LSU-HSC School of Public HealthBiostatistics
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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 arejudged.
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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 treatedalike.
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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 ofexpt’l error and treatment effects we relyon ...
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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 systematicdifferences due to uncontrolled variables
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Randomized Experimental Designs
Without randomization, we mayneed 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 systematicdifferences
- e.g., clinic site, gender, or age.
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Randomized Experimental Designs
•Blocking
•leads to increased sensitivity of statisticaltests 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 oftreatment 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 achievenumbness
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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 aremultiple sites measured?
•Is one blood pressure reading obtained or aremultiple blood pressure readings taken?
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Randomized Experimental Designs
3. Error Control Design
•concerned with actual arrangement ofthe 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 amistake 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 errorrate, put what you think is likely to happen in thealternative.
Then, you are more likely to reject H0, since youknow 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 todetermining whether the null or the alternativehypothesis is more likely true.
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Steps in Hypothesis Testing
test statistic
When testing hypotheses about means, teststatistics usually take the form of a standardizedifference between the sample and hypothesizedmeans.
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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 hypothesisdecision.
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) atleast as extreme as that observed, given the null istrue.
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Hypothesis Testing
P-values
Probability of obtaining a result at least as extremegiven 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 bloodpressure of an exercise regimen consisting of walking30 minutes twice a day.
Let the outcome of interest be resting systolic BP.
Our research hypothesis is that following the exerciseregimen will result in a reduction of systolic BP.
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Hypothesis Testing
Study Design #1: Take baseline SBP (beforetreatment) and at the end of the therapy period.
Primary analysis variable = difference in SBP betweenthe 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 fromzero.
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Hypothesis Testing
Test Statistic:
The mean change in SBP (pre – post) divided by thestandard error of the differences.
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Hypothesis Testing
Study Design #2: Randomly assign patients to controland experimental treatments. Take baseline SBP(before treatment) and at the end of the therapy period(post-treatment).
Primary analysis variable = difference in SBP betweenthe 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 bothgroups.
Alternative Hypothesis:
The mean change in SBP (pre – post) is different betweenthe 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 errorof the differences.
Interval Estimation
Statistics such as the sample mean,median, and variance are called
point estimates
-vary from sample to sample
-do not incorporate precision
Interval Estimation
Take as an example the sample mean:
X ——————>
(popn mean)
Or the sample variance:
S2 ——————> 2
(popn variance)
Estimates
Estimates
Interval Estimation
Recall, a one-sample t-test on thepopulation mean. The test statistic was
This can be rewritten to yield:
Interval Estimation
Confidence Interval for :
The basic form of most CI :
Estimate ± Multiple of Std Error of the Estimate
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)