why Z score values are correlated in opposite direction of raw value?

Question

I am trying to find the correlation between age at onset of diabetes and Body Mass Index of patients. When i performed correlation test in R I get a positive "r" between age and BMI.. but when the same BMI is converted to z-scores I am getting negative correlation. How would interpret this result. Can someone please help.. Following is my results from R

WITH BMI

Call:
lm(formula = age ~ bmi, data = temp)

Residuals:
    Min      1Q  Median      3Q     Max 
-4.3649 -1.8292 -0.2815  1.3361  8.6716 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  5.76891    0.45737  12.613   <2e-16 ***
bmi          0.23689    0.02382   9.944   <2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 2.427 on 508 degrees of freedom
Multiple R-squared:  0.1629,    Adjusted R-squared:  0.1613 
F-statistic: 98.89 on 1 and 508 DF,  p-value: < 2.2e-16

WITH BMI Z SCORE

Call:
lm(formula = age ~ z, data = temp)

Residuals:
    Min      1Q  Median      3Q     Max 
-4.4553 -1.9565 -0.1407  1.4838  7.0065 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  10.0316     0.1188  84.433  < 2e-16 ***
z            -0.7271     0.1439  -5.054 6.05e-07 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 2.588 on 508 degrees of freedom
Multiple R-squared:  0.04787,   Adjusted R-squared:  0.046 
F-statistic: 25.54 on 1 and 508 DF,  p-value: 6.055e-07

This is the code to calculate Z-score. The code is in PERL

http://pastie.org/10843530

Following is the regression result of log transformed age vs z , actual BMI and gender

lm(formula = log(age) ~ NZ + bmi + gender, data = temp)

Residuals:
     Min       1Q   Median       3Q      Max 
-0.50191 -0.07392  0.01485  0.08512  0.46986 

Coefficients:
             Estimate Std. Error t value Pr(>|t|)    
(Intercept)  0.731636   0.046103  15.869   <2e-16 ***
NZ          -0.305887   0.010358 -29.532   <2e-16 ***
bmi          0.079247   0.002353  33.681   <2e-16 ***
genderM     -0.022121   0.012759  -1.734   0.0836 .  
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.1416 on 506 degrees of freedom
Multiple R-squared:  0.6981,    Adjusted R-squared:  0.6963 
F-statistic:   390 on 3 and 506 DF,  p-value: < 2.2e-16

Answer 1

I will first address a reformulation of your question, which is statistically relevant: if $\def\cov{\text{cov}}\cov(X_1, X_2) > 0$, $\cov(X_2, Y) > 0$, is it possible that $\cov(X_1, Y) < 0$ ?

The answer is yes, although it can seem counterintuitive. Here is an example of positive definite matrix exhibiting the desired behavior:

> D <- matrix( c(1, 0.2, -0.1, 0, 0.9, 0.3, 0, 0, 0.8), nrow=3 ) 
> D
     [,1] [,2] [,3]
[1,]  1.0  0.0  0.0
[2,]  0.2  0.9  0.0
[3,] -0.1  0.3  0.8
> D %*% t(D)
     [,1] [,2]  [,3]
[1,]  1.0 0.20 -0.10
[2,]  0.2 0.85  0.25
[3,] -0.1 0.25  0.74

So you can have $\cov(X_1, X_2) = 0.2$, $\cov(X_2, Y) = 0.25$ and $\cov(X_1, Y) = -0.1$.

This case corresponds to a situation where

• $X_1$ has variance $1$,
• $X_2 = 0.2\> X_1 + \varepsilon$ with $\varepsilon$ independent of $X_1$ and $\def\var{\text{var}}\var(\varepsilon) = 0.9^2$
• $Y = -0.1 \> X_1 + 0.3 \>\varepsilon + \varepsilon' = -0.16 \> X_1 + 0.3 \> X_2 + \varepsilon'$ with $\varepsilon'$ independent of $X_1, \varepsilon$ and $\var(\varepsilon') = 0.8^2$.

More generally, this will not be possible if $Y$ is independent of $X_1$ conditionally to $X_2$.

Now back to your problem: in your sample, early onset implies high $z$ value (i.e. high BMI for an individual of this age) but low BMI -- well, yes, the younger you are, the lower your BMI (cf this chart for the evolution of the BMI in the age range in your sample). So the (slightly overweight) guys you have in your incident diabetes sample have a BMI correlated with their age. The fact that the younger they are, the chubbier, doesn’t counterbalance this.

I think it could be interesting to try to produce a chart of BMI for incident diabetes cases, as a function of age, and to compare it to the above linked chart. I think this will help to grasp what’s happening in these data.