Thomas Songer, PhDwith acknowledgment to several slides provided by

M Rahbar and Moataza Mahmoud Abdel Wahab

Introduction to Research MethodsIn the Internet Era

Inferential StatisticsHypothesis Testing

Introduction to Biostatistics

Key Lecture Concepts

• Assess role of random error (chance) as an influence on the validity of the statistical association

• Identify role of the p-value in statistical assessments

• Identify role of the confidence interval in statistical assessments

• Briefly introduce tests to undertake2

Research Process

Research question

Hypothesis

Identify research design

Data collection

Presentation of data

Data analysis

Interpretation of data

Polgar, Thomas 3

Interpreting Results

When evaluating an association between disease and exposure, we need guidelines

to help determine whether there is a true difference in the frequency of disease

between the two exposure groups, or perhaps just random variation from the study sample.

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Random Error (Chance)

1. Rarely can we study an entire population, soinference is attempted from a sample ofthe population

2. There will always be random variation from sample to sample

3. In general, smaller samples have lessprecision, reliability, and statistical power(more sampling variability)

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Hypothesis Testing

• The process of deciding statistically whether the findings of an investigation reflect chance or real effects at a given level of probability.

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Elements of Testing hypothesis

• Null Hypothesis

• Alternative hypothesis

• Identify level of significance

• Test statistic

• Identify p-value / confidence interval

• Conclusion

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H0: There is no association between theexposure and disease of interest

H1: There is an association between theexposure and disease of interest

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Hypothesis Testing

Note: With prudent skepticism, the null hypothesis is given the benefit of the doubt until the data convince us otherwise.

Hypothesis Testing

• Because of statistical uncertainty regarding inferences about population parameters based upon sample data, we cannot prove or disprove either the null or alternate hypotheses as directly representing the population effect.

• Thus, we make a decision based on probability and accept a probability of making an incorrect decision.

9Chernick

Associations

• Two types of pitfalls can occur that affect the association between exposure and disease

– Type 1 error: observing a difference when in truth there is none

– Type 2 error: failing to observe a difference where there is one.

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Interpreting Epidemiologic Results

YOUR

DECISION

H0 True

(No assoc.)

H1 True

(Yes assoc.)

Do not reject H0

(not stat. sig.)

Correct decision

Type II

(beta error)

Reject H0

(stat. sig.)

Type I

(alpha error)

Correct decision

Four possible outcomes of any epidemiologic study:

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REALITY

YOUR

DECISION

H0 True

(No assoc.)

H1 True

(Yes assoc.)

Do not reject H0

(not stat. sig.)

Correct decision

Failing to find a difference when

one exists

Reject H0

(stat. sig.)

Finding a

difference when there is none

Correct decision

Four possible outcomes of any epidemiologic study:

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REALITY

Type I and Type II errors

is the probability of committing type I error.

is the probability of committing type II error.

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DECISION H0 True H1 True

Do not reject H0

(not stat. sig.)

Reject H0

(stat. sig.)

Type I

(alpha error)

“Conventional” Guidelines:

• Set the fixed alpha level (Type I error) to 0.05This means, if the null hypothesis is true, theprobability of incorrectly rejecting it is 5% or less.

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Stu

dy

Res

ult

Empirical Rule

For a Normal distribution approximately, a) 68% of the measurements fall within one

standard deviation around the mean

b) 95% of the measurements fall within two standard deviations around the mean

c) 99.7% of the measurements fall within three standard deviations around the mean

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34.13%34.13%

13.59%

2.28%2.28%

Normal Distribution

50 %50%

13.59%

16• usually set at 5%)

4. A test statistic to assess “statistical significance” is performed to assess the degree to which the data are compatible with the null hypothesis of no

association

5. Given a test statistic and an observed value, you can compute the probability of observing a value as extreme or more extreme than the observed value under the null hypothesis of no association.This probability is called the “p-value”

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Random Error (Chance)

6. By convention, if p < 0.05, then the association between the exposure and disease is considered to be “statistically significant.”

(e.g. we reject the null hypothesis (H0) andaccept the alternative hypothesis (H1))

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Random Error (Chance)

Random Error (Chance)• p-value

– the probability that an effect at least as extreme as that observed could have occurred by chance alone, given there is truly no relationship between exposure and disease (Ho)

– the probability the observed results occurred by chance

– that the sample estimates of association differ only because of sampling variability.

Sever 19

What does p < 0.05 mean?

Indirectly, it means that we suspect that the magnitude of effect observed (e.g. odds ratio) is not due to chance alone (in the absence of biased data collection or analysis)

Directly, p=0.05 means that one test result out of twenty results would be expected to occur due to chance (random error) alone

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Random Error (Chance)

D+ D-

E+ 15 85

E- 10 90

Example:

IE+ = 15 / (15 + 85) = 0.15IE- = 10 / (10 + 90) = 0.10

RR = IE+/IE- = 1.5, p = 0.30

Although it appears that the incidence of disease may behigher in the exposed than in the non-exposed (RR=1.5),the p-value of 0.30 exceeds the fixed alpha level of 0.05.This means that the observed data are relativelycompatible with the null hypothesis. Thus, we do notreject H0 in favor of H1 (alternative hypothesis).

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Random Error (Chance)Take Note:

The p-value reflects both the magnitude of thedifference between the study groups AND the sample size

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• The size of the p-value does not indicate the importance of the results

• Results may be statistically significant but be clinically unimportant

• Results that are not statistically significant may still be important

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Sometimes we are more concerned with estimating the true difference than the

probability that we are making the decision that the difference between

samples is significant

Random Error (Chance)

A related, but more informative, measure knownas the confidence interval (CI) can also be calculated.

CI = a range of values within which the truepopulation value falls, with a certain degree ofassurance (probability).

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Confidence Interval - Definition

A range of values for a variable constructed so that this range has a specified probability of including the true value of the variable

A measure of the study’s precision

Sever

Lower limit Upper limit

Point estimate

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Statistical Measures of Chance

• Confidence interval

– 95% C.I. means that true estimate of effect (mean, risk, rate) lies within 2 standard errors of the population mean 95 times out of 100

Sever 26

Interpreting Results

Confidence Interval: Range of values for a point estimate that has a specified probability of including the true value of the parameter.

Confidence Level: (1.0 – ), usually expressedas a percentage (e.g. 95%).

Confidence Limits: The upper and lower endpoints of the confidence interval.

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Hypothetical Example of 95% Confidence Interval

Exposure: Caffeine intake (high versus low)Outcome: Incidence of breast cancerRisk Ratio: 1.32 (point estimate)p-value: 0.14 (not statistically significant)95% C.I.: 0.87 - 1.98

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_____________________________________________________0.0 0.5 1.0 1.5 2.0

(null value)

95% confidence interval

INTERPRETATION:

Our best estimate is that women with high caffeineintake are 1.32 times (or 32%) more likely to developbreast cancer compared to women with low caffeineintake. However, we are 95% confident that thetrue value (risk) of the population lies between0.87 and 1.98 (assuming an unbiased study).

_____________________________________________0.0 0.5 1.0 1.5 2.0

(null value)

95% confidence interval

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Random Error (Chance)

If the 95% confidence interval does NOT includethe null value of 1.0 (p < 0.05), then we declare a“statistically significant” association.

If the 95% confidence interval includes the null value of 1.0, then the test result is “not statistically significant.”

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Random Error (Chance)

Interpretation:

Interpretation of C.I. For OR and RR:

The C.I. provides an idea of the likely magnitude of the effect and the random variability of the point estimate.

On the other hand, the p-value reveals nothing about the magnitude of the effect or the random variability of the point estimate.

In general, smaller sample sizes have larger C.I.’s dueto uncertainty (lack of precision) in the point estimate.

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Interpreting Results

Selection of Tests of Significance

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Scale of Data1. Nominal: Data do not represent an amount or

quantity (e.g., Marital Status, Sex)

2. Ordinal: Data represent an ordered series of relationship (e.g., level of education)

3. Interval: Data are measured on an interval scale having equal units but an arbitrary zero point. (e.g.: Temperature in Fahrenheit)

4. Interval Ratio: Variable such as weight for which we can compare meaningfully one weight versus another (say, 100 Kg is twice 50 Kg) 33

Which Test to Use?

Scale of Data

Nominal Chi-square test

Ordinal Mann-Whitney U test

Interval (continuous)

- 2 groupsT-test

Interval (continuous)

- 3 or more groupsANOVA

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Protection against Random Error

• Test statistics provide protection from type 1 error due to random chance

• Test statistics do not guarantee protection against type 1 errors due to bias or confounding.

• Statistics demonstrate association, but not causation.

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