Chapter 27

Comparative Statistics

Overview

• Comparative statistics compare groups of participants by sex or age, by exposure or disease status, or by other characteristics.

FIGURE 27- 1 Analytic Plan for Comparing Groups

Hypotheses for Statistical Tests

• Comparative statistical tests usually are designed to test for difference rather than for sameness.

• Statistical test questions are usually phrased in terms of differences: – Are the means different? – Are the proportions different? – Are the distributions different?

Hypotheses for Statistical Tests

• The null hypothesis (H0) describes the expected result of a statistical test if there is no difference between the two values being compared.

• The alternative hypothesis (Ha) describes the expected result if there is a difference.

FIGURE 27-2 Examples of Hypotheses for Statistical Tests

Rejecting the Null Hypothesis

• Rejecting the null hypothesis means concluding that the values are different by rejecting the claim that the values are not different.

• Failing to reject the null hypothesis means concluding that the there is no evidence that the values are different. Functionally, this is like saying that the values are close enough to be considered similar, but failing to reject H0 should never be taken as evidence that the values are the same.

Rejecting the Null Hypothesis

The decision to reject or fail to reject the null hypothesis is based on the likelihood that the result of a test was due to chance.

Rejecting the Null Hypothesis

• One way to understand the concept of chance is to consider the variability in sample populations.

• When a sample population is drawn from a source population, the mean age in the sample population is usually not exactly the mean age of the source population.

• The range of expected values for the mean age of sample populations drawn from a source population can be estimated using statistics.

Rejecting the Null Hypothesis

• Some sample populations will have mean ages that are very close to the mean in the source population; some will be quite far from the mean.

• The 5% of sample means farthest from the true mean are designated “extreme.”

• Thus, by chance, 5% of the samples drawn from a source population will be expected to have an extreme mean.

FIGURE 27- 3 Example of the Distribution of Mean Ages for Sample Populations Drawn from a Larger Source

Population

Rejecting the Null Hypothesis

• Comparative statistical tests accommodate expected variability in sample populations when testing whether 2 groups in a study population are different.

• When the difference in mean ages is great, the statistical test will show that it is highly unlikely that the group means are not significantly different → reject H0.

• If the statistical test shows that the mean ages of cases and controls are fairly close → fail to reject H0.

Interpreting P-values

• A p-value, or probability value, determines whether the null hypothesis (H0) will be rejected.

• The standard is to use a significance level of α = 0.05, or 5%.

• Any statistical test with a result that is in the 5% of most extreme responses expected by chance will result in the rejection of the null hypothesis.

FIGURE 27-4 Interpreting p -Values

FIGURE 27- 5 Examples of One-Sided and Two-Sided Alternative Hypotheses

Interpreting Confidence Intervals

• Confidence intervals (CIs) provide information about the expected value of a measure in a source population based on the value of that measure in a study population.

• The width of the interval is related to the sample size of the study. A larger sample size will yield a narrower confidence interval.

FIGURE 27- 6 Interpreting Confidence Intervals (CIs)

FIGURE 27-7 90%, 95%, and 99% Confidence Intervals (CIs) for the Same Odds Ratio (OR)

Measures of Association

• Some of the most common types of comparative analysis are the odds ratio (OR) used for case-control studies and the rate ratio (RR) used for cohort studies.

• The reference group for an OR or RR should be well-defined.

• The 95% confidence interval provides information about the statistical significance of the tests.

FIGURE 27-8 Example of Odds Ratios for a Case-Control Study of Acute Myocardial Infarction

Selecting an Appropriate Test

Statistical analysts must select a test that is appropriate to the goal of the analysis and the types of variables being analyzed.

Selecting an Appropriate Test

• Parametric tests assume that the variables being examined have particular (usually normal) distributions and that the variances for the variables being examined be similar in the population groups being compared.

• Nonparametric tests do not make assumptions about the distributions of responses.

Selecting an Appropriate Test

• Parametric tests are typically used for ratio and interval variables with relatively normal distributions of responses.

• Nonparametric tests are used for ranked variables, categorical variables, and when the distribution of a ratio or interval variable is non-normal.

One-Sample Tests

• The goal of some statistical tests is to compare the value of a statistic in a study population to some set value.

Two-Sample Tests

• Independent populations: populations in which each individual can be a member of only one of the population groups being compared

• A variety of statistical tests can be used to compare independent populations.

• The appropriate test to use depends on the type of variable being examined.

FIGURE 27-11 Tests for Comparing Two or More Groups

FIGURE 27-12 Examples of Tests for Comparing Males and

Females in a Study Population

FIGURE 27-13 Simplified Version of Figure 27-12

Paired Tests

A different set of tests is used when the goal is to compare before-and-after results in the same individuals.

FIGURE 27-14 Tests for Comparing Matched Populations

FIGURE 27-15 Examples of Tests for Comparing Pretest and Post-Test Results for Participants in a 3-Month Exercise Program