+ Reply to Thread
Results 1 to 9 of 9

Thread: Variance of a sum of predictions

  1. #1
    Points: 2,462, Level: 30
    Level completed: 8%, Points required for next Level: 138

    Posts
    200
    Thanks
    20
    Thanked 48 Times in 43 Posts

    Variance of a sum of predictions




    Hi,

    I predict the spatial distribution of a species on a regular grid using a regression model.
    Species numbers vary with environmental covariates. Now I want to predict the overall size of the population within a given area, which means technically that I sum up all predicted values. But how can I calculate the standard error / confidence interval for this sum? Just by summing up all pointwise standard errors?

    Thank you in advance

  2. #2
    Super Moderator
    Points: 13,151, Level: 74
    Level completed: 76%, Points required for next Level: 99
    Dragan's Avatar
    Location
    Illinois, US
    Posts
    2,014
    Thanks
    0
    Thanked 223 Times in 192 Posts

    Re: Variance of a sum of predictions

    If you sum up the predicted values (Yhats), then this sum will be equal to the summation of the actual values of the dependent variable (Y). In short, the mean of the predicted values will be equal the mean of the actual values. (That is the purpose of the intercept term in a regression model.)

  3. #3
    Points: 2,462, Level: 30
    Level completed: 8%, Points required for next Level: 138

    Posts
    200
    Thanks
    20
    Thanked 48 Times in 43 Posts

    Re: Variance of a sum of predictions

    Sorry, I can't relate this answer to my question. The question is: Does it make sense to sum up the pointwise given standard deviations of the prediction in order to get the standard deviation of the sum? In other words: My predict function gives me outcome values and standard deviations for different values of covariates. How doe I get the standard error of the sum of these values?

  4. #4
    Points: 1,741, Level: 24
    Level completed: 41%, Points required for next Level: 59

    Posts
    230
    Thanks
    37
    Thanked 68 Times in 59 Posts

    Re: Variance of a sum of predictions

    Quote Originally Posted by mmercker View Post
    The question is: Does it make sense to sum up the pointwise given standard deviations of the prediction in order to get the standard deviation of the sum?
    To help you answer your question here, first answer this question. Are standard deviations additive? The answer to this can help point you towards the right direction, or away from the wrong direction.

  5. #5
    Super Moderator
    Points: 13,151, Level: 74
    Level completed: 76%, Points required for next Level: 99
    Dragan's Avatar
    Location
    Illinois, US
    Posts
    2,014
    Thanks
    0
    Thanked 223 Times in 192 Posts

    Re: Variance of a sum of predictions

    But how can I calculate the standard error / confidence interval for this sum? Just by summing up all pointwise standard errors?

    Quote Originally Posted by mmercker View Post
    How doe I get the standard error of the sum of these values?

    Your overall query is rather vague. That said, I do know this, you cannot get the standard error by simply summing the standard errors. The way to approach this would be to sum the Variance Estimates and then take the square root of that sum (to get the standard error that you're looking for).
    Last edited by Dragan; 04-01-2017 at 07:30 PM.

  6. #6
    Points: 1,741, Level: 24
    Level completed: 41%, Points required for next Level: 59

    Posts
    230
    Thanks
    37
    Thanked 68 Times in 59 Posts

    Re: Variance of a sum of predictions

    Quote Originally Posted by Dragan View Post
    You overall query is rather vague. That said, I do know this, you cannot get the standard error by simply summing the standard errors. The way to approach this would be to sum the Variance Estimates and then take the square root of that sum (to get the standard error that you're looking for).
    Well, you spoiled my surprise for the OP! I was hoping he or she would see that the variances would be additive, rather than the standard deviations/standard errors.

  7. #7
    Super Moderator
    Points: 13,151, Level: 74
    Level completed: 76%, Points required for next Level: 99
    Dragan's Avatar
    Location
    Illinois, US
    Posts
    2,014
    Thanks
    0
    Thanked 223 Times in 192 Posts

    Re: Variance of a sum of predictions

    Quote Originally Posted by ondansetron View Post
    Well, you spoiled my surprise for the OP! I was hoping he or she would see that the variances would be additive, rather than the standard deviations/standard errors.
    Well, yes, but someone needs to be proactive (with knowledge) to spell it out.

  8. #8
    Points: 1,741, Level: 24
    Level completed: 41%, Points required for next Level: 59

    Posts
    230
    Thanks
    37
    Thanked 68 Times in 59 Posts

    Re: Variance of a sum of predictions

    Quote Originally Posted by Dragan View Post
    Well, yes, but someone needs to be proactive (with knowledge) to spell it out.
    I'd favor on the side of teaching a man to fish, I suppose. Different approaches are always good!

  9. #9
    Points: 2,462, Level: 30
    Level completed: 8%, Points required for next Level: 138

    Posts
    200
    Thanks
    20
    Thanked 48 Times in 43 Posts

    Re: Variance of a sum of predictions


    Hi, sorry for formulating it wrong, it was clear to me that I have to sum up variances instead of standard errors, that was not my concern. My main concern is: if we sum up the poinwise given variances in order to geht the variance of the sum, this is (as far as I know) only valid if covariances are zero. I am used to the concept of covariance calculation based on different random variables. However, in the case of a prediction from a regression model (where I have only predicted values and it's standard errors for each covariate value) can I always assume that covariances between different predicted points are zero, if I specified the regression model correctly? It is even not clear to me how and if the definition of covariance exist in this context.

    Thanks in advance

+ Reply to Thread

           




Tags for this Thread

Posting Permissions

  • You may not post new threads
  • You may not post replies
  • You may not post attachments
  • You may not edit your posts






Advertise on Talk Stats