Basic Elements of Testing Hypothesis

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Basic Elements of Testing Hypothesis

• Dr. M. H. Rahbar
• Professor of Biostatistics
• Department of Epidemiology
• Director, Data Coordinating Center
• College of Human Medicine
• Michigan State University

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Inferential Statistics

• Estimation This includes point and interval
  estimation of certain characteristics in the
  population(s).
• Testing Hypothesis about population parameter(s)
  based on the information contained in the
  sample(s).

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Important Statistical Terms

• Population A set which includes all
  measurements of interest to the researcher
• Sample Any subset of the population
• Parameter of interest The characteristic of
  interest to the researcher in the population is
  called a parameter.

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Estimation of Parameters

• Point Estimation
• Interval Estimation (Confidence Intervals)
• Bound on the error of estimation (????)
• The width of a confidence interval is directly
  related to the bound on the error.

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Factors influencing the Bound on the error of
estimation

• Narrow confidence intervals are preferred
• As the sample size increases the bound on the
  error of estimation decreases.
• As the confidence level increases the bound on
  the error of estimation increases.
• You need to plan a sample size to achieve the
  desired level of error and confidence.

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Testing hypothesis about population parameters

• OR or RR
• Mean ?
• Standard deviation ?
• Difference between two population means
• Proportion p
• Difference between two population proportions
• Incidence

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Testing Hypothesis about a Population Prevalence
p
Suppose the Government report that prevalence of
hypertension among adults in Pakistan is at most
0.20 but you as a researcher believe that such
prevalence is greater than 0.20 Now we want to
formally test these hypothesis. Null Hypothesis
H0 P?0.20 vs Alternative Hypothesis Ha
Pgt0.20
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A sample of n100 adults is selected from
Pakistan. In this sample 28 adults are
hypertensive. Do the data provide sufficient
evidence that the Governments figure is wrong,
i.e., Pgt0.20? Test at 5 level of significance,
that is, ?0.05.

• Question
• Estimate prevalenceÞ0.28
• Hypothesized prevalence 0.20
• Is the gap of 0.08 0.28-0.20 considered
  statistically significant at 5 level?

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Testing hypothesis about P

• We need to calculate a test statistic
• How many standard deviations have we deviated if
  the null hypothesis p0.20 was true?

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What is the likelihood of observing a Z2.0 or
more extreme if the Governments figure was
correct?
P-value PZ gt 2.0 0.025 How does this
p-value as compared with ?0.05? If p-value lt
?, then reject the null hypothesis H0 in favor of
the alternative hypothesis Ha. In this
situation we reject the Governments claim in
favor of the alternative hypothesis.??
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Elements of Testing hypothesis

• Null Hypothesis
• Alternative hypothesis
• Level of significance
• Test statistics
• P-value
• Conclusion
• Power of the test

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Is there an association between Drinking and Lung
Cancer?

• What is the most appropriate and feasible study
  design in order to test the above research
  hypothesis?

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Case Control Study of Smoking and Lung Cancer

• Null Hypothesis There is no association between
  Smoking and Lung cancer, P1P2

Alternative Hypothesis There is some kind of
association between Smoking and Lung cancer,
P1?P2.
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In the following contingency table estimate the
proportion and odds of drinkers among those who
develop Lung Cancer and those without the disease?
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QUESTION Is there a difference between the
proportion of drinkers among cases and controls?
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Test Statistic

• A statistical yard stick which is computed based
  on the information contained in the sample under
  the assumption that the null hypothesis is true.
• Knowledge about the sampling distribution of the
  test statistics is needed in determining the
  likelihood of observing extreme values for the
  test statistics in a given situation.

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P-value

• An indicator which measures the likelihood of
  observing values as extreme as the one observed
  based on the sample information, assuming the
  null hypothesis is true.
• P-value is also known as the observed level of
  significance.

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The level of significance (? )

• ? is known as the nominal level of significance.
• If p-value lt ?, then we reject the null
  hypothesis in favor of the alternative
  hypothesis.
• Most of statistical packages give P-value in
  their computer output.
• ? needs to be pre-determined. (Usually 5)

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Type I and Type II errors

• Type I error is committed when a true null
  hypothesis is rejected.
• ? is the probability of committing type I error.
• Type II error is committed when a false null
  hypothesis is not rejected.
• ? is the probability of committing type II error.

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Power of a test

• The power of a test is the probability that a
  false null hypothesis is rejected.
• Power 1 - ?, where ? is the probability of
  committing type II error.
• More powerful tests are preferred. At the design
  stage one should identify the desired level of
  power in the given situation.

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Factors influencing the Power

• The power of a test is influenced by the
  magnitude of the difference between the null
  hypothesis and the true parameter.
• The power of a test could be improved by
  increasing the sample size.
• The power of a test could be improved by
  increasing ?. (this is a very artificial way)

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Minimum Required Sample Size

• Usually a Sample size calculation formula is
  available for most of the well known study
  designs. Some software packages such as Epi-Info
  could also be utilized for the sample size
  calculation purpose.
• It is extremely important to consult a
  biostatistician at the design phase to ensure
  adequate sample is considered for the study.

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Testing hypothesis about one population mean

• H0 ? 16 vs Ha ? gt16
• Z (sample mean hypothesized mean)
• SE of the Mean
• Under the null hypothesis and when n is large,
  (ngt30), the distribution of Z is standard normal.
• P-value
• Conclusion