Polit ln ch06

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Statistics and Data Analysis for Nursing Research, Second EditionDenise F. Polit

Statistics and Data Analysisfor Nursing Research

Second Edition

CHAPTER

t-Tests: Testing Two Mean Differences

6




Two-Sample t-Tests

• Two-sample t-tests are used when the independent variable is a nominal-level variable with two levels—that is, when two groups are being compared

• Examples:– Experimentals versus controls– Men versus women– Children with versus without cystic fibrosis




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

scale)




Null Hypotheses for t-Tests

• The null hypothesis is that the two group means are the same—i.e., that the independent variable and dependent variable are not related

• Formally stated:– H0: µ1 = µ2 where

– µ1 = the mean for Group 1 and

– µ2 = the mean for Group 2




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

– µ1 = the mean for Group 1 and

– µ2 = the mean for Group 2




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




Sampling Distribution

• The mean of this sampling distribution of the difference between two means always equals the population mean difference

• The standard error of this distribution is the standard error of the difference (SED) and is a key component in computing the t statistic




Assumptions for t-Tests

• To use a t-test, it is assumed that:– Cases have been independently and

randomly sampled from the two populations– The outcome variable is normally distributed

in the two populations– The two population variances are equal: This

is the assumption of the homogeneity of variance




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)




Tests for Assumptions

• With small samples, or severely unequal group sizes, violation of assumptions should be tested

• Test for normality: the Kolgomorov-Smirnov test

• Test for equal variances: Levene’s F test




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

Pooled variance formula Separate variance formula

– Dependent groups t-test




Independent Groups t-Tests

• Independent groups t-test: Used to test the difference between means for two unrelated groups

• Examples: Men versus women Teenagers versus adults Lung cancer patients versus pancreatic cancer

patients

• The people in the two groups are not the same people, and they are not related to each other in any systematic way




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




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




Levene’s Test

• Tests the null hypothesis that the variances in the two populations are equal

• SPSS printout below: The null hypothesis is retained, p = .719– Thus, it’s OK to use the pooled variance estimate

Levene’s Test for Equality of

Variances

F Sig.

Child’s Pulse Rate

Equal variances assumed

.134 .719

Equal variances not assumed




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




t- Test Printout

• SPSS printout for independent groups t-test• Same result for both formulas

– Means are not significantly different when α = .05, two-tailed test: p = .081

t-Test for Equality of Means

t df Sig. (Two-Tailed)

Mean Difference

Std. Error Difference

Equal variances assumed

-1.847 18 .081 -10.00 5.41

Equal variances not assumed

-1.847 17.9 .081 -10.00 5.41




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)




Dependent Groups t-Tests (cont’d)

• Dependent groups t-test formula for computing t :– Numerator: The mean difference between

pairs of values (MD)

– Denominator: Estimated standard error of the difference

– df = number of pairs - 1




t- Test Printout

• SPSS printout (selected segments) for dependent groups t-test• Mean differences in direct and indirect measurement of SBP

for the same people are not statistically significant (p = .090) Null hypothesis is retained

Paired Differences

Mean SD Std Error Mean

t df Sig. Two-Tailed

Direct SBP-Indirect SBP

-1.30 2.16 .68 1.901 9 .090




Other Statistical Issues

• Existence of a relationship: t-tests

• Nature of relationship: Direct comparison of the two means

• Precision of estimates: Confidence intervals around the mean difference

• Magnitude of estimates: Effect size




Confidence Intervals

• A CI can be constructed around the difference between the two group means, to indicate precision of sample estimate

• Formula for CIs: – Compute the mean group difference (M1 – M2)

– Then add/subtract SED × the tabled value of t for correct CI (e.g., 95%) and df




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




Cohen’s d

• In a two-group mean-difference situation (like when a t-test is called for), the effect size is Cohen’s d– Sometimes called the standardized mean

difference (SMD)

• The d statistic captures the amount of difference in the two means, in standard deviation units– d = (M1 – M2) ÷ pooled SD




Cohen’s Guidelines

• Cohen provided these qualitative descriptors for d: d = .20 = Small effect

d = .50 = Moderate effect

d = .80 = Large effect

Most effect sizes in nursing research are in the vicinity of .35 (small to moderate)




Effect Size and Power

• Effect size strongly affects the power of a statistical test—i.e., the power to correctly reject the null hypothesis when it is false

• Power is the inverse of β, the risk of a Type II error– Power = 1 – β




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




Power Analysis Components

• There are four components to a power analysis—three must be known or estimated to solve for the fourth:

1. Significance criterion (.05)

2. Power (.80)

3. Population effect size

4. Sample size




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)




Sample Size Estimation (cont’d)

• Effect size estimates:– Prior similar studies– A pilot study

– Cohen’s guidelines (assume no more than small-to-moderate unless there is evidence to the contrary)

– Minimal effect that would be cost effective and/or clinically relevant, and that seems feasible




Power Analysis Examples

• Assume α = .05 and β = .20• Sample size needs (per group) for different

ds:– d = .20 (small effect): 394/group– d = .35 (small-to-moderate effect): 129/group– d = .50 (moderate effect): 64/group– d = .80 (large effect): 25/group

• When group ns are not equal, samples must be larger, all else equal




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




Post Hoc Power Analysis: Some Examples

• Assume α = .05 • β = Risk of committing a Type II error:

n (per group) d Power β25 .20 (small) .11 .89

25 .50 (moderate) .42 .58

25 .80 (large) .81 .19

50 .20 (small) .17 .83

50 .50 (moderate) .71 .29

50 .80 (large) .98 .02




Uses of t-Tests

• Primary use of t-tests—Substantive:– To answer research questions and test

substantive research hypotheses involving comparisons between two groups

• Other uses—Methodologic




Uses of t-Tests (cont’d)

• Evaluations of a study’s internal validity through bias assessment– Internal validity concerns the strength of inferences

about the degree to which the independent variable really affected the dependent variable





• Bias assessment examples:– Selection bias: Are groups being compared

really comparable (e.g., experimentals and controls)?

– Attrition bias: Are people who dropped out similar to those who stayed in a study?





• To contribute evidence about the construct validity of a scale, through the use of the known-groups technique– Involves comparing two groups who are

hypothesized, based on theory or prior research, to differ on the new instrument




t-Tests in SPSS: Independent Groups

• Select Analyze Compare Means Independent Samples T Test

• Move dependent variable from list into slot for Test Variables




t-Tests in SPSS: Independent Groups (cont’d)

• Move independent variable into slot for Grouping Variable

• Click Define Groups pushbutton




t-Tests in SPSS: Independent Groups (cont’d)

• In the next dialog box, enter the values associated with the two groups being compared




t-Tests in SPSS: Dependent Groups

• Select Analyze Compare Means Paired Samples T Test

• Select one or more pairs of variables, moving them from list into Paired Variables slot

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