LSU-HSC School of Public Health Biostatistics 1 Statistical Core Didactic Introduction to Biostatistics Donald E. Mercante, PhD.

LSU-HSC School of Public Health Biostatistics

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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.

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 the population mean. The test statistic was

This can be rewritten to yield:

0xt

sn

Interval Estimation

Confidence Interval for :

CC sx tn

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)