Testing Statistical Hypothesis for Dependent Samples

Testing Hypotheses about Two Dependent MeansDependent Groups t-testPaired Samples t-testCorrelated Groups t-test

Steps in Test of HypothesisDetermine the appropriate test Establish the level of significance:Determine whether to use a one tail or two tail testCalculate the test statisticDetermine the degree of freedomCompare computed test statistic against a tabled/critical valueSame as Before

1. Determine the appropriate testWhen 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 wivesFirst-born children versus younger siblingsAIDS patients versus their primary caretakers

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.

Example: Two Interventions in Same PatientsSuppose 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 catheterX2 = Indirect method: the bell component of the stethoscope

2. Establish Level of Significance is a predetermined valueThe convention = .05 = .01 = .001

3. Determine Whether to Use a One or Two Tailed TestH0 : D = 0Ha : D 0

Meanof differencesacross patientsTwo Tailed Test if no direction is specified

3. Determine Whether to Use a One or Two Tailed TestH0 : D = 0Ha : D 0

One Tailed Test if direction is specified

4. Calculating Test StatisticsHow to calculatestandard deviationof differences

4. Calculating Test Statistics Defining FormulaCalculating Formula

4. Calculating Test Statistics

4. Calculating Test StatisticsCalculatetotals

4. Calculating Test Statistics

4. Calculating Test StatisticsCalculate t-statistic from average of differences and standard error of differences

5. Determine Degrees of FreedomDegrees 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 estimatedDf for paired t-test is n minus 1

6. Compare the Computed Test Statistic Against a Tabled Value = .05Df = n-1 = 9

t(df = 9) = 2.26 Two tailedt(df = 9) = 1.83One tailedReject H0 if tc is greater than t

Alternative ApproachEstimating Standard deviation of differences from sample standard deviations

Variance / Covariance matrix X1 X2

X1

X2Variance of the first measureVariance of the second measureCo-variance ofMeasures of 1 and 2

Variance / Covariance matrix X1 X2X1X2Standard error of differencecan be calculated from above table

S21S22

Alternative Approach for Calculating standard ErrorStandard error ofDifferences Correlationbetweentwo measures

Correlation Matrix DirectIndirectDirectPearson Correlation1.996(**) Sum of Squares and Cross-products4496.1004611.00 Covariance499.567512.333 N1010IndirectPearson Correlation.996(**)1 Sum of Squares and Cross-products4611.004768.00 Covariance512.333529.778 N1010

Alternative Approach for Calculating standard ErrorSame value as before

SPSS output for Paired Sample t-testPaired Samples Statistics

Paired Samples CorrelationsPaired Samples Test

Take Home LessonHow to compare means of paired dependent samples

Mean difference between pairs of values**All the matrix has to be added upRemove the correlation because it is pairedThere is really only one sample but two information (pre-test and post-test)