• . . . Randomization

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Randomized Experimental Designs

• 2. Randomization
• leads to unbiased estimates of
• treatment effects

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Randomized Experimental Designs

• Randomization
• leads to unbiased estimates of
• treatment effects
• i.e., estimates free from systematic differences
  due to uncontrolled variables

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Randomized Experimental Designs

• Without randomization, we may need to adjust
  analysis by
• stratifying
• covariate adjustment

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Randomized Experimental Designs

• 3. Blocking
• Arranging subjects into similar groups to
• account for systematic differences

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Randomized Experimental Designs

• Blocking
• Arranging subjects into similar groups (i.e.,
  blocks) to account for systematic differences
• - e.g., clinic site, gender, or age.

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Randomized Experimental Designs

• Blocking
• leads to increased sensitivity of statistical
  tests by reducing exptl error.

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Randomized Experimental Designs

• Blocking
• Result More powerful statistical test

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Randomized Experimental Designs

• Summary
• Replication allows us to estimate Exptl Error
  
• Randomization ensures unbiased estimates of
  treatment effects
• Blocking increases power of statistical tests

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Randomized Experimental Designs

• Three Aspects of Any Statistical Design
• Treatment Design
• Sampling Design
• Error Control Design

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Randomized Experimental Designs

• 1. Treatment Design
• How many factors
• How many levels per factor
• Range of the levels
• Qualitative vs quantitative factors

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Randomized Experimental Designs

• One Factor Design Examples
• Comparison of multiple bonding agents
• Comparison of dental implant techniques
• Comparing various dose levels to achieve numbness

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Randomized Experimental Designs

• Multi-Factor Design Examples
• Factorial or crossed effects
• Bonding agent and restorative compound
• Type of perio procedure and dose of antibiotic
• Nested or hierarchical effects
• Surface disinfection procedures within clinic type

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Randomized Experimental Designs

• 2. Sampling or Observation Design
• Is observational unit experimental unit ?
• or,
• is there subsampling of EU ?

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Randomized Experimental Designs

• Sampling or Observation Design
• For example,
• Is one measurement taken per mouth, or are
  multiple sites measured?
• Is one blood pressure reading obtained or are
  multiple blood pressure readings taken?

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Randomized Experimental Designs

• 3. Error Control Design
• concerned with actual arrangement of the exptl
  units
• How treatments are assigned to eus

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Randomized Experimental Designs

• 3. Error Control Design
• Goal Decrease experimental error

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Randomized Experimental Designs

• 3. Error Control Design
• Examples
• CRD Completely Randomized Design
• RCB Randomized Complete Block Design
• Split-mouth designs (whole incomplete block)
• Cross-Over Design

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Inferential Statistics

• Hypothesis Testing
• Confidence Intervals

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Hypothesis Testing

• Start with a research question
• Translate this into a testable hypothesis

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Hypothesis Testing

• Specifying hypotheses
• H0 null or no effect hypothesis
• H1 research or alternative hypothesis
• Note Only the null is tested.

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Errors in Hypothesis Testing

• When testing hypotheses, the chance of making a
  mistake always exists.
• Two kinds of errors can be made
• Type I Error
• Type II Error

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Errors in Hypothesis Testing
Reality ? ? Decision H0 True H0 False
Fail to Reject H0 ? Type II (?)
Reject H0 Type I (?) ?
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Errors in Hypothesis Testing

• Type I Error
• Rejecting a true null hypothesis
• Type II Error
• Failing to reject a false null hypothesis

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Errors in Hypothesis Testing

• Type I Error
• Experimenter controls or explicitly sets this
  error rate - ?
• Type II Error
• We have no direct control over this error rate - ?

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Randomized Experimental Designs

• When constructing an hypothesis
• Since you have direct control over Type I error
  rate, put what you think is likely to happen in
  the alternative.
• Then, you are more likely to reject H0, since
  you know the risk level (?).

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Errors in Hypothesis Testing

• Goal of Hypothesis Testing
• Simultaneously minimize chance of making either
  error

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Errors in Hypothesis TestingIndirect Control of ß

• Power
• Ability to detect a false null hypothesis
• POWER 1 - ?

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Steps in Hypothesis Testing

• General framework
• Specify null alternative hypotheses
• Specify test statistic and ?-level
• State rejection rule (RR)
• Compute test statistic and compare to RR
• State conclusion

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Steps in Hypothesis Testing

• test statistic
• Summary of sample evidence relevant to
  determining whether the null or the alternative
  hypothesis is more likely true.

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Steps in Hypothesis Testing

• test statistic
• When testing hypotheses about means, test
  statistics usually take the form of a standardize
  difference between the sample and hypothesized
  means.

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Steps in Hypothesis Testing

• test statistic
• For example, if our hypothesis is
• Test statistic might be

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Steps in Hypothesis Testing

• Rejection Rule (RR)
• Rule to base an Accept or Reject null
  hypothesis decision.
• For example,
• Reject H0 if t gt 95th percentile of
  t-distribution

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Hypothesis Testing

• P-values
• Probability of obtaining a result (i.e., test
  statistic) at least as extreme as that observed,
  given the null is true.

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Hypothesis Testing

• P-values
• Probability of obtaining a result at least as
  extreme given the null is true.
• P-values are probabilities
• 0 lt p lt 1 lt-- valid range
• Computed from distribution of the test
  statistic

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Hypothesis Testing

• P-values
• Generally, plt0.05 considered significant

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Hypothesis Testing
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Hypothesis Testing

• Example
• Suppose we wish to study the effect on blood
  pressure of an exercise regimen consisting of
  walking 30 minutes twice a day.
• Let the outcome of interest be resting systolic
  BP.
• Our research hypothesis is that following the
  exercise regimen will result in a reduction of
  systolic BP.

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Hypothesis Testing

• Study Design 1 Take baseline SBP (before
  treatment) and at the end of the therapy period.
• Primary analysis variable difference in SBP
  between the baseline and final measurements.

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Hypothesis Testing

• Null Hypothesis
• The mean change in SBP (pre post) is equal to
  zero.
• Alternative Hypothesis
• The mean change in SBP (pre post) is
  different from zero.

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Hypothesis Testing

• Test Statistic
• The mean change in SBP (pre post) divided by
  the standard error of the differences.

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Hypothesis Testing

• Study Design 2 Randomly assign patients to
  control and experimental treatments. Take
  baseline SBP (before treatment) and at the end of
  the therapy period (post-treatment).
• Primary analysis variable difference in SBP
  between the baseline and final measurements in
  each group.

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Hypothesis Testing

• Null Hypothesis
• The mean change in SBP (pre post) is equal in
  both groups.
• Alternative Hypothesis
• The mean change in SBP (pre post) is
  different between the groups.

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Hypothesis Testing

• Test Statistic
• The difference in mean change in SBP (pre
  post) between the two groups divided by the
  standard error of the differences.

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Interval Estimation

• Statistics such as the sample mean, median, and
  variance are called
• point estimates
• -vary from sample to sample
• -do not incorporate precision

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Interval Estimation

• Take as an example the sample mean
• X gt ?
• (popn mean)
• Or the sample variance
• S2 gt ?2
• (popn variance)

Estimates
Estimates
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Interval Estimation

• Recall, a one-sample t-test on the population
  mean. The test statistic was
• This can be rewritten to yield

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Interval Estimation
Confidence Interval for ?
The basic form of most CI Estimate
Multiple of Std Error of the Estimate
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Interval Estimation

• Example Standing SBP
• Mean 140.8, S.D. 9.5, N 12
• 95 CI for ?
• 140.8 2.201 (9.5/sqrt(12))
• 140.8 6.036
• (134.8, 146.8)