# Does the regression line tell anything about mean

I am trying to find the association of BMI on the age at onset of hypertension in diabetes vs non-diabetes patients. Or in other words, does diabetes (if the patient is already diabetic before the hypertension diagnosis) modify the association of BMI and age at onset of hypertension.

To find this, I have tried a multiple linear regression model with onset age of hypertension as dependent variable and BMI as independent variables adjusted for other covariates.

I am getting a slope of -0.33 * BMI for non-diabetic hypertensives and -0.65 * BMI for diabetic hypertensives. i.e. slope is almost two fold for the diabetic hypertensives compared to non-diabetics.

So I was concluding that the average rate of change in onset age of hypertension with respect to increase in each unit of BMI is more in diabetic patients than in non-diabetic patients.

Now, counter intuitive to this result the mean onset age of diabetic hypertensive patients (55 years) is higher than non-diabetic hypertensives (46 years).

Shouldn't the diabetic hypertensive patients have lower mean onset age than the non-diabetic hypertensives since the slope is steeper for diabetic hypertensives?

Can anyone tell me if there is any flaw in this result and conclusion.

Thanks much

Here is the plot • try to post plot in here, it will be helpfull. – Ladislav Naďo Jan 7 '14 at 6:34
• My first guess is you've run into Simpsons Paradox -- e.g. where in this case the distribution of BMIs is not the same across groups -- but there's not enough information given to be sure. Note that you have four variables here, so there are several ways the paradox can come in. – Glen_b Jan 7 '14 at 7:42
• What is the distribution of BMI in the diabetes vs non-diabetic groups? Are your diabetes patients aleady "treated" with lifestyle, diet interventions? There are probably several reasonable explanations, and it would be impossible to know exactly what was going on without more information. Can you give the estimates for the coefficients? How did you calculate the mean values in diabetics vs non-diabetic? Are these estimated from the model coefficients, or just the means observed in the data? – D L Dahly Feb 6 '14 at 17:16

Does the regression line tell anything about mean?

Yes...

One part of the definition of the best-fit straight line (regression line) in scatter plot is that it passes through the point determined by the mean values of x and y.

• Thanks for your comments. In this scenario, do you think the female diabetic hypertensive patients should have lower mean onset age than the non-diabetic female hypertensives since the slope is steeper for female diabetic hypertensives? – arshad Jan 7 '14 at 6:46
• The more steeper slope of the regression line does not necessarily means that the mean value must be lower. So my answer is no. – Ladislav Naďo Jan 7 '14 at 7:02

Yes it does.

The mean value x_bar and y_bar is always on the regression line, which is, the definition of linear regression.

• This says exactly what the previous answer does. Did you intend to add some more information to that? – whuber Feb 6 '14 at 16:47