My residuals, oop, my residuals are my response y minus the predicted values,
beta nought plus beta1 x, so here I've just carried the subtraction through.
There's my residuals.
My estimate of the standard deviate of the,
standard deviation around the regression line,
the variability around the regression is the average of the residuals,
except remember, we're dividing by n minus 2 instead of n and to get it to be
a standard deviation rather than a variance, I'm going to square root it.
There's my sigma.
That's my estimate of sigma from the previous several pages.
My sums of my squares of x's is just the numerator of the variance calculation,
but let's just do it directly.
It's x minus its mean squared and sum of the, adding those up.
So let me get that and
then my standard error from my beta nought from my intercept.
If I just simply plug into the formula, there, I've written it out.
And my standard error for my beta1, which is I think the more useful and,
and mentally important formula to commit to memory is the sigma,
the variance, the standard deviation around the regression
line divided by the sum of the squares of the x's.
The sum of the square deviations of the x's around their mean.
Okay.
There we go.
There's the standard error for beta1, I'm going to create the two t statistics.
Which if you're testing a hypothesis that beta1 is zero or
beta nought is zero, then that's just going to be the estimate.
Here's my estimate divided by its standard error.
And so we don't have to subtract off the true value,
because the true value is assumed to be zero under this hypothesis.
And so for my beta1 hypothesis, it's going to be beta1 divided
by its standard error and then I'm going to calculate my two p values.
Again, if you've taken the inference class,
then how you go from a t statistic to a p value should not be a great leap.
But in this case, it's going to be twice my t probability where
in both these cases, the estimates are larger than zero so the,
so I'm going to look at the probability of being this statistic or
larger and I'm going to double those two t probabilities [SOUND] and
them I'm going to create my coefficient table created manually by
me without having done any lm or any built in higher level R function.
And I'm going to give it, its rownames and its column names, so
that it looks like R's output.
And then I'm, now I'm going to show you the easy way to do it.
Okay.
So here let me print it out.
There it is printed out and then now, I'm just going to do fit, lm,
my response y is a linear relationship with my predictor x and
then I'm going to get the summary of the coefficients.
You'll see everything is exactly the same.
So, if you didn't follow all this, it's, it's not too big of a deal.
I just wanted to show you for
once that the formulas that I'm giving you are exactly the right formulas and
we verified them computationally by checking them out on a dataset.
Maybe check them out on another dataset just to verify for yourself.
Let's go through getting a confidence interval.
So my fit, the variable fit was the output of lm.
Let's see what happens if I type summary fit.
You get the full output from lm.
So it gives you facts about the residuals, it give you the coefficients,
the estimated coefficients.
The Standard Errors, the t values and the associated p values.
These are all for test of whether or not the intercept is 0 or the slope is 0.
Now of course 0 intercept may not be of interest, but
almost always a test of a 0 slope is of interest.
But it also gives you the residual standard deviation,
the degrees of freedom The r-squared, the adjusted r-squared,
which we'll talk about later on in the class.
Let's go ahead and generate our confidence intervals for
our intercept and for our slope.
So now I've assigned that summary.
The summary from the lm statement to an object in R, but
I just wanted the table part, so I'm grabbing the, just the coefficient.
So, it's right here.
See, I'm just grabbing the coefficient table.
So let me just print out that variable, just to show you what it looks like.
It's just that table part, just the estimates, the standard errors,
the t values, and the p values.
Okay.
So, our confidence interval is just going to be estimate.
So here's for the intercept, estimate plus or minus
the 97.5 t quantile degrees of freedom, just to make
sure I get the degrees of freedom right, so I don't even have to think at all,
I'm grabbing fit$ degrees of freedom and then times the standard error.
There, I'm grabbing the standard error.
Brian Caffo, PhDProfessor, Biostatistics
Roger D. Peng, PhDAssociate Professor, Biostatistics
Jeff Leek, PhDAssociate Professor, Biostatistics
