One question regarding to the simple regression

[liujx80]

there are 3 variables. X,Y,Z.
Assuming
Y = a + b*X + errorTerm
Z = c + d*X + errorTerm,

Y = g + h*Z + errorTerm;

what can we say about the relationtion between h and (b,d)? under what
condition so we can have "h = b/d"?

thanks

 

[haruspex]

If h, b and d represent the actual parameter values then clearly you have:
Y = a + b*X + errorTerm_xy
Z = c + d*X + errorTerm_xz,

Y = g + h*(c + d*X + errorTerm_xy) + errorTerm_xz;
whence:
g = a - h*c
h = b/d
errorTerm_zy = h*errorTerm_xy + errorTerm_xz
But perhaps you mean b, d and h to be the estimated parameters. That gets more subtle.

 

[chiro]

Hey liujx80 and welcome to the forums.

Following on from haruspex's post above, are you trying to estimate them or do you know them already? If you are estimating them, do you have a specific distribution or analysis in mind? Do you have priors?

 

[liujx80]

Thanks guys. The parameters are all estimated. I don't have distributions in mind. In general cases, can I use h=b/d using estimates? How wrong it could be? I had the same steps with haruspex's. However I'm not sure we then can say all assumptions are met. As you guys pointed out, it depends on estimations, which I have no clue .

Thanks

 

[blue_raver22]

I would first clarify that the provided information describes a multiple regression model, as there are multiple independent variables (X and Z) being used to predict a dependent variable (Y). In this case, the relationship between h and (b,d) can be interpreted as the effect of Z on Y, while controlling for the effect of X. In other words, h represents the change in Y for every unit change in Z, while keeping X constant.

The condition for h to equal b/d would be when the effect of X on Y is equal to the effect of X on Z. This means that X has the same impact on both Y and Z, and therefore, the relationship between Y and Z is directly proportional. In other words, as X increases, both Y and Z will increase by the same amount.

However, it is important to note that this condition is not always met in real-world data and it is possible for h to be different from b/d. This could indicate that there are other factors at play that are affecting the relationship between Y and Z, and the effect of X on each of them may not be the same.

Overall, the relationship between h and (b,d) depends on the specific data and context of the study. As a scientist, it is important to carefully examine the data and consider all possible factors before making any conclusions about the relationship between variables.