Comparing Population Parameters (Z-test, t-tests and Chi-Square test)

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Comparing Population Parameters (Z-test, t-tests
and Chi-Square test)

• 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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Is there an association between Drinking and Lung
Cancer?

• Suppose a case-control study is conducted to test
  the above hypothesis?

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QUESTION Is there a difference between the
proportion of drinkers among cases and controls?
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Elements of Testing hypothesis

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

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Case Control Study of Drinking and Lung Cancer

• Null Hypothesis There is no association between
  Drinking and Lung cancer, P1P2 or P1-P20

Alternative Hypothesis There is some kind of
association between Drinking and Lung cancer,
P1?P2 or P1-P2?0
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Based on the data in the following contingency
table we estimate the proportion of drinkers
among those who develop Lung Cancer and those
without the disease?
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Test Statistic

• How many standard deviations has our estimate
  deviated from the hypothesized value if the null
  hypothesis was true?

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P-value for a two tailed test
P-value 2 PZ gt 2.003 2(.024)0.048 How
does this p-value compared with ?0.05? Since
p-value0.048 lt ?0.05, reject the null
hypothesis H0 in favor of the alternative
hypothesis Ha. Conclusion There is an
association between drinking and lung cancer.
Is this relationship causal?
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Chi-Square Test of Independence(based on a
Contingency Table)
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In the following contingency table estimate the
proportion of drinkers among those who develop
Lung Cancer and those without the disease?
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E111700(60)/400025.5 E1234.5
E211674.5 E222265.5
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How do we calculate P-value?

• SPSS, Epi-Info statistical packages could be used
  to calculate the p-value for various tests
  including the Chi-Square Test
• If p-value is less than 0.05, then reject the
  null hypothesis that rows and column variables
  are independent

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Testing Hypothesis When Two Population Means are
Compared

• H0 ?1 ?2
• Ha ?1? ?2

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QUESTION Is there an association between age
and Lung Cancer?
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Use Two-sample t-test when both samples are
independent

• H0 ?1 ?2 vs Ha ?1 ? ?2
• H0 ?1 - ?2 0 vs Ha ?1 - ?2 ? 0
• t difference in sample means hypothesized
  diff.
• SE of the Difference in Means
• Statistical packages provide p-values and degrees
  of freedom
• Conclusion If p-value is less than 0.05, then
  reject the equality of the means

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Paired t-test for Matched case control study

• H0 ?1 ?2 vs Ha ?1 ? ?2
• H0 ?1 - ?2 0 vs Ha ?1 - ?2 ? 0
• Paired t-test Mean of the differences 0
• SE of the Differences in Means
• Statistical packages provide p-values for paired
  t-test
• Conclusion If p-value is less than 0.05, then
  reject the equality of the means