Calculation of prediction interval for a new observation

Hello,

does anyone know how the calculation of the 95% prediction interval for a new observation in the "grpstats"-function is performed? I can't find anything in the documentation. Is there any documentation availible?

If I have a set of values (x), is there an other way for calculating the prediction interval?

I can only find the calculation for regression models, but not for single vectors.

x = [95 84 105 96 86 86 95 94 75 93]';
Group = (ones(10,1));
T = table(x,Group,'VariableNames',{'Data','Group'});
statarray = grpstats(T,{'Group'},{'predci','meanci'},'DataVars',{'Data'});

Answer 1

[Star Strider]

The code is not doing a regression, so the ±95% confidence intervals on the vector would be the confidence intervals on the mean, calculated here as ‘MeanCI’ and corresponding to 'meanci'. I am not certain that 'predci' is appropriate here because no regression is being calculated, as would be the situation in something like Plot Prediction Intervals for a New Observation in Each Group.

x = [95 84 105 96 86 86 95 94 75 93]';
Group = (ones(10,1));
T = table(x,Group,'VariableNames',{'Data','Group'})

T = 10×2 table
    Data    Group
    ____    _____

     95       1  
     84       1  
    105       1  
     96       1  
     86       1  
     86       1  
     95       1  
     94       1  
     75       1  
     93       1  

statarray = grpstats(T,{'Group'},{'predci','meanci'},'DataVars',{'Data'})

statarray = 1×4 table
         Group    GroupCount      predci_Data         meanci_Data   
         _____    __________    ________________    ________________

    1      1          10        71.194    110.61    84.958    96.842

MeanCI = std(x)/sqrt(numel(x)) * tinv([0.025 0.975],numel(x)-1) + mean(x)

SEMCI = 1×2

84.9583 96.8417

.

[Paul Müller]

Thanks for your answer.

I am not entirely sure about the statistic-calculation here. But 'MeanCI' is calculating the 95% confidence intervall of the mean, like you said.

The calculation can be performed "manually" with the equation:

MeanCI = std(x)/sqrt(numel(x)) * tinv([0.025 0.975],numel(x)-1) + mean(x)

The 'predci' argument is calculating the 95% prediction intervall for the mean of each group, which naturally involves a greater degree of uncertainty.

Just as for the confidence intervall, I am looking for the principle calculation or the equation for the prediction intervall. I hope my problem is getting clearer...

[Star Strider]

I understand, however a ‘prediction interval’ is not appropriate unless it is in the context of a regression, because there is actually nothing to ‘predict’. Here, the next observation will simply extend the ‘Data’ vector and there is a 95% probability of that value being within the 'meanci' range. It can be added at any time and the interpretation and values remain the same (although the subsequent values may change slightly after it is added and included in the subsequent calculation).

To use a prediction interval requires that a value of the dependent variable be ‘predicted’ with respect to some value of the independent variable, and the uncertainty of the ‘predicted’ value depends on the value of the independent variable (so for example for a time series, the prediction interval for a value to be acquired in the next day will be less than the prediciton interval for a value to be acquired next week).

If the intent is to do a regression, then the prediciton interval is appropriate. If the intent is to add values at random times to a vector, then the standard error of the mean is appropriate.

[Paul Müller]

I think i understand the problem.

In the context of a regression, the prediction intervall makes perfect sence to me.

In the context with the grpstats function, the 'meanci'-statistic calculates the confidence intervall for the mean with the given significance level.

What makes sense from my point of view for 'predci': It could calculate the prediction of the 95%-quantile with the given significance level. So if I don't want to predict the mean, but the 95%-quantile or the range of 'meanci', the uncertainty will get greater.

If 'predci' is not appropriate for a simple set of grouped values, it would not make sence that you can call it in the grpstats function?

It would be very helpfull if there is a more detailed documentation besides the one from the "grpstats" function which says "95% prediction interval for a new observation. You can specify different significance levels using the Alpha name-value pair argument."

[Star Strider]

‘If 'predci' is not appropriate for a simple set of grouped values, it would not make sence that you can call it in the grpstats function?’

In the context of grpstats, the 'predci' option makes sense for regeressions, as the documentation demonstrates. There is no regression here. (This also applies to the paragraph that follows in your previous Comment.)

So 'meanci' applies to vectors of observations that are not regressions. Prediction intervals are appropriate only in the situation of a regression that actually predicts a future dependent variable oin the basis of a new independnet variable. That does not apply here.