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Testing Statistical Hypothesis for Dependent Samples

Testing Statistical Hypothesis for Dependent Samples. Testing Hypotheses about Two Dependent Means. Dependent Groups t-test Paired Samples t-test Correlated Groups t-test. Steps in Test of Hypothesis. Determine the appropriate test Establish the level of significance: α

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Testing Statistical Hypothesis for Dependent Samples

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  1. Testing Statistical Hypothesisfor Dependent Samples

  2. Testing Hypotheses about Two Dependent Means • Dependent Groups t-test • Paired Samples t-test • Correlated Groups t-test

  3. Steps in Test of Hypothesis • Determine the appropriate test • Establish the level of significance:α • Determine whether to use a one tail or two tail test • Calculate the test statistic • Determine the degree of freedom • Compare computed test statistic against a tabled/critical value Same as Before

  4. 1. Determine the appropriate test • When means are computed for the same group of people at two different points in time (e.g., before and after intervention) • When subjects in one group are paired to subjects in the second group on the basis of some attribute. Examples: • Husbands versus wives • First-born children versus younger siblings • AIDS patients versus their primary caretakers

  5. Continued 1. Determine the appropriate test • Researchers sometimes deliberately pair-match subjects in one group with unrelated subjects in another group to enhance the comparability of the two groups. • For example, people with lung cancer might be pair-matched to people without lung cancer on the basis of age, education, and gender, and then the smoking behavior of the two groups might be compared.

  6. Example: Two Interventions in Same Patients • Suppose that we wanted to compare direct and indirect methods of blood pressure measurement in a sample of trauma patients. Blood pressure values (mm Hg) are obtained from 10 patients via both methods: • X1 = Direct method: radial arterial catheter • X2 = Indirect method: the bell component of the stethoscope

  7. 2. Establish Level of Significance • α is a predetermined value • The convention • α = .05 • α = .01 • α = .001

  8. 3. Determine Whether to Use a One or Two Tailed Test • H0 : µD = 0 • Ha : µD 0 Two Tailed Test if no direction is specified Mean of differences across patients

  9. Continued 3. Determine Whether to Use a One or Two Tailed Test • H0 : µD = 0 • Ha : µD 0 One Tailed Test if direction is specified

  10. Standard Deviation of differences Average of differences Standard Error of differences Sample size 4. Calculating Test Statistics How to calculate standard deviation of differences

  11. Continued 4. Calculating Test Statistics Defining Formula Calculating Formula

  12. Observations 1 and 2 on same patient Difference of observations Squared differences Continued 4. Calculating Test Statistics

  13. Continued 4. Calculating Test Statistics Calculate totals

  14. Continued 4. Calculating Test Statistics

  15. Continued 4. Calculating Test Statistics • Calculate t-statistic from average of differences and standard error of differences

  16. 5. Determine Degrees of Freedom • Degrees of freedom, df, is value indicating the number of independent pieces of information a sample can provide for purposes of statistical inference. • Df = Sample size – Number of parameters estimated • Df for paired t-test is n minus 1

  17. 6. Compare the Computed Test Statistic Against a Tabled Value • α = .05 • Df = n-1 = 9 • tα(df = 9) = 2.26 Two tailed • tα(df = 9) = 1.83 One tailed • RejectH0 if tc is greater than tα

  18. Alternative Approach Estimating Standard deviation of differences from sample standard deviations

  19. Correlation of measures 1 and 2 Variance / Covariance matrix X1 X2 X1 X2 Variance ofthe first measure Variance ofthe second measure Co-variance of Measures of 1 and 2

  20. Variance / Covariance matrix X1 X2 X1 X2 Standard error of difference can be calculated from above table

  21. Variance of second measure Standard deviation of second measure Variance of First measure Standard deviation of First measure Alternative Approach for Calculating standard Error Standard error of Differences Correlation between two measures

  22. Direct Indirect Direct Pearson Correlation 1 .996(**) Sum of Squares and Cross-products 4496.100 4611.00 Covariance 499.567 512.333 N 10 10 Indirect Pearson Correlation .996(**) 1 Sum of Squares and Cross-products 4611.00 4768.00 Covariance 512.333 529.778 N 10 10 Correlation Matrix

  23. Alternative Approach for Calculating standard Error Same value as before

  24. Mean N Std. Deviation Std. Error Mean Pair 1 Direct Method 129.30 10 22.351 7.068 Indirect Method 128.00 10 23.017 7.279 N Correlation Sig. Pair 1 Direct Method & Indirect Method 10 .996 .000 Paired Differences t df Sig. Level (p-value) Mean Std. Deviation Std. Error Mean 95% Confidence Interval of the Difference Lower Upper Pair 1 Direct Method - Indirect Method 1.300 2.163 .684 -.247 2.847 1.901 9 .090 SPSS output for Paired Sample t-test  Paired Samples Statistics Paired Samples Correlations Paired Samples Test

  25. Take Home Lesson How to compare means of paired dependent samples

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