statistical core didactic
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Statistical Core Didactic. Introduction to Biostatistics Donald E. Mercante, PhD. Randomized Experimental Designs. Randomized Experimental Designs. Three Design Principles: 1. Replication 2. Randomization 3. Blocking. Randomized Experimental Designs. 1. Replication - PowerPoint PPT PresentationTRANSCRIPT
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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 expt’l 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 don’t replicate . . .
. . . You can’t estimate!
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Randomized Experimental Designs
To ensure the validity of our
estimates of expt’l 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 expt’l
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 Expt’l 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 expt’l units
•How treatments are assigned to eu’s
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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 β
• 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:
N N0 1H : pop mean = 120 vs. H : pop mean 120
sample mean hypothesized mean
standard errort
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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| > 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 < p < 1 <-- valid range
Computed from distribution of the test
statistic
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Hypothesis Testing
P-values
Generally, p<0.05 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 ——————> (popn mean)
Or the sample variance:
S2 ——————> 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:
0xt
sn
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Interval Estimation
Confidence Interval for :
CC sx tn
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)