Biostatistics in Practice

Session 5: Associations and Confounding

Peter D. ChristensonBiostatistician

http://research.LABioMed.org/Biostat

Session 5 Preparation #1

1. We often hear news reports of "seasonally adjusted unemployment rates". Can you think of a logical way that this adjustment could be made?

Session 5 Preparation #2

Unadjusted

Adjusted

What does “adjusted” mean?

How is it done?

From Table 3

Goal One of Session 5Earlier: Compare means for a single measure among groups.

Use t-test, ANOVA.

Session 5: Relate two or more measures.

Use correlation or regression.

Qu et al(2005), JCEM 90:1563-1569.

ΔΔY/ΔX

Goal Two of Session 5

Try to isolate the effects of different characteristics on an outcome.

Previous slide:

Gender

BMI

GH Peak

CorrelationVisualize Y (vertical) by X (horizontal) scatter plot.

Pearson correlation, r, is used to measure association between two measures X and Y

Ranges from -1 (perfect inverse association) to 1 (perfect direct association)

Value of r does not depend on:

scales (units) of X and Ywhich role X and Y assume, as in a X-Y plot

Value of r does depend on: the ranges of X and Yvalues chosen for X, if X is fixed & Y is measured

Graphs and Values of Correlations

Logic for Value of Correlation

Σ (X-Xmean) (Y-Ymean)

√Σ(X-Xmean)2 Σ(Y-Ymean)2

r =

+

+-

-

Statistical software gives r.

Correlation Depends on Ranges of X & Y

Graph B contains only the graph A points in the ellipse.

Correlation is reduced in graph B.

Thus: correlations for the same quantities X and Y may be quite different in different study populations.

BA

Correlation and Measurement Precision

A lack of correlation for the subpopulation with 5<x<6 may be due to inability to measure x and y well.

Lack of evidence of association is not evidence of lack of association.

B

A

r=0 for s

Boverall

5 6

12

10

Regression

Again: Y (vertical) by X (horizontal) scatterplot, as with correlation. See next slide.

X and Y now assume unique roles: Y is an outcome, response, output, dependent

variable. X is an input, predictor, independent variable. Regression analysis is used to:

Measure X-Y association, as with correlation. Fit a straight line through the scatter plot, for:

Prediction of Y from X. Estimation of Δ in Y for a unit change in X (slope = “effect” of X on Y).

Regression Example

ei

MinimizesΣei

2

Range for Individuals

Range for mean

Statistical software gives all this info.

X-Y Association

If slope=0 then X and Y are not associated.

But the slope measured from a sample will never be 0. How different from 0 does a measured slope need to be in order to claim X and Y are associated?

[ Side note: It turns out that slope=0 is equivalent to correlation r = 0. ]

X-Y Association

Test slope=0 vs. slope≠0, with the rule:

Claim association (slope≠0) if

tc=|slope/SE(slope)| > t ≈ 2.

There is a 5% chance of claiming an X-Y association that really does not exist.

Note similarity to t-test for means:

tc=|mean/ SE(mean)|

Formula for SE(slope) is in statistics books.

Example Software OutputThe regression equation is: Y = 81.6 + 2.16 X

Predictor Coeff StdErr T PConstant 81.64 11.47 7.12 <0.0001X 2.1557 0.1122 19.21 <0.0001

S = 21.72 R-Sq = 79.0%

Predicted Values:

X: 100Fit: 297.21SE(Fit): 2.1795% CI: 292.89 - 301.5295% PI: 253.89 - 340.52

Predicted y = 81.6 + 2.16(100)

Range of Ys with 95% assurance for:

Mean of all subjects with x=100.

Individual with x=100.

19.21=2.16/0.112 should be between ~ -2 and 2 if “true” slope=0.

Refers to Intercept

Multiple Regression

We now generalize to prediction from multiple characteristics.

The next slide gives a geometric view of prediction from two factors simultaneously.

Multiple Regression: Geometric View

LHCY is the Y (homocysteine) to be predicted from the two X’s: LCLC (folate) and LB12 (B12).

LHCY = b0 + b1LCLC + b2LB12 is the equation of the plane

Suppose multiple predictors are continuous.

Geometrically, this is fitting a slanted plane to a cloud of points:

www.StatisticalPractice.com

Multiple Regression: Software

Multiple Regression: Software

Output: Values of b0, b1, and b2 for

LHCY = b0 + b1LCLC + b2LB12

How Are Coefficients Interpreted?

LHCY = b0 + b1LCLC + b2LB12

OutcomePredictors

LHCY

LCLC

LB12

LB12 may have both an independent and an indirect (via LCLC) association with LHCY

Correlation

b1 ?

b2 ?

Coefficients: Meaning of their Values

LHCY = b0 + b1LCLC + b2LB12

OutcomePredictors

LHCY increases by b2 for a 1-unit increase in LB12

… if other factors (LCLC) remain constant, or

… adjusting for other factors in the model (LCLC)

May be physiologically impossible to maintain one predictor constant while changing the other by 1 unit.

Another Example: HDL Cholesterol Std Coefficient Error t Pr > |t|

Intercept 1.16448 0.28804 4.04 <.0001 AGE -0.00092 0.00125 -0.74 0.4602 BMI -0.01205 0.00295 -4.08 <.0001BLC 0.05055 0.02215 2.28 0.0239PRSSY -0.00041 0.00044 -0.95 0.3436DIAST 0.00255 0.00103 2.47 0.0147GLUM -0.00046 0.00018 -2.50 0.0135SKINF 0.00147 0.00183 0.81 0.4221LCHOL 0.31109 0.10936 2.84 0.0051

The predictors of log(HDL) are age, body mass index, blood vitamin C, systolic and diastolic blood pressures, skinfold thickness, and the log of total cholesterol. The equation is:

Log(HDL) = 1.16 - 0.00092(Age) +…+ 0.311(LCHOL)

www.

Statistical

Practice

.com

Output:

HDL Example: Coefficients

Interpretation of coefficients on previous slide:

1. Need to use entire equation for making predictions.

2. Each coefficient measures the difference in expected LHDL between 2 subjects if the factor differs by 1 unit between the two subjects, and if all other factors are the same. E.g., expected LHDL is 0.012 lower in a subject whose BMI is 1 unit greater, but is the same as the other subject on other factors.

Continued …

HDL Example: Coefficients

Interpretation of coefficients two slides back:

3. P-values measure the association of a factor with Log(HDL) , if other factors do not change.

This is sometimes expressed as “after accounting for other factors” or “adjusting for other factors”, and is called independent association.

SKINF probably is associated. Its p=0.42 says that it has no additional info to predict LogHDL, after accounting for other factors such as BMI.

Special Cases of Multiple Regression

So far, our predictors were all measured over a continuum, like age or concentration.

This is simply called multiple regression.

When some predictors are grouping factors like gender or ethnicity, regression has other special names:

ANOVA

Analysis of Covariance

Analysis of Variance

• All predictors are grouping factors.

• One-way ANOVA: Only 1 predictor that may have only 2 “levels”, such as gender, or more levels, such as ethnicity.

• Two-way ANOVA: Two grouping predictors, such as decade of age and genotype.

Analysis of Variance

• Interaction in 2-way ANOVA: Measures whether the effect of one factor depends on the other factor. Difference of a difference in outcome. E.g.,

(Female – Male)Asian – (Female – Male)Hispanic

• The effect of gender, adjusted for ethnicity, is a weighted average of gender differences within ethnic subgroups, i.e., of :

(Female – Male)Asian and (Female – Male)Hispanic

Analysis of Covariance

• At least one predictor is a grouping factor, such as ethnicity, and at least one predictor is continuous, such as age, called a “covariate”.

• Interest is often on comparing the groups.

• The covariate is often a nuisance.

Confounder: A covariate that both co-varies with the outcome and is distributed differently in the groups.

A Study Using Analysis of Covariance

J Clin Endocrin Metab 2006 Nov; 91(11):4424-32.

Potential doping test for athletes.

Study Goal: Outcomes are IGF-1 and Collagen Markers

Determine the relative and combined explanatory power of age, gender, BMI, ethnicity, and sport type on the markers.

*

* for age, gender, and BMI.

Figure 2.One conclusion is lack of differences between ethnic IGF-1 means, after adjustment for age, gender, and BMI (Fig 2).

How are these adjustments made?

Adjustment: For a Single Continuous Characteristic

We simulate data for Caucasians and Africans only for simplicity, to demonstrate attenuation of a large 155-140=15 μg/L ethnic difference to a small 160-157=3 μg/L age-adjusted ethnic difference.

157155 160

140

15 = Diff

3 = Adj Diff

Adjustment: For a Single Continuous Characteristic

Problem:

Want to compare groups on IGF-1.

Groups to be compared (ethnicities) have different mean ages, and IGF-1 tends to decrease with age.

Solution:

Make groups appear to have the same mean age.

250

200

150

100

Age (Years)

IGF

1 (

ug

/L)

IGF1 Adjustment for Age - Simulated Data

(Mean)

140

155

15 = Diff

160157

Diff = 3

Unadjusted 22.2 Adjusted

CaucasianAfrican

15 30

Adjustment: Software Output

Unadjusted Group Difference:

Estimated IGF-1 = intercept + b0(indicator)

Age-adjusted Group Difference :

Estimated IGF-1 = intercept + b0(indicator) +b1(age)

Indicator = 0 if African, 1 if Caucasian.

15

3

Adjustment: Software Input

Select:

Regression or Analysis of Covariance.

Usually menu such as

Output:

Values of b0, b1, and b2 (and p-values and CIs) for

IGF1 = b0 + b1(indicator) + b2(age)

Adjustment: Software Output

Unadjusted Group Difference:

Estimated IGF-1 = intercept + b0(indicator)

Age-adjusted Group Difference :

Estimated IGF-1 = intercept + b0(indicator) +b1(age)

Indicator = 0 if African, 1 if Caucasian.

15

3

Δ in IGF-1 per year(weighted average of the 2 groups)