Confidence Intervals

Nancy D. Barker, M.S.

Statistical Inference

Statistical Inference

Statistical Inference

Statistical Inference

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)].

variancedeviate normal standard estimatepoint

*variance deviate normal standardestimate)nt exp[ln(poi

Confidence Interval

• Mean:

n

szx

2

Confidence Level Z-value

90% 1.645

95% 1.960

99% 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

n

szx

2

28.0 4.24 32

15.596.12.26

32

15.596.12.26

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

27.7 7.24 32

15.5645.12.26

32

15.5645.12.26

28.6 9.23 32

15.5576.22.26

32

15.5576.22.26

Confidence Interval

• Proportion:

Example: Calculate a 95% CI for the proportion

Sample proportion: 0.34

Sample size: n=400

n

ppzp

)1(

2

28.0 4.24 400

)66.0)(34.0(96.1.340

400

)66.0)(34.0(96.134.0

p

p

95% Confidence Interval

• Difference between proportions:

95% CI for difference between proportions

• Example

Interpretation of CI

Large Sample95% 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 Sample95% 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 Sample95% Confidence Interval for IDR

• Incidence Density Ratio (Multiplicative scale)

Which is equivalent to

*Uses a Taylor Series approximation for the variance

Large Sample95% 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 Ratio 95% CI P value

1 2.0 (1.2, 3.3) 0.007

2 7.0 (1.2, 40.8) 0.03

3 7.0 (0.8, 61.3) 0.08

4 0.98 (0.9, 1.07) 0.65

5 0.98 (0.97, 0.99) 0.0001

Confidence Interval vs. P-Value

Study # Risk Ratio 95% CI P value

6 0.6 (0.5, 0.7) <<0.00001

7 4.0 (0.94, 17.0) 0.06

8 4.0 (0.99, 16.2) 0.052

9 4.0 (1.01, 15.8) 0.048

10 4.0 (2.0, 8.0) 0.00009