Confidence Intervals Nancy D. Barker, M.S.

Statistical Inference

Hypothesis Testing –Is there evidence that the population parameter, e.g., RR, OR, IDR is different from the null value? Interval Estimation –How do we determine the precision of the point estimate by accounting for sampling variability?

Confidence Intervals The goal: Use sample information to compute two numbers, L and U, about which we can claim with a certain amount of confidence, say 95%, that they surround the true value of the parameter.

General CI Formulas Arithmetic scale measures: Multiplicative scale measures: *Note: The variance in this formula refers to the variance [ln (point estimate)].

Confidence Interval Mean: Confidence Level Z-value 90% % % 2.576

Confidence Interval Mean: Example: Calculate a 95% CI for the mean Sample Mean: 26.2 Sample standard deviation: s=5.15 Sample size: n=32

Confidence Interval Calculate a 90% CI for the mean Sample Mean: 26.2 Sample standard deviation: s=5.15 Sample size: n=32 Calculate a 99% CI for the mean Sample Mean: 26.2 Sample standard deviation: s=5.15 Sample size: n=32

Confidence Interval Proportion: Example: Calculate a 95% CI for the proportion Sample proportion: 0.34 Sample size: n=400

95% Confidence Interval Difference between proportions:

95% CI for difference between proportions Example

Interpretation of CI

Large Sample 95% Confidence Interval for RR Risk Ratio (multiplicative scale) Which is equivalent to: Where, * Uses a Taylor Series approximation for the variance

Large Sample 95% Confidence Interval for RR

Large Sample 95% Confidence Interval for OR Odds Ratio (multiplicative scale) Which is equivalent to *Uses a Taylor Series approximation for the variance

Large Sample 95% Confidence Interval for OR

Large Sample 95% Confidence Interval for IDR Incidence Density Ratio (Multiplicative scale) Which is equivalent to *Uses a Taylor Series approximation for the variance

Large Sample 95% Confidence Interval for IDR

Properties of Confidence Intervals The wider the CI, the less precise the estimate. The more narrow the CI, the more precise the estimate. Note: The confidence interval does not address the issue of bias.

What affects the Confidence Interval The level of confidence Sample Size Variation in the data For RR, OR, IDR, the strength of the association

Confidence Interval vs. P-value Similarities –Multiple formulas, (approximate and exact) –Neither account for bias –Statistically equivalent (Theoretically!) Differences –CI provides same information as a statistical test, plus more –CI reminds reader of variability –CI provides range of compatible values (interval estimation) –CI more clearly shows influence of sample size

Confidence Interval vs. P-Value Study #Risk Ratio95% CIP value 12.0(1.2, 3.3) (1.2, 40.8) (0.8, 61.3) (0.9, 1.07) (0.97, 0.99)0.0001

Confidence Interval vs. P-Value Study #Risk Ratio95% CIP value 60.6(0.5, 0.7)<< (0.94, 17.0) (0.99, 16.2) (1.01, 15.8) (2.0, 8.0)