1. Correlation, Regression Covariate-Adjusted Group Comparisons
and
Covariate-Adjusted Group Comparisons
Robert Boudreau, PhD
Co-Director of Methodology Core
PITT-Multidisciplinary Clinical Research Center
for Rheumatic and Musculoskeletal Diseases
Core Director for Biostatistics
Center for Aging and Population Health
Dept. of Epidemiology, GSPH

2. Flow chart for group comparisons
Measurements to be compared
continuous
discrete
( binary, nominal, ordinal with few values)
Distribution approx normal
or N ≥ 20?
Chi-square
Fisher’s Exact No
Yes
Non-parametrics
T-tests

3. Flow chart for regression models (includes adjusted group comparisons)
Outcome variable continuous or dichotomous?
continuous
dichotomous
Predictor variable categorical?
Time-to-event available? No
Yes
(e.g. groups) No
Yes
Multiple linear
regression
ANCOVA
(Multiple linear regression -
using dummy
variable(s) for
categorical var(s)
Multiple logistic
regression
Cox proportional
hazards regression

4. Health, Aging and Body Composition Study (HABC)
Observational study of 3075 men and women (now in year 13 followup)
age 70-79
45% African-American
Pittsburgh, PA and Memphis, TN
Able to walk 1/4 mile and climb 10 steps (study eligibility criteria)
Designed to assess the relationship of weight and body composition to
incident weight related diseases and
disability
Funded by National Institute on Aging 1997-continuing
University of Pittsburgh
University of Tennessee, Memphis
Coordinating Center: University of California, San Francisco
Laboratory for Epidemiology, Demography and Biometry, NIA

5. HABC: Knee OA Substudy
Year 2: Knee x-rays and MRIs were done on participants with “qualifying knee pain”
“knee OA”
Cases (N=862): Sx (Knee-Pain) SxRxKOA (KL ≥ 2)
African-American female
African-American male
White female
White male

6. Today’s Objective
In HABC: Examine Association between SxRxKOA (knee OA) and CRP adjusted for BMI.
Sowers M, Hochberg M et. al. C-reactive protein as a biomarker of emergent osteoarthritis. Osteoarthritis and Cartilage
Volume 10, Issue 8, August 2002, Pages
N=1025 women aged 27-53; 18% had Knee OA of those
Higher CRP associated with prevalent KOA (also incident KOA)
Bilateral KOA had higher CRP than unilateral
Conclusion: “CRP is highly associated with Knee OA; however, its high correlation with obesity limits its utility as an exclusive marker for knee OA”

7. All White Females in HABC (N=844)
[includes SxRxKOA (n=93); also rest of parent study cohort]
N=5 had CRP > 30 (max=63.2)
N=5

8. log CRP

9. White Females Knee OA P-value No (n=752) Yes (n=92) Mean (SD)
Equal vars
Unequal
logCRP
0.43 (0.83)
0.76 (0.58)
0.0002
<
BMI
25.4 (4.3)
28.8 (5.2)
logCRP SD’s were signif diff (p<0.0001)
=> Use Satterthwaite unequal variance test
Difference in average logCRP: – 0.43 = 0.33

10. Two-Group Unadjusted Comparison Of Means Using Regression with Dummy-coded Groups
proc reg data=kneeOA_vs_noOA; model logCRP= KneeOA; where female=1 and white=1; run;
* No OA is “referent” group (i.e. kneeOA=0)
HABCID logCRP kneeOA BMI

11. proc reg data=kneeOA_vs_noOA; model logCRP= KneeOA; where female=1 and white=1; run;
(intercept)
KneeOA=0
 logCRP= *0 =
KneeOA=1
 logCRP= *1 =

12. White Females: 2-Group Comparison Using Dummy-coded Groups
* No OA is “referent” group (KneeOA=0);
proc reg data=kneeOA_vs_noOA;
model logCRP= KneeOA;
where female=1 and white=1;
run;
“No OA” mean
“kneeOA” mean
difference
from referent
Note: Regression using Dummy (0, 1) for group variable (e.g. KneeOA=0,1)
In regression, equal (pooled) variance is assumed

13. White Females: 2-Group Comparison Using Dummy-coded Groups
* No OA is “referent” group (KneeOA=0);
proc reg data=kneeOA_vs_noOA;
model logCRP= KneeOA;
where female=1 and white=1;
run;
“No OA” mean
“kneeOA” mean
difference
from referent
Same p-value as equal
variance t-test
Note: Regression using Dummy (0, 1) for group variable (e.g. KneeOA=0,1)
In regression, equal (pooled) variance is assumed

15. Pearson Correlation
Pearson Correlation = a measure of linear association

16. Pearson vs Spearman Correlation
A measure of rank order correlation
Works for any general trend that is increasing or
decreasing and not necessarily linear

17. Pearson vs Spearman Correlation
A measure of rank order correlation
Works for any general trend that is increasing or
decreasing and not necessarily linear
Equals Pearson Correlation using the ranks of the
observations instead of actual values
Heuristically: Spearman measures degree that
low goes with low, middle with middle, high with high

18. Regression on a continuous independent variable (BMI)
proc reg data=kneeOA_vs_noOA;
model logCRP=bmi;
where female=1 and white=1 and kneeOA=1;
run; 
logCRP= *BMI
Exact same p-value
as test of
H0: Correlation=0

19. Effect of Centering BMI at 25
proc reg data=kneeOA_vs_noOA;
model logCRP=bmi_minus25;
where female=1 and white=1 and kneeOA=1;
run; 
logCRP= *(BMI-25)
= at BMI=25 (see graphic)

20. Effect of Centering BMI at 25
Model 2: logCRP= *(BMI-25)
= * *BMI
= *BMI

22. Unadjusted
Mean Difference {

23. Notice: At any BMI level,
the mean logCRP difference
between KneeOA vs Not
is smaller than the
unadjusted difference
Unadjusted
Mean Difference {

24. ANCOVA (Analysis of Covariance) Compare logCRP adjusted for BMI
proc reg data=kneeOA_vs_noOA;
model logCRP=KneeOA bmi;
where female=1 and white=1;
run;
Unadjusted diff
Was 0.33
BMI partially
“explains” this
difference 
Note: Equal BMI slopes in each group is being modeled

25. ANCOVA (Analysis of Covariance) Centering BMI at 25
proc reg data=kneeOA_vs_noOA;
model logCRP=KneeOA bmi_minus25;
where female=1 and white=1;
run; 
Note: Equal BMI slopes in each group is being modeled

26. ANCOVA (Analysis of Covariance) Compare logCRP adjusted for BMI

27. Check of ANCOVA Assumption: Equality of BMI slopes: KneeOA vs Not
proc reg data=knee_vs_noOA;
model logCRP=KneeOA bmi BMI_x_KneeOA;
where female=1 and white=1;
run; (“interaction term”)
HABCID logCRP kneeOA BMI BMI_x_KneeOA

28. Check of ANCOVA Assumption: Equality of BMI slopes: KneeOA vs Not
proc reg data=knee_vs_noOA;
model logCRP=KneeOA bmi BMI_x_KneeOA;
where female=1 and white=1;
run;
The “BMI” slopes are not signif different (p=0.8019) => they are parallel

29. logCRP between KneeOA vs Not Adjusted for BMI, Age and Anti-inflammatory Meds

30. Thank you Questions, comments, suggestions or insights?
Remaining time: Open consultation …