If I have a linear model and want to use predict() to predict the mean and confidence interval of multiple ($m$) new observations of a given x-value ($x_h$), how do I account for the $m$ in the formula for the $s$ of the predicted mean?
$$
{\rm MSE}(\frac 1 m + \frac 1 n + \frac{(x_h - \bar x)^2}{S_{xx}})
$$
Is it the weight argument? Or do I have to include the desired $x_h$ $m$ times in the newdata?
The weights argument will work -- just set it equal to $m$ in your formula. It will not work to put multiple instances of $x_h$ in newdata -- you'll just get several copies of the same interval.
Another approach is to use pred.var = summary(mod)$sigma^2 / m where mod is your model and m is your value of $m$. The reason this works is that pred.var is used to set the variance of future observations; by default it is assumed to be the same as in the data ($\sigma^2$, estimated by $MSE$). By pretending it is $\sigma^2/m$ (estimated as $MSE/m$), you are using the variance of the average of $m$ predictions and will produce the correct result.
res.var is the same as summary(mod)$sigma^2?
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weights will work, do you want to consider changing the top part of your answer for others' future reference?
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res.var is a variable internal to predict.lm that is equal to that value.
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Commented
May 14, 2015 at 23:02