1 Testing Statistical Hypothesis for 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 1. Determine the appropriate test 2. Establish the level of significance: α 3. Determine whether to use a one tail or two tail test 4. Calculate the test statistic 5. Determine the degree of freedom 6. Compare computed test statistic against a tabled/critical value Same as Before

4 1. Determine the appropriate test 1. When means are computed for the same group of people at two different points in time (e.g., before and after intervention) 2. 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 1. Determine the appropriate test 3. 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 H 0 : µ D = 0 H a : µ D  0 Mean of differences across patients Two Tailed Test if no direction is specified

9 3. Determine Whether to Use a One or Two Tailed Test H 0 : µ D = 0 H a : µ D  0 One Tailed Test if direction is specified

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

11 Defining FormulaCalculating Formula 4. Calculating Test Statistics

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

13 4. Calculating Test Statistics Calculate totals

14 4. Calculating Test Statistics

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

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 t α (df = 9) = 2.26 Two tailed t α (df = 9) = 1.83One tailed Reject H 0 if t c is greater than t α

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

19 Variance / Covariance matrix S21S21 S 12 S22S22 X 1 X 2 X1X2X1X2 Variance of the first measure Variance of the second measure Co-variance of Measures of 1 and 2 Correlation of measures 1 and 2

20 Variance / Covariance matrix X1 X2 S21S21 S22S22 X1X2X1X2 Standard error of difference can be calculated from above table

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

22 Correlation Matrix DirectIndirect DirectPearson Correlation1.996(**) Sum of Squares and Cross-products Covariance N1010 IndirectPearson Correlation.996(**)1 Sum of Squares and Cross-products Covariance N1010

23 Alternative Approach for Calculating standard Error Same value as before

24 Paired Samples Statistics Direct Method MeanN Std. Deviation Std. Error Mean Pair 1 Indirect Method Paired Samples Correlations NCorrelationSig. Pair 1 Direct Method & Indirect Method Paired Samples Test Paired Differencest dfdf Sig. Level (p-value) Mean Std. Deviation Std. Error Mean 95% Confidence Interval of the Difference LowerUpper Pair 1 Direct Method - Indirect Method SPSS output for Paired Sample t- test

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