basic elements of testing hypothesis dr. m. h. rahbar professor of biostatistics department of...
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![Page 1: Basic Elements of Testing Hypothesis Dr. M. H. Rahbar Professor of Biostatistics Department of Epidemiology Director, Data Coordinating Center College](https://reader036.vdocuments.net/reader036/viewer/2022062516/56649d485503460f94a2461b/html5/thumbnails/1.jpg)
Basic Elements of Testing Hypothesis
Dr. M. H. RahbarProfessor 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: P0.20 vs Alternative Hypothesis Ha: P>0.20
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A sample of n=100 adults is selected from Pakistan. In this sample 28 adults are hypertensive. Do the data provide sufficient evidence that the Government’s figure is wrong, i.e., P>0.20? Test at 5% level of
significance, that is, =0.05. Question:
Estimate prevalence=Þ=0.28
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 p=0.20 was true?
(0.28 0.20) /(( (0.20)(1 0.20) 100))
2.0
Z
Z
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What is the likelihood of observing a Z=2.0 or more extreme if the
Government’s figure was correct?
P-value= P[Z > 2.0] = 0.025
How does this p-value as compared with =0.05?
If p-value < , then reject the null hypothesis H0 in favor of the alternative hypothesis Ha.
In this situation we reject the Government’s 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, P1=P2
Alternative Hypothesis: There is some kind of association between Smoking and Lung cancer, P1P2.
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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?
Lung Cancer Total Case Control Drinker Yes A=33 B=27 60 No C=1667 D= 2273 3940
P1=33/1700 P2=27/2300 Odds1=33/1667 Odds2=27/2273
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QUESTION: Is there a difference between the proportion of drinkers among cases and controls?
G rou p 1D isease
P 1 = p rop ortion o f d rin kers
G rou p 2N o D isease
P 2 = p rop ortion o f d rin k rs
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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 < , 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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Decision made about the validity of null hypothesis
Rejected Not rejected Null Hypothesis
True Type I Error
No Error
False No Error
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: >16 • Z= (sample mean – hypothesized mean)
SE of the Mean
• Under the null hypothesis and when n is large, (n>30), the distribution of Z is standard normal.
• P-value• Conclusion
n