Error propagation in combined linear models

Cross Validated Asked by Chochot on January 5, 2022

I have a set of observed values ($y_{1Obs}$) and 3 predictive variables (n = 27). I use multiple linear regression to create a linear model:

$Z_1=alpha_0 + alpha_1 W_1 + alpha_2 X_1 + alpha_3 Y_1$

Where $Z_1$ is my response variable, $alpha_0$ is the intercept, $alpha_1 , alpha_2 , alpha_3$ are regression coefficients and $W_1, X_1, Y_1$ are predictive variables.

I also have a second set of observed values ($y_{2Obs}$) and 3 different predictive variables (n = 27). I again use multiple linear regression to create a second linear model of the same form:

$Z_2=beta_0 + beta_1 W_2 + beta_2 X_2 + beta_3 Y_2$

Where $Z_2$ is my response variable, $beta_0$ is the intercept, $beta_1 , beta_2 , beta_3$ are regression coefficients and $W_2, X_2, Y_2$ are predictive variables.

With 27 predicted values from each model I then calculate the predicted change between the two:

$Delta Z_{Pred}=Z_2 – Z_1$

While my observed change comes from the two sets of observed values used to create the two linear models:

$Delta Z_{Obs}=y_{2Obs} – y_{1Obs}$

My question is: what formula do I use to calculate the RMSE of my predictions ($Delta Z_{Pred}$) that will propagate the errors from predictions of $Z_1$ and $Z_2$?

I’ve tried using the following formula though I’m not convinced this is correct:

$RMSE_{Delta Z_{Pred}} = sqrt{{RMSE_1}^2 + {RMSE_2}^2}$

Where $RMSE_1$ and $RMSE_2$ are the RMSEs from the first two models shown above.

One Answer

Not a complete answer but a couple of hopefully useful comments:

  • The formula $RMSE_{Delta Z_{Pred}} = sqrt{{RMSE_1}^2 + {RMSE_2}^2}$ may work for independent sets of responses and predictors, but is likely to be false if the two sets are correlated. For example, if your two sets of predictors and responses are the same $RMSE_{Delta Z_{Pred}}=0$, while $sqrt{{RMSE_1}^2 + {RMSE_2}^2} > 0$.
  • If you are interested in $Delta Z_{Pred}$, I suggest regressing it on the six predictors - although this is not what you are asking, it might server your goal.

Answered by Pere on January 5, 2022

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