Biostatistics in Practice Session 5: Associations and confounding Youngju Pak, Ph.D. Biostatistician...
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Transcript of Biostatistics in Practice Session 5: Associations and confounding Youngju Pak, Ph.D. Biostatistician...
Biostatistics in PracticeSession 5:
Associations and confoundingYoungju Pak, Ph.D.
Biostatistician
http://research.LABioMed.org/Biostat
1
Revisiting the Food Additives Study
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
5
Correlation
Standard English word correlate• to establish a mutual or reciprocal relation
between <correlate activities in the lab and the field> b: to show correlation or a causal relationship between
In statistics, it has a more precise meaning
6
Correlation in Statistics Correlation: measure of the strength of LINEAR
association
Positive correlation: two variables move to the same direction As one variable increase, other variables also tends to increase LINEARLY or vice versa.• Example: Weight vs Height
Negative correlation: two variables move opposite of each other. As one variable increases, the other variable tends to decrease LINEARLY or vice versa (inverse relationship).• Example: Physical Activity level vs. Abdominal height (Visceral Fat)
7
Pearson r correlation coefficient
r can be any value from -1 to +1 r = -1 indicates a perfect negative LINEAR
relationship between the two variables
r = 1 indicates a perfect positive LINEAR
relationship between the two variables
r = 0 indicates that there is no LINEAR
relationship between the two variables
8
Scatter Plot: r= 1.0
0
2
4
6
8
10
12
14
0 2 4 6 8 10
9
Scatter Plot: r= -1.0
0
1
2
3
4
5
6
7
1 2 3 4 5 6
10
Scatter Plot: r= 0
0
2
4
6
8
10
12
14
0 2 4 6 8 10 12 14
Anemic women: Anemia.sav n=20
Hb(g/dl) PCV(%)
11.1 3510.7 4512.4 4713.1 3110.5 309.6 2512.5 3313.5 35… …
r expresses how well the data fits in a straight
line. Here, Pearson’s r =0.673
Correlations in real data
Logic for Value of Correlation
Σ (X-Xmean) (Y-Ymean)
√Σ(X-Xmean)2 Σ(Y-Ymean)2Pearson’s 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
Simple Linear Regression (SLR) X and Y now assume unique roles: Y is an outcome, response, output, dependent
variable. X is an input, predictor, explanatory, independent
variable. Regression analysis is used to:
Measure more than X-Y association, as with correlation.
Fit a straight line through the scatter plot, for:Prediction of Ymean from X. Estimation of Δ in Ymean for a unit change in X
= Rate of change of Ymean as a unit change in X
(slope = regression coefficient measure “effect” of X on Y).
SLR Example
ei
Minimizes
Σei2
Range for Individuals
Range for mean
Statistical software gives all this info.
Range for Individuals
Range for individuals
Hypothesis testing for the true slope=0
H0: true slope = 0 vs. Ha: true 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: Ymean = 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 Lienar 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
LHCYmean = b0 + b1LCLC + b2LB12
How Are Coefficients Interpreted?
LHCYmean = 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
Mean 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.
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
*
* for age, gender, and BMI.
Figure 2.
Determine the relative and combined explanatory power of age, gender, BMI, ethnicity, and sport type on the markers.
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) mean = 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 mean 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: CoefficientsInterpretation of coefficients two slides back:
3. P-values measure how strong the association of a factor with Log(HDL) is , 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.
Two way ANOVA
• 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.,
(Trt.-– control)Female – (Trt. – control)Male
• The effect of treatment, adjusted for gender, is a weighted average of group differences over two gender group, i.e., of :
(Trt.– control)Female and (Trt. – control)Male
Analysis of Covariance
• At least one primary predictor is a grouping factor, such as treatment group , 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.