1. Session 4: Measuring Associations Among Factors in Research Studies
Basic Biostatistics: Session 4
Session 4: Measuring Associations Among Factors in Research Studies
Types of Associations:
between 2 factors among different subjects   i.e., do subjects high on one factor tend to be high on another factor?
by predicting one factor from another
between replicated measures of the same factor on a subject
Reliability
Repeatability

2. Correlation
Pearson correlation, r, used when X and Y are different measures
Ranges from -1 (perfect inverse association) to 1 (direct association)
Value of r does not change:
if scales (units) of X and Y are changed
if roles of X and Y are interchanged
Value of r does change:
if the range of X or Y is restricted
if X is fixed and Y is measured (depends on values chosen for X)
i.e., subsets of possible X,Y are selected
p-value for r measures whether r is sig'ly different from 0; small p indicates X-Y association

3. Regression Regression:
fits a straight line to the scatterplot of X and Y
measures association between X and Y, and
quantifies predicting Y from X

4. Regression, continued Association: Prediction:
using r or slope of fitted line are equivalent for assessing whether X and Y are related
significance (p-value) for r and slope are the same
Prediction:
slope = r*(sy/sx) specifies expected ΔY for 1-unit change in X
software gives the fitted line, r (as 100*r2), and:
95% CI for slope [ for data above: ~2.16 ±2SE(slope) = 2.16 ± 0.22 = 1.94 to 2.38 = within red dashed lines]
95% CIs for predictions [ within blue dashed lines vertically for a fixed X ]

5. Reliability
Only used with replicated data, i.e., same quantity measured multiple times on the same subject under identical conditions
With 2 replications, reliability = pearson correlation between the 2 replicates
Dependent on the range of values, i.e., often different for different types of subjects
Dependent on the ordering of the replicates (i.e., change order for half of the subjects, and r can change)
Intraclass correlation, rI, is a more general measure of reliability:
rI is the average r, among all replicate interchanges within subjects
is usually close to r, if the two sets of conditions for the two reps were really equivalent
from an ANOVA table (Session 5), it is calculated as rI = [MSa - MSw]/[MSa + (#reps-1)MSw]

6. Measurement Error
Estimate of measurement error is denoted by sw, and is calculated by finding replicate variability within many subjects and averaging over subjects
ANOVA Table (session 5) gives sw. With 2 replications, sw = sqrt([ΣΔi2]/2n), summed over n subjects
measures how close a single value for an individual is expected to be to his "true" (mean) value
With 95% certainty, a single measurement is within about ±2sw of the true value

7. Repeatability
Measures how close together a pair of values for an individual are expected to be
Is ±1.96*Sqrt(2)*sw = 2.77sw
With 95% certainty, two values are within about ±2.77sw

8. Example of Repeatability
See next slide for graph:
Two pain assessments (from 0 to 100) were made 1 minute apart in each subject
Intraclass correlation r = 0.97
Repeatability = ±2.77sw is about ±18
Note that ±18 is large for a range of 100, even though correlation is almost 1.

9. Repeatability of Visual Analog Pain Scale
Reference: Bijur PE, Silver W, Gallagher EG. Reliability of the Visual Analog Scale for Measurement of Acute Pain, Acad Emerg Med, 2001;8: