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بسم الله الرحمن الرحيم

بسم الله الرحمن الرحيم. Hypothesis Testing. Two-samples tests, X 2. Dr. Mona Hassan Ahmed Prof. of Biostatistics HIPH, Alexandria University. Z-test (two independent proportions). P1= proportion in the first group P2= proportion in the second group n1= first sample size

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بسم الله الرحمن الرحيم

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  1. بسم الله الرحمن الرحيم

  2. Hypothesis Testing Two-samples tests, X2 Dr. Mona Hassan Ahmed Prof. of Biostatistics HIPH, Alexandria University

  3. Z-test(two independent proportions) P1= proportion in the first group P2= proportion in the second group n1= first sample size n2= second sample size

  4. Critical z = • 1.96 at 5% level of significance • 2.58 at 1% level of significance

  5. Example Researchers wished to know if urban and rural adult residents of a developing country differ with respect to prevalence of a certain eye disease. A survey revealed the following information Test at 5% level of significance, the difference in the prevalence of eye disease in the 2 groups

  6. P1 = 24/300 = 0.08 p2 = 15/500 = 0.03 Answer 2.87 > Z* The difference is statistically significant

  7. t-Test (two independent means) = mean of the first group = mean of the second group S2p = pooled variance

  8. Critical t from table is detected • at degree of freedom = n1+ n2 - 2 • level of significance 1% or 5%

  9. Example Sample of size 25 was selected from healthy population, their mean SBP =125 mm Hg with SD of 10 mm Hg . Another sample of size 17 was selected from the population of diabetics, their mean SBP was 132 mmHg, with SD of 12 mm Hg . Test whether there is a significant difference in mean SBP of diabetics and healthy individual at 1% level of significance

  10. Answer S1 = 12 S2 =11 State H0H0 : 1=2 State H1H1 : 12 Choose αα = 0.01

  11. Answer Critical t at df = 40 & 1% level of significance = 2.58 Decision: Since the computed t is smaller than critical t so there is no significant difference between mean SBP of healthy and diabetic samples at 1 %.

  12. Paired t- test(t- difference) Uses: To compare the means of two paired samples. Example, mean SBP before and after intake of drug.

  13. di = difference (after-before) Sd = standard deviation of difference n = sample size Critical t from table at df = n-1

  14. Example The following data represents the reading of SBP before and after administration of certain drug. Test whether the drug has an effect on SBP at 1% level of significance.

  15. Answer

  16. Answer

  17. Critical t at df = 6-1 = 5 and 1% level of significance = 4.032 Decision: Since t is < critical t so there is no significant difference between mean SBP before and after administration of drug at 1% Level. Answer

  18. Chi-Square test It tests the association between variables... The data is qualitative . It is performed mainly on frequencies. It determines whether the observed frequencies differ significantly from expected frequencies.

  19. Where E = expected frequency O = observed frequency

  20. Critical X2 at df = (R-1) ( C -1) Where R = raw C = column • I f 2 x 2 table X2*=3.84 at 5 % level of significance X2* = 6.63 at 1 % level of significance

  21. In a study to determine the effect of heredity in a certain disease, a sample of cases and controls was taken: Example Using 5% level of significance, test whether family history has an effect on disease

  22. Answer X2 = (80-88)2/88 + (120-112)2/112 + (140-132)2/132 + (160-168)2/168 = 2.165 < 3.84 Association between the disease and family history is not significant

  23. Odds Ratio (OR) • The odds ratio was developed to quantify exposure – disease relations using case-control data • Once you have selected cases and controls  ascertain exposure • Then, cross-tabulate data to form a 2-by-2 table of counts

  24. 2-by-2 Crosstab Notation • Disease status A+C = no. of cases B+D = no. of non-cases • Exposure status A+B = no. of exposed individuals C+D = no. of non-exposed individuals

  25. The Odds Ratio (OR) Cross-product ratio

  26. Example • Exposure variable = Smoking • Disease variable = Hypertension

  27. Interpretation of the Odds Ratio • Odds ratios are relative risk estimates • Relative risk are risk multipliers • The odds ratio of 9.3 implies 9.3× risk with exposure

  28. Interpretation Positive association Higher risk OR > 1 OR = 1 No association OR < 1 Negative association Lower risk (Protective)

  29. OR Confidence level • In the previous example OR = 9.3 • 95% CI is 1.20 – 72.14

  30. Multiple Levels of Exposure

  31. Multiple Levels of Exposure • k levels of exposure  break up data into (k – 1) 22 tables • Compare each exposure level to non-exposed • e.g., heavy smokers vs. non-smokers

  32. Multiple Levels of Exposure Notice the trend in OR (dose-response relationship)

  33. Small Sample Size Formula For the Odds Ratio It is recommend to add ½ to each cell before calculating the odds ratio when some cells are zeros

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