statistics in biomedical research rise program 2012 los angeles biomedical research institute at...

Statistics in Biomedical Research

RISE Program 2012

Los Angeles Biomedical Research Institute

at Harbor-UCLA Medical Center

January 19, 2012

Peter D. Christenson

Statistics in Biomedical Research

We will not cover all of these slides in class.

Skipped slides are side points.

Complete slides available at:

www.research.labiomed.org/biostat

http://research.labiomed.org/biostat

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other biostat talks

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Scientific Decision Making

Setting:

Two groups of animals: one gets a new molecule you have created, the other group doesn't.

Measure the relevant outcome in all animals.

How do we decide if the molecule has an effect?

Balancing Risk: Business

You have a vending machine business.

The machines have a dollar bill reader.

You can set the reader to be loose or strict.

Setting too high → rejects valid bills, lose customers.

Setting too low → accepts bogus bills, lose $.

Need to balance both errors in some way.

Balancing Risk: U.S. Legal System

Need to decide guilty or innocent.

Jury or judge measures degree of guilt.

Setting degree high → frees suspects who are guilty.

Setting degree low → jails suspects who are innocent.

Need to balance both errors in some way.

Civil case: lower degree needed than legal case.

Balancing Risk: Personal

Investments have expected returns of 1% to 20%.

As expected returns ↑, volatility ↑, chance of loss ↑.

You choose your degree of risk.

Setting degree high → chances of losing ↑.

Setting degree low → chances of missing big $ ↑.

Need to balance both errors in some way.

Balancing Risk: Scientific Research

Perform experiment on 20 mice to measure an effect.

Only 20 mice may not be representative.

Need to decide if observed effect is real or random.

You choose a minimal degree of effect, call it Min.

If effect > Min then claim effect is real.

Setting degree high → chance* of missing effect ↑.

Setting degree low → chance** of wrong + result ↑.

Need to balance both errors in some way.

What would you want chances for * and for ** to be?

Scientific Decision MakingSetting: Two groups, one gets drug A, one gets placebo (B). Measure outcome.

Subjects may respond very differently.

How do we decide if the drug has “an effect”?

Perhaps: Say yes if the mean outcome of those receiving drug is greater than the mean of the others? Or twice as great? Or the worst responder on drug was better than the best on placebo? Other?

Meaning or Randomness?

Meaning or Randomness?

This is the goal of science in general.

The role of statistics is to give an objective way to make those decisions.

Meaning or Randomness?

Scientific inference: Perform experiment.

Make a decision. Is it correct?

Quantify chances that our decision is correct or not.

Other areas of life:

Suspect guilty? Nobel laureate's opinion?

Make a decision. Is it correct?

Usually cannot quantify.

Specialness of Scientific ResearchScientific method:

Assume the opposite of what we think.

Design the experiment so that our opinions cannot influence the outcome.

Say exactly how we will make a conclusion, i.e. making a decision from the experiment.

Tie our hands behind our back. Do experiment.

Make decision. Find the chances (from calculations, not opinion) that we are wrong.

Experimental conclusions are not expert opinion.

Decision Making

We first discuss using a medical device to make decisions about a patient.

These decisions could be right or wrong.

We then make an analogy to using an experiment to make decisions about a scientific question.

These decisions could be right or wrong.

Decision Making

The next nine slides will make an analogy to how conclusions or decisions from

experiments are made.

The numbers are made-up.

Mammograms are really better than this.

Decision Making: Diagnosis

Mammogram Spot Darkness 100

Definitely Not Cancer

Definitely Cancer

How is the decision made for intermediate darkness?

A particular woman with cancer may not have a 10. Another woman without cancer may not have 0.

Intermediate darkness ↔ Considerable overlap of CA and non-CA:

True Non-CA Patients

True CA Patients

Mammogram Spot Darkness0 2 4 6 8 10

Area under curve =

Probability

↑

# of women

Decision Making: Overlap

Decision Making: Diagnosis

Mammogram Spot Darkness100

Suppose a study found the mammogram rating (0-10) for 1000 women who definitely have cancer by biopsy (truth).

Proportion of 1000 Women:1000/1000

0/1000

100/1000600/1000

900/1000990/1000

2 4 6 8

Use What Cutoff?

Decision Making: Sensitivity

Cutoff for Spot Darkness Mammogram Sensitivity

≥0 100%

>2 99%

>4 90%

>6 60%

>8 10%

>10 0%

Sensitivity = Chances of correctly detecting disease.

Why not just choose a low cutoff and detect almost everyone with disease?

Decision Making Continued

Mammogram Spot Darkness100

Suppose a study found the mammogram rating (0-10) for 1000 women who definitely do NOT have cancer by biopsy (truth).

Proportion of 1000 Women: 1000/1000

0/1000

350/1000700/1000

900/1000950/1000

2 4 6 8

Use What Cutoff?

Decision Making: Specificity

Cutoff for Spot Darkness Mammogram Specificity

<0 0%

≤2 35%

≤4 70%

≤6 90%

≤8 95%

≤10 100%

Specificity=Chances of correctly NOT detecting disease.

Decision Making: Tradeoff

Cutoff Sensitivity Specificity

0 100% 0%

2 99% 35%

4 90% 70%

6 60% 90%

8 10% 95%

10 0% 100%

Choice of cutoff depends on whether the diagnosis is a screening or a final one. For example:

Cutoff=6 : Call disease in 60% with it and 10% without.

Make Decision: If Spot>6, Decide CA. If Spot≤6, Decide Not CA.

True Non-CA Patients

True CA Patients

Mammogram Spot Darkness0 2 4 6 8 10

\\\ = Specificity = 90%.

/// = Sensitivity = 60%.

Graphical Representation of Tradeoffs

Area under curve =

Probability

↑

# of women

90% 60%

cutoff

Decide not CA Decide CA

95%

10%

Tradeoffs From a Stricter Cutoff

cutoff

0 2 4 6 8 10

Mammogram Spot Darkness

Decide not CA Decide CA

Decision Making for Diagnosis: Summary

As sensitivity increases, specificity decreases and vice-versa.

Cannot increase both sensitivity and specificity together.

We now develop sensitivity and specificity to test or decide scientific claims. Analogy:

True Disease ↔ True claim, real effect.

Decide Disease ↔ Decide effect is real.

But, can both increase sensitivity and specificity together.

Decision Making

End of analogy.

Back to our original problem in experiments.

Scientific Decision Making

Setting: Two groups, one gets drug A, one gets placebo (B). Measure outcome.

Subjects may respond very differently.

How do we decide if the drug has “an effect”?

Perhaps: Say yes if the mean outcome of those receiving drug is greater than the mean of the others? Or twice as great? Or the worst responder on drug was better than the best on placebo? Other?

Scientific Decision Making

Setting: Two groups, one gets drug A, one gets placebo (B). Measure outcome.

How do we decide if the drug has an effect?

Perhaps: Say yes if the mean of those receiving drug is greater than the mean of the placebo group? Other decision rules?

Let’s just try an arbitrary decision rule:

Let Δ = Group A Mean minus Group B Mean

Decide that A is effective if Δ>2. [Not just Δ>0.]

Make Decision: If Δ>2, then Decide Effective. If Δ≤2, then Decide Not.

True No Effect (A=B)

True Effect (A≈B+2.2)

Eventual Graphical Representation

1. Where do these curves come from?

2. What are the consequences of using cutoff=2?

Δ = Group A Mean minus Group B Mean -2 0 2 4 6

90% 60%

Decide Not Effective Decide Effective

Question 2 First

2. What are the consequences of using cutoff=2?

Answer:

If the effect is real (A≠B), there is a 60% chance of deciding so. [Actually, if in particular A is 2.2 more than B.] This is the experiment’s sensitivity, more often called power.

If effect is not real (A=B), there is a 90% of deciding so. This is the experiment’s specificity. More often, 100-specificity is called the level of significance.

Question 2 ContinuedWhat if cutoff=1 was used instead?

If the effect is real (Δ=A-B=2.2), there is about a 85% chance of deciding so. Sensitivity ↑ (from 60%).

If effect is not real (Δ=A-B=0), there is about a 60% of deciding so. Specificity ↓ (from about 90%).

Δ: 0 1 2.2

60%85%

Typical Choice of Cutoff

Δ = Group A Mean minus Group B Mean -2 0 2 4 6

Require specificity to be 95%. This means there is only a 5% chance of wrongly declaring an effect. → Need overwhelming evidence, beyond a reasonable (5%) doubt, to make a claim.

~45% Power

95% Specificity

Strength of the Scientific MethodScientists (and their journals and FDA) require overwhelming evidence, beyond a reasonable (5%) doubt, not just “preponderance of the truth” which would be specificity=50%.

So much stronger than expert opinion.

~45% Power

Only 5% chance of a false positive claim

How can we increase power above this 45%, but maintain the chances of a false positive conclusion at ≤5%?

Are we just stuck with knowing that many true conjectures will be thrown away as collateral damage to this rigor?

How to Increase Power

How can we increase power above this 45%, but maintain the chances of a false positive conclusion at ≤5%?

Are we just stuck with knowing that many true conjectures will be thrown away as collateral damage to this rigor?

To answer this, we need to go into how the curves are made:

So, we take a detour for the next 9 slides to show this.

Short Answer – Skip Next 8 Slides

The curves are for the means of the groups, not individuals.

The spread of a curve depends on the natural variability in subject response (SD) and the # of subjects (N).

As N ↑, the spread ↓. Mean is less extreme with more subjects.

Smaller N Larger N

Back to Question 1

1. Where do the curves in the last figure come from?

Answer:

You specify three quantities: (1) where their peaks are (the experiment’s detectable difference), and how wide they are (which is determined by (2) natural variation and (3) the # of subjects or animals or tissue samples, N).

Those specifications give a unique set of “bell-shaped” curves. How?

A “Law of Large Numbers”

Suppose individuals have values ranging from Lo to Hi, but the % with any particular value could be anything, say:

You choose a sample of 2 of these individuals, and find their average. What value do you expect the average to have?

Lo LoHi Hi

Prob ↑

N = 1

A “Law of Large Numbers”

In both cases, values near the center will be more likely:

Now choose a sample of 4 of these individuals, and find their average. What value do you expect the average to have?

Lo LoHi Hi

Prob ↑

N = 2

A “Law of Large Numbers”

In both cases, values near the center will be more likely:

Now choose a sample of 10 of these individuals, and find their average. What value do you expect the average to have?

Lo LoHi Hi

Prob ↑

N = 4

A “Law of Large Numbers”

In both cases, values near the center will be more likely:

Now choose a sample of 50 of these individuals, and find their average. What value do you expect the average to have?

Lo LoHi Hi

Prob ↑

N = 10

A “Law of Large Numbers”

In both cases, values near the center will be more likely:

A remarkable fact is that not only is the mean of the sample is expected to be close to the mean of “everyone” if N is large enough, but we know exact probabilities of how close, and the shape of the curve.

Lo LoHi Hi

Prob ↑

N = 50

Summary: Law of Large Numbers

Lo LoHi Hi

Prob ↑

N = 1

SD↔ ↔SD

Lo Value of the mean of N subjects Hi↔

SD(Mean) = SD/√N

SD is about 1/6 of the total range.

SD ≈ 1.25 x average deviation from the center.

Large N

Law of Large Numbers: Another View

rescaled

You can make the range of possible values for a mean as small as you like by choosing a large enough sample.

Also the shape will always be a bell curve if the sample is large enough.

Scientific Decision Making

So, where are we?

We can now answer the basic dilemma we raised.

Repeat earlier slide:

Strength of the Scientific MethodScientists (and their journals and FDA) require overwhelming evidence, beyond a reasonable (5%) doubt, not just “preponderance of the truth” which would be specificity=50%.

Similar to US court of law. So much stronger than expert opinion.

~45% Power

Only 5% chance of a false positive claim

How can we increase power, but maintain the chances of a false positive conclusion at ≤5%?

Are we just stuck with knowing that many true conjectures will be thrown away as collateral damage to this rigor?

N = 50

Scientific Decision Making

So, the answer is that by choosing N large enough, the mean has to be in a small range.

That narrow the curves.

That in turn increases the chances that we will find the effect in our study, i.e., its power.

The next slide shows this.

Fix the max chances of a false positive claim at 5%

80% Power

N = 75

N = 50

95% 45%

95%

95%

N = 88

74%

80%

74% Power

45% Power

Find N that gives the power you want.

In many experiments, five factors are inter-related. Specifying four of these determines the fifth:

1. Study size, N.

2. Power, usually 80% to 90% is used.

3. Acceptable false positive chance, usually 5%.

4. Magnitude of the effect to be detected (Δ).

5. Heterogeneity among subjects or units (SD).

The next 2 slides show how these factors are typically examined, and easy software to do the calculations.

Putting it All Together

Quote from An LA BioMed ProtocolThe following table presents detectable differences, with p=0.05 and 80% power, for different study sizes.

Total Number

of Subjects

Detectable Difference in

Change in Mean MAP (mm Hg)(1)

Detectable Difference in

Change in Mean Number

of Vasopressors(2)

20 10.9 0.77 40 7.4 0.49 60 6.0 0.39 80 5.2 0.34 100 4.6 0.30 120 4.2 0.27

Thus, with a total of the planned 80 subjects, we are 80% sure to detect (p<0.05) group differences if treatments actually differ by at least 5.2 mm Hg in MAP change, or by a mean 0.34 change in number of vasopressors.

Software for Previous SlidePilot data: SD=8.19 for ΔMAP in 36 subjects.

For p-value<0.05, power=80%, N=40/group, the detectable Δ of 5.2 in the previous table is found as:

Study Size : May Not be Based on Power

Precision refers to how well a measure is estimated.

Margin of error = the ± value (half-width) of the 95% confidence interval (sorry – not discussed here).

Smaller margin of error ←→ greater precision.

To achieve a specified margin of error, solve the CI formula for N.Polls: N ≈ 1000→ margin of error on % ≈ 1/√N ≈ 3%.

Pilot Studies, Phase I, Some Phase II: Power not relevant; may have a goal of obtaining an SD for future studies.

Study Design Considerations Statistical Components of Protocols

• Target population / source of subjects.• Quantification of aims, hypotheses.• Case definitions, endpoints quantified. • Randomization plan, if one will be used.• Masking, if used.• Study size: screen, enroll, complete.• Use of data from non-completers.• Justification of study size (power, precision, other).• Methods of analysis.• Mid-study analyses.

Resources, Software, and References

Professional Statistics Software Package

Output

Enter code; syntax.

Stored data; access-ible.

Comprehensive, but steep learning curve: SAS, SPSS, Stata.

Microsoft Excel for Statistics

• Primarily for descriptive statistics.

• Limited output.

Typical Statistics Software PackageSelect Methods from Menus

Output after menu selection

Data in spreadsheet

www.ncss.com

www.minitab.com

www.stata.com

$100 - $500

Free Statistics Software: Mystatwww.systat.com

Free Study Size Software

www.stat.uiowa.edu/~rlenth/Power

http://research.labiomed.org/biostat

This and

other biostat talks

posted

Recommended Textbook: Making Inference

Design issues

Biases

How to read papers

Meta-analyses

Dropouts

Non-mathematical

Many examples

Thank You

Nils Simonson, in

Furberg & Furberg,

Evaluating Clinical Research

Outline

Meaning or randomness?

Decisions, truth and errors.

Sensitivity and specificity.

Laws of large numbers.

Experiment size and study power.

Study design considerations.

Resources, software, and references.