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Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit
Two-Sample t-Tests (cont’d)
• Two-sample t-tests are used when the dependent variable is an interval- or ratio-level variable—that is, a variable for which it is appropriate to compute a mean
• Examples:– Weight (pounds or kilos)– Scores on a stress scale– Fatigue (measured on a 100-point visual analog
Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit
Alternative Hypotheses for t-Tests
• The alternative (research) hypothesis is that the two group means are different—that is, that the independent variable and dependent variable are related
• Formally stated (nondirectional):– H0: µ1 ≠ µ2 where
Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit
Testing the Null Hypothesis
• In a two-sample t-test, testing the null hypothesis involves computing a test statistic and comparing it to values of what is “improbable,” if the null hypothesis were true – What is “improbable” lies in the tails of a theoretical
sampling distribution
• Here, the relevant distribution is the sampling distribution of the difference between two means
Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit
Assumptions for t-Tests (cont’d)
• Robustness of the t-test—violations of the last two assumptions do not affect statistical decision making if:– Sample sizes are large (40+ per group)– Sample sizes in the two groups are similar
(less than 1.5 times the number in one group as in the other group)
Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit
Types of t-Tests
• There are three different formulas for computing a t statistic, all of which share the goal of testing mean differences between two groups:– Independent groups t-test
Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit
Pooled Variance Formula
• The basic formula for the independent groups t-test uses a pooled variance estimate for the standard error of the difference– The numerator in the equation to compute t is the
difference in sample means (M1 – M2)
– The denominator is the estimated SED
– Degrees of freedom = n1 + n2 – 2
• If calculated t (absolute value) > tabled t (for appropriate df and α), the result is statistically significant
Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit
Pooled Variance Formula (cont’d)
• The pooled variance formula can be used if the population variances are equal–tested via Levene’s test– If the F from this test is significant, the pooled
variance formula should not be used
• Levene’s test is automatically run within SPSS when doing an independent groups t-test
Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit
Separate Variance Formula
• If Levene’s test indicates statistically significant differences in variances, the separate variance formula should be used to estimate SED in computing the t statistic
Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit
Dependent Groups t-Tests
• Dependent groups t-test: Used to test the difference between means for two related groups, or for the same people measured twice – Also called a paired t-test or correlated groups t-test
• Examples: Preintervention versus postintervention Husbands and wives Twins
• The people in the two groups are systematically connected, which lowers variability (the SED)
Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit
Effect Size
• Effect size is a measure of the strength (magnitude) of the relationship between variables in the population– When calculated with sample data, an effect size is an
estimate of “how wrong” the null hypothesis is
• Effect size is a critical construct in meta-analyses—analyses that integrate the results of multiple studies on a given question statistically
Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit
Power Analysis
• Power analysis is most often used during the planning phase of a study to estimate how many participants are needed to minimize the risk of a Type II error
• Just as .05 is the standard acceptable risk for a Type I error, .20 is the standard for a Type II error– So, minimum acceptable power = .80
Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit
Sample Size Estimation
• To estimate the sample size needed to reduce the risk of a Type I error to .05 and the risk of a Type II error to .20, you need an estimate of effect size (d in a two-group mean difference situation)
Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit
Post Hoc Power Analysis
• It is sometimes useful to estimate what power actually was in a study, especially if results are not significant– Somewhat controversial, but can be useful in
interpretation and in thinking about “next steps” for moving an area of research forward
• Here, effect size and sample size are known, and so the analysis solves for power