Multiple regression parameters confidence limit

[Taliah]

We are asked to use a multiple regression equation constructed from the lower 90% confidence limit value of each parameter estiamte. I can get these values from SAS.  Would appriciate it if anyone can explain the validity and meaning of such a regression line -

Does SAS take into consideration the parameter estimates of all other variables when computing the confidence limits for each parameter estimate, so that the resulting regression equation is a valid regression equation for that data?

Would the predictions of that regression equation represent the 90% lower limit level of the predictions for that data?

Thank you!

[Rick_SAS]

1. I've never seen this before. That doesn't necessarily mean it is invalid, but it seems unusual.

2. The parameter estimates are assumed to be MVN. For large data, this is usually a reasonable assumption, as shown by the last image in the article "Simulate many samples from a linear regression model." If you take the lower 90% limit, there is no reason to expect that those parameters fit the data at all. To give a simple example, you might estimate the slope and intercept of a one-variable regression model. If the covariance of the estimates is negative, that means that if you decrease one estimate you should INCREASE the other. But the process of taking the lower 90% decreases both from their OLS values, which probably results in a model that does not fit the data.

3. No, the "predictions of that regression equation" do not "represent the 90% lower limit level of the predictions for that data."

Is it possible you misunderstood? Could the client have asked you to predict the 90th percentile of the response? You can do that by using PROC QUANTREG to perform quantile regression.

[Taliah]

Thank you very much for your detailed answer. What you wrote makes perfect sense. The client wanted the regression equation I wrote about, which didn't make sense to us. Checking back with the client, what would satifies their needs is the 90% lower confidence interval values of what the regression would predict. Do you know if there is a way to get that form SAS?

I just started looking up quantile regression and proc quatreg, trying to understand what we would be predicting using it.

We also thought of taking the predictions we get from the (usual) linear regression computed on the data, and computing the lower 90% confidence limit values of them. We did similar things (not exactly the same as here) in the past with excel's NORMINV. Is there a way to do this in SAS?

[Taliah]

Thank you vety much for your detailed answer. It makes perfect sense. The client wanted the regression equation I asked about, which did not seem right to us. Checking back with the client, what would satisfy their needs is the values of the 90% lower confidence limit of the predictions of the noramlly computed regression equation.

I just started looking into quantile regression and proc quantreg to see what we would be predicting with it and if we can get from it the predictions of the 90% lower confidence limit.

We thought of another solution, getting the predicted values from the (usally computed) linear regression equation, and then getting the values of the 90% lower confidence interval. In the past we did something similar (not the same) with excel's NORMINV. Is there a way to do this in SAS:

Thank you

[Rick_SAS]

You don't mention what SAS procedure you are using, but in many procedures, you can request a confidence interval for the predicted values. You would use ALPHA=0.1 to request a 90% CL. The syntax varies a little between procedures, but here is an example of using PROC REG. The OUTPUT statement uses the LCL= to output the lower prediction limit :

proc reg data=Sashelp.Class alpha=0.1;
model Weight = Height;
output out=RegOut pred=Pred LCL=Lower90;
quit;

/* for one-variable regression, you can draw a graph of the results */
proc sort data=RegOut; by Height; run;

proc sgplot data=RegOut;
scatter x=Height y=Weight;
series x=Height y=Pred;
series x=Height y=Lower90;
run;

[Taliah · Accepted solution]

Thank you, this sounds like a great solution. We are using proc reg and have several veriables (predictors). If we can use the proc reg code with the output statement you wrote to get the 90% CL for the predictions we can get the graphs we need from that.